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IB Maths Applications & Interpretation · SL · Topic 1

Number and algebra

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Original questions in the style of each exam boardFull worked mark schemes with method and accuracy marksCreated by Evan Kotronis with AI assistanceUpdated 16 October 2026 · new questions added continuouslyFree access until 31 December 2026How every question is checked →

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+ Add to test✓ In your test1IBMI-030-00-01Approximation, percentage error and standard form Paper 1Easy5 marks

Light from the Sun takes 499 seconds to reach the Earth. The speed of light is taken to be 3.00×10⁸ m s⁻¹.

(a)

Calculate the distance from the Sun to the Earth, in metres. Give your answer exactly, in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(2)

(b)

Write down your answer to part (a) correct to two significant figures.

(1)

(c)

The accepted mean Sun–Earth distance is 1.496×10¹¹ m. Calculate the percentage error in your answer to part (a).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for distance = 3.00×10⁸ × 499	✓ 1
A1 1.497×10¹¹ m	✓ 1
Part (b)
A1 1.5×10¹¹ m	✓ 1
Part (c)
M1 for $\left\|\frac{1.497\times10^{11}-1.496\times10^{11}}{1.496\times10^{11}}\right\|\times100$	✓ 1
A1 0.0668 %	✓ 1

Exam tip: In a percentage error, always divide by the exact (accepted) value, not by the approximate value.

+ Add to test✓ In your test2IBMI-030-00-02Approximation, percentage error and standard form Paper 1Easy4 marks

A surveyor estimates by eye that a footbridge is 85 m long. The actual length of the footbridge is 82.4 m.

(a)

Calculate the percentage error in the surveyor's estimate.

(2)

(b)

A second person underestimates the length of the footbridge with a percentage error of exactly 5 %. Find this person's estimate.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for $\left\|\frac{85-82.4}{82.4}\right\|\times100$	✓ 1
A1 3.16 % (accept 3.155…)	✓ 1
Part (b)
M1 for 82.4 × 0.95 (or 82.4 − 0.05 × 82.4)	✓ 1
A1 78.28 m (accept 78.3 m)	✓ 1

Common error: An underestimate means the estimate is below the true value, so multiply the true value by 0.95, not 1.05.

+ Add to test✓ In your test3IBMI-030-00-03Approximation, percentage error and standard form Paper 1Easy6 marks

A rectangular vegetable garden measures 12.47 m by 8.53 m.

(a)

Calculate the area of the garden. Write down your full calculator answer and then give it correct to three significant figures.

(2)

(b)

A gardener estimates the area by first rounding each length to the nearest metre. Find the gardener's estimate.

(2)

(c)

Using the full calculator value of the area as the exact value, calculate the percentage error in the gardener's estimate.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 106.3691 m²	✓ 1
A1 106 m²	✓ 1
Part (b)
M1 for 12 × 9	✓ 1
A1 108 m²	✓ 1
Part (c)
M1 for $\left\|\frac{108-106.3691}{106.3691}\right\|\times100$	✓ 1
A1 1.53 %	✓ 1

Exam tip: Use the unrounded area in the percentage error calculation. Using 106 instead gives 1.89%, which is wrong.

+ Add to test✓ In your test4IBMI-030-00-04Approximation, percentage error and standard form Paper 1Easy5 marks

A country has a population of 6.72×10⁷ people and an annual gross domestic product (GDP) of 2.85×10¹² US dollars (USD). Its land area is 2.43×10⁵ km².

(a)

Calculate the GDP per person, giving your answer in USD correct to three significant figures.

(2)

(b)

Write your answer to the previous part in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(1)

(c)

Calculate the population density of the country in people per km², giving your answer correct to three significant figures in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for $\frac{2.85\times10^{12}}{6.72\times10^{7}}$	✓ 1
A1 42 400 USD	✓ 1
Part (b)
A1 4.24×10⁴ USD	✓ 1
Part (c)
M1 for $\frac{6.72\times10^{7}}{2.43\times10^{5}}$	✓ 1
A1 2.77×10² people per km²	✓ 1

Common error: Writing 42.4×10³ is not standard form: the number in front must satisfy 1 ≤ a < 10.

+ Add to test✓ In your test5IBMI-030-05-01Approximation, percentage error and standard form Paper 1Easy6 marks

A rectangular solar panel is 1.65 m long and 0.992 m wide. A solar farm has 2400 of these panels.

(a)

Calculate the exact area of one panel.

(1)

(b)

Write your answer to part (a) correct to two significant figures.

(1)

(c)

Using your answer to part (a), find the total area of all 2400 panels. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$, to three significant figures.

(2)

(d)

A planner uses 1.6 m² for the area of each panel. Calculate the percentage error in this value.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.6368 m²	✓ 1
Part (b)
A1 1.6 m²	✓ 1
Part (c)
M1 for 2400 × 1.6368 = 3928.32	✓ 1
A1 3.93×10³ m²	✓ 1
Part (d)
M1 for $\left\|\frac{1.6-1.6368}{1.6368}\right\| \times 100\%$	✓ 1
A1 2.25%	✓ 1

Exam tip: In a percentage error the exact value always goes in the denominator. Here that is 1.6368, not the rounded value 1.6.

+ Add to test✓ In your test6IBMI-030-05-02Approximation, percentage error and standard form Paper 1Easy4 marks

The speed of light in a vacuum is 299 792 458 m s⁻¹.

(a)

Write this speed in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$, correct to three significant figures.

(2)

(b)

The mean distance from the Sun to Neptune is 4.50×10¹² m. Use the unrounded speed to find the time light takes to travel this distance. Give your answer in seconds in the form $a \times 10^k$, to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 for 3.00	✓ 1
A1 for ×10⁸ m s⁻¹	✓ 1
Part (b)
M1 for 4.50×10¹² ÷ 299 792 458	✓ 1
A1 1.50×10⁴ s	✓ 1

Common error: 3.00×10⁸ is correct to three significant figures. Writing 3×10⁸ gives only one significant figure and loses the accuracy mark.

+ Add to test✓ In your test7IBMI-030-05-05Approximation, percentage error and standard form Paper 1Easy4 marks

A type of bacterium has a diameter of 0.000 002 3 m.

(a)

Write this diameter in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$.

(1)

(b)

Bacteria are placed in a single line, touching end to end, across a slide that is 1.5 cm wide. Find the number of complete bacteria that fit across the slide.

(2)

(c)

Write your answer to the previous part in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$, correct to three significant figures.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 2.3×10⁻⁶ m	✓ 1
Part (b)
M1 converts 1.5 cm to 0.015 m and divides: 0.015 ÷ 2.3×10⁻⁶ = 6521.7…	✓ 1
A1 6521 complete bacteria	✓ 1
Part (c)
A1 6.52×10³	✓ 1

Common error: Convert 1.5 cm to 0.015 m before you divide. If you divide 1.5 by 2.3×10⁻⁶, the answer is 100 times too large.

+ Add to test✓ In your test8IBMI-030-05-11Approximation, percentage error and standard form Paper 1Easy5 marks

The attendance at a football match was 58 437. Each ticket cost $47.85.

(a)

Write the attendance correct to the nearest thousand.

(1)

(b)

Write the attendance correct to three significant figures.

(1)

(c)

Estimate the total ticket revenue by rounding both the attendance and the ticket price to one significant figure.

(1)

(d)

Calculate the percentage error in your estimate from the previous part.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 58 000	✓ 1
Part (b)
A1 58 400	✓ 1
Part (c)
A1 60 000 × 50 = $3 000 000	✓ 1
Part (d)
M1 for actual revenue 58 437 × 47.85 = $2 796 210.45 used in $\frac{\|3\,000\,000-2\,796\,210.45\|}{2\,796\,210.45} \times 100\%$	✓ 1
A1 7.29% (7.288…%)	✓ 1

Common error: Rounding 58 437 to 3 s.f. gives 58 400, not 584. The zeros are placeholders and must be kept.

+ Add to test✓ In your test9IBMI-030-06-r1-01Financial mathematics, loans and annuities Paper 1Easy6 marks

Priya invests €4500 in a savings account that pays a nominal annual interest rate of 3.2%, compounded monthly. She makes no further deposits or withdrawals.

(a)

Find the minimum number of complete months before the value of the investment first exceeds €6000.

(3)

(b)

Find the value of the investment after 6 years. Give your answer to the nearest cent.

(2)

(c)

Hence find the total interest earned in the first 6 years.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 equation or TVM entry: 4500(1 + 0.032/12)^N = 6000, or N unknown, I% = 3.2, PV = −4500, FV = 6000, C/Y = 12	✓ 1
A1 N = 108.02…	✓ 1
A1 109 months (must be rounded up, because after 108 months the value is about €5999.61)	✓ 1
Part (b)
M1 4500(1 + 0.032/12)⁷², or N = 72 entered in TVM	✓ 1
A1 €5451.12	✓ 1
Part (c)
A1 5451.12 − 4500 = €951.12	✓ 1

Common error: A solver value of N = 108.02 does not mean 108 months. The value only exceeds €6000 after a whole extra month of interest, so round up to 109.

+ Add to test✓ In your test10IBMI-030-06-r1-02Financial mathematics, loans and annuities Paper 1Easy5 marks

Tomasz takes out a loan of €18 000 to buy a car. The loan has a nominal annual interest rate of 6.5%, compounded monthly. He repays it in equal monthly instalments at the end of each month over 5 years.

(a)

Find the amount of each monthly instalment. Give your answer to the nearest cent.

(3)

(b)

Find the total interest Tomasz pays over the 5 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 N = 60 (or 5 × 12) and I% = 6.5 with C/Y = P/Y = 12	✓ 1
M1 PV = 18000 and FV = 0, with PV and PMT of opposite signs	✓ 1
A1 €352.19	✓ 1
Part (b)
M1 total repaid = 60 × 352.19 = €21 131.40 (accept 21 131.44 from the unrounded instalment)	✓ 1
A1 interest = 21 131.40 − 18 000 = €3131.40 (accept €3131.44)	✓ 1

Exam tip: Write down the values you enter into the financial solver (N, I%, PV, PMT, FV, P/Y, C/Y). The method marks come from these values, even if the final answer is slightly wrong.

+ Add to test✓ In your test11IBMI-030-01-r2-01Financial mathematics, loans and annuities Paper 1Easy6 marks

Priya invests 4500 euros in a savings account that pays a nominal annual interest rate of 3.2%, compounded monthly. She makes no further deposits or withdrawals.

(a)

Find the minimum number of complete months needed for the value of the investment to exceed 6000 euros.

(3)

(b)

Find the interest Priya earns in the first 6 years. Give your answer to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Sets up $4500\left(1+\frac{0.032}{12}\right)^n > 6000$ or equivalent TVM entry (PV = −4500, FV = 6000, I% = 3.2, C/Y = 12) M1	✓ 1
$n = 108.02...$ A1	✓ 1
109 months A1	✓ 1
Part (b)
$4500\left(1+\frac{0.032}{12}\right)^{72}$ or TVM entry with N = 72 M1	✓ 1
Value = 5451.12 euros A1	✓ 1
Interest = 5451.12 − 4500 = 951.12 euros A1	✓ 1

Common error: Writing 108 months loses the final mark. After 108 months the value is still just under 6000 euros, so the answer must be rounded up to 109 complete months.

+ Add to test✓ In your test12IBMI-030-02-r1-01Arithmetic and geometric sequences in context Paper 1Easy6 marks

A new community theatre is designed so that the front row (row 1) has 18 seats. Each row behind it has 3 more seats than the row in front. The theatre has 25 rows.

(a)

Find the number of seats in row 25.

(2)

(b)

Find the first row that has more than 60 seats.

(2)

(c)

Find the total number of seats in the theatre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Uses $u_{25}=18+(25-1)\times3$ M1 ; 90 seats A1	✓ 2
Part (b)
Forms $18+3(n-1)>60$ or lists terms. Row 15 has exactly 60 seats M1 ; row 16, with 63 seats A1	✓ 2
Part (c)
Uses $S_{25}=\frac{25}{2}(18+90)$ or an equivalent formula M1 ; 1350 seats A1	✓ 2

Common error: In part (b) the condition is 'more than 60'. Row 15 has exactly 60 seats, so it does not qualify. Always check the boundary term against the strict inequality.

+ Add to test✓ In your test13IBMI-030-02-r1-02Arithmetic and geometric sequences in context Paper 1Easy7 marks

A fitness app was downloaded 2400 times in its first week. The number of downloads in each week is modelled as being 15% more than in the previous week.

(a)

Find the number of downloads in week 8.

(2)

(b)

Find the first week in which the number of downloads is more than 10 000.

(3)

(c)

Find the total number of downloads in the first 12 weeks.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Uses $2400\times1.15^{7}$ M1 ; 6380 downloads (6384.05…) A1	✓ 2
Part (b)
Forms $2400\times1.15^{n-1}>10000$ or uses a table on a GDC M1 ; gives $n-1>10.2$, or week 11 ≈ 9709 and week 12 ≈ 11 166 A1 ; week 12 A1	✓ 3
Part (c)
Uses $S_{12}=\frac{2400(1.15^{12}-1)}{1.15-1}$ M1 ; 69 600 downloads (69 604.0…) A1	✓ 2

Exam tip: Week 1 is the first term, so week n uses the power n − 1. A quick table of values on a GDC is a reliable way to confirm the first week that passes a threshold.

+ Add to test✓ In your test14IBMI-030-06-r2-01Financial mathematics, loans and annuities Paper 1Easy6 marks

Priya invests 4500 euros (EUR) in a savings account. The account pays a nominal annual interest rate of 3.2%, compounded monthly. She makes no further deposits or withdrawals.

(a)

Find the least number of complete months after which the value of Priya's investment first exceeds EUR 6000.

(3)

(b)

Find the interest Priya has earned after exactly 6 years.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Solve $4500\left(1+\frac{0.032}{12}\right)^{n}=6000$, or TVM with PV = −4500, FV = 6000, I% = 3.2, P/Y = C/Y = 12	✓ 1
n = 108.02…	✓ 1
109 months, since after 108 months the value is still just below EUR 6000	✓ 1
Part (b)
$4500\left(1+\frac{0.032}{12}\right)^{72}$	✓ 1
= EUR 5451.12	✓ 1
Interest = 5451.12 − 4500 = EUR 951.12	✓ 1

Exam tip: When n is not a whole number, decide in context whether to round up or down. Here the value has not yet reached EUR 6000 at month 108, so the answer is 109 months.

+ Add to test✓ In your test15IBMI-030-07-r1-01Arithmetic and geometric sequences in context Paper 1Easy6 marks

A community theatre has seating arranged in 24 rows. The first row has 18 seats, and each row after the first has 3 more seats than the row in front of it.

(a)

Find the number of seats in the 24th row.

(2)

(b)

Find the total number of seats in the theatre.

(2)

(c)

Find the number of the first row that has more than 60 seats.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$u_{24} = 18 + (24-1)(3)$ M1	✓ 1
$= 87$ seats A1	✓ 1
Part (b)
$S_{24} = \frac{24}{2}(18 + 87)$ or $\frac{24}{2}(2\times18 + 23\times3)$ M1	✓ 1
$= 1260$ seats A1	✓ 1
Part (c)
$18 + 3(n-1) > 60$, or noting that row 15 has exactly 60 seats M1	✓ 1
Row 16, which has 63 seats A1	✓ 1

Common error: Row 15 has exactly 60 seats, and that is not more than 60. Check the boundary value against the exact wording of the inequality.

+ Add to test✓ In your test16IBMI-030-01-r1-02Financial mathematics, loans and annuities Paper 1Easy6 marks

Tomás pays £150 into a savings account at the end of every month for 8 years. The account pays a nominal annual interest rate of 3%, compounded monthly.

(a)

Find the amount in the account immediately after the final payment. Give your answer to the nearest penny.

(2)

(b)

Find the total interest Tomás earns over the 8 years.

(2)

(c)

Tomás would like to have £20 000 in the account after 8 years, with the same interest conditions. Find the monthly payment he would need to make.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
TVM with N = 96, I% = 3, PV = 0, PMT = −150, P/Y = C/Y = 12, or $150\times\frac{1.0025^{96}-1}{0.0025}$	✓ 1
£16 252.11	✓ 1
Part (b)
Total paid in: 96 × 150 = £14 400	✓ 1
Interest = £1852.11	✓ 1
Part (c)
TVM with N = 96, I% = 3, PV = 0, FV = 20 000, solving for PMT	✓ 1
£184.59 (accept £184.60, which guarantees the target is reached)	✓ 1

Exam tip: When you use the TVM solver, money paid in and money received must have opposite signs. If PMT and FV have the same sign, the solver gives an error or a meaningless answer.

+ Add to test✓ In your test17IBMI-030-08-r2-01Exponents and logarithms Paper 1Easy5 marks

The sound level, L decibels (dB), of a noise with intensity I W m⁻² is modelled by
$L = 10\log_{10}\left(\frac{I}{I_0}\right)$
where $I_0 = 1.0 \times 10^{-12}$ W m⁻².

(a)

A hairdryer produces a sound of intensity 3.2×10⁻⁵ W m⁻². Find its sound level.

(2)

(b)

A motorbike passing a pedestrian produces a sound level of 88 dB. Find the intensity of this sound.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes correctly, $L = 10\log_{10}\left(\frac{3.2\times10^{-5}}{1.0\times10^{-12}}\right) = 10\log_{10}(3.2\times10^{7})$	✓ 1
A1: $L = 75.1$ dB	✓ 1
Part (b)
M1: substitutes correctly, $88 = 10\log_{10}\left(\frac{I}{10^{-12}}\right)$	✓ 1
M1: rearranges correctly, $\frac{I}{10^{-12}} = 10^{8.8}$, or solves with GDC	✓ 1
A1: $I = 6.31\times10^{-4}$ W m⁻²	✓ 1

Common error: In part (b), divide 88 by 10 before undoing the logarithm. Writing $10^{88}$ instead of $10^{8.8}$ gives an intensity that is far too large.

+ Add to test✓ In your test18IBMI-030-08-r2-02Exponents and logarithms Paper 1Easy6 marks

After a person drinks a large coffee, the amount of caffeine, C mg, remaining in their body t hours later is modelled by
$C(t) = 200e^{-0.139t}, \quad t \ge 0$.

(a)

Write down the amount of caffeine in the coffee.

(1)

(b)

Find the amount of caffeine remaining after 5 hours.

(2)

(c)

The person can sleep well once the amount of caffeine has fallen below 25 mg. Find the least time after drinking the coffee at which this happens.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: 200 mg	✓ 1
Part (b)
M1: substitutes $t = 5$, $200e^{-0.139\times5}$	✓ 1
A1: 99.8 mg	✓ 1
Part (c)
M1: sets up the equation $200e^{-0.139t} = 25$ (or an inequality)	✓ 1
M1: uses logarithms, $-0.139t = \ln(0.125)$, or solves with GDC	✓ 1
A1: $t = 15.0$ hours	✓ 1

Exam tip: For a 'find the time' question, write the equation you are solving before using the GDC. The equation earns the method mark even if a keying error spoils the final answer.

+ Add to test✓ In your test19IBMI-030-02-r2-01Arithmetic and geometric sequences in context Paper 1Easy6 marks

A new outdoor theatre has 25 rows of seats. The front row (row 1) has 18 seats. Each row after the first has 3 more seats than the row in front of it.

(a)

Find the number of seats in row 25.

(2)

(b)

Find the first row that has more than 60 seats.

(2)

(c)

Find the total number of seats in the theatre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$u_{25} = 18 + (25-1)\times 3$ M1	✓ 1
= 90 seats A1	✓ 1
Part (b)
Sets up $18 + 3(n-1) > 60$, or lists terms using a GDC table M1	✓ 1
$n > 15$, so row 16 (which has 63 seats) A1	✓ 1
Part (c)
$S_{25} = \frac{25}{2}(18 + 90)$ or $\frac{25}{2}(2\times 18 + 24\times 3)$ M1	✓ 1
= 1350 seats A1	✓ 1

Exam tip: When a question asks for the first term to exceed a value, an inequality gives a non-integer or boundary value. Round up to the next whole term number and check that term actually exceeds the value.

+ Add to test✓ In your test20IBMI-030-02-r2-02Arithmetic and geometric sequences in context Paper 1Easy5 marks

A short video is shared 40 times on the first day after it is posted online. The number of shares on each day is modelled as being 15% greater than the number of shares on the previous day.

(a)

Write down the common ratio of the geometric sequence that models the number of shares per day.

(1)

(b)

Find the number of shares on day 10.

(2)

(c)

Find the total number of shares over the first 10 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$r = 1.15$ A1	✓ 1
Part (b)
$u_{10} = 40 \times 1.15^{9}$ M1	✓ 1
= 140.7… ≈ 141 shares A1	✓ 1
Part (c)
$S_{10} = \frac{40(1.15^{10} - 1)}{1.15 - 1}$ M1	✓ 1
= 812.1… ≈ 812 shares A1	✓ 1

Common error: Using $1.15^{10}$ for the 10th term. The first term is day 1, so only 9 increases have happened by day 10: the power is $n-1$.

+ Add to test✓ In your test21IBMI-030-03-r2-02Exponents and logarithms Paper 1Easy5 marks

The value, in dollars, of a new electric car $t$ years after it is bought is modelled by
$V = 28\,500\times0.84^{t}$, $t \ge 0$.

(a)

Find the value of the car 4 years after it is bought. Give your answer to two decimal places.

(2)

(b)

Find the value of $t$ when the value of the car first falls to $8000.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes $t=4$, $28\,500\times0.84^{4}$	✓ 1
A1: $14 189.33	✓ 1
Part (b)
M1: sets up the equation $28\,500\times0.84^{t} = 8000$	✓ 1
M1: uses logarithms or technology, e.g. $t = \dfrac{\log(8000/28\,500)}{\log 0.84}$, or finds the intersection on a GDC	✓ 1
A1: $t = 7.29$ years	✓ 1

Exam tip: Money answers should be given to two decimal places. Other answers should be given to 3 significant figures unless the question asks for something else. Keep full calculator values in your working and round only at the end.

+ Add to test✓ In your test22IBMI-030-08-r1-01Exponents and logarithms Paper 1Easy6 marks

The acidity of a liquid is measured on the pH scale, where $\text{pH} = -\log_{10} H$ and $H$ is the hydrogen ion concentration in mol dm⁻³.

(a)

A sample of rainwater has $H = 3.2$ ×10⁻⁴ mol dm⁻³. Calculate its pH.

(2)

(b)

A sample of seawater has a pH of 8.1. Find its hydrogen ion concentration.

(2)

(c)

Lemon juice has a pH of 2.3 and tomato juice has a pH of 4.1. Find how many times greater the hydrogen ion concentration of lemon juice is than that of tomato juice.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes correctly, pH = −log₁₀(3.2×10⁻⁴)	✓ 1
A1: pH = 3.49	✓ 1
Part (b)
M1: rearranges to H = 10^−8.1	✓ 1
A1: H = 7.94×10⁻⁹ mol dm⁻³	✓ 1
Part (c)
M1: forms the ratio 10^−2.3 ÷ 10^−4.1 or 10^1.8	✓ 1
A1: 63.1 times	✓ 1

Common error: A difference of 1.8 on the pH scale is not a ratio of 1.8. Each unit on a log₁₀ scale is a factor of 10, so the ratio is 10¹.⁸.

+ Add to test✓ In your test23IBMI-030-08-r1-02Exponents and logarithms Paper 1Easy6 marks

After drinking a large coffee, the amount of caffeine, $C$ mg, in Dana's bloodstream $t$ hours later is modelled by $C = 200(0.87)^t$, for $t \ge 0$.

(a)

Interpret, in context, the values 200 and 0.87 in the model.

(2)

(b)

Find the amount of caffeine in Dana's bloodstream 5 hours after drinking the coffee.

(2)

(c)

Dana can sleep well once the amount of caffeine falls below 25 mg. Find how long after drinking the coffee this happens.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
R1: 200 mg is the initial amount of caffeine in the bloodstream	✓ 1
R1: the amount of caffeine decreases by 13% each hour	✓ 1
Part (b)
M1: substitutes t = 5, giving 200(0.87)⁵	✓ 1
A1: 99.7 mg	✓ 1
Part (c)
M1: sets up 200(0.87)^t = 25, for example by using a graph, a solver or logarithms	✓ 1
A1: t = 14.9 hours	✓ 1

Exam tip: When you use your GDC to solve an equation, write the equation you entered first. This earns the method mark even if the final value is misread.

+ Add to test✓ In your test24IBMI-030-03-r1-01Exponents and logarithms Paper 1Easy5 marks

A pharmacologist models the mass of caffeine, C mg, remaining in a person's bloodstream t hours after drinking a large coffee by
$C(t) = 200(0.87)^t$, $t \ge 0$.

(a)

State the percentage by which the mass of caffeine decreases each hour.

(1)

(b)

Find the mass of caffeine remaining 3 hours after the coffee is drunk.

(2)

(c)

Find the time taken for the mass of caffeine to fall to 50 mg.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
13% decrease per hour A1	✓ 1
Part (b)
Substituting t = 3: 200(0.87)³ M1	✓ 1
= 132 mg (131.70…) A1	✓ 1
Part (c)
Setting 200(0.87)^t = 50 and solving, e.g. t = ln 0.25 / ln 0.87 or using GDC M1	✓ 1
t = 9.95 hours (9.9546…) A1	✓ 1

Common error: The multiplier 0.87 means 87% remains each hour, so the decrease is 13%, not 87%.

+ Add to test✓ In your test25IBMI-030-03-r1-02Exponents and logarithms Paper 1Easy6 marks

The sound level, L decibels (dB), of a sound of intensity I W m⁻² is given by
$L = 10\log_{10}\left(\frac{I}{I_0}\right)$, where $I_0 = 1.0 \times 10^{-12}$ W m⁻².

(a)

A vacuum cleaner produces sound of intensity 3.2 ×10⁻⁵ W m⁻². Find its sound level.

(2)

(b)

The sound level at a concert is 95 dB. Find the intensity of the sound.

(2)

(c)

Find the factor by which the intensity of a sound must be multiplied to increase its sound level by 6 dB.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
L = 10 log₁₀(3.2×10⁻⁵ / 1.0×10⁻¹²) = 10 log₁₀(3.2×10⁷) M1	✓ 1
L = 75.1 dB (75.05…) A1	✓ 1
Part (b)
I = 1.0×10⁻¹² × 10^9.5 (or equivalent rearrangement) M1	✓ 1
I = 3.16×10⁻³ W m⁻² A1	✓ 1
Part (c)
10 log₁₀(k) = 6, so k = 10^0.6 M1	✓ 1
k = 3.98 A1	✓ 1

Exam tip: An increase in decibels corresponds to multiplying intensity, not adding to it: a rise of ΔL dB multiplies the intensity by 10ΔL/¹⁰.

+ Add to test✓ In your test26IBMI-030-01-r1-01Financial mathematics, loans and annuities Paper 1Easy6 marks

Priya invests $4500 in a savings account that pays a nominal annual interest rate of 3.6%, compounded monthly. She makes no further deposits or withdrawals.

(a)

Find the value of Priya's investment after 6 years. Give your answer to the nearest cent.

(2)

(b)

Find the least number of complete months after which the value of the investment first exceeds $6000.

(3)

(c)

Find the value of the investment at the end of the month found in the previous part. Give your answer to the nearest cent.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Correct substitution: $4500\left(1+\frac{0.036}{12}\right)^{72}$, or TVM with N = 72, I% = 3.6, PV = −4500, PMT = 0, P/Y = C/Y = 12	✓ 1
$5583.16	✓ 1
Part (b)
Sets up $4500(1.003)^n > 6000$ or equivalent, e.g. TVM solving for N with FV = 6000	✓ 1
$n = 96.04...$	✓ 1
97 months, because the value has not reached $6000 after 96 months	✓ 1
Part (c)
$4500(1.003)^{97}$ = $6017.32	✓ 1

Common error: Writing 96 months loses the final mark. Solving gives n = 96.04, so the balance is still below $6000 after 96 months. Round up to the next whole month.

+ Add to test✓ In your test27IBMI-030-03-r2-01Exponents and logarithms Paper 1Easy6 marks

The sound intensity level $L$, in decibels (dB), of a sound of intensity $I$ W m⁻² is modelled by
$L = 10\log_{10}\left(\dfrac{I}{I_0}\right)$, where $I_0 = 1\times10^{-12}$ W m⁻².
At an outdoor music festival, the intensity near the front of the stage is measured as 0.25 W m⁻².

(a)

Calculate the sound intensity level near the front of the stage.

(2)

(b)

Hearing protection is recommended for sound intensity levels above 85 dB. Find the intensity of a sound with a sound intensity level of 85 dB.

(2)

(c)

Find how many times greater the intensity near the front of the stage is than the intensity of a sound at 85 dB.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes correctly, $L = 10\log_{10}\left(\dfrac{0.25}{1\times10^{-12}}\right)$	✓ 1
A1: $L = 114$ dB (113.979…)	✓ 1
Part (b)
M1: $85 = 10\log_{10}\left(\dfrac{I}{10^{-12}}\right) \Rightarrow I = 10^{-12}\times10^{8.5}$	✓ 1
A1: $I = 3.16\times10^{-4}$ W m⁻²	✓ 1
Part (c)
M1: $\dfrac{0.25}{3.162\ldots\times10^{-4}}$ or $10^{(L-85)/10}$ using their $L$	✓ 1
A1: 791 times (790.57…); accept any answer from 790 to 794 that comes from using rounded values, e.g. $10^{2.9} \approx 794$	✓ 1

Common error: A difference of about 29 dB does not mean the intensity is 29 times greater. Each 10 dB is a factor of 10 in intensity, so convert back to intensities before comparing.

+ Add to test✓ In your test28IBMI-030-07-r2-02Arithmetic and geometric sequences in context Paper 1Easy5 marks

The population of a coastal town was 42 000 at the start of 2020. A planning model assumes that the population increases by 2.5% each year. Let $u_1$ be the population at the start of 2020, $u_2$ the population at the start of 2021, and so on.

(a)

Write down the common ratio of this geometric sequence.

(1)

(b)

Find the year in which the population is first predicted to exceed 60 000 at the start of the year.

(2)

(c)

Find the predicted population at the start of 2030.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $r=1.025$	✓ 1
Part (b)
M1: sets up $42000(1.025)^{n-1}>60000$ and solves it, e.g. with a GDC or logarithms, giving $n-1>14.4$	✓ 1
A1: $n=16$, so the start of 2035	✓ 1
Part (c)
M1: $42000\times1.025^{10}$, i.e. the correct power for $u_{11}$	✓ 1
A1: 53764 (accept 53800 or 53763)	✓ 1

Common error: Mixing up $n$ and $n-1$ in the power. 2030 is 10 years after 2020, so the factor 1.025 is applied 10 times, even though 2030 is the 11th term.

+ Add to test✓ In your test29IBMI-030-07-r2-01Arithmetic and geometric sequences in context Paper 1Easy6 marks

A new outdoor amphitheatre is designed with curved rows of seats. The front row (row 1) has 18 seats, and each row after that has 3 more seats than the row in front of it.

(a)

Find the number of seats in row 20.

(2)

(b)

The designers consider a larger version in which the same pattern continues for as many rows as needed. Find the number of the first row that would have more than 100 seats.

(2)

(c)

The amphitheatre that is actually built has exactly 20 rows. Find the total number of seats.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: use of $u_n=u_1+(n-1)d$ with $u_1=18$, $d=3$, $n=20$, giving $18+19\times3$	✓ 1
A1: $u_{20}=75$ seats	✓ 1
Part (b)
M1: sets up $18+3(n-1)>100$, or an equivalent equation or table of values	✓ 1
A1: $n>28.3$, so row 29 (which has 102 seats)	✓ 1
Part (c)
M1: use of $S_n=\frac{n}{2}(2u_1+(n-1)d)$ or $\frac{n}{2}(u_1+u_n)$ with $n=20$	✓ 1
A1: 930 seats	✓ 1

Exam tip: When the answer must be a whole row number, solve the inequality first and then round up. Check your answer by working out the number of seats in the row before it (row 28 has 99 seats).

+ Add to test✓ In your test30IBMI-030-06-r2-02Financial mathematics, loans and annuities Paper 1Easy5 marks

Daniel buys a car costing 21 500 Australian dollars (AUD). He pays a deposit of AUD 3500 and borrows the rest. The loan is at a nominal annual interest rate of 6.5%, compounded monthly. It is repaid with equal monthly payments, made at the end of each month, over 5 years.

(a)

Find Daniel's monthly payment, giving your answer to the nearest cent.

(3)

(b)

Find the total amount of interest Daniel pays over the 5 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Loan amount PV = 21 500 − 3500 = AUD 18 000	✓ 1
Correct financial-solver entries: N = 60, I% = 6.5, PV = ±18 000, FV = 0, P/Y = C/Y = 12, solve for PMT. Alternatively, uses the formula $18000 = P\cdot\frac{1-(1+\frac{0.065}{12})^{-60}}{\frac{0.065}{12}}$	✓ 1
Monthly payment = AUD 352.19	✓ 1
Part (b)
Total repaid = 60 × 352.19 = AUD 21 131.40	✓ 1
Interest = 21 131.40 − 18 000 = AUD 3131.40	✓ 1
Also accept AUD 3131.48, which comes from the unrounded payment of 352.1913…. Follow through from the answer to (a).	—

Common error: Using the full price of the car, AUD 21 500, as the loan amount. Only the amount borrowed after the deposit earns interest.

+ Add to test✓ In your test31IBMI-030-01-r2-02Financial mathematics, loans and annuities Paper 1Easy7 marks

A new electric car is bought for 28 000 dollars. Its value depreciates by 15% of its value at the start of each year.

(a)

Find the value of the car after 4 years.

(2)

(b)

Find the number of complete years after which the value of the car first falls below 5000 dollars.

(3)

(c)

A different model of car loses exactly half of its value in 4 years. Assuming a constant annual rate of depreciation, find this rate as a percentage.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$28000 \times 0.85^4$ M1	✓ 1
14 616.18 dollars (accept 14 600 dollars to 3 s.f.) A1	✓ 1
Part (b)
$28000 \times 0.85^n < 5000$ or equivalent, e.g. a table of values or a GDC solver M1	✓ 1
$n = 10.6...$, OR values after 10 and 11 years found as 5512 and 4686 dollars A1	✓ 1
11 years A1	✓ 1
Part (c)
$(1-r)^4 = 0.5$ M1	✓ 1
$r = 1 - 0.5^{0.25} = 0.159$, so 15.9% A1	✓ 1

Exam tip: For depreciation, the multiplier is 1 − r, not 1 + r. If you use a TVM solver, enter the rate as negative, for example I% = −15.

+ Add to test✓ In your test32IBMI-030-07-r1-02Arithmetic and geometric sequences in context Paper 1Easy5 marks

A short video gets 1200 views on the day it is posted (day 1). The number of views each day is 15% more than on the previous day. Assume this pattern continues.

(a)

Find the number of views on day 8.

(2)

(b)

Find the total number of views over the first 14 days.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$u_8 = 1200 \times 1.15^{7}$ M1	✓ 1
$= 3190$ views (3192.02…; accept 3192) A1	✓ 1
Part (b)
Recognises a geometric series with $u_1 = 1200$ and $r = 1.15$ M1	✓ 1
$S_{14} = \frac{1200(1.15^{14}-1)}{1.15-1}$ A1	✓ 1
$= 48600$ views (48605.6…; accept 48606) A1	✓ 1

Exam tip: An increase of 15% gives a common ratio of 1.15, not 0.15. Day 8 is the 8th term, so the power is 7.

+ Add to test✓ In your test33IBMI3S-1-01-01Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

A pharmacist weighs one tablet and records its active ingredient as 2.4 mg. The label states the exact value is 2.5 mg per tablet.

(a)

Calculate the percentage error in the pharmacist's measurement.

(2)

(b)

A batch contains 12 000 tablets. Using the label value, find the total mass of active ingredient in the batch, in grams. Give your answer in the form $a\times10^{k}$, where $1\le a<10$ and $k\in\mathbb{Z}$.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to use $\frac{\|v_A - v_E\|}{\|v_E\|}\times 100$ with 2.4 and 2.5	✓ 1
A1 4 %	✓ 1
Part (b)
M1 $12\,000\times2.5 = 30\,000$ mg, then convert to grams (divide by 1000)	✓ 1
A1 $3\times10^{1}$ g	✓ 1

Common error: Divide by the exact (true) value in a percentage error, not by the measured value, and check that standard form has a coefficient from 1 up to (but not including) 10.

+ Add to test✓ In your test34IBMI3S-1-01-02Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

Mei invests $4500 in an account paying a nominal annual interest rate of 3.2%, compounded quarterly.

(a)

Find the value of the investment after 6 years. Give your answer to the nearest cent.

(2)

(b)

Find the number of years, to 3 significant figures, for the investment to grow to $6000.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to use compound interest with $n=24$ quarters (or financial solver: N = 6, I% = 3.2, PV = ±4500, P/Y = C/Y = 4)	✓ 1
A1 $5448.35	✓ 1
Part (b)
M1 setting up $4500\left(1+\frac{0.032}{4}\right)^{n}=6000$ or solver with FV = ∓6000	✓ 1
A1 $n \approx 36.1$ quarters	✓ 1
A1 9.03 years	✓ 1

Exam tip: With quarterly compounding the solver gives time in years only if you enter N as years and set C/Y = 4; otherwise convert quarters to years yourself.

+ Add to test✓ In your test35IBMI3S-1-01-03Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

A runner follows a training plan. In week 1 she runs 3.0 km in total. In each following week she runs 0.4 km more than in the previous week.

(a)

Find the distance she runs in week 10.

(2)

(b)

Find the total distance she runs in the first 15 weeks.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_{10}=3.0+9(0.4)$	✓ 1
A1 6.6 km	✓ 1
Part (b)
M1 $S_{15}=\frac{15}{2}\left(2(3.0)+14(0.4)\right)$	✓ 1
A1 87 km	✓ 1

Common error: The nth term uses $(n-1)d$, not $nd$. Writing $u_{10}=3.0+10(0.4)$ gives 7.0 km.

+ Add to test✓ In your test36IBMI3S-1-01-04Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

The concentration of a drug in a patient's blood is modelled by $C = 80\times0.85^{t}$ mg per litre, where $t$ is the time in hours after the drug is given.

(a)

State the concentration at the moment the drug is given.

(1)

(b)

Find the concentration after 5 hours.

(2)

(c)

Find the time, in hours to 3 significant figures, at which the concentration first falls below 10 mg per litre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 80 mg per litre	✓ 1
Part (b)
M1 $80\times0.85^{5}$	✓ 1
A1 35.5 mg per litre	✓ 1
Part (c)
M1 $80\times0.85^{t}=10$ and attempt to solve (logarithms or graph)	✓ 1
A1 $t=12.8$ hours	✓ 1

Exam tip: Take logarithms of both sides after isolating the power: $0.85^t=0.125$ gives $t=\frac{\ln0.125}{\ln0.85}$.

+ Add to test✓ In your test37IBMI3S-1-01b-01Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

A farmer estimates that a field has an area of 4.8 hectares. A land survey later gives the actual area as 4.53 hectares. 1 hectare = 10 000 m².

(a)

Write the actual area in m² in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Calculate the percentage error in the farmer's estimate.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 multiplies 4.53 by 10000 (or 10⁴)	✓ 1
A1 4.53 × 10⁴ m²	✓ 1
Part (b)
M1 \|4.8 − 4.53\| ÷ 4.53 × 100	✓ 1
A1 5.96 %	✓ 1

Common error: The percentage error is found relative to the actual value (4.53), not the estimate (4.8).

+ Add to test✓ In your test38IBMI3S-1-01b-02Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

A light aircraft climbs in a series of one-minute stages. In the first minute it gains 180 m of height. In each later minute it gains 15 m more than in the minute before.

(a)

Find the height gained during the 10th minute.

(2)

(b)

The pilot needs to gain at least 5000 m in total. Find the smallest whole number of minutes this takes.

(3)

(c)

Find the total height gained after this number of minutes.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₀ = 180 + 9 × 15	✓ 1
A1 315 m	✓ 1
Part (b)
M1 sets up S_n = n/2 (2 × 180 + (n − 1) × 15) ≥ 5000, or uses a table of values	✓ 1
A1 S₁₆ = 4680 (< 5000) and S₁₇ = 5100 (≥ 5000)	✓ 1
A1 17 minutes	✓ 1
Part (c)
A1 5100 m (ECF from (b))	✓ 1

Exam tip: When a sequence question asks for the smallest whole number of terms, check the value just below your answer to show it is too small.

+ Add to test✓ In your test39IBMI3S-1-01b-03Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

Musical notes are tuned so that the frequency f hertz of a note n semitones above the note A (440 Hz) is given by f = 440 × 2^n/12. A note below A has a negative value of n.

(a)

Find the frequency of the note 7 semitones above A. Give your answer to the nearest hertz.

(2)

(b)

A note has a frequency of 587 Hz. Find n, giving your answer first to 3 significant figures and then as a whole number.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 440 × 2^7/12	✓ 1
A1 659 Hz	✓ 1
Part (b)
M1 587 = 440 × 2^n/12, solved using a calculator or logarithms	✓ 1
A1 n = 4.99	✓ 1
A1 n = 5 (a whole number of semitones)	✓ 1

Common error: Writing 2ⁿ/¹² as 2n/12 changes the model completely: keep the whole fraction in the exponent.

+ Add to test✓ In your test40IBMI3S-1-01b-04Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

A tractor is bought for €92 000. Its value decreases by 11% each year.

(a)

Find the value of the tractor after 5 years. Give your answer to the nearest cent.

(2)

(b)

Find the least number of whole years after which the value is below €40 000.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 92000 × 0.89⁵	✓ 1
A1 €51373.35	✓ 1
Part (b)
M1 92000 × 0.89ⁿ < 40000 solved, giving n > 7.15	✓ 1
A1 8 years	✓ 1

Exam tip: A fall of 11% each year means multiplying by 0.89 each year, not subtracting 11% of the original price each time.

+ Add to test✓ In your test41IBMI3S-1-01r5-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A wind farm company installs turbines in stages. It installs 6 turbines in the first year, and in each later year it installs 4 more turbines than in the year before.

(a)

Find the number of turbines installed in year 10.

(2)

(b)

Find the total number of turbines installed during the first 10 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of u_n = u₁ + (n − 1)d with u₁ = 6, d = 4	✓ 1
A1 u₁₀ = 42 turbines	✓ 1
Part (b)
M1 use of S_n = (n/2)(u₁ + u_n) or equivalent, with n = 10 (ECF from (a))	✓ 1
A1 240 turbines	✓ 1

Exam tip: Write down which sequence formula you are using and the values of u₁, d and n before substituting.

+ Add to test✓ In your test42IBMI3S-1-01r5-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

A swimmer plans her training so that she swims 2.0 km in week 1. Her distance in each later week is 10% greater than in the previous week.

(a)

Write down the common ratio of this geometric sequence.

(1)

(b)

Find the distance she swims in week 8. Give your answer to 3 significant figures.

(2)

(c)

Find the total distance she swims in the first 8 weeks. Give your answer to 2 decimal places.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 common ratio = 1.1	✓ 1
Part (b)
M1 use of u_n = u₁rⁿ⁻¹, i.e. 2.0 × 1.1⁷	✓ 1
A1 3.90 km (3 s.f.)	✓ 1
Part (c)
M1 use of S_n = u₁(rⁿ − 1)/(r − 1) with n = 8	✓ 1
A1 22.87 km	✓ 1

Common error: A 10% increase means multiplying by 1.1, not by 0.1 or 1.10 added each week.

+ Add to test✓ In your test43IBMI3S-1-01r5-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

A library holds 2.4 × 10⁵ books. The average thickness of a book is 2.8 cm. All the books are placed side by side on shelves.

(a)

Calculate the total length of shelving needed. Give your answer in kilometres, in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

(3)

(b)

The librarian estimates that 7 km of shelving is needed. Calculate the percentage error in her estimate. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2.4 × 10⁵ × 2.8 = 672000 cm (6.72 × 10⁵ cm)	✓ 1
M1 conversion of cm to km (divide by 100000)	✓ 1
A1 6.72 × 10⁰ km	✓ 1
Part (b)
M1 percentage error = \|estimate − exact\| / exact × 100	✓ 1
A1 4.17%	✓ 1

Exam tip: Convert units before writing the answer in standard form, and keep the exact value in the denominator of a percentage error.

+ Add to test✓ In your test44IBMI3S-1-01r5-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A community fund linked to a wind farm wants to have €5000 available in 6 years' time to buy a new notice board. The money is invested at a rate of 2.8% per year, compounded annually.

(a)

Calculate the amount that must be invested now. Give your answer to the nearest cent.

(3)

(b)

State how much of the €5000 is interest.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 FV = PV × (1 + 2.8/100)⁶ with FV = 5000	✓ 1
M1 PV = 5000 ÷ 1.028⁶	✓ 1
A1 €4236.54	✓ 1
Part (b)
A1 €763.46 (ECF from (a))	✓ 1

Common error: When the future value is known, divide by the growth factor; multiplying 5000 by 1.028⁶ finds a different quantity.

+ Add to test✓ In your test45IBMI3S-1-02r7-01Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

A marine biologist studies plankton in a tank holding 4.8 × 10⁵ litres of water. A sample shows 3.5 × 10³ plankton cells per litre.

(a)

Calculate the total number of plankton cells in the tank. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

In a separate survey, the biologist estimates the length of a whale shark as 12.4 m. Its actual length is 12.65 m. Calculate the percentage error in the estimate, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4.8 × 10⁵ × 3.5 × 10³	✓ 1
A1 1.68 × 10⁹	✓ 1
Part (b)
M1 \|12.4 − 12.65\| / 12.65 × 100	✓ 1
A1 1.98%	✓ 1

Common error: Percentage error is found relative to the exact value, not the estimate.

+ Add to test✓ In your test46IBMI3S-1-02r7-02Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

An architecture firm invests €18 000 in an account that pays a nominal annual interest rate of 3.6%, compounded monthly. No money is added or withdrawn.

(a)

Find the value of the investment after 6 years, correct to the nearest euro.

(2)

(b)

Calculate the interest earned over the 6 years. Use the unrounded value from part (a) and give your answer correct to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18000(1 + 0.036/12)⁷²	✓ 1
A1 €22333	✓ 1
Part (b)
M1 FV − 18000	✓ 1
A1 €4333	✓ 1

Exam tip: With monthly compounding there are 12 periods per year, so 6 years means 72 periods at a rate of 0.036/12 per period.

+ Add to test✓ In your test47IBMI3S-1-02r7-03Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

The seating in a lecture theatre is arranged in 18 rows. The first row has 22 seats and each row has 2 more seats than the row in front of it, so the numbers of seats form an arithmetic sequence.

(a)

Find the number of seats in the 18th row.

(2)

(b)

Find the total number of seats in the theatre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₈ = 22 + 17 × 2	✓ 1
A1 56	✓ 1
Part (b)
M1 S₁₈ = (18/2)(22 + 56)	✓ 1
A1 702	✓ 1

Common error: Check which term the question asks for before substituting: the 18th term uses 17 common differences.

+ Add to test✓ In your test48IBMI3S-1-02r7-04Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

The number of bacteria in a sample from a coral reef is modelled by P = 120 × 2^t/3, where t is the time in hours after the sample is taken.

(a)

Find the number of bacteria after 9 hours.

(2)

(b)

Find the time, in hours, for the number of bacteria to reach 3000. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 120 × 2³	✓ 1
A1 960	✓ 1
Part (b)
M1 2^t/3 = 25	✓ 1
M1 t/3 = log 25 / log 2	✓ 1
A1 13.9 hours	✓ 1

Exam tip: Isolate the exponential term first, then take logarithms of both sides.

+ Add to test✓ In your test49IBMI3S-1-02r10-01Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

A fibre-optic cable of length 8.6 × 10⁴ m links two telephone exchanges. A light signal travels along the cable at a speed of 2.04 × 10⁸ m s⁻¹.

(a)

Calculate the time taken by the signal to travel along the cable. Give your answer in seconds in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ, with a correct to 3 significant figures.

(2)

(b)

Convert your answer to part (a) into microseconds, giving your answer to the nearest whole number. (1 microsecond = 10⁻⁶ s)

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Time = distance ÷ speed = 8.6 × 10⁴ ÷ 2.04 × 10⁸	✓ 1
4.22 × 10⁻⁴ s	✓ 1
Part (b)
Multiply by 10⁶ (or 4.22 × 10⁻⁴ ÷ 10⁻⁶)	✓ 1
422 microseconds	✓ 1

Common error: When converting to a smaller unit the numerical value gets larger: 10^-4 seconds is a hundred microseconds, not a hundredth of one.

+ Add to test✓ In your test50IBMI3S-1-02r10-02Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

The true height of a mountain pass is 6180 m. A climber's altimeter shows a reading of 6250 m.

(a)

Write 6180 m in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

(1)

(b)

The reading of 6250 m is given correct to the nearest 10 m. Write down the lower bound of the actual altimeter reading.

(1)

(c)

Calculate the absolute error in the reading of 6250 m.

(1)

(d)

Find the percentage error in the reading of 6250 m. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
6.18 × 10³ m	✓ 1
Part (b)
6245 m	✓ 1
Part (c)
6250 − 6180 = 70 m	✓ 1
Part (d)
Use of (error ÷ true value) × 100 = 70 ÷ 6180 × 100	✓ 1
1.13%	✓ 1

Exam tip: Percentage error is measured against the true (exact) value, not against the measured value.

+ Add to test✓ In your test51IBMI3S-1-02r10-03Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

On a mountaineering expedition, camp 1 is at an altitude of 3200 m. The altitudes of successive camps form an arithmetic sequence in which each camp is 450 m higher than the one below.

(a)

Find the altitude of camp 6.

(2)

(b)

A camp is at an altitude of 6800 m. Find the number of this camp.

(2)

(c)

Find the height difference between camp 2 and camp 9.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Use of u_n = u₁ + (n − 1)d with n = 6	✓ 1
3200 + 5 × 450 = 5450 m	✓ 1
Part (b)
3200 + (n − 1) × 450 = 6800	✓ 1
n = 9	✓ 1
Part (c)
7 × 450 = 3150 m	✓ 1

Common error: To get from the first term to term n there are n − 1 steps, not n.

+ Add to test✓ In your test52IBMI3S-1-02r10-04Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

In food science the acidity of a liquid is measured by its pH, where pH = −log₁₀[H⁺] and [H⁺] is the concentration of hydrogen ions in mol dm⁻³.

(a)

The concentration of hydrogen ions in a sample of lemon juice is 6.3 × 10⁻³ mol dm⁻³. Calculate its pH. Give your answer to 3 significant figures.

(2)

(b)

A yoghurt has a pH of 3.4. Find the concentration of hydrogen ions in the yoghurt. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitution into −log₁₀(6.3 × 10⁻³)	✓ 1
pH = 2.20	✓ 1
Part (b)
[H⁺] = 10^−3.4	✓ 1
3.98 × 10⁻⁴ mol dm⁻³ (= 0.000398)	✓ 1

Exam tip: Because pH uses a negative logarithm, a lower pH means a higher concentration of hydrogen ions.

+ Add to test✓ In your test53IBMI3S-1-02r12-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy6 marks

A space probe travels at a constant speed of 17.0 km s⁻¹ and must cover a distance of 2.2 × 10¹⁰ km.

(a)

Calculate the time needed, in seconds. Give your answer in the form a × 10^k where 1 ≤ a < 10, with a correct to 3 significant figures.

(2)

(b)

Convert the time to years, using 1 year = 3.156 × 10⁷ s and the unrounded time from (a). Give your answer to 3 significant figures.

(2)

(c)

A planner estimates the time as 1.3 × 10⁹ s. Calculate the percentage error in this estimate, using the unrounded time from (a) as the exact value. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 time = distance ÷ speed = 2.2 × 10¹⁰ ÷ 17.0	✓ 1
A1 1.29 × 10⁹ s	✓ 1
Part (b)
M1 divides the time in seconds by 3.156 × 10⁷	✓ 1
A1 41.0 years	✓ 1
Part (c)
M1 percentage error = \|1.3 × 10⁹ − exact\| ÷ exact × 100	✓ 1
A1 0.455%	✓ 1

Exam tip: Keep the unrounded value on your calculator between parts. Rounding early can change the third significant figure of a later answer.

+ Add to test✓ In your test54IBMI3S-1-02r12-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy6 marks

The vertical hangers on a suspension bridge decrease in length along the span. The lengths form an arithmetic sequence. The first hanger is 24.6 m long and each hanger is 1.8 m shorter than the one before it.

(a)

Find the length of the 12th hanger.

(2)

(b)

Find the number of the first hanger that is shorter than 3.0 m.

(2)

(c)

Find the total length of the first 12 hangers.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₂ = 24.6 + 11(−1.8)	✓ 1
A1 4.8 m	✓ 1
Part (b)
M1 24.6 − 1.8(n − 1) < 3.0, giving n > 13 (the 13th hanger is exactly 3.0 m, so it is not shorter)	✓ 1
A1 n = 14	✓ 1
Part (c)
M1 uses S₁₂ = 12/2 (24.6 + 4.8), or the sum formula with d = −1.8	✓ 1
A1 176.4 m	✓ 1

Common error: Treating 'shorter than' as 'at most'. When a term equals the limit exactly, it does not satisfy a strict inequality.

+ Add to test✓ In your test55IBMI3S-1-02r12-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy5 marks

During the drying stage of coffee roasting, the moisture content of the beans falls by 15% each minute. At the end of the first minute the moisture content is 10.4% (of the mass of the beans). The moisture contents at the end of each minute form a geometric sequence.

(a)

Find the number of the first minute at the end of which the moisture content is below 2.0%.

(3)

(b)

Find the moisture content at the end of the fifth minute. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10.4(0.85)ⁿ⁻¹ < 2.0	✓ 1
M1 solves, e.g. n − 1 > 10.1, or compares u₁₁ = 2.05 with u₁₂ = 1.74	✓ 1
A1 n = 12	✓ 1
Part (b)
M1 10.4 × 0.85⁴	✓ 1
A1 5.43%	✓ 1

Common error: Using 0.15 as the common ratio. A 15% fall each minute means the ratio is 1 − 0.15 = 0.85.

+ Add to test✓ In your test56IBMI3S-1-02r12-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy6 marks

The strength L of a radio signal received from a space probe is measured in decibels (dB) and is given by L = 10 log₁₀(P ÷ 10⁻¹⁸), where P is the received power in watts.

(a)

Calculate L when P = 3.2 × 10⁻¹⁷ W. Give your answer to 3 significant figures.

(2)

(b)

Find the power received when L = 9.0 dB. Give your answer in the form a × 10^k where 1 ≤ a < 10, with a correct to 3 significant figures.

(2)

(c)

Show that when the received power is divided by 4, the signal strength decreases by approximately 6.0 dB.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 L = 10 log₁₀(32)	✓ 1
A1 15.1 dB	✓ 1
Part (b)
M1 P ÷ 10⁻¹⁸ = 10^0.9	✓ 1
A1 7.94 × 10⁻¹⁸ W	✓ 1
Part (c)
M1 change in L = 10 log₁₀(1/4), or 10 log₁₀(P) − 10 log₁₀(4P) = −10 log₁₀4	✓ 1
A1 −6.02 dB, a decrease of 6.02 dB, which is approximately 6.0 dB	✓ 1

Exam tip: The logarithm of a quotient becomes a difference. A fixed factor on the power gives a fixed change in decibels, whatever the starting power.

+ Add to test✓ In your test57IBMI3S-1-02r14-01Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

The probe Kestrel travels at a constant speed of 17.2 km s⁻¹. It still has 6.4 × 10⁹ km to travel to reach its target. Use 1 year = 3.156 × 10⁷ s.

(a)

Calculate the time needed, in seconds. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(2)

(b)

Convert your answer to part (a) into years. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\frac{6.4\times10^{9}}{17.2}$	✓ 1
A1 3.72 × 10⁸ s	✓ 1
Part (b)
M1 $\frac{3.72\times10^{8}}{3.156\times10^{7}}$ (or the unrounded value from (a))	✓ 1
A1 11.8 years	✓ 1

Common error: Check the exponent when dividing numbers in standard form: 10⁹ ÷ 10⁰ stays 10⁹, and the unit conversion then moves it to 10⁸ in seconds.

+ Add to test✓ In your test58IBMI3S-1-02r14-02Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

A surveyor measures the length of a bridge deck as 148 m, correct to the nearest metre. The design specification for the deck is 147.6 m.

(a)

Write down the lower bound and the upper bound for the length of the deck, given the surveyor's measurement.

(2)

(b)

Show that the design length of 147.6 m is consistent with the surveyor's measurement.

(1)

(c)

Treating 147.6 m as the exact value, find the percentage error in the surveyor's measurement of 148 m. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 lower bound 147.5 m	✓ 1
A1 upper bound 148.5 m	✓ 1
Part (b)
R1 147.5 ≤ 147.6 < 148.5, so 147.6 m lies within the bounds	✓ 1
Part (c)
M1 $\frac{\|148-147.6\|}{147.6}\times100$	✓ 1
A1 0.271%	✓ 1

Exam tip: Percentage error divides by the exact value, not by the measured value.

+ Add to test✓ In your test59IBMI3S-1-02r14-03Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

During a coffee roast, the bean temperature at the end of minute 1 is 165 °C and rises by 7.5 °C each minute after that, so the temperatures at the end of each minute form an arithmetic sequence.

(a)

Find the first minute at whose end the bean temperature exceeds 215 °C.

(3)

(b)

Find the bean temperature at the end of minute 10.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $165+7.5(n-1)>215$	✓ 1
A1 $n>7.67$ (or $n-1>6.67$)	✓ 1
A1 $n=8$ (temperature 217.5 °C at the end of minute 8)	✓ 1
Part (b)
M1 $165+9\times7.5$	✓ 1
A1 232.5 °C	✓ 1

Common error: The nth term is u₁ + (n − 1)d. Using nd instead of (n − 1)d overshoots by one step.

+ Add to test✓ In your test60IBMI3S-1-02r14-04Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

The electrical power output of a probe's radioisotope generator is modelled as decreasing by 1% each year. At launch the output is 480 W. Let P_n be the output, in watts, after n years.

(a)

Write down the common ratio of the sequence of yearly power outputs and find the power output after 10 years. Give your answer to 3 significant figures.

(2)

(b)

Find the number of years after launch at which the output has fallen to 400 W. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 common ratio 0.99 (so $P_n=480\times0.99^n$)	✓ 1
A1 434 W	✓ 1
Part (b)
M1 $480\times0.99^{t}=400$	✓ 1
M1 take logarithms: $t=\frac{\ln(400/480)}{\ln0.99}$	✓ 1
A1 18.1 years	✓ 1

Common error: A 1% decrease multiplies by 0.99, not 0.01. Check that the ratio is below 1 for decay.

+ Add to test✓ In your test61IBMI3S-1-01r18-01Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A fish farm estimates that its reservoir holds 3.60×10⁴ fish. A full count later shows that the actual number of fish is 3.95×10⁴.

(a)

Calculate the percentage error in the estimate. Give your answer to 3 significant figures.

(2)

(b)

Each fish is fed 0.040 kg of food per day. Calculate the total mass of food needed per day for the actual number of fish. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 \|3.60×10⁴ − 3.95×10⁴\| ÷ 3.95×10⁴ × 100	✓ 1
A1 8.86 %	✓ 1
Part (b)
M1 3.95×10⁴ × 0.040	✓ 1
A1 1.58×10³ kg	✓ 1

Exam tip: Percentage error is always found relative to the exact (actual) value, not the estimate.

+ Add to test✓ In your test62IBMI3S-1-01r18-02Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy7 marks

A warehouse robot travels 1.20 km on its first day of operation. On each following day it travels 0.15 km further than on the previous day, so the daily distances form an arithmetic sequence.

(a)

Find the distance the robot travels on day 20.

(2)

(b)

Find the first day on which the robot travels more than 6.1 km.

(3)

(c)

Find the total distance travelled by the robot in the first 20 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₂₀ = 1.2 + 19 × 0.15	✓ 1
A1 4.05 km	✓ 1
Part (b)
M1 1.2 + 0.15(n − 1) > 6.1	✓ 1
A1 n > 33.67 (or n − 1 > 32.67)	✓ 1
A1 day 34	✓ 1
Part (c)
M1 S₂₀ = (20/2)(2 × 1.2 + 19 × 0.15) or (20/2)(1.2 + 4.05)	✓ 1
A1 52.5 km	✓ 1

Common error: Solving 1.2 + 0.15n > 6.1 instead of 1.2 + 0.15(n − 1) > 6.1 gives the wrong day, because the first term is day 1, not day 0.

+ Add to test✓ In your test63IBMI3S-1-01r18-03Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

A fashion boutique wants to have €15 000 available in 5 years to fund a new collection. It invests a sum of money now in an account that pays a nominal annual interest rate of 3.0 %, compounded monthly. No further deposits or withdrawals are made.

(a)

Calculate the amount that must be invested now. Give your answer to the nearest euro.

(3)

(b)

The owner notes that the same growth could come from an account that is compounded once a year at an annual rate of r %. Find r, giving your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 15000 = PV(1 + 0.030/12)⁶⁰ (or TVM solver with N = 60, I% = 3.0, FV = 15000, C/Y = 12)	✓ 1
A1 correct values N = 60 and r = 0.030/12 used	✓ 1
A1 €12913	✓ 1
Part (b)
M1 (1 + r/100) = (1 + 0.030/12)¹²	✓ 1
A1 3.04 %	✓ 1

Exam tip: For compounding questions, write down the number of compounding periods and the rate per period before substituting.

+ Add to test✓ In your test64IBMI3S-1-01r18-04Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy7 marks

Algae are growing on the surface of a fish pond. The area covered by algae, A cm², after t days is modelled by
A = 40 × 2^t/3, t ≥ 0.

(a)

State the initial area covered by algae and the number of days it takes for the area to double.

(2)

(b)

Find the area covered after 10 days. Give your answer to the nearest cm².

(2)

(c)

Find the time taken for the area to reach 1000 cm². Give your answer in days to 4 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 40 cm²	✓ 1
A1 3 days	✓ 1
Part (b)
M1 40 × 2^10/3	✓ 1
A1 403 cm²	✓ 1
Part (c)
M1 40 × 2^t/3 = 1000, so 2^t/3 = 25	✓ 1
M1 correct use of logarithms, e.g. t/3 = ln 25 ÷ ln 2	✓ 1
A1 13.93 days	✓ 1

Exam tip: Isolate the exponential term first, then take logarithms of both sides; the base of the logarithm does not affect the answer.

+ Add to test✓ In your test65IBMI3S-1-01r19-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy8 marks

A sprinter's coach plans a training block in which the distance run each day increases by the same amount. On day 1 the sprinter runs 400 m, and on each following day she runs 50 m further than on the day before.

(a)

Find the distance she runs on day 12.

(2)

(b)

Find the first day on which she runs more than 2000 m.

(3)

(c)

Find the total distance she runs in the first 12 days.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use u_n = u₁ + (n − 1)d with u₁ = 400, d = 50, n = 12	✓ 1
A1 950 m	✓ 1
Part (b)
M1 set up 400 + 50(n − 1) > 2000	✓ 1
A1 n − 1 > 32 (or n > 33; day 33 gives exactly 2000 m, which does not exceed 2000)	✓ 1
A1 day 34	✓ 1
Part (c)
M1 use S_n = (n/2)(u₁ + u_n) or (n/2)(2u₁ + (n − 1)d)	✓ 1
M1 substitute n = 12 with 400 and 950 (or d = 50) (ECF from (a))	✓ 1
A1 8100 m	✓ 1

Common error: In 'more than' questions, check the boundary term: if u₃₃ equals the limit exactly it does not satisfy the inequality.

+ Add to test✓ In your test66IBMI3S-1-01r19-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy7 marks

A cyclist rides 2.35 × 10⁴ m in 1.8 × 10³ s at a constant speed.

(a)

Calculate her speed in m s⁻¹, to 3 significant figures.

(2)

(b)

A coach estimates the speed using the rounded values 2.4 × 10⁴ m and 2 × 10³ s, giving 12 m s⁻¹. Taking the exact speed as 23 500 ÷ 1800, find the percentage error in the coach's estimate. Give your answer to 3 significant figures.

(3)

(c)

Write the time of 1.8 × 10³ s in hours, giving your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 speed = distance ÷ time = 23 500 ÷ 1800	✓ 1
A1 13.1 m s⁻¹	✓ 1
Part (b)
M1 correct use of \|estimate − exact\| / exact × 100	✓ 1
M1 substitute 12 and 13.0555… (unrounded)	✓ 1
A1 8.09%	✓ 1
Part (c)
M1 divide by 3600: 1800 ÷ 3600 = 0.5	✓ 1
A1 5 × 10⁻¹ h	✓ 1

Exam tip: Percentage error is always measured relative to the exact value, not the approximate one.

+ Add to test✓ In your test67IBMI3S-1-01r19-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy7 marks

The seating at a small athletics stadium is arranged in rows. The first row has 24 seats and each row has 2 more seats than the row in front of it. The stadium owners want at least 2000 seats in total.

(a)

Find the least number of rows needed.

(3)

(b)

Find the number of seats in the last of these rows.

(2)

(c)

Find the total number of seats in these rows.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 set up S_n = (n/2)(2 × 24 + (n − 1) × 2) ≥ 2000, i.e. n(n + 23) ≥ 2000	✓ 1
A1 n ≥ 34.6… (or S₃₄ = 1938 and S₃₅ = 2030)	✓ 1
A1 35 rows	✓ 1
Part (b)
M1 u₃₅ = 24 + 34 × 2	✓ 1
A1 92 seats	✓ 1
Part (c)
M1 S₃₅ = (35/2)(24 + 92) (ECF from (a) and (b))	✓ 1
A1 2030 seats	✓ 1

Exam tip: Keep the form Sn = (n/2)(2u₁ + (n − 1)d) when the last term is unknown.

+ Add to test✓ In your test68IBMI3S-1-01r19-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy8 marks

Astronomers compare the apparent brightness of stars using magnitudes. If star 1 has apparent brightness b₁ and star 2 has apparent brightness b₂, their magnitudes satisfy m₂ − m₁ = 2.5 log₁₀(b₁ / b₂).

(a)

Star A appears 25 times brighter than star B. Find m_B − m_A, to 3 significant figures.

(2)

(b)

Two stars differ in magnitude by 6.0. Find the ratio of the brighter star's brightness to the fainter star's brightness, to 3 significant figures.

(3)

(c)

Sirius has magnitude −1.46 and Polaris has magnitude 1.98. Find how many times brighter Sirius appears than Polaris, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2.5 log₁₀(25)	✓ 1
A1 3.49	✓ 1
Part (b)
M1 set up 2.5 log₁₀(R) = 6.0	✓ 1
M1 R = 10^2.4	✓ 1
A1 251	✓ 1
Part (c)
M1 1.98 − (−1.46) = 3.44 = 2.5 log₁₀(ratio)	✓ 1
M1 ratio = 10^3.44/2.5 = 10^1.376	✓ 1
A1 23.8	✓ 1

Common error: A smaller (or more negative) magnitude means a brighter star; check the sign before substituting.

+ Add to test✓ In your test69IBMI3S-1-01r21-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy7 marks

Calculator

A beekeeper records the honey collected from an apiary each week. In week 1 the apiary yields 18 kg, and the yield rises by 2.5 kg in each following week. The weekly yields form an arithmetic sequence.

(a)

Find the yield in week 12.

(2)

(b)

The beekeeper wants the total honey collected since week 1 to exceed 600 kg. Find the first week in which this happens.

(3)

(c)

The yield in week 12 is greater than the yield in week 1. Find this increase as a percentage of the week 1 yield.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $u_n = u_1 + (n-1)d$ with $n=12$, $d=2.5$	✓ 1
A1 $u_{12} = 45.5$ kg	✓ 1
Part (b)
M1 use of $S_n=\frac{n}{2}(2u_1+(n-1)d)$ with an inequality or by trying values of $n$	✓ 1
A1 $S_{16}=588$ (below 600) or $S_{17}=646$	✓ 1
A1 week 17 (since \(S_{16}<600	✓ 1
Part (c)
M1 $\frac{45.5-18}{18}\times100$	✓ 1
A1 $153\%$ (3 s.f.)	✓ 1

Exam tip: Writing both $S_{16}$ and $S_{17}$ shows clearly why the answer is week 17 rather than week 16.

+ Add to test✓ In your test70IBMI3S-1-01r21-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

Calculator

The pressure in a Formula 1 spare tyre slowly falls as air leaks out. The pressure $P$ bar, $t$ hours after inflation, is modelled by $P = 1.85\times 0.992^{t}$.

(a)

Find the pressure after 6 hours. Give your answer to 3 significant figures.

(2)

(b)

The team will not use the tyre once the pressure drops to 1.60 bar. Find how long this takes. Give your answer in hours to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute $t=6$ into $P=1.85\times0.992^{t}$	✓ 1
A1 $P=1.76$ bar	✓ 1
Part (b)
M1 $1.85\times0.992^{t}=1.60$	✓ 1
M1 take logarithms: $t=\frac{\ln(1.60/1.85)}{\ln 0.992}$	✓ 1
A1 $t=18.1$ hours	✓ 1

Common error: Do not divide $\ln(1.60)$ by $\ln(1.85\times0.992)$; first divide both sides by 1.85 so that the base is only 0.992.

+ Add to test✓ In your test71IBMI3S-1-01r21-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

Calculator

An adult has 5.1 litres of blood. The blood contains 5.2 × 10⁶ red blood cells in each microlitre. Treat the given values as exact. (1 litre = 10⁶ microlitres)

(a)

Calculate the total number of red blood cells in the blood. Give your answer in the form $a\times10^{k}$ where $1\le a<10$ and $k$ is an integer, with $a$ correct to 3 significant figures.

(3)

(b)

A technician quickly estimates the total by using $5\times10^{6}$ for both values. Find the percentage error in the technician's estimate, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 convert: 5.1 litres = $5.1\times10^{6}$ microlitres	✓ 1
M1 $5.2\times10^{6}\times5.1\times10^{6}$	✓ 1
A1 $2.65\times10^{13}$	✓ 1
Part (b)
M1 $\frac{\|2.5\times10^{13}-2.652\times10^{13}\|}{2.652\times10^{13}}\times100$	✓ 1
A1 $5.73\%$	✓ 1

Exam tip: Percentage error is measured against the exact value, so use the unrounded $2.652\times10^{13}$ in the denominator.

+ Add to test✓ In your test72IBMI3S-1-02r21-01Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy4 marks

Calculator

A hospital outpatient clinic starts a vaccination programme. It sees 45 patients on day 1, and on each later day it sees 3 more patients than on the previous day. The daily numbers form an arithmetic sequence.

(a)

Find the number of patients seen on day 20.

(2)

(b)

Find the total number of patients seen during the first 20 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_{20}=45+19\times3$	✓ 1
A1 102 patients	✓ 1
Part (b)
M1 $S_{20}=\frac{20}{2}(45+102)$	✓ 1
A1 1470 patients	✓ 1

Exam tip: Check that $u_{20}$ is the 20th term, which uses $19$ common differences, not 20.

+ Add to test✓ In your test73IBMI3S-1-02r21-02Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy4 marks

Calculator

A beekeepers' association invests £8000 in a savings account that pays 3.2% per year, compounded quarterly. No money is added or withdrawn.

(a)

Find the value of the investment after 6 years, to the nearest penny.

(2)

(b)

A different account pays simple interest of 3.2% per year on the original £8000. Find how much more interest the compound account earns over the 6 years, to the nearest penny.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance solver with $N=24$ (or $N=6$ with $C/Y=4$), $I=3.2$, $PV=-8000$, $PMT=0$, $P/Y=C/Y=4$	✓ 1
A1 £9685.96	✓ 1
Part (b)
M1 simple interest $=8000\times0.032\times6=1536$ and compound interest $=9685.96-8000$	✓ 1
A1 £149.96	✓ 1

Common error: Simple interest is always calculated on the original amount; compound interest is calculated on the growing balance, which is why the values differ.

+ Add to test✓ In your test74IBMI3S-1-02r21-03Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy5 marks

Calculator

A hospital laboratory monitors a virus sample. The number of virus copies is modelled by $V=200\times3^{t}$, where $t$ is the time in days after monitoring starts.

(a)

Find the number of virus copies after 4 days.

(2)

(b)

Find the time when the sample contains 150 000 copies. Give your answer in days to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $200\times3^{4}$	✓ 1
A1 16 200	✓ 1
Part (b)
M1 $200\times3^{t}=150000$, so $3^{t}=750$	✓ 1
M1 $t=\frac{\ln750}{\ln3}$	✓ 1
A1 6.03 days	✓ 1

Exam tip: Isolate the exponential term first ($3^{t}=750$) before taking logarithms.

+ Add to test✓ In your test75IBMI3S-1-03r22-01Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy6 marks

A rail operator places €250 000 in an account that pays a nominal annual interest rate of 3.2%, compounded quarterly. No further deposits or withdrawals are made.

(a)

Find the value of the account after 6 years. Give your answer to the nearest cent.

(2)

(b)

Calculate the interest earned over the 6 years.

(1)

(c)

Find the number of years, to 3 significant figures, for the account to grow to €300 000. Assume the formula still applies when the time is not a whole number of quarters.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitution into compound interest formula with rate 0.8% per quarter and 24 quarters, e.g. 250000(1 + 0.032/4)²⁴	✓ 1
A1 302686.31	✓ 1
Part (b)
A1 302686.31 − 250000 = 52686.31 (use the unrounded value from (a))	✓ 1
Part (c)
M1 equation 250000(1.008)^4t = 300000	✓ 1
M1 taking logarithms (or using a finance solver) to solve for t, e.g. 4t = ln 1.2 / ln 1.008	✓ 1
A1 5.72 years	✓ 1

Common error: The interest rate per period is the nominal rate divided by the number of compounding periods per year, and the number of periods is years × periods per year; mixing these up is the most frequent slip.

+ Add to test✓ In your test76IBMI3S-1-03r22-04Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy6 marks

A rail freight company buys a wagon for €96 000. Each year the value of the wagon decreases by 12% of its value at the start of that year.

(a)

Find the value of the wagon after 5 years, to the nearest cent.

(2)

(b)

Find the total loss in value over these 5 years, to the nearest cent.

(1)

(c)

The company will sell the wagon when its value first falls to €30 000. Find how many years this takes, to 3 significant figures. Assume the model applies for non-integer times.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 96000 × 0.88⁵	✓ 1
A1 €50662.26	✓ 1
Part (b)
A1 96000 − 50662.26 = €45337.74	✓ 1
Part (c)
M1 equation 96000 × 0.88^t = 30000	✓ 1
M1 logarithms: $t = \dfrac{\ln(30000/96000)}{\ln 0.88}$ (or finance solver)	✓ 1
A1 9.10 years	✓ 1

Common error: A 12% decrease each year multiplies the value by 0.88, not by 0.12 and not by 1.12.

+ Add to test✓ In your test77IBMI3S-1-03r22-05Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy6 marks

The mean distance from the Sun to the Earth is 1.496 × 10⁸ km. Light travels at 3.00 × 10⁵ km s⁻¹.

(a)

Calculate the time taken for light to travel from the Sun to the Earth. Give your answer in seconds, in the form a × 10^k where 1 ≤ a < 10 and a is given to 3 significant figures.

(2)

(b)

Convert your unrounded time from (a) to minutes, giving your answer to 3 significant figures.

(1)

(c)

A student uses 500 s as the time. Using the unrounded time from (a) as the exact value, find the percentage error in the student's value. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 time = distance ÷ speed = 1.496 × 10⁸ ÷ 3.00 × 10⁵	✓ 1
A1 4.99 × 10² s (unrounded 498.67 s)	✓ 1
Part (b)
A1 498.67 ÷ 60 = 8.31 minutes	✓ 1
Part (c)
M1 absolute error \|500 − 498.67\| (using unrounded time)	✓ 1
M1 divides by exact value and multiplies by 100	✓ 1
A1 0.267 %	✓ 1

Exam tip: Percentage error compares the error with the exact value, not with the approximate value.

+ Add to test✓ In your test78IBMI3S-1-03r22-08Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy7 marks

An athlete returns from injury. In week 1 of her training programme she runs 4 km in total. In each following week she runs 0.75 km more than in the previous week, so the weekly distances form an arithmetic sequence.

(a)

Find the distance she runs in week 12.

(2)

(b)

Find the first week in which she runs more than 15 km.

(3)

(c)

Find the total distance she runs in the weeks 1 to 16.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_{12} = 4 + 11 \times 0.75$	✓ 1
A1 12.25 km	✓ 1
Part (b)
M1 4 + 0.75(n − 1) > 15	✓ 1
A1 n > 14.67 (14.666…)	✓ 1
A1 week 16	✓ 1
Part (c)
M1 $S_{16} = \dfrac{16}{2}(2 \times 4 + 15 \times 0.75)$	✓ 1
A1 154 km	✓ 1

Common error: Write 4 + 0.75(n − 1), not 4 + 0.75n: the first term has no common difference added to it.

+ Add to test✓ In your test79IBMI3S-1-02r23-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 1Easy4 marks

A virus particle is modelled as a sphere of diameter 1.2 × 10⁻⁷ m. A scientist wants to know how many such particles, placed side by side in a straight line, would fit across a length of 1.0 mm.

(a)

Calculate the number of particles, giving your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(2)

(b)

The scientist estimates the number as 8000. Find the percentage error in this estimate, taking the exact value as 1.0 × 10⁻³ ÷ 1.2 × 10⁻⁷.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1.0 × 10⁻³ ÷ 1.2 × 10⁻⁷	✓ 1
A1 8.33 × 10³	✓ 1
Part (b)
M1 percentage error = (8333.33… − 8000) ÷ 8333.33… × 100	✓ 1
A1 4.00%	✓ 1

Exam tip: Keep the unrounded value in your calculator when you use it as the exact value in a percentage error.

+ Add to test✓ In your test80IBMI3S-1-02r23-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 1Easy4 marks

A shop sells cheese at €7.95 per kilogram. A customer buys 2.35 kg.

(a)

Calculate the exact cost of the cheese, and state it to the nearest cent.

(2)

(b)

At the till the shop charges €18.70, which is the exact cost rounded to the nearest 5 cents. Find the percentage error in the amount charged, relative to the exact cost, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 7.95 × 2.35 = 18.6825	✓ 1
A1 €18.68	✓ 1
Part (b)
M1 (18.70 − 18.6825) ÷ 18.6825 × 100	✓ 1
A1 0.0937%	✓ 1

Exam tip: Percentage error is divided by the exact value, not by the approximate value.

+ Add to test✓ In your test81IBMI3S-1-02r23-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 1Easy5 marks

A supermarket builds a display of tins in rows. The top row has 4 tins and each row below it has 3 more tins than the row above. The bottom row has 49 tins.

(a)

Show that the display has 16 rows.

(2)

(b)

Find the total number of tins in the display.

(2)

(c)

Each tin sells for €1.35. Find the total selling price of all the tins in the display.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4 + 3(n − 1) = 49	✓ 1
A1 n = 16 (shown)	✓ 1
Part (b)
M1 16/2 × (4 + 49)	✓ 1
A1 424	✓ 1
Part (c)
A1 €572.40	✓ 1

Exam tip: Check the number of terms before using the sum formula; the sum of an arithmetic sequence needs n, the first term and the last term.

+ Add to test✓ In your test82IBMI3S-1-03r23-01Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

An archaeologist measures the length of a buried stone wall. The true length of the wall is 12.84 m. Before excavation she estimated the length to be 12.5 m.

(a)

Write the true length of the wall in millimetres, in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Calculate the percentage error in her estimate. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 multiplies by 1000 to obtain 12 840 mm	✓ 1
A1 1.284 × 10⁴ mm	✓ 1
Part (b)
M1 finds the difference \|12.5 − 12.84\| = 0.34	✓ 1
M1 divides by the true value 12.84 and multiplies by 100: (0.34 / 12.84) × 100	✓ 1
A1 2.65 %	✓ 1

Common error: Percentage error is measured relative to the true (exact) value, not relative to the estimate.

+ Add to test✓ In your test83IBMI3S-1-03r23-02Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

A cyclist begins a training plan by riding 40 km in week 1. In each following week she rides 5% further than in the previous week, so the weekly distances form a geometric sequence.

(a)

Find the first week in which she rides more than 80 km.

(3)

(b)

Find the total distance she rides in the first 16 weeks. Give your answer to the nearest kilometre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses u_n = 40 × 1.05ⁿ⁻¹ with ratio 1.05	✓ 1
M1 solves 40 × 1.05ⁿ⁻¹ > 80, e.g. n − 1 > ln 2 / ln 1.05 ≈ 14.2 (or from a table: week 15 gives 79.2 km, week 16 gives 83.2 km)	✓ 1
A1 week 16	✓ 1
Part (b)
M1 uses S₁₆ = 40(1.05¹⁶ − 1) / (1.05 − 1)	✓ 1
A1 946 km (FT from (a) if a different number of weeks was found)	✓ 1

Exam tip: When a sequence question asks for the first term that passes a value, check the terms either side of your solution to confirm the answer.

+ Add to test✓ In your test84IBMI3S-1-03r23-03Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy7 marks

Isabel invests €8000 in an account that pays a nominal annual interest rate of 3.6%, compounded monthly. No money is added or withdrawn.

(a)

Find the value of the investment after 7 years. Give your answer to the nearest cent.

(3)

(b)

Find the smallest whole number of months after which the investment is worth more than €16 000.

(3)

(c)

Calculate the interest earned during the first 7 years. Use your answer to part (a).

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses a compound interest formula or finance solver with r = 3.6, C/Y = 12	✓ 1
A1 N = 7 × 12 = 84 compounding periods	✓ 1
A1 €10 288.89	✓ 1
Part (b)
M1 sets up 8000 × (1.003)ⁿ = 16 000 or FV = −16 000 in a solver	✓ 1
M1 solves for n to get 231.4… (or table: 231 months gives €15 981.06, 232 months gives €16 029.00)	✓ 1
A1 232 months	✓ 1
Part (c)
A1 €10 288.89 − €8000 = €2288.89	✓ 1

Common error: Dividing the annual rate by 12 and also using 7 compounding periods mixes monthly and yearly units; use 84 monthly periods with 0.3% per month.

+ Add to test✓ In your test85IBMI3S-1-03r23-07Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy7 marks

The seating in an ancient amphitheatre is arranged in rows. The first row has 36 seats and every row has 4 more seats than the row in front of it, so the numbers of seats form an arithmetic sequence.

(a)

Find the number of seats in row 20.

(2)

(b)

A reconstruction plan requires a total capacity of at least 2500 seats. Find the smallest number of rows needed.

(3)

(c)

Find the total number of seats when the amphitheatre has the number of rows found in part (b).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses u_n = 36 + (n − 1) × 4 with n = 20	✓ 1
A1 112 seats	✓ 1
Part (b)
M1 uses S_n = (n/2)(2 × 36 + (n − 1) × 4) = 2n² + 34n ≥ 2500	✓ 1
M1 solves the inequality, e.g. by table or graph (27 rows give 2376, 28 rows give 2520)	✓ 1
A1 28 rows	✓ 1
Part (c)
M1 uses S₂₈ = (28/2)(2 × 36 + 27 × 4) (FT from (b))	✓ 1
A1 2520 seats	✓ 1

Common error: In a sum formula, n is the number of terms, not the last term's value; check that your n is a whole number of rows.

+ Add to test✓ In your test86IBMI3S-1-01r24-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

During a descent, the altitude of a passenger aircraft, recorded at the end of each minute, forms an arithmetic sequence. The first three readings are 9800 m, 9350 m and 8900 m.

(a)

State the common difference of the sequence.

(1)

(b)

Find the altitude of the aircraft at the end of the 10th minute.

(2)

(c)

The aircraft must be at an altitude of 2600 m to begin its final approach. Find the minute in which the reading first equals 2600 m.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 d = −450 (m per minute)	✓ 1
Part (b)
M1 u₁₀ = 9800 + 9(−450)	✓ 1
A1 5750 m	✓ 1
Part (c)
M1 setting 9800 + (n − 1)(−450) = 2600	✓ 1
A1 (n − 1) = 16 (or 450(n − 1) = 7200)	✓ 1
A1 n = 17	✓ 1

Exam tip: In a decreasing arithmetic sequence the common difference is negative; keep the sign when you substitute into un = u₁ + (n − 1)d.

+ Add to test✓ In your test87IBMI3S-1-01r24-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

A farmer places €18 000 in a fund to replace a grain dryer. The fund pays a nominal annual interest rate of 3.6 %, compounded monthly. No money is added or withdrawn.

(a)

Find the value of the fund after 5 years. Give your answer to the nearest euro.

(3)

(b)

Hence find the interest earned over the 5 years. Use the unrounded value from part (a) and give your answer to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of compound interest formula or TVM solver with monthly rate 3.6 ÷ 12 = 0.3 %	✓ 1
A1 correct values: n = 60 (or N = 5, k = 12), PV = 18 000, r = 3.6	✓ 1
A1 €21 544	✓ 1
Part (b)
M1 21 544.106… − 18 000	✓ 1
A1 €3544 (ECF from (a) if the unrounded value is used)	✓ 1

Common error: Do not use 3.6 % as the rate for each month; the nominal rate must be divided by 12 when it is compounded monthly.

+ Add to test✓ In your test88IBMI3S-1-01r24-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

A grain silo holds 1.8 × 10⁵ kg of wheat. The mass of one grain of wheat is 3.5 × 10⁻⁵ kg.

(a)

Calculate the number of grains in the silo. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(2)

(b)

A farm has 24 silos, each holding this mass of wheat. Find the total number of grains on the farm. Use the unrounded answer from part (a) and give your answer in the form a × 10^k with a correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (1.8 × 10⁵) ÷ (3.5 × 10⁻⁵)	✓ 1
A1 5.14 × 10⁹	✓ 1
Part (b)
M1 24 × 5.142857… × 10⁹	✓ 1
A1 1.23 × 10¹¹	✓ 1

Exam tip: Keep the full calculator value for the next part and round only the final answer; rounding early can change the third significant figure.

+ Add to test✓ In your test89IBMI3S-1-01r24-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

The frequency f of a note that is n semitones above the note A₄ (frequency 440 Hz) on a piano is modelled by
f = 440 × 2^n/12, where f is in hertz.

(a)

Calculate the frequency of the note that is 7 semitones above A₄. Give your answer to the nearest hertz.

(2)

(b)

A synthesiser is set to produce a tone of frequency 1000 Hz. Find the value of n that corresponds to this frequency. Give your answer to 3 significant figures.

(3)

(c)

Explain why a piano cannot play a note of exactly 1000 Hz according to this model.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 440 × 2^7/12	✓ 1
A1 659 Hz	✓ 1
Part (b)
M1 440 × 2^n/12 = 1000	✓ 1
M1 correct method to solve, e.g. n = 12 log₂(1000 ÷ 440) or use of a solver	✓ 1
A1 n = 14.2	✓ 1
Part (c)
R1 n is not a whole number, but piano notes are a whole number of semitones apart	✓ 1

Exam tip: Isolate the power first by dividing both sides by 440, then take logarithms or use your solver; do not divide the exponent by 440.

+ Add to test✓ In your test90IBMI3S-1-01r26-01Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

The summit of Mount Ardel is 4 250 m above sea level. A climber's handheld altimeter shows the summit height as 4 300 m.

(a)

Write 4 250 m in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

(1)

(b)

Calculate the percentage error in the altimeter reading. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4.25 × 10³ (m)	✓ 1
Part (b)
M1 attempt at the difference \|4300 − 4250\| = 50	✓ 1
M1 dividing by the exact value 4250 and multiplying by 100	✓ 1
A1 1.18 (%)	✓ 1

Common error: Percentage error is found relative to the exact value, not the measured value.

+ Add to test✓ In your test91IBMI3S-1-01r26-02Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

A guided expedition sets up a series of camps on its route. Camp 1 is at an altitude of 3 200 m. Each later camp is 350 m higher than the previous one, so the camp altitudes form an arithmetic sequence.

(a)

Find the altitude of Camp 6.

(2)

(b)

The highest camp is at an altitude of 6 350 m. Find the total number of camps on the route.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₆ = 3200 + 5 × 350	✓ 1
A1 4950 (m)	✓ 1
Part (b)
M1 3200 + (n − 1) × 350 = 6350	✓ 1
M1 correct manipulation, (n − 1) = 9	✓ 1
A1 10 (camps)	✓ 1

Exam tip: State the sequence formula you are using before substituting; it earns the method mark even if the arithmetic slips.

+ Add to test✓ In your test92IBMI3S-1-01r26-08Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

Air pressure decreases as altitude increases. At sea level the pressure is 101 kPa. At each successive altitude of 1 000 m, 2 000 m, 3 000 m, … the pressure is 0.88 times the pressure 1 000 m lower. The pressures at 0 m, 1 000 m, 2 000 m, … form a geometric sequence.

(a)

Find the pressure at an altitude of 4 000 m. Give your answer to 3 significant figures.

(2)

(b)

Show that 8 000 m is the first of these altitudes at which the pressure is below 40 kPa.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 101 × 0.88⁴	✓ 1
A1 60.6 (kPa)	✓ 1
Part (b)
M1 pressure at 7 000 m = 101 × 0.88⁷	✓ 1
A1 41.3 (kPa), which is greater than 40	✓ 1
A1 pressure at 8 000 m = 101 × 0.88⁸ = 36.3 (kPa), which is less than 40, so 8 000 m is the first	✓ 1

Common error: The sea-level pressure is the first term, u₀ here, so the pressure at 4 000 m uses the power 4, not 3.

+ Add to test✓ In your test93IBMI3S-1-02r26-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A space probe is 1.62 × 10¹⁰ km from Earth. Radio signals travel at 3.00 × 10⁵ km s⁻¹.

(a)

Calculate the time taken for a signal to travel from the probe to Earth. Give your answer in seconds in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

(2)

(b)

Convert your answer to part (a) into hours.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1.62 × 10¹⁰ ÷ 3.00 × 10⁵	✓ 1
A1 5.4 × 10⁴ (s)	✓ 1
Part (b)
M1 54000 ÷ 3600	✓ 1
A1 15 (hours)	✓ 1

Exam tip: Write the unit with every intermediate value; here km cancels but seconds must be converted before hours.

+ Add to test✓ In your test94IBMI3S-1-02r26-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

A coffee roaster buys a roasting machine for €26 000. Its value falls by 12 % each year.

(a)

Find the value of the machine after 5 years. Give your answer to the nearest euro.

(2)

(b)

The roaster will replace the machine when its value is below €10 000. Show that this happens after 8 years.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 26000 × 0.88⁵	✓ 1
A1 €13721	✓ 1
Part (b)
M1 value after 7 years = 26000 × 0.88⁷ = €10625.57, which is greater than 10000	✓ 1
A1 value after 8 years = 26000 × 0.88⁸ = €9350.50, which is less than 10000	✓ 1
A1 so the value first falls below €10000 after 8 years (AG)	✓ 1

Common error: A fall of 12 % multiplies by 0.88, not 0.12.

+ Add to test✓ In your test95IBMI3S-1-02r26-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A batch of green coffee beans of mass 12.0 kg loses 15 % of its mass during roasting.

(a)

Calculate the mass of the roasted beans.

(2)

(b)

The scale in the roastery displays the roasted mass as 10.0 kg. Find the percentage error in this reading. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 12.0 × 0.85	✓ 1
A1 10.2 (kg)	✓ 1
Part (b)
M1 \|10.0 − 10.2\| ÷ 10.2 × 100	✓ 1
A1 1.96 (%)	✓ 1

Exam tip: Use the exact value in the denominator of a percentage error, not the measured one.

+ Add to test✓ In your test96IBMI3S-1-02r28-03Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy7 marks

A heritage fund invests €15 000 at a nominal annual interest rate of 3.2%, compounded quarterly.

(a)

Find the value of the investment after 8 years, giving your answer in € to 2 decimal places.

(2)

(b)

The same amount is invested at a nominal annual rate of 3.2%, compounded annually. Find the value after 8 years, in € to 2 decimal places.

(2)

(c)

With quarterly compounding, find how many years it takes for the investment to reach €20 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 15 000 × (1 + 0.032/4)³²	✓ 1
A1 € 19 356.56	✓ 1
Part (b)
M1 15 000 × (1.032)⁸	✓ 1
A1 € 19 298.73	✓ 1
Part (c)
M1 set 15 000 × 1.008^4t = 20 000	✓ 1
M1 solve using logarithms or GDC	✓ 1
A1 9.03 years	✓ 1

Exam tip: State the compounding periods per year clearly: a quarterly rate is the nominal annual rate divided by 4, and the number of periods is 4 × the number of years.

+ Add to test✓ In your test97IBMI3S-1-02r28-06Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy7 marks

A cyclist's training plan has the distance ridden in week 1 equal to 45 km, and each following week the distance increases by 6 km.

(a)

Find the distance ridden in week 10.

(2)

(b)

Find the first week in which the distance ridden is greater than 200 km.

(3)

(c)

Find the total distance ridden in the first 12 weeks.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₀ = 45 + 9 × 6	✓ 1
A1 99 km	✓ 1
Part (b)
M1 45 + 6(n − 1) > 200	✓ 1
A1 n > 26.83…	✓ 1
A1 week 27	✓ 1
Part (c)
M1 S₁₂ = 12/2 (2 × 45 + 11 × 6)	✓ 1
A1 936 km	✓ 1

Common error: For the nth term of an arithmetic sequence use u₁ + (n − 1)d; using n × d shifts every term by one step.

+ Add to test✓ In your test98IBMI3S-1-02r28-09Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy7 marks

A hard drive in a data centre stores 1.8 × 10¹² bytes. A storage hall contains 2.4 × 10³ such drives.

(a)

Find the total number of bytes the hall can store, giving your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

(2)

(b)

A manager estimates the capacity as 4 × 10¹⁵ bytes. Find the percentage error in the estimate, to 3 significant figures.

(2)

(c)

All the data in the full hall is copied at a steady rate of 2.5 × 10⁹ bytes per second. Find the time taken, in days.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1.8 × 10¹² × 2.4 × 10³	✓ 1
A1 4.32 × 10¹⁵ bytes	✓ 1
Part (b)
M1 \|4 × 10¹⁵ − 4.32 × 10¹⁵\| / (4.32 × 10¹⁵) × 100	✓ 1
A1 7.41%	✓ 1
Part (c)
M1 time = 4.32 × 10¹⁵ / 2.5 × 10⁹ = 1.728 × 10⁶ s	✓ 1
M1 divide by 86 400 seconds per day	✓ 1
A1 20 days	✓ 1

Common error: When multiplying numbers in standard form, multiply the coefficients and add the powers of ten, then check the coefficient is between 1 and 10.

+ Add to test✓ In your test99IBMI3S-1-02r28-10Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy6 marks

The pH of a solution is given by pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol dm⁻³. Soil water sample A, taken near a dig site, has [H⁺] = 3.2 × 10⁻⁵ mol dm⁻³.

(a)

Find the pH of sample A, to 3 significant figures.

(2)

(b)

Sample B has pH 6.3. Find [H⁺] for sample B, in the form a × 10^k with a to 3 significant figures.

(2)

(c)

Using the unrounded value of [H⁺] for sample B, find how many times greater the hydrogen ion concentration of sample A is than that of sample B. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 −log₁₀(3.2 × 10⁻⁵)	✓ 1
A1 4.49	✓ 1
Part (b)
M1 [H⁺] = 10^−6.3	✓ 1
A1 5.01 × 10⁻⁷ mol dm⁻³	✓ 1
Part (c)
M1 3.2 × 10⁻⁵ / 10^−6.3	✓ 1
A1 63.8	✓ 1

Exam tip: A lower pH means a higher hydrogen ion concentration; the pH scale is logarithmic, so a change of 1 unit is a factor of 10.

+ Add to test✓ In your test100IBMI3S-1-01r29-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A warehouse robot assembles 12 units in its first hour of operation. Each following hour it assembles 5 more units than in the previous hour, so the numbers of units assembled each hour form an arithmetic sequence.

(a)

Find the number of units assembled in the 8th hour.

(2)

(b)

Find the total number of units assembled during the first 8 hours.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $u_n=u_1+(n-1)d$ with $u_1=12$, $d=5$, $n=8$	✓ 1
A1 47 units	✓ 1
Part (b)
M1 use of $S_n=\frac{n}{2}(2u_1+(n-1)d)$ or $\frac{8}{2}(12+47)$	✓ 1
A1 236 units	✓ 1

Exam tip: Write the formula and the values you substitute before using your calculator, so a slip in entering numbers still earns the method mark.

+ Add to test✓ In your test101IBMI3S-1-01r29-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

The biomass of a fish stock in a coastal bay is 4800 tonnes. Each year the biomass falls by 6% of its value at the start of that year.

(a)

Find the biomass of the stock after 5 years. Give your answer to the nearest tonne.

(2)

(b)

Find the number of years, to 3 significant figures, after which the biomass first reaches 3000 tonnes, assuming the model continues to apply.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4800 × 0.94⁵	✓ 1
A1 3523 tonnes	✓ 1
Part (b)
M1 4800 × 0.94^t = 3000	✓ 1
M1 attempt to solve for t, for example using logarithms or a graph	✓ 1
A1 t = 7.60 years	✓ 1

Common error: A fall of 6% each year means multiplying by 0.94 each year, not subtracting 6% of the original amount each time.

+ Add to test✓ In your test102IBMI3S-1-01r29-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A fashion designer estimates that one dress needs 3.2 m of fabric. The actual length of fabric used for one dress is 3.45 m. The designer makes 240 dresses.

(a)

Calculate the total length of fabric used for the 240 dresses. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Calculate the percentage error in the designer's estimate of the length of fabric needed for one dress. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 240 × 3.45 = 828	✓ 1
A1 8.28 × 10² m	✓ 1
Part (b)
M1 $\frac{\|3.2-3.45\|}{3.45}\times100$	✓ 1
A1 7.25%	✓ 1

Exam tip: Percentage error is divided by the exact (actual) value, not by the approximate value.

+ Add to test✓ In your test103IBMI3S-1-01r29-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

Mina invests €6000 in an account that pays a nominal annual interest rate of 3.2%, compounded monthly. She leaves the money in the account for 7 years.

(a)

Calculate the value of the investment after 7 years. Give your answer to the nearest cent.

(3)

(b)

Calculate the effective annual interest rate of the account. Give your answer as a percentage to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of compound interest formula or TVM solver with r = 3.2, n = 7, k = 12	✓ 1
M1 6000(1 + 0.032/12)⁸⁴	✓ 1
A1 €7,504.19	✓ 1
Part (b)
M1 (1 + 0.032/12)¹² − 1	✓ 1
A1 3.25%	✓ 1

Common error: Do not use 3.2% as the monthly rate; the monthly rate is the nominal annual rate divided by 12.

+ Add to test✓ In your test104IBMI3S-1-02r29-01Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

The prices of items on a boutique's price ladder increase by the same amount from one item to the next, so the prices form an arithmetic sequence. The 3rd item costs €31 and the 8th item costs €61.

(a)

Find the common difference of the sequence.

(2)

(b)

Find the price of the 1st item.

(1)

(c)

Find the price of the 20th item.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 5d = 61 − 31 or equivalent equation	✓ 1
A1 d = €6	✓ 1
Part (b)
A1 €19	✓ 1
Part (c)
A1 €133	✓ 1

Exam tip: Between the 3rd and 8th terms there are 5 steps, not 6; subtract the term numbers to count the steps.

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exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

A marine biologist samples plankton in a fish farm. The mass of plankton is 3.2 × 10⁻⁵ kg in each cubic metre of water. The farm contains 7.5 × 10⁶ m³ of water.

(a)

Calculate the total mass of plankton in the farm. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

A quick estimate gives the total mass as 200 kg. Calculate the percentage error in this estimate. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3.2 × 10⁻⁵ × 7.5 × 10⁶ = 240	✓ 1
A1 2.4 × 10² kg	✓ 1
Part (b)
M1 error = \|200 − 240\| = 40	✓ 1
M1 $\frac{40}{240}\times100$	✓ 1
A1 16.7%	✓ 1

Common error: Writing 240 as 24 × 10¹ is not standard form; the number in front of the power of ten must be at least 1 and less than 10.

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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

The amount of data stored by a learning robot triples every month. The amount of data, D gigabytes, after t months is modelled by D = 40 × 3^t.

(a)

Calculate the amount of data stored after 4 months.

(2)

(b)

Find the time, in months to 3 significant figures, for the amount of data to reach 1000 gigabytes.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 40 × 3⁴	✓ 1
A1 3240 gigabytes	✓ 1
Part (b)
M1 3^t = 25 and attempt to solve, for example using logarithms	✓ 1
A1 t = 2.93 months	✓ 1

Exam tip: Divide both sides by 40 before using logarithms; taking logs of 40 × 3t directly is a common source of errors.

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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

A delivery robot costs €18 000 when new. Its value falls by 14% of its value at the start of each year.

(a)

Calculate the value of the robot after 5 years. Give your answer to the nearest euro.

(2)

(b)

Find the total fall in value over the 5 years, using your answer to part (a).

(1)

(c)

Express the value after 5 years as a percentage of the original cost. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18000 × 0.86⁵	✓ 1
A1 €8,468	✓ 1
Part (b)
A1 €9,532 (ECF from (a))	✓ 1
Part (c)
M1 0.86⁵ × 100	✓ 1
A1 47.0%	✓ 1

Common error: Falling by 14% a year for 5 years does not give a total fall of 5 × 14% = 70%; each year's fall is applied to a smaller value.

+ Add to test✓ In your test108IBMI3S-1-01r30-01Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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A cycling club buys a fleet of e-bikes with a loan of €18 000. The loan has a nominal annual interest rate of 4.8%, compounded monthly, and is repaid by equal payments made at the end of each month for 5 years.

(a)

Calculate the monthly payment, giving your answer to the nearest cent.

(3)

(b)

Using your answer to (a), calculate the total interest paid over the 5 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a finance solver (or annuity formula) with N = 60	✓ 1
A1 correct inputs: I% = 4.8, PV = 18000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 monthly payment = €338.04	✓ 1
Part (b)
M1 total repaid = 60 × 338.04 = 20282.40	✓ 1
A1 interest = 20282.40 − 18000 = €2282.40	✓ 1
ECF from (a).	—

Common error: Using 5 as the number of payments instead of 60. Count the payments, not the years, when payments are monthly.

+ Add to test✓ In your test109IBMI3S-1-01r30-02Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy7 marks

A heritage trust invests €25 000 in an account that pays a nominal annual interest rate of 3.6%, compounded quarterly. No money is added or withdrawn.

(a)

Find the value of the investment after 5 years. Give your answer to the nearest euro.

(2)

(b)

The trust needs the account to reach €40 000 to pay for an excavation. Find the time, in years, for the investment to reach €40 000. Give your answer to 3 significant figures.

(3)

(c)

Explain why the trust cannot actually use the €40 000 at the time found in (b).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 25000(1 + 0.036/4)²⁰	✓ 1
A1 €29906	✓ 1
Part (b)
M1 25000(1.009)^4t = 40000 (or equivalent equation in the number of quarters)	✓ 1
A1 4t = 52.46… quarters (or solver output)	✓ 1
A1 t = 13.1 years	✓ 1
Part (c)
R1 interest is added only at the end of each quarter, so the balance is below €40000 until the end of a whole quarter	✓ 1
A1 the first time is the end of the 53rd quarter, which is 13.25 years	✓ 1

Exam tip: When the question gives a nominal rate, divide by the number of compounding periods and multiply the years by the same number.

+ Add to test✓ In your test110IBMI3S-1-01r30-05Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy6 marks

A data centre has 2460 servers. Each server has a storage capacity of 1.6 × 10¹³ bytes.

(a)

Calculate the total storage capacity of the data centre. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(2)

(b)

A planner estimates the total by rounding each of the two given numbers to 1 significant figure and multiplying. Find the planner's estimate in the form a × 10^k.

(2)

(c)

Find the percentage error in the planner's estimate, using the exact value 3.936 × 10¹⁶ bytes as the true value. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2460 × 1.6 × 10¹³	✓ 1
A1 3.94 × 10¹⁶ bytes	✓ 1
Part (b)
M1 2000 × 2 × 10¹³	✓ 1
A1 4 × 10¹⁶ bytes	✓ 1
Part (c)
M1 \|4 × 10¹⁶ − 3.936 × 10¹⁶\| ÷ (3.936 × 10¹⁶) × 100	✓ 1
A1 1.63%	✓ 1

Common error: Dividing by the estimate instead of the exact value when finding a percentage error. The error is always measured relative to the true value.

+ Add to test✓ In your test111IBMI3S-1-01r30-12Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A cyclist deposits €4200 in a savings account that pays compound interest at an annual rate of r%, added once a year. After 3 years the account contains €4710.81.

(a)

Find the interest earned over the 3 years.

(1)

(b)

Find the value of r. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4710.81 − 4200 = €510.81	✓ 1
Part (b)
M1 4200(1 + r/100)³ = 4710.81	✓ 1
M1 cube root, or use of solver: $1 + r/100 = \left(\frac{4710.81}{4200}\right)^{1/3}$	✓ 1
A1 r = 3.90	✓ 1

Common error: Dividing the total growth by 3 to find the annual rate. Compound growth needs a cube root, not a division.

+ Add to test✓ In your test112IBMI3S-1-03r30-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy4 marks

A patient is prescribed a course of a painkiller. On day 1 the dose is 50 mg and on each following day the dose is 15 mg larger than on the previous day. The doses form an arithmetic sequence.

(a)

Find the dose, in mg, on day 12.

(2)

(b)

Find the total mass of the drug, in mg, taken during the first 12 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $u_n = u_1 + (n-1)d$ with $d=15$ (or listing terms)	✓ 1
A1 215 mg	✓ 1
Part (b)
M1 use of $S_n=\frac{n}{2}(u_1+u_n)$ or $\frac{n}{2}(2u_1+(n-1)d)$ with $n=12$	✓ 1
A1 1590 mg (ECF from (a) if the formula is used correctly)	✓ 1

Common error: Remember that day 12 is the 12th term, so the common difference is added 11 times, not 12 times.

+ Add to test✓ In your test113IBMI3S-1-03r30-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A hospital gives a patient a single 240 mg dose of an antibiotic. The body removes 18% of the drug still present in the blood each hour, so the mass remaining after t hours is $M = 240 \times 0.82^{t}$ mg.

(a)

Calculate the mass of the drug remaining after 3 hours. Give your answer to the nearest mg.

(2)

(b)

Find the time, in hours, for the mass of the drug to fall to 20 mg. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute $t=3$: $240\times0.82^3$	✓ 1
A1 132 mg (132.3...)	✓ 1
Part (b)
M1 set up $240\times0.82^t=20$ or $0.82^t=\frac{1}{12}$	✓ 1
M1 take logarithms: $t=\frac{\ln(1/12)}{\ln 0.82}$ (or use of a graph or solver)	✓ 1
A1 12.5 hours	✓ 1

Exam tip: When the unknown is in the exponent, use logarithms or your calculator's solver; do not try to take a root.

+ Add to test✓ In your test114IBMI3S-1-03r30-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A water-storage tank is a cylinder of radius 3.4 m and height 2.15 m.

(a)

Calculate the volume of the tank in m³. Give your answer correct to 3 significant figures.

(2)

(b)

Write the volume in litres in the form $a\times10^{k}$, where $1\le a<10$ and $k$ is an integer. Use 1 m³ = 1000 litres and your answer to part (a).

(1)

(c)

A technician estimates the volume as 80 m³. Calculate the percentage error in her estimate. Use the unrounded volume from your calculator and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $V=\pi\times3.4^2\times2.15$	✓ 1
A1 78.1 m³ (78.08...)	✓ 1
Part (b)
A1 $7.81\times10^{4}$ litres	✓ 1
Part (c)
M1 $\frac{\|80-78.08\ldots\|}{78.08\ldots}\times100$	✓ 1
A1 2.46%	✓ 1

Common error: Percentage error is measured relative to the exact value, not relative to the estimate.

+ Add to test✓ In your test115IBMI3S-1-03r30-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy4 marks

A gamer saves €1200 in an account that pays a nominal annual interest rate of 4%, compounded quarterly. No other money is paid in or taken out.

(a)

Calculate the amount in the account after 5 years. Give your answer to 2 decimal places.

(3)

(b)

Find the interest earned over the 5 years. Use the value from part (a) given to 2 decimal places.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of compound interest formula or TVM solver with $n=5$, $k=4$, $r=4$	✓ 1
M1 $1200\left(1+\frac{4}{400}\right)^{20}$ (or N = 20, I% = 4, PV = −1200, P/Y = C/Y = 4)	✓ 1
A1 €1464.23	✓ 1
Part (b)
A1 €264.23	✓ 1

Exam tip: With compounding quarterly, the number of compounding periods is 4 times the number of years, and the periodic rate is the nominal rate divided by 4.

+ Add to test✓ In your test116IBMI3S-1-03r32-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A cyclist plans a training block. In week 1 she rides 120 km and in each following week she rides 8% further than in the previous week, so the weekly distances form a geometric sequence.

(a)

Find the distance she rides in week 6. Give your answer to the nearest kilometre.

(2)

(b)

Find the total distance she rides in the first 10 weeks. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_6 = 120\times 1.08^{5}$	✓ 1
A1 176.3 so 176 km	✓ 1
Part (b)
M1 use of $S_n = \dfrac{u_1(r^n-1)}{r-1}$ with $r = 1.08$, $n = 10$	✓ 1
A1 $\dfrac{120(1.08^{10}-1)}{0.08} = 1738.4$ (accept sum of ten terms on a GDC)	✓ 1
A1 1740 km	✓ 1

Exam tip: An increase of 8% each week means the common ratio is 1.08, not 0.08.

+ Add to test✓ In your test117IBMI3S-1-03r32-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A data centre contains 2.4 × 10⁵ storage drives. Each drive has a capacity of 1.6 × 10¹³ bytes.

(a)

Calculate the total capacity of the data centre. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

A planner estimates the capacity as 4 × 10¹⁸ bytes. Calculate the percentage error in the planner's estimate. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $2.4\times10^{5}\times1.6\times10^{13}$	✓ 1
A1 3.84 × 10¹⁸ bytes	✓ 1
Part (b)
M1 $\dfrac{\|v_A - v_E\|}{v_E}\times100$ with $v_A = 4\times10^{18}$ and $v_E = 3.84\times10^{18}$ (ECF from (a))	✓ 1
A1 $\dfrac{0.16\times10^{18}}{3.84\times10^{18}}\times100$	✓ 1
A1 4.17%	✓ 1

Common error: Percentage error is measured relative to the exact value, not the estimate: dividing by 4 × 10¹⁸ gives 4.00%, which is wrong.

+ Add to test✓ In your test118IBMI3S-1-03r32-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy6 marks

The amount of carbon-14 in an organic sample from an archaeological site halves every 5730 years. The fraction of the original carbon-14 that remains after t years is modelled by $F = \left(\tfrac{1}{2}\right)^{t/5730}$.

(a)

Find the percentage of the original carbon-14 remaining in a sample that is 2000 years old. Give your answer to 3 significant figures.

(3)

(b)

A wooden beam is found to contain 62% of its original carbon-14. Find the age of the beam. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $F = 0.5^{2000/5730}$	✓ 1
A1 0.7851	✓ 1
A1 78.5%	✓ 1
Part (b)
M1 $0.5^{t/5730} = 0.62$	✓ 1
M1 take logarithms: $\dfrac{t}{5730} = \dfrac{\ln 0.62}{\ln 0.5}$ (or use a GDC solver)	✓ 1
A1 $t = 3952$, so 3950 years	✓ 1

Exam tip: If an exponential equation has the unknown in the power, take logarithms of both sides (any base) and bring the power down.

+ Add to test✓ In your test119IBMI3S-1-03r33-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy4 marks

The seating in a new concert hall is arranged in 22 rows. The first row has 14 seats and each row has 3 more seats than the row in front of it.

(a)

Find the number of seats in the 22nd row.

(2)

(b)

Find the total number of seats in the concert hall.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $u_{22}=14+(22-1)\times 3$	✓ 1
A1 77 seats	✓ 1
Part (b)
M1 use of $S_{22}=\frac{22}{2}(14+77)$ (or $\frac{22}{2}(2\times14+21\times3)$), ECF from (a)	✓ 1
A1 1001 seats	✓ 1

Exam tip: Check whether a sum formula needs the last term or the common difference, and use whichever you already know.

+ Add to test✓ In your test120IBMI3S-1-03r33-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A colony of marine plankton contains 2400 cells at the start of an experiment. The number of cells increases by 15% each day.

(a)

Find the number of cells after 5 days. Give your answer to the nearest whole number.

(2)

(b)

Assuming the growth continues at the same rate, find the time, in days, for the colony to reach 10 000 cells. Give your answer to 1 decimal place.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $2400\times1.15^5$	✓ 1
A1 4827 cells	✓ 1
Part (b)
M1 set up $2400\times1.15^{t}=10\,000$	✓ 1
M1 take logarithms (or use a solver or table) to solve for $t$	✓ 1
A1 10.2 days	✓ 1

Common error: Writing 15% growth as a multiplier of 0.15 instead of 1.15 gives a decaying value; add the 15% to 100% first.

+ Add to test✓ In your test121IBMI3S-1-03r33-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A culture of marine bacteria doubles in number every 40 minutes. The number of bacteria, N, after t minutes is modelled by $N=500\times2^{t/40}$.

(a)

Find the number of bacteria after 2 hours.

(2)

(b)

Find the time taken for the number of bacteria to reach 50 000. Give your answer in minutes, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $t=120$ in $500\times2^{120/40}$	✓ 1
A1 4000 bacteria	✓ 1
Part (b)
M1 set up $500\times2^{t/40}=50\,000$ (so $2^{t/40}=100$)	✓ 1
M1 use of logarithms, e.g. $\frac{t}{40}=\frac{\ln100}{\ln2}$	✓ 1
A1 266 minutes	✓ 1

Exam tip: Isolate the exponential term before taking logarithms, and remember to convert the exponent back to a time.

+ Add to test✓ In your test122IBMI3S-1-04r34-01Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Easy4 marks

A fish farm uses a rectangular net pen measuring 12.4 m by 8.7 m.

(a)

Calculate the area of the pen.

(1)

(b)

Write the area from part (a) in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a given to 2 significant figures.

(1)

(c)

Find the percentage error when the area is approximated by your answer to part (b). Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 107.88 m²	✓ 1
Part (b)
A1 1.1 × 10² m²	✓ 1
Part (c)
M1 \|110 − 107.88\| ÷ 107.88 × 100	✓ 1
A1 1.97 %	✓ 1

Common error: Percentage error is measured relative to the exact value, not the approximate one.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Easy6 marks

A robotic assembly line makes 35 units on day 1. Each following day it makes 6 more units than the day before, so the daily output forms an arithmetic sequence.

(a)

Find the number of units made on day 12.

(2)

(b)

Find the first day on which the daily output is greater than 200 units.

(2)

(c)

Find the total number of units made during the first 12 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₂ = 35 + 11 × 6	✓ 1
A1 101 units	✓ 1
Part (b)
M1 35 + 6(n − 1) > 200, giving n > 28.5 (or a table / list check)	✓ 1
A1 day 29	✓ 1
Part (c)
M1 S₁₂ = 12/2 × (2 × 35 + 11 × 6)	✓ 1
A1 816 units	✓ 1

Exam tip: In a ‘first term to exceed’ question, a whole-number answer must be checked with the term before and after, since a decimal solution of the inequality is not a term number.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A livestream of a fashion show has 400 viewers in the first hour. In each following hour the number of viewers is 15% lower than in the previous hour. The numbers of viewers form a geometric sequence.

(a)

Write down the common ratio.

(1)

(b)

Find the number of viewers in the 6th hour, to the nearest whole number.

(2)

(c)

Find the total number of viewer-hours (the sum of the hourly numbers) over the first 8 hours. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.85	✓ 1
Part (b)
M1 400 × 0.85⁵	✓ 1
A1 177	✓ 1
Part (c)
M1 S₈ = 400(1 − 0.85⁸)/(1 − 0.85)	✓ 1
A1 1940	✓ 1

Common error: A 15% decrease gives a common ratio of 0.85; using 0.15 as the ratio is a frequent slip.

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The sound level L, in decibels (dB), of a machine with sound intensity I W m⁻² is given by L = 10 log₁₀(I/10⁻¹²).

(a)

A robotic arm produces a sound intensity of 3.2 × 10⁻⁵ W m⁻². Calculate its sound level, to 3 significant figures.

(2)

(b)

A grinding machine has a sound level of 92 dB. Find its sound intensity, giving your answer in the form a × 10^k where 1 ≤ a < 10, a to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(3.2 × 10⁻⁵/10⁻¹²)	✓ 1
A1 75.1 dB	✓ 1
Part (b)
M1 92 = 10 log₁₀(I/10⁻¹²), so log₁₀(I/10⁻¹²) = 9.2	✓ 1
M1 I = 10⁻¹² × 10^9.2	✓ 1
A1 1.58 × 10⁻³ W m⁻²	✓ 1

Exam tip: To undo a base-10 logarithm, raise 10 to the power of both sides; do this before dividing or multiplying by the other constants.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy4 marks

A community energy group deposits €8500 in an account that pays a nominal annual interest rate of 3.2%, compounded quarterly. No further deposits or withdrawals are made.

(a)

Calculate the value of the account after 6 years. Give your answer to the nearest euro.

(3)

(b)

Find the interest earned over the 6 years, to the nearest euro.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 identifies 24 compounding periods (or N = 24, C/Y = 4)	✓ 1
M1 8500 × (1 + 0.032/4)²⁴ or correct TVM entries (I% = 3.2, PV = ±8500, P/Y = 1, C/Y = 4)	✓ 1
A1 €10291	✓ 1
Part (b)
A1 €1791 (ECF from (a))	✓ 1

Common error: Dividing the annual rate by 4 but then using n = 6 instead of 24 compounding periods gives a value that is far too small.

+ Add to test✓ In your test127IBMI3S-1-03r35-02Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy4 marks

During an outbreak of seasonal flu, a clinic records 18 new cases on day 1. The number of new cases recorded each day then increases by 7 per day, so the daily counts form an arithmetic sequence.

(a)

Find the number of new cases recorded on day 14.

(2)

(b)

Find the total number of new cases recorded during the first 14 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₄ = 18 + 13 × 7	✓ 1
A1 109	✓ 1
Part (b)
M1 S₁₄ = 14/2 × (18 + 109) or 14/2 × (2 × 18 + 13 × 7)	✓ 1
A1 889 (ECF from (a) if the sum formula is used correctly)	✓ 1

Exam tip: Check that you use n − 1 common differences to reach the nth term, not n.

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A flu virus particle is a sphere with a diameter of 1.2 × 10⁻⁷ m.

(a)

Find how many virus particles placed side by side in a straight line would span 3 mm. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

A student estimates the diameter as 1 × 10⁻⁷ m. Calculate the percentage error in the student's estimate. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (3 × 10⁻³) / (1.2 × 10⁻⁷) (accept 0.003 / 0.00000012)	✓ 1
A1 2.5 × 10⁴	✓ 1
Part (b)
M1 \|1 × 10⁻⁷ − 1.2 × 10⁻⁷\| / (1.2 × 10⁻⁷) × 100	✓ 1
A1 16.7%	✓ 1

Exam tip: A calculator may display 2.5E4; always rewrite it as 2.5 × 10⁴ when asked for standard form.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Easy4 marks

A shop buys a card payment terminal for €6200. Its value falls by 15% each year.

(a)

Find the value of the terminal after 4 years, to the nearest euro.

(2)

(b)

Find the time, in years, after which the value of the terminal is €2500. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 6200 × 0.85⁴	✓ 1
A1 €3236	✓ 1
Part (b)
M1 6200 × 0.85^t = 2500 (or table or graph)	✓ 1
A1 5.59 years	✓ 1

Common error: A 15% fall each year means multiplying by 0.85 each year, not subtracting 15% of the original value each year.

+ Add to test✓ In your test130IBMI3S-1-01r37-02Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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Calculator

A pharmaceutical company produces tablets. The label on each tablet states that it contains 3.2 × 10⁻⁴ g of the active ingredient. A batch contains 6.8 × 10⁶ tablets.

(a)

Calculate the total mass of active ingredient stated for the batch. Give your answer in grams in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a given to three significant figures.

(2)

(b)

A laboratory test finds that one tablet actually contains 3.17 × 10⁻⁴ g. Find the percentage error in the labelled value. Give your answer to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3.2 × 10⁻⁴ × 6.8 × 10⁶ = 2176	✓ 1
A1 2.18 × 10³ g	✓ 1
Part (b)
M1 \|3.2 × 10⁻⁴ − 3.17 × 10⁻⁴\| ÷ 3.17 × 10⁻⁴ × 100	✓ 1
A1 0.946%	✓ 1

Common error: Percentage error is measured against the true (exact) value, so it goes in the denominator.

+ Add to test✓ In your test131IBMI3S-1-01r37-05Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Easy4 marks

Calculator

A reservoir holds 4.75 × 10⁷ litres of water. A town uses 3.2 × 10⁵ litres per day, and no water is added to the reservoir.

(a)

Find the number of complete days for which the reservoir can supply the town.

(2)

(b)

Given that 1 m³ = 1000 litres, write the volume of the reservoir in m³ in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4.75 × 10⁷ ÷ 3.2 × 10⁵ = 148.4…	✓ 1
A1 148 days	✓ 1
Part (b)
M1 4.75 × 10⁷ ÷ 1000	✓ 1
A1 4.75 × 10⁴ m³	✓ 1

Common error: Round down when asked for complete days: a partial day is not a complete day.

+ Add to test✓ In your test132IBMI3S-1-01r37-06Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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Calculator

In a video game, a player needs a set amount of experience points (XP) to complete each level. Level 1 needs 120 XP, and each later level needs 35 XP more than the level before it.

(a)

Find the XP needed to complete level 12.

(2)

(b)

Find the total XP needed to complete levels 1 to 12.

(2)

(c)

Find the XP needed for the first level that requires more than 800 XP.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₂ = 120 + 11 × 35	✓ 1
A1 505 XP	✓ 1
Part (b)
M1 use of S₁₂ = 12/2 (120 + 505) (or 12/2 (2 × 120 + 11 × 35))	✓ 1
A1 3750 XP (ECF from (a))	✓ 1
Part (c)
M1 120 + (n − 1) × 35 > 800 giving n > 20.4… so n = 21	✓ 1
A1 820 XP	✓ 1

Exam tip: When solving an inequality for a term number, round up to the next whole number of terms.

+ Add to test✓ In your test133IBMI3S-1-01r37-10Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Calculator

Bacteria in a sample of water from a storage tank grow so that the number of bacteria after t days is modelled by N = 400 × 1.35^t.

(a)

Find the number of bacteria after 6 days, to the nearest whole number.

(2)

(b)

Find the time taken for the number of bacteria to reach 5000. Give your answer in days to three significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 400 × 1.35⁶	✓ 1
A1 2421	✓ 1
Part (b)
M1 400 × 1.35^t = 5000, so 1.35^t = 12.5	✓ 1
M1 t = log 12.5 ÷ log 1.35 (or graphical/solver method)	✓ 1
A1 8.42 days	✓ 1

Common error: Divide by the initial value before taking logarithms; taking logs of 400 × 1.35t directly loses the structure.

+ Add to test✓ In your test134IBMI3S-1-03r38-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

Calculator

A library starts a reading challenge. On day 1, 36 books are borrowed by participants. On each following day, 5 more books are borrowed than on the previous day, so the numbers borrowed each day form an arithmetic sequence.

(a)

Find the number of books borrowed on day 12.

(2)

(b)

Find the total number of books borrowed during the first 12 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $u_n=u_1+(n-1)d$ with $u_1=36$, $d=5$, $n=12$	✓ 1
A1 $u_{12}=91$	✓ 1
Part (b)
M1 use of $S_n=\frac{n}{2}(u_1+u_n)$ or $S_n=\frac{n}{2}(2u_1+(n-1)d)$ with $n=12$	✓ 1
A1 762 books (ECF from (a) if $u_{12}$ is used)	✓ 1

Exam tip: Write down the formula you are using before substituting; the method mark is awarded for the correct substitution even if the arithmetic slips.

+ Add to test✓ In your test135IBMI3S-1-03r38-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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Calculator

The blades of a wind turbine sweep out a circle of radius 48 m. The area of this circle is $A=\pi r^2$.

(a)

Calculate the swept area. Give your answer in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$, with $a$ correct to three significant figures.

(2)

(b)

An engineer approximates $\pi$ by 3.14 and calculates the area as $A'=3.14\times48^2$. Taking the unrounded value of $A$ from part (a) as the exact value, find the percentage error in $A'$. Give your answer correct to three significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $A=\pi\times48^2$ $(=7238.23\ldots)$	✓ 1
A1 $7.24\times10^{3}\ \text{m}^2$	✓ 1
Part (b)
A1 $A'=7234.56$	✓ 1
M1 $\left\|\frac{7238.23\ldots-7234.56}{7238.23\ldots}\right\|\times100$	✓ 1
A1 $0.0507\%$	✓ 1

Common error: Percentage error is measured against the exact value, not against the approximate one; dividing by the wrong value changes the third significant figure.

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Calculator

A swimming pool contains 250 g of fine impurities at the start of a filtering cycle. During each hour the filter removes 12% of the impurities that are in the pool at the start of that hour. The mass remaining at the start of each hour forms a geometric sequence.

(a)

Find the number of complete hours of filtering needed for the mass remaining to fall below 50 g.

(3)

(b)

Find the mass of impurities remaining after 5 hours. Give your answer correct to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $250\times0.88^n<50$ (or equality, or a table/graph approach)	✓ 1
M1 correct use of logarithms, noting that dividing by $\ln0.88$ reverses the inequality ($n>12.59\ldots$)	✓ 1
A1 13 hours	✓ 1
Part (b)
M1 $250\times0.88^5$	✓ 1
A1 $132$ g	✓ 1

Exam tip: When solving an inequality with logarithms, check the sign of $\ln r$: for a decreasing sequence it is negative, so the inequality sign reverses.

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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Easy6 marks

A swimmer follows a 20-week training plan. In week 1 she swims 1200 m in total. In each following week she swims 150 m more than in the previous week, so the weekly distances form an arithmetic sequence.

(a)

Find the distance she swims in week 10.

(2)

(b)

Find the total distance she swims in the first 10 weeks.

(2)

(c)

In which week does her weekly distance reach 3300 m?

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of u_n = u₁ + (n − 1)d with n = 10	✓ 1
A1 2550 m	✓ 1
Part (b)
M1 use of S_n = n/2 (u₁ + u_n) (or the formula with d)	✓ 1
A1 18 750 m (ECF from (a))	✓ 1
Part (c)
M1 1200 + (n − 1) × 150 = 3300	✓ 1
A1 week 15	✓ 1

Exam tip: The first term is week 1, so the common difference is added n − 1 times, not n times.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Easy4 marks

The number of loans made by a town library fell by 4% each year after 2024, when 12 500 loans were made. Assume the loans form a geometric sequence.

(a)

Find the number of loans made in 2030, to the nearest whole number.

(2)

(b)

Find the overall percentage decrease in the number of loans from 2024 to 2030, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 12 500 × 0.96⁶	✓ 1
A1 9784 loans	✓ 1
Part (b)
M1 (1 − 0.96⁶) × 100 (use the unrounded value)	✓ 1
A1 21.7%	✓ 1

Common error: A 4% fall per year is a common ratio of 0.96; the overall fall over 6 years is not 6 × 4%.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Easy6 marks

A wind farm generates 3.6 × 10⁸ kWh of electricity in one year. An average household uses 4.5 × 10³ kWh in one year.

(a)

Calculate the number of households the wind farm can supply. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Given that 1 kWh = 3.6 × 10⁶ J, find the energy generated in one year in joules, in the form a × 10^k with a given to 3 significant figures.

(2)

(c)

A planner estimates that the farm can supply 7.5 × 10⁴ households. Calculate the percentage error in this estimate, using your answer to (a) as the exact value.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (3.6 × 10⁸) ÷ (4.5 × 10³)	✓ 1
A1 8 × 10⁴	✓ 1
Part (b)
M1 3.6 × 10⁸ × 3.6 × 10⁶	✓ 1
A1 1.30 × 10¹⁵ J (1.296 × 10¹⁵)	✓ 1
Part (c)
M1 \|7.5 × 10⁴ − 8 × 10⁴\| / (8 × 10⁴) × 100	✓ 1
A1 6.25%	✓ 1

Common error: Percentage error is divided by the exact value; dividing by the estimate gives a different answer.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Easy6 marks

The concentration of chlorine in a swimming pool after the chlorinator is switched off is modelled by C(t) = 2.8 × 0.88^t mg/L, where t is the time in hours.

(a)

Write down the initial concentration and the percentage by which the concentration falls each hour.

(2)

(b)

Find the concentration after 5 hours, to 3 significant figures.

(2)

(c)

The pool must be retested when the concentration falls to 1.0 mg/L. Find the time at which this happens, in hours to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 2.8 mg/L	✓ 1
A1 12%	✓ 1
Part (b)
M1 2.8 × 0.88⁵	✓ 1
A1 1.48 mg/L	✓ 1
Part (c)
M1 2.8 × 0.88^t = 1 (or logs: t = ln(1/2.8) / ln 0.88)	✓ 1
A1 t = 8.05 hours	✓ 1

Exam tip: Solve exponential equations either by GDC (intersection) or by taking logarithms of both sides; show the equation you are solving.

+ Add to test✓ In your test141IBMI3S-1-03r40-01Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

A passenger jet flies a route of 5.62 × 10³ km at an average ground speed of 8.4 × 10² km h⁻¹. The jet burns fuel at a constant rate of 2.7 × 10³ kg per hour.

(a)

Calculate the flight time in hours. Give your answer correct to 3 significant figures.

(2)

(b)

Calculate the mass of fuel burnt on the flight. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, correct to 3 significant figures. Use the unrounded flight time.

(2)

(c)

Regulations require the jet to be loaded with 15% more fuel than the flight burns. Calculate the mass of fuel to be loaded, correct to the nearest 100 kg.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 time = 5620 ÷ 840	✓ 1
A1 6.69 hours	✓ 1
Part (b)
M1 6.69048 × 2700 (or 5620 ÷ 840 × 2700)	✓ 1
A1 1.81 × 10⁴ kg	✓ 1
Part (c)
M1 18064.3 × 1.15 (or 18100 × 1.15, accept ECF from (b))	✓ 1
A1 20800 kg (20800 kg)	✓ 1

Common error: Keep full calculator precision between parts and round only the final answer; rounding the time to 6.69 before multiplying changes the later digits.

+ Add to test✓ In your test142IBMI3S-1-03r40-02Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

A music student increases the length of her daily scale practice as part of an exam plan. In session 1 she practises for 20 minutes, and each later session is 5 minutes longer than the one before. The session lengths form an arithmetic sequence.

(a)

Find the length of session 12.

(2)

(b)

Find the total time she practises in the first 12 sessions.

(2)

(c)

In which session does her practice time first reach 2 hours?

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 20 + 11 × 5	✓ 1
A1 75 minutes	✓ 1
Part (b)
M1 S₁₂ = 12/2 (20 + 75) (or 12/2 (2 × 20 + 11 × 5))	✓ 1
A1 570 minutes	✓ 1
Part (c)
M1 20 + 5(n − 1) = 120	✓ 1
A1 session 21	✓ 1

Exam tip: Convert all quantities to the same unit before forming the equation: here 2 hours must be written as 120 minutes.

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The number of aphids, N, on a section of a barley crop is modelled by N = 120e^0.18t, where t is the time in days since the aphids were first counted.

(a)

Write down the number of aphids when they were first counted.

(1)

(b)

Find the number of aphids after 10 days. Give your answer to the nearest whole number.

(2)

(c)

The farmer must spray when there are 2000 aphids. Find, to 3 significant figures, the number of days after the first count at which this happens.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 120	✓ 1
Part (b)
M1 120e^{0.18 × 10} = 120e^1.8	✓ 1
A1 726	✓ 1
Part (c)
M1 120e^0.18t = 2000 (or e^0.18t = 16.67…)	✓ 1
M1 t = ln(2000/120) ÷ 0.18	✓ 1
A1 15.6 days	✓ 1

Common error: Divide both sides by 120 first, then take the natural logarithm of both sides and divide by 0.18.

+ Add to test✓ In your test144IBMI3S-1-03r40-04Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

A farmer invests €18 000 from the sale of a harvest in an account that pays a nominal annual interest rate of 3.6%, compounded monthly. No money is added or withdrawn.

(a)

Calculate the value of the investment after 7 years. Give your answer to the nearest cent.

(2)

(b)

Find the interest earned over the 7 years.

(1)

(c)

Calculate the effective annual interest rate, as a percentage correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18000 × (1 + 0.036/12)⁸⁴ (or TVM with N = 84, I% = 3.6, PV = −18000, P/Y = C/Y = 12)	✓ 1
A1 €23 149.99	✓ 1
Part (b)
A1 €5 149.99	✓ 1
Part (c)
M1 (1 + 0.036/12)¹² − 1	✓ 1
A1 3.66%	✓ 1

Exam tip: Nominal rates are quoted per year: divide by the number of compounding periods per year to find the rate per period, and match N to the same periods.

+ Add to test✓ In your test145IBMI3S-1-01r41-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A beekeeper's hive produces 18 kg of honey in its first year. As the colony grows, the harvest increases by 2.5 kg each year, so the yearly harvests form an arithmetic sequence.

(a)

Find the harvest in the 8th year.

(2)

(b)

Find the total harvest over the first 8 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_8 = 18 + 7 \times 2.5$	✓ 1
A1 35.5 kg	✓ 1
Part (b)
M1 use of $S_8 = \frac{8}{2}(18+35.5)$ or $\frac{8}{2}(2\times18+7\times2.5)$ (ECF from (a))	✓ 1
A1 214 kg	✓ 1

Exam tip: Write down the formula you use and the values substituted; method marks are awarded even if the final value is slightly off.

+ Add to test✓ In your test146IBMI3S-1-01r41-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

After a disinfectant is applied to a hospital ward surface, there are 5000 bacteria on the surface. Each minute the number of bacteria falls by 15% of the number present at the start of that minute.

(a)

Find the number of bacteria after 6 minutes. Give your answer to 3 significant figures.

(2)

(b)

Find the time, in minutes, for the number of bacteria to fall to 10% of its initial value. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $5000\times0.85^6$	✓ 1
A1 1890	✓ 1
Part (b)
M1 $5000\times0.85^t = 500$ or $0.85^t=0.1$	✓ 1
M1 use of logarithms or a GDC solver to find $t$	✓ 1
A1 14.2 minutes	✓ 1

Common error: A 15% fall means multiplying by 0.85, not by 0.15.

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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

A Formula 1 circuit has length 5412 m. A driver completes one lap in 1 minute 32.416 seconds.

(a)

Calculate the driver's average speed, in m s⁻¹, for the lap. Give your answer to 3 significant figures.

(2)

(b)

A commentator estimates the speed using a length of 5400 m and a time of 92 s. Using the unrounded value of the speed from part (a) as the exact value, find the percentage error in the commentator's estimate. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 convert time to $92.416$ s and use speed = distance ÷ time	✓ 1
A1 58.6 m s⁻¹	✓ 1
Part (b)
M1 $\frac{5400}{92} = 58.6956\ldots$	✓ 1
M1 $\left\|\frac{58.6956\ldots - 58.5612\ldots}{58.5612\ldots}\right\| \times 100$	✓ 1
A1 0.229 %	✓ 1

Exam tip: Percentage error is always measured relative to the exact value, not the estimate.

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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

The sound level, L decibels (dB), of a noise of intensity I W m⁻² is given by $L = 10\log_{10}\left(\frac{I}{10^{-12}}\right)$. The noise in a hospital ward has intensity $3.2\times10^{-8}$ W m⁻².

(a)

Find the sound level in the ward. Give your answer to 3 significant figures.

(2)

(b)

A piece of equipment is switched on and the sound level rises to 52.0 dB. Find the new intensity. Give your answer in the form $a\times10^{k}$ with $a$ correct to 3 significant figures.

(2)

(c)

Using the unrounded value of the new intensity, find how many times greater it is than the original intensity. Give your answer to 3 significant figures.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $10\log_{10}\left(\frac{3.2\times10^{-8}}{10^{-12}}\right)$	✓ 1
A1 45.1 dB	✓ 1
Part (b)
M1 $52.0=10\log_{10}\left(\frac{I}{10^{-12}}\right)$ leading to $I=10^{5.2}\times10^{-12}$	✓ 1
A1 $1.58\times10^{-7}$ W m⁻²	✓ 1
Part (c)
A1 $\frac{10^{5.2}\times10^{-12}}{3.2\times10^{-8}}=4.95$	✓ 1

Exam tip: Use the unrounded answer from an earlier part when a later part says so; rounded values can change the third significant figure.

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Calculator

A pharmacist records the mass of the active ingredient in one tablet as 2.5 × 10⁻⁴ g. A laboratory balance later shows the true mass is 2.64 × 10⁻⁴ g.

(a)

Calculate the percentage error in the pharmacist's recorded mass. Give your answer correct to 3 significant figures.

(3)

(b)

Using the true mass, calculate the total mass of active ingredient in 500 tablets. Give your answer in the form a × 10^k g, where 1 ≤ a < 10 and k is an integer.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses \|recorded − true\| = 0.14 × 10⁻⁴ g	✓ 1
M1 divides by the true value and multiplies by 100	✓ 1
A1 5.30 %	✓ 1
Part (b)
A1 500 × 2.64 × 10⁻⁴ = 0.132 g = 1.32 × 10⁻¹ g	✓ 1

Common error: Percentage error is divided by the true (exact) value, not by the approximate value.

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Calculator

In a role-playing game, reaching level 1 needs 120 experience points (XP). Each later level needs 45 XP more than the level before, so the XP needed for each level forms an arithmetic sequence.

(a)

Find the XP needed to complete level 12.

(2)

(b)

Find the number of the first level that needs more than 1000 XP.

(3)

(c)

Find the total XP needed to complete all the levels from level 1 up to the level found in part (b).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₂ = 120 + 11 × 45 (1 mark for correct use of u_n = u₁ + (n − 1)d)	✓ 1
A1 615 XP	✓ 1
Part (b)
M1 120 + 45(n − 1) > 1000	✓ 1
A1 n > 20.55… (accept equality solved, n = 20.56)	✓ 1
A1 level 21	✓ 1
Part (c)
M1 S₂₁ = 21/2 (2 × 120 + 20 × 45) (ECF from (b))	✓ 1
A1 11970 XP	✓ 1

Exam tip: When an inequality gives a non-integer value of n, state clearly which whole number satisfies the condition and check it.

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Calculator

A reservoir holds 8.5 × 10⁶ litres of water. Because of evaporation and leakage, the volume falls by 3% each week.

(a)

Find the volume of water in the reservoir after 10 weeks. Give your answer correct to 3 significant figures.

(2)

(b)

Find the time, in weeks, for the volume to fall to half of its original value. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 8.5 × 10⁶ × 0.97¹⁰	✓ 1
A1 6.27 × 10⁶ litres	✓ 1
Part (b)
M1 0.97^t = 0.5	✓ 1
M1 t = ln 0.5 / ln 0.97 (or equivalent use of logarithms, or a graph or table)	✓ 1
A1 22.8 weeks	✓ 1

Common error: A 3% fall each week means multiplying by 0.97 each week, not subtracting 3% of the original volume each time.

+ Add to test✓ In your test152IBMI3S-1-03r41-04Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; 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Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy4 marks

Calculator

Maya invests €3500 in an account that pays a nominal annual interest rate of 2.4%, compounded monthly. No money is added or withdrawn.

(a)

Calculate the value of the investment after 6 years. Give your answer to the nearest cent.

(3)

(b)

Hence find the interest Maya has earned over the 6 years. Use your answer to part (a) rounded to the nearest cent.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 identifies 72 compounding periods and monthly rate 0.024/12 (or TVM entries: N = 6, I% = 2.4, PV = −3500, P/Y = C/Y = 12)	✓ 1
M1 3500(1 + 0.024/12)⁷²	✓ 1
A1 €4041.51	✓ 1
Part (b)
A1 4041.51 − 3500 = €541.51	✓ 1

Exam tip: Always match the number of compounding periods to the compounding frequency: monthly for 6 years is 72 periods.

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A marine biologist estimates that an algal bloom contains 3.8 × 10⁶ cells per litre of seawater. A more precise laboratory count gives 4.1 × 10⁶ cells per litre.

(a)

Calculate the percentage error in the biologist’s estimate. Give your answer to 3 significant figures.

(2)

(b)

The bloom fills 3.2 × 10⁴ litres of water. Using the laboratory count, find the total number of cells in the bloom. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k ∈ ℤ, with a given to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to use \|estimate − exact\| / \|exact\| × 100, i.e. (0.3 × 10⁶)/(4.1 × 10⁶) × 100	✓ 1
A1 7.32 %	✓ 1
Part (b)
M1 4.1 × 10⁶ × 3.2 × 10⁴ (= 1.312 × 10¹¹)	✓ 1
A1 1.31 × 10¹¹ cells	✓ 1

Common error: In percentage error the exact (accurate) value goes in the denominator, not the estimate.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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A coral colony covers an area of 2.4 m² at the start of a survey. Its area increases by 8 % each year, so that the area at the end of year n is 2.4 × 1.08ⁿ m².

(a)

Find the smallest whole number of years after which the area of the colony exceeds 6 m².

(3)

(b)

Find the area of the colony at the end of year 15. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forming the equation or inequality 2.4 × 1.08ⁿ = 6 (or > 6)	✓ 1
A1 n = 11.9 (from logarithms, table or graph)	✓ 1
A1 12 years (area after 11 years is 5.60 m², after 12 years 6.04 m²)	✓ 1
Part (b)
M1 2.4 × 1.08¹⁵	✓ 1
A1 7.61 m²	✓ 1

Exam tip: When the unknown is an exponent, take logarithms or use the table of values on your calculator, and then round up to a whole number of years if the question asks when a value is first exceeded.

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The first row of an open-air amphitheatre has 18 seats. Each following row has 4 more seats than the row in front of it, so the numbers of seats in the rows form an arithmetic sequence. The amphitheatre has 25 rows.

(a)

Find the number of seats in row 25.

(2)

(b)

Find the number of the first row that has more than 100 seats.

(3)

(c)

Find the total number of seats in the amphitheatre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₂₅ = 18 + 24 × 4	✓ 1
A1 114 seats	✓ 1
Part (b)
M1 18 + 4(n − 1) > 100 (or = 100)	✓ 1
A1 n > 21.5 (or n = 21.5)	✓ 1
A1 row 22 (row 21 has 98 seats, row 22 has 102)	✓ 1
Part (c)
M1 S₂₅ = (25/2)(2 × 18 + 24 × 4) or (25/2)(18 + 114)	✓ 1
A1 1650 seats	✓ 1

Common error: In 18 + 4(n − 1) the common difference is multiplied by n − 1, not by n.

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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy5 marks

The number of bacteria, P, in a sample of seawater is modelled by P = 500 × 2^t/3, where t is the time in hours since the sample was taken.

(a)

Find the number of bacteria after 12 hours.

(2)

(b)

Find the time taken for the number of bacteria to reach 20 000. Give your answer in hours to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 500 × 2^12/3 = 500 × 2⁴	✓ 1
A1 8000 bacteria	✓ 1
Part (b)
M1 20 000 = 500 × 2^t/3, so 2^t/3 = 40	✓ 1
M1 correct use of logarithms (or a graph or solver), e.g. t/3 = log₂ 40 = ln 40 / ln 2	✓ 1
A1 t = 16.0 hours	✓ 1

Exam tip: Isolate the exponential term first, then take logarithms of both sides. Do not round the intermediate value 5.32… before multiplying by 3.

+ Add to test✓ In your test157IBMI3S-1-02r43-01Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

A welding robot moves along a straight seam of length 18 m at a constant speed. The speed is quoted as 2.4 × 10⁻² m s⁻¹, correct to 2 significant figures.

(a)

Write down the least possible value of the speed. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(1)

(b)

Using the quoted speed of 2.4 × 10⁻² m s⁻¹, calculate the time taken to weld the seam. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(c)

A technician estimates the time as 700 s. Calculate the percentage error in the technician's estimate, compared with your answer to (b). Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 2.35 × 10⁻² m s⁻¹	✓ 1
Part (b)
M1 time = 18 ÷ 0.024	✓ 1
A1 7.5 × 10² s	✓ 1
Part (c)
M1 \|700 − 750\| / 750 × 100 (ECF from (b))	✓ 1
A1 6.67%	✓ 1

Exam tip: Percentage error is found relative to the accepted value, so the denominator is 750 here, not 700.

+ Add to test✓ In your test158IBMI3S-1-02r43-02Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

A fashion boutique arranges scarves in a triangular display. The number of scarves in row n forms an arithmetic sequence. Row 1 has 6 scarves and row 5 has 22 scarves.

(a)

Find the common difference of the sequence.

(2)

(b)

The boutique has 500 scarves. Find the least number of rows for which the total number of scarves needed is at least 500.

(3)

(c)

Find the number of scarves in the last row of this display.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 6 + 4d = 22 (or (22 − 6) ÷ 4)	✓ 1
A1 d = 4	✓ 1
Part (b)
M1 sum formula: S_n = n/2 (2 × 6 + (n − 1) × 4) = 2n² + 4n	✓ 1
M1 solving 2n² + 4n ≥ 500, or testing values (S₁₄ = 448, S₁₅ = 510; ECF from (a))	✓ 1
A1 15 rows	✓ 1
Part (c)
A1 6 + 14 × 4 = 62 (ECF from (a) and (b))	✓ 1

Common error: Do not confuse the nth term with the sum of the first n terms: the question about total stock needs the sum formula.

+ Add to test✓ In your test159IBMI3S-1-02r43-03Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

A fishing cooperative invests €18 000 in an account that pays a nominal annual interest rate of 3.2%, compounded quarterly. No money is added or withdrawn.

(a)

Calculate the value of the investment after 6 years. Give your answer to the nearest cent.

(2)

(b)

Find the interest earned over the 6 years.

(1)

(c)

Find the number of years it takes for the investment to first reach €25 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18000 × (1 + 0.032/4)²⁴ (or TVM with N = 24 and 0.8% per quarter, or N = 6, I = 3.2, C/Y = 4)	✓ 1
A1 €21 793.41	✓ 1
Part (b)
A1 €3793.41 (ECF from (a))	✓ 1
Part (c)
M1 18000 × 1.008^4t = 25000 (or the equivalent in quarters)	✓ 1
M1 correct method to solve (logarithms, graph or TVM solver)	✓ 1
A1 t = 10.3 years (10.3068… before rounding)	✓ 1

Exam tip: Divide the nominal annual rate by the number of compounding periods per year, and multiply the number of years by the same number.

+ Add to test✓ In your test160IBMI3S-1-02r43-10Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

A fish hatchery collects 3.6 × 10⁶ eggs. Each egg has a mass of 1.2 × 10⁻⁴ kg.

(a)

Calculate the total mass of the eggs. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

The hatchery's scale shows a total mass of 450 kg. Calculate the percentage error in this reading. Give your answer to 3 significant figures.

(2)

(c)

Each egg survives to maturity with probability 3.5 × 10⁻³. Find the expected number of eggs that survive. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3.6 × 10⁶ × 1.2 × 10⁻⁴	✓ 1
A1 4.32 × 10² kg	✓ 1
Part (b)
M1 \|450 − 432\| / 432 × 100 (ECF from (a))	✓ 1
A1 4.17%	✓ 1
Part (c)
M1 3.6 × 10⁶ × 3.5 × 10⁻³	✓ 1
A1 1.26 × 10⁴ (12 600)	✓ 1

Exam tip: Standard form needs 1 ≤ a < 10: if your calculator shows 12 600, write it as 1.26 × 10⁴ to earn the accuracy mark.

+ Add to test✓ In your test161IBMI3S-1-04r44-01Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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A mountaineering club records the height of a peak as 5642 m. A climber's altimeter shows 5600 m at the summit.

(a)

Write the recorded height in millimetres in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Find the percentage error in the altimeter reading, taking the recorded height as the exact value. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 5642 × 1000 = 5 642 000 mm	✓ 1
A1 5.642 × 10⁶ mm	✓ 1
Part (b)
M1 \|5600 − 5642\|/5642 × 100	✓ 1
A1 0.744%	✓ 1

Common error: Percentage error is divided by the exact value (5642 m), not by the approximate reading.

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A telecoms engineer transfers a file of size 7.5 × 10⁹ bytes over a link with constant transfer rate 3.2 × 10⁷ bytes per second.

(a)

Calculate the time taken for the transfer. Give your answer in seconds in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a given to 3 significant figures.

(2)

(b)

A storage drive has capacity 1.0 × 10¹² bytes. Find the greatest number of complete files of this size that the drive can hold.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 time = (7.5 × 10⁹)/(3.2 × 10⁷) = 234.375…	✓ 1
A1 2.34 × 10² s	✓ 1
Part (b)
M1 (1.0 × 10¹²)/(7.5 × 10⁹) = 133.3…	✓ 1
A1 133 files	✓ 1

Exam tip: When counting complete items, round down: a fraction of a file cannot be stored.

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On a mountain expedition, Camp 1 is at an altitude of 3200 m. Each higher camp is 450 m above the previous camp, so the altitudes of the camps form an arithmetic sequence.

(a)

Find the altitude of Camp 7.

(2)

(b)

The team plans to set up a camp above 6500 m. Find the number of the first camp that is above this altitude.

(3)

(c)

Find the altitude of this camp.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₇ = 3200 + 6 × 450	✓ 1
A1 5900 m	✓ 1
Part (b)
M1 3200 + (n − 1)450 > 6500	✓ 1
M1 n − 1 > 7.33… (or comparison of Camp 8 = 6350 m with Camp 9)	✓ 1
A1 Camp 9	✓ 1
Part (c)
A1 3200 + 8 × 450 = 6800 m	✓ 1

Exam tip: For the nth term of an arithmetic sequence, use un = u₁ + (n − 1)d; there are n − 1 steps, not n.

+ Add to test✓ In your test164IBMI3S-1-04r44-04Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context Paper 1Easy5 marks

A food scientist starts a culture with 250 yeast cells. The number of cells increases by 12% every hour, so after n hours there are 250 × 1.12ⁿ cells.

(a)

Find the number of cells after 8 hours, to the nearest whole number.

(2)

(b)

Find the time, in hours, for the number of cells to reach 1000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 250 × 1.12⁸	✓ 1
A1 619	✓ 1
Part (b)
M1 250 × 1.12^t = 1000 (or 1.12^t = 4)	✓ 1
M1 use of logarithms or a graphical or numerical solver	✓ 1
A1 12.2 hours	✓ 1

Common error: A 12% increase has a multiplier of 1.12, not 0.12.

+ Add to test✓ In your test165IBMI3S-1-02r45-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy6 marks

A beekeeper records the mass of honey collected from a hive each day. On day 1 she collects 2.4 kg, and the amount collected increases by 0.35 kg each day, forming an arithmetic sequence.

(a)

Find the mass of honey collected on day 12.

(2)

(b)

Find the first day on which more than 10 kg of honey is collected.

(2)

(c)

Find the total mass of honey collected during the first 12 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₂ = 2.4 + 11(0.35)	✓ 1
A1 6.25 kg	✓ 1
Part (b)
M1 2.4 + (n − 1)(0.35) > 10 (or a table showing 9.75 on day 22 and 10.1 on day 23)	✓ 1
A1 day 23	✓ 1
Part (c)
M1 S₁₂ = (12/2)(2.4 + 6.25) (ECF from (a))	✓ 1
A1 51.9 kg	✓ 1

Common error: The nth term uses (n − 1) common differences, not n.

+ Add to test✓ In your test166IBMI3S-1-02r45-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy6 marks

A hospital laboratory grows a bacterial culture that starts with 800 cells. The number of cells increases by 15% every hour.

(a)

Write down the common ratio of the geometric sequence that models the number of cells at the end of each hour.

(1)

(b)

Find the least whole number of hours after which the culture contains more than 3000 cells.

(3)

(c)

Find the number of cells after 6 hours. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.15	✓ 1
Part (b)
M1 800(1.15)^t = 3000 (or a table of values)	✓ 1
M1 attempt to solve, for example using logarithms, giving t = 9.46… (or 2814 cells at 9 hours)	✓ 1
A1 10 hours (3236 cells)	✓ 1
Part (c)
M1 800 × 1.15⁶	✓ 1
A1 1850 cells	✓ 1

Exam tip: When a question asks for the least whole number of time periods, round the solution up, then check the result with the neighbouring values.

+ Add to test✓ In your test167IBMI3S-1-02r45-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy6 marks

The acidity of blood is measured by pH, where pH = −log₁₀[H⁺] and [H⁺] is the concentration of hydrogen ions in mol per litre. A hospital laboratory records a healthy blood sample with [H⁺] = 4.0 × 10⁻⁸.

(a)

Calculate the pH of the sample, correct to 3 significant figures.

(2)

(b)

A second sample has pH 7.30. Find [H⁺] for this sample, giving your answer in the form a × 10^k with a correct to 3 significant figures.

(2)

(c)

Find the percentage increase in [H⁺] from the first sample to the second. Use unrounded values and give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 −log₁₀(4.0 × 10⁻⁸)	✓ 1
A1 7.40	✓ 1
Part (b)
M1 [H⁺] = 10^−7.30	✓ 1
A1 5.01 × 10⁻⁸	✓ 1
Part (c)
M1 (10^−7.3 − 4.0 × 10⁻⁸) ÷ (4.0 × 10⁻⁸) × 100	✓ 1
A1 25.3%	✓ 1

Exam tip: Lower pH means higher hydrogen-ion concentration: a decrease of 0.1 in pH is an increase of about 26% in [H⁺], not 0.1.

+ Add to test✓ In your test168IBMI3S-1-02r45-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Easy4 marks

A hospital pharmacist weighs a vial of medicine. The exact mass is 3.264 g. The balance displays the mass correct to 1 decimal place, and a technician records the reading.

(a)

Write down the lower and upper bounds for a mass that the balance displays as 3.3 g.

(2)

(b)

The technician records 3.3 g. Find the percentage error in the recorded value. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 lower bound 3.25 g	✓ 1
A1 upper bound 3.35 g	✓ 1
Part (b)
M1 \|3.3 − 3.264\| ÷ 3.264 × 100	✓ 1
A1 1.10%	✓ 1

Common error: A reading to 1 d.p. covers values up to 0.05 either side of it; the bounds are not 3.2 and 3.4.

+ Add to test✓ In your test169IBMI3S-1-01r47-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy7 marks

A hospital vaccination clinic gives 40 jabs on day 1. On each following day it gives 6 more jabs than on the previous day, so the daily numbers form an arithmetic sequence.

(a)

Find the number of jabs given on day 15.

(2)

(b)

Find the first day on which the clinic gives more than 200 jabs.

(3)

(c)

Find the total number of jabs given during the first 28 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of u_n = u₁ + (n − 1)d with n = 15	✓ 1
A1 124	✓ 1
Part (b)
M1 sets up 40 + 6(n − 1) > 200 (or equation, or table of values)	✓ 1
A1 n > 27.67 (or u₂₇ = 196 and u₂₈ = 202)	✓ 1
A1 day 28	✓ 1
Part (c)
M1 use of S_n = (n/2)(2u₁ + (n − 1)d) with n = 28	✓ 1
A1 3388	✓ 1

Common error: Write down which formula you are using and the values of u₁, d and n before using your calculator; mixing up n and n − 1 is the most common slip.

+ Add to test✓ In your test170IBMI3S-1-01r47-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy6 marks

A Formula 1 car completes a lap of a 5.8 km circuit in 82.4 seconds.

(a)

Calculate the average speed of the car in m s⁻¹. Give your answer to 3 significant figures.

(2)

(b)

Write the length of the circuit in metres in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(1)

(c)

An engineer estimates the average speed as 70 m s⁻¹. Using the unrounded value from part (a) as the exact value, calculate the percentage error in the estimate. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 5800 ÷ 82.4	✓ 1
A1 70.4	✓ 1
Part (b)
A1 5.8 × 10³	✓ 1
Part (c)
M1 use of \|approximate − exact\| ÷ exact × 100	✓ 1
A1 correct substitution \|70 − 70.388…\| ÷ 70.388…	✓ 1
A1 0.552 %	✓ 1

Common error: Percentage error is measured relative to the exact value, not the estimate. Keep unrounded values in your calculator until the final step.

+ Add to test✓ In your test171IBMI3S-1-01r47-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

A hospital charity invests €8500 in an account that pays a nominal annual interest rate of 3.2%, compounded quarterly. No money is added or withdrawn.

(a)

Find the value of the investment after 5 years. Give your answer correct to 2 decimal places.

(3)

(b)

Calculate the interest earned over the 5 years.

(1)

(c)

State what would happen to the final value if interest were compounded monthly instead of quarterly. Give a reason.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of compound interest with r = 3.2, k = 4, n = 5 (or TVM: N = 20, I% = 3.2, PV = −8500, PMT = 0, P/Y = 4, C/Y = 4)	✓ 1
A1 correct substitution 8500(1 + 3.2/(4×100))²⁰	✓ 1
A1 €9968.49	✓ 1
Part (b)
A1 €1468.49 (ECF from (a))	✓ 1
Part (c)
R1 it would be greater (slightly), because interest is added to the balance more often so interest is earned on interest sooner	✓ 1

Exam tip: If you use the finance solver, state each value you enter (N, I%, PV, PMT, FV, P/Y, C/Y); the sign of PV and FV must be opposite.

+ Add to test✓ In your test172IBMI3S-1-01r48-01Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

Calculator

Archaeologists sieving a burial site record the number of pottery shards found each day. On day 1 they find 14 shards, and on each following day they find 6 more shards than on the previous day. The numbers of shards found each day form an arithmetic sequence.

(a)

Find the number of shards found on day 15.

(2)

(b)

Find the total number of shards found during the first 15 days.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $u_n = u_1+(n-1)d$ with $u_1=14$, $d=6$	✓ 1
A1 98 shards	✓ 1
Part (b)
M1 use of $S_n=\frac{n}{2}(u_1+u_n)$ or $\frac{n}{2}(2u_1+(n-1)d)$	✓ 1
M1 correct substitution, e.g. $\frac{15}{2}(14+98)$	✓ 1
A1 840 shards	✓ 1

Common error: Using $n$ instead of $n-1$ in $u_n=u_1+(n-1)d$ gives the term after the one you want.

+ Add to test✓ In your test173IBMI3S-1-01r48-02Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

Calculator

A data centre has a storage capacity of 240 terabytes (TB) at the start of 2025. Its capacity is increased by 15% at the start of each later year, so the capacity at the start of each year forms a geometric sequence.

(a)

Write down the common ratio of the sequence.

(1)

(b)

Find the number of years after the start of 2025 when the capacity first exceeds 1000 TB.

(2)

(c)

Find the capacity, in TB to the nearest TB, at the start of that year.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.15	✓ 1
Part (b)
M1 attempt to solve $240\times1.15^n>1000$, e.g. by table, graph or logarithms	✓ 1
A1 $n=11$	✓ 1
Part (c)
M1 $240\times1.15^{11}$	✓ 1
A1 1117 TB	✓ 1

Exam tip: A 15% increase is multiplication by 1.15, not by 0.15. Write the ratio down before you start.

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arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

Calculator

A bicycle wheel has a diameter of 0.70 m. A cyclist estimates the circumference of the wheel as 2.1 m.

(a)

Calculate the circumference of the wheel, giving your answer correct to 3 significant figures.

(1)

(b)

Calculate the percentage error in the cyclist's estimate. Give your answer to 3 significant figures.

(2)

(c)

The wheel turns 5000 times during a training ride. Using the accurate circumference, find the distance travelled in metres. Give your answer in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$, with $a$ correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 2.20 m	✓ 1
Part (b)
M1 $\left\|\frac{2.1-2.19911}{2.19911}\right\|\times100$	✓ 1
A1 4.51%	✓ 1
Part (c)
M1 $5000\times\pi\times0.70$ (10,995.6 m)	✓ 1
A1 $1.10\times10^4$ m	✓ 1

Common error: Percentage error is measured relative to the exact value, not relative to the estimate.

+ Add to test✓ In your test175IBMI3S-1-01r48-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Easy5 marks

Calculator

The percentage of carbon-14 remaining in a sample of ancient wood after $t$ years is modelled by $P=100\times0.5^{\frac{t}{5730}}$.

(a)

Find the percentage of carbon-14 remaining after 2000 years. Give your answer to 3 significant figures.

(2)

(b)

A wooden beam from a dig contains 62% of its original carbon-14. Find the age of the beam, to the nearest 10 years.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute $t=2000$: $100\times0.5^{2000/5730}$	✓ 1
A1 78.5%	✓ 1
Part (b)
M1 set up $100\times0.5^{t/5730}=62$	✓ 1
M1 use of logarithms or a solver to isolate $t$, e.g. $t=5730\times\frac{\ln0.62}{\ln0.5}$	✓ 1
A1 3952 years, so 3950 years to the nearest 10 years	✓ 1

Exam tip: When the unknown is in an exponent, take logarithms of both sides or use the graph or solver on your calculator and quote the result in context.

+ Add to test✓ In your test176IBMI-030-00-05Approximation, percentage error and standard form Paper 1Medium6 marks

A spherical weather balloon has a radius of 1.236 m.
An engineer quickly estimates its volume using $V=\frac{4}{3}\pi r^3$ with $\pi$ taken as 3 and the radius rounded to 1.2 m.

(a)

Calculate the actual volume of the balloon.

(2)

(b)

Find the engineer's estimate of the volume.

(1)

(c)

Calculate the percentage error in the engineer's estimate.

(2)

(d)

Explain why the engineer's estimate is smaller than the actual volume.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for $\frac{4}{3}\pi(1.236)^3$	✓ 1
A1 7.91 m³	✓ 1
Part (b)
A1 $\frac{4}{3}\times3\times1.2^3=6.912$ m³	✓ 1
Part (c)
M1 for $\left\|\frac{6.912-7.909...}{7.909...}\right\|\times100$	✓ 1
A1 12.6 %	✓ 1
Part (d)
R1 both approximations (3 for $\pi$ and 1.2 for the radius) are smaller than the true values, so the errors do not cancel and the estimate is too small	✓ 1

Exam tip: Keep the full calculator value of the actual volume when finding the percentage error; rounding to 7.91 first can change the third significant figure.

+ Add to test✓ In your test177IBMI-030-00-06Approximation, percentage error and standard form Paper 1Medium8 marks

A car travels 248 km between two cities in 2 hours 47 minutes.

(a)

Calculate the average speed of the car in km h⁻¹.

(2)

(b)

Convert the average speed you found into m s⁻¹.

(2)

(c)

Before the journey, the driver estimated the journey time by assuming an average speed of 90 km h⁻¹.
Find the driver's estimated journey time in hours. Hence, using unrounded values throughout, calculate the percentage error in this estimate of the journey time.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for converting the time to $2\frac{47}{60}=2.783...$ h and dividing 248 by it	✓ 1
A1 89.1 km h⁻¹	✓ 1
Part (b)
M1 for $\frac{89.10...\times1000}{3600}$	✓ 1
A1 24.8 m s⁻¹	✓ 1
Part (c)
M1 for $\frac{248}{90}$	✓ 1
A1 2.76 h	✓ 1
M1 for $\left\|\frac{2.7555...-2.7833...}{2.7833...}\right\|\times100$	✓ 1
A1 0.998 %	✓ 1

Common error: 2 hours 47 minutes is not 2.47 hours. Divide the minutes by 60 before using the time in a calculation, and keep unrounded values in your calculator for later parts.

+ Add to test✓ In your test178IBMI-030-00-08Approximation, percentage error and standard form Paper 1Medium7 marks

A rectangular solar panel has length 1.65 m and width 0.99 m, each measured correct to two decimal places.

(a)

Write down the lower and upper bounds of the length of the panel.

(1)

(b)

Calculate the upper bound of the area of the panel.

(2)

(c)

Calculate the lower bound of the area of the panel.

(1)

(d)

The area is calculated using the measured values 1.65 m and 0.99 m. The exact area of the panel could be any value between the bounds found in parts (b) and (c). Using percentage error $= \left|\frac{v_A - v_E}{v_E}\right| \times 100$, where $v_A$ is the calculated area and $v_E$ is the exact area, find the maximum possible percentage error in the calculated area.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 lower bound 1.645 m and upper bound 1.655 m, i.e. 1.645 ≤ length < 1.655	✓ 1
Part (b)
M1 for 1.655 × 0.995	✓ 1
A1 1.65 m² (1.646725)	✓ 1
Part (c)
A1 1.645 × 0.985 = 1.62 m² (1.620325)	✓ 1
Part (d)
M1 for calculated area 1.65 × 0.99 = 1.6335 m²	✓ 1
M1 for substituting the lower bound as $v_E$: $\frac{1.6335-1.620325}{1.620325}\times100$. Also award this mark if both bounds are substituted and the two results are compared.	✓ 1
A1 0.813 % (accept 0.81 %)	✓ 1

Exam tip: The exact value always goes in the denominator. Try both bounds. The lower bound gives 0.813 % and the upper bound gives 0.803 %, so the maximum is 0.813 %.

+ Add to test✓ In your test179IBMI-030-00-09Approximation, percentage error and standard form Paper 1Medium5 marks

A person's heart beats on average 72 times per minute. Assume that a year has 365.25 days and that the person lives for 80 years.

(a)

Calculate the total number of heartbeats in the person's lifetime. Give your answer in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(3)

(b)

A student estimates the total by using 70 beats per minute and 500 000 minutes per year. Calculate the percentage error in the student's estimate.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for minutes in a year 60 × 24 × 365.25 = 525 960	✓ 1
M1 for 72 × 525 960 × 80	✓ 1
A1 3.03×10⁹	✓ 1
Part (b)
M1 for estimate 70 × 500 000 × 80 = 2.8×10⁹ and $\left\|\frac{2.8\times10^9-3.0295...\times10^9}{3.0295...\times10^9}\right\|\times100$	✓ 1
A1 7.58 %	✓ 1

Exam tip: When an answer will be used again, store the unrounded value in your calculator to avoid accumulated rounding errors.

+ Add to test✓ In your test180IBMI-030-00-10Approximation, percentage error and standard form Paper 1Medium6 marks

The dose of a medicine for a child is 0.025 mg per kilogram of body mass. A child has a mass of 18.6 kg.

(a)

Calculate the correct dose for this child in mg. Give your answer in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(2)

(b)

Write down the dose in grams, in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(1)

(c)

A nurse rounds the child's mass to 20 kg before calculating the dose. Calculate the percentage error in the dose given.

(2)

(d)

Hospital policy requires the percentage error in any dose to be less than 5 %. State, with a reason, whether the nurse's method meets this policy.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for 0.025 × 18.6 = 0.465	✓ 1
A1 4.65×10⁻¹ mg	✓ 1
Part (b)
A1 4.65×10⁻⁴ g (follow through from their answer to part (a))	✓ 1
Part (c)
M1 for a dose of 0.025 × 20 = 0.5 mg substituted into $\left\|\frac{0.5-0.465}{0.465}\right\|\times100$	✓ 1
A1 7.53 % (accept 7.5 %)	✓ 1
Part (d)
R1 No, because 7.53 % > 5 %. Follow through from their part (c) answer if the conclusion is consistent with it	✓ 1

Common error: 1 g = 1000 mg, so converting mg to g means dividing by 10³. The power of ten becomes more negative, not less.

+ Add to test✓ In your test181IBMI-030-05-03Approximation, percentage error and standard form Paper 1Medium6 marks

A surveyor estimates by eye that a radio tower is 48 m tall. A laser measurement then gives the height as 52.6 m, correct to the nearest 0.1 m.

(a)

Taking 52.6 m as the exact height, calculate the percentage error in the surveyor's estimate.

(2)

(b)

Write down the lower bound and the upper bound for the true height of the tower.

(2)

(c)

Find the largest possible percentage error when 52.6 m is used in place of the true height.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for $\frac{\|48-52.6\|}{52.6} \times 100\%$	✓ 1
A1 8.75%	✓ 1
Part (b)
A1 lower bound 52.55 m	✓ 1
A1 upper bound 52.65 m	✓ 1
Part (c)
M1 for using the lower bound as the true value, $\frac{0.05}{52.55} \times 100\%$	✓ 1
A1 0.0951%	✓ 1

Exam tip: The maximum error is 0.05 m for both bounds. Dividing by the smaller true value, 52.55, gives the larger percentage.

+ Add to test✓ In your test182IBMI-030-05-06Approximation, percentage error and standard form Paper 1Medium7 marks

A car travels 386 km, measured to the nearest km. It uses 27.4 litres of fuel, measured to the nearest 0.1 litre.

(a)

Write down the lower bound and the upper bound for the distance travelled.

(1)

(b)

Calculate the upper bound for the fuel efficiency of the car, in km per litre.

(2)

(c)

Calculate the lower bound for the fuel efficiency of the car, in km per litre.

(2)

(d)

Hence state the fuel efficiency to an appropriate degree of accuracy, giving a reason.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 385.5 km and 386.5 km, both required	✓ 1
Part (b)
M1 for upper distance ÷ lower fuel, 386.5 ÷ 27.35	✓ 1
A1 14.1 km/L (14.131…)	✓ 1
Part (c)
M1 for lower distance ÷ upper fuel, 385.5 ÷ 27.45	✓ 1
A1 14.0 km/L (14.043…)	✓ 1
Part (d)
A1 14 km/L	✓ 1
R1 both bounds round to 14 at two significant figures, but they differ at three significant figures (14.1 and 14.0); award only with the correct value 14	✓ 1

Common error: For the upper bound of a quotient, divide the upper bound of the numerator by the LOWER bound of the denominator. Using both upper bounds does not give the maximum.

+ Add to test✓ In your test183IBMI-030-05-07Approximation, percentage error and standard form Paper 1Medium6 marks

The distance from Earth to Proxima Centauri is 4.02×10¹³ km. A spacecraft travels at a constant speed of 6.1×10⁴ km h⁻¹.

(a)

Find the time, in hours, that the spacecraft takes to reach Proxima Centauri. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$.

(2)

(b)

Taking one year as 365.25 days, convert your answer to part (a) into years. Give your answer in the form $a \times 10^k$.

(2)

(c)

A student repeats the calculation in part (a) with the distance rounded to 4×10¹³ km. Find the percentage error in the student's time.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for 4.02×10¹³ ÷ 6.1×10⁴	✓ 1
A1 6.59×10⁸ hours	✓ 1
Part (b)
M1 for dividing by 24 × 365.25	✓ 1
A1 7.52×10⁴ years	✓ 1
Part (c)
M1 for recognising that time is proportional to distance, so the percentage error is $\frac{0.02}{4.02} \times 100\%$, or for comparing the two times directly	✓ 1
A1 0.498%	✓ 1

Exam tip: Time is directly proportional to distance at constant speed. The percentage error in the time is therefore the same as the percentage error in the distance, so you do not need to redo the full calculation.

+ Add to test✓ In your test184IBMI-030-05-09Approximation, percentage error and standard form Paper 1Medium5 marks

A streaming company stores video files that are each 2.35×10⁹ bytes. One of its servers has a capacity of 4.8×10¹⁵ bytes.

(a)

Find the number of complete video files that the server can store.

(2)

(b)

Write the number of complete video files found in the previous part in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$, correct to three significant figures.

(1)

(c)

An advert claims that the server holds "2 million videos". Taking the number of complete video files as the exact value, calculate the percentage error in this claim.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for (4.8×10¹⁵) ÷ (2.35×10⁹) = 2 042 553.19…	✓ 1
A1 2 042 553 (rounded down to a whole number of complete files)	✓ 1
Part (b)
A1 2.04×10⁶	✓ 1
Part (c)
M1 for $\frac{\|2\,000\,000-2\,042\,553\|}{2\,042\,553} \times 100\%$	✓ 1
A1 2.08%	✓ 1

Common error: When dividing in standard form, divide the front numbers (4.8 ÷ 2.35) and subtract the indices as 15 − 9 = 6. Subtracting them the wrong way round gives 10⁻⁶, which is clearly unrealistic for a number of files. Also remember to round down: a partial file cannot be stored.

+ Add to test✓ In your test185IBMI-030-02-r1-04Arithmetic and geometric sequences in context Paper 1Medium7 marks

A new car costs $28 500. Its value decreases by 12% of its value at the start of each year.

(a)

Find the value of the car after 5 years.

(2)

(b)

Find the number of complete years after which the value of the car first falls below $10 000.

(3)

(c)

A different car of the same original price is worth $13 000 after 5 years. Find its annual rate of depreciation, assuming the rate is constant.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Uses $28500\times0.88^{5}$ M1 ; $15 040.36 A1	✓ 2
Part (b)
Forms $28500\times0.88^{n}<10000$ or uses a table M1 ; $n>8.19$, or value after 8 years ≈ $10 249.58 and after 9 years ≈ $9019.63 A1 ; 9 years A1	✓ 3
Part (c)
Forms $28500(1-r)^{5}=13000$ M1 ; $r=14.5\%$ A1	✓ 2

Common error: Depreciation of 12% means the multiplier is 0.88, not 0.12. In part (c), solve for (1 − r) first and then subtract from 1 to get the rate.

+ Add to test✓ In your test186IBMI-030-06-r2-04Financial mathematics, loans and annuities Paper 1Medium7 marks

Amara needs a mortgage of 320 000 US dollars (USD), repaid with equal monthly payments, made at the end of each month, over 25 years. Two banks make offers.
Bank A: a nominal annual interest rate of 4.8%, compounded monthly, with no fees.
Bank B: a nominal annual interest rate of 4.5%, compounded monthly, with a USD 6500 arrangement fee added to the amount borrowed.

(a)

Find the monthly payment for Bank A.

(2)

(b)

Find the monthly payment for Bank B.

(2)

(c)

Determine which bank offers the lower total cost, and by how much.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
N = 300, I% = 4.8, PV = 320 000, FV = 0, P/Y = C/Y = 12	✓ 1
USD 1833.59	✓ 1
Part (b)
PV = 326 500 with N = 300, I% = 4.5	✓ 1
USD 1814.79	✓ 1
Part (c)
Totals: A = 300 × 1833.59 = USD 550 077.00 and B = 300 × 1814.79 = USD 544 437.00	✓ 1
Difference = USD 5640.00 (accept 5639.19 from unrounded payments)	✓ 1
Bank B costs less in total, even after the fee, because the lower rate saves more than USD 6500 in interest over 25 years	✓ 1

Common error: Leaving out the arrangement fee, or adding it to the total cost without interest. When the fee is added to the loan, interest is charged on it for the full term.

+ Add to test✓ In your test187IBMI-030-07-r1-03Arithmetic and geometric sequences in context Paper 1Medium7 marks

Two athletes follow different training plans.
Amara runs 12 km in week 1 and increases her weekly distance by 2.5 km each week.
Ben runs 10 km in week 1 and increases his weekly distance by 8% each week.

(a)

Find the distance Amara runs in week 10.

(2)

(b)

Find the total distance Ben runs over the first 10 weeks.

(2)

(c)

Find the first week in which Ben's weekly distance is greater than Amara's.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$u_{10} = 12 + 9(2.5)$ M1	✓ 1
$= 34.5$ km A1	✓ 1
Part (b)
$S_{10} = \frac{10(1.08^{10}-1)}{0.08}$ M1	✓ 1
$= 145$ km (144.865…) A1	✓ 1
Part (c)
Sets up $10(1.08)^{n-1} > 12 + 2.5(n-1)$ M1	✓ 1
Uses a table or graph on the GDC, e.g. week 27: Ben 73.96 km, Amara 77 km; week 28: Ben 79.88 km, Amara 79.5 km M1	✓ 1
Week 28 A1	✓ 1

Exam tip: When one side of an inequality is linear and the other exponential, use a GDC table or a graph intersection. Quote the values on either side of the change to support the answer.

+ Add to test✓ In your test188IBMI-030-07-r1-04Arithmetic and geometric sequences in context Paper 1Medium6 marks

A patient takes a 200 mg tablet of a medicine every 24 hours. By the time the next tablet is due, 35% of the medicine present immediately after the previous dose is still in the body.

(a)

Find the amount of medicine in the body immediately after the 3rd tablet.

(2)

(b)

Show that the amount immediately after the $n$th tablet is $\frac{200(1-0.35^{n})}{0.65}$ mg.

(2)

(c)

Find the amount of medicine in the body immediately after the 7th tablet.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$200 + 200(0.35) + 200(0.35)^2$ M1	✓ 1
$= 294.5$ mg (accept 295 mg) A1	✓ 1
Part (b)
The amount is $200 + 200(0.35) + \dots + 200(0.35)^{n-1}$, a geometric series with $u_1 = 200$ and $r = 0.35$ R1	✓ 1
$S_n = \frac{200(1-0.35^{n})}{1-0.35} = \frac{200(1-0.35^{n})}{0.65}$ A1	✓ 1
Part (c)
$\frac{200(1-0.35^{7})}{0.65}$ M1	✓ 1
$= 307$ mg (307.494…) A1	✓ 1

Common error: The first tablet has been in the body the longest, so it has shrunk the most. The newest tablet is the full 200 mg. Do not apply the factor 0.35 to every term.

+ Add to test✓ In your test189IBMI-030-01-r1-03Financial mathematics, loans and annuities Paper 1Medium7 marks

Helena retires with a fund of $250 000. The fund earns a nominal annual interest rate of 4.2%, compounded quarterly. She plans to withdraw an equal amount at the end of every quarter for 20 years, so that the fund is exactly used up.

(a)

Find the amount of each quarterly withdrawal.

(3)

(b)

Find the total interest earned by the fund over the 20 years.

(2)

(c)

Helena decides instead to withdraw $5000 at the end of every quarter. Find the number of full $5000 withdrawals she can make.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
N = 80 with quarterly compounding (P/Y = C/Y = 4)	✓ 1
PV = ±250 000 and FV = 0, with I% = 4.2	✓ 1
$4634.58	✓ 1
Part (b)
Total withdrawn: 80 × 4634.58 = $370 766.40	✓ 1
Interest = 370 766.40 − 250 000 = $120 766.40	✓ 1
Part (c)
TVM with PV = 250 000, PMT = −5000, FV = 0, I% = 4.2, solving for N	✓ 1
N = 71.3, so 71 full withdrawals (followed by a smaller final one)	✓ 1

Exam tip: In context, a non-integer N means the last withdrawal is only part of a full payment. State the number of complete withdrawals, not the decimal.

+ Add to test✓ In your test190IBMI-030-01-r1-04Financial mathematics, loans and annuities Paper 1Medium7 marks

Jonas has €12 000 to invest for 10 years and is comparing two accounts.
Option A: 5.1% per annum, compounded annually.
Option B: 4.95% nominal annual rate, compounded monthly.

(a)

Find the value of the investment after 10 years under Option A.

(2)

(b)

Find the value of the investment after 10 years under Option B.

(2)

(c)

State which option gives the greater return, and by how much.

(1)

(d)

The annual inflation rate is expected to remain at 2.3% for the 10 years. Find the value of the Option A investment after 10 years in terms of today's money. Give your answer to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$12000(1.051)^{10}$	✓ 1
€19 733.69	✓ 1
Part (b)
$12000\left(1+\frac{0.0495}{12}\right)^{120}$	✓ 1
€19 665.95	✓ 1
Part (c)
Option A, by €67.74	✓ 1
Part (d)
Divides by $1.023^{10}$: $\frac{19733.69}{1.023^{10}}$	✓ 1
€15 720	✓ 1

Common error: A higher compounding frequency does not always give a better return. Compare the final values or the effective annual rates. Here monthly compounding at 4.95% does not make up for the lower nominal rate.

+ Add to test✓ In your test191IBMI-030-08-r2-05Exponents and logarithms Paper 1Medium7 marks

A mug of tea is left to cool in a room. Its temperature, T °C, t minutes after it is made is modelled by
$T(t) = 22 + 68e^{-kt}$
where $k$ is a positive constant. After 10 minutes the temperature of the tea is 55 °C.

(a)

Write down the temperature of the tea when it is made.

(1)

(b)

Find the value of k.

(3)

(c)

Find the time it takes for the tea to cool to 30 °C.

(2)

(d)

State the meaning of the value 22 in this context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: 90 °C	✓ 1
Part (b)
M1: substitutes, $55 = 22 + 68e^{-10k}$	✓ 1
M1: isolates the exponential and uses logarithms, $-10k = \ln\left(\frac{33}{68}\right)$, or solves with GDC	✓ 1
A1: $k = 0.0723$	✓ 1
Part (c)
M1: forms the equation $22 + 68e^{-kt} = 30$	✓ 1
A1: $t = 29.6$ minutes	✓ 1
Part (d)
R1: the temperature of the room, which the tea approaches in the long term but does not go below	✓ 1

Common error: Subtract 22 before taking logarithms. Taking logarithms of $22 + 68e^{-kt}$ term by term is invalid, because the logarithm of a sum is not the sum of the logarithms.

+ Add to test✓ In your test192IBMI-030-07-r2-04Arithmetic and geometric sequences in context Paper 1Medium7 marks

A small gold mine produces 12 000 kg of ore in its first year of operation. A geologist models the output in each following year as 6% less than the output in the year before.

(a)

Find the output in the 8th year.

(2)

(b)

Find the total output over the first 15 years.

(2)

(c)

The mine's owners claim that, if the model continues indefinitely, the total output will eventually exceed 200 000 kg. Use the model to show that this claim is incorrect.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $12000\times0.94^{7}$	✓ 1
A1: 7780 kg (7781.7)	✓ 1
Part (b)
M1: use of $S_{15}=\frac{12000(1-0.94^{15})}{1-0.94}$	✓ 1
A1: 121 000 kg (120 942)	✓ 1
Part (c)
M1: use of $S_\infty=\frac{u_1}{1-r}=\frac{12000}{0.06}$	✓ 1
A1: $S_\infty=200000$ kg	✓ 1
R1: since $\|r\|=0.94<1$, the partial sums increase towards 200 000 kg but never reach it, so the total can never exceed 200 000 kg and the claim is incorrect	✓ 1

Common error: Using $r=0.06$ instead of $r=0.94$. A decrease of 6% leaves 94% of the previous value, so the common ratio is 0.94.

+ Add to test✓ In your test193IBMI-030-02-r2-03Arithmetic and geometric sequences in context Paper 1Medium7 marks

A ball is dropped from a height of 2.5 m onto a hard floor. After each bounce it rises to 70% of the height from which it last fell.

(a)

Find the maximum height reached by the ball after its 4th bounce.

(2)

(b)

Find the number of the first bounce after which the maximum height is less than 5 cm.

(2)

(c)

Find the total vertical distance travelled by the ball from the moment it is dropped until it hits the floor for the 5th time.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$2.5 \times 0.7^{4}$ M1	✓ 1
= 0.600 m (0.60025 m) A1	✓ 1
Part (b)
Sets up $2.5 \times 0.7^{n} < 0.05$ and solves (logs or GDC), giving $n > 10.97$ M1	✓ 1
11th bounce A1	✓ 1
Part (c)
Sum of first four rebound heights: $\frac{1.75(1 - 0.7^{4})}{1 - 0.7} = 4.43275$ M1	✓ 1
Total $= 2.5 + 2 \times 4.43275$ (each rebound is travelled up and down) M1	✓ 1
= 11.4 m (11.3655 m) A1	✓ 1

Common error: Forgetting that each rebound height is travelled twice (up and down) while the initial drop is travelled only once, or including a fifth rebound that has not happened before the 5th impact.

+ Add to test✓ In your test194IBMI-030-02-r2-04Arithmetic and geometric sequences in context Paper 1Medium7 marks

Priya is offered two job contracts, each lasting 10 years.
Contract A: a starting annual salary of $42 000, increasing by $1500 each year.
Contract B: a starting annual salary of $40 000, increasing by 4.5% each year.

(a)

Find Priya's annual salary in year 8 under each contract.

(3)

(b)

Determine which contract gives Priya the greater total earnings over the 10 years. Justify your answer.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Contract A: $42000 + 7 \times 1500 = $52\,500.00$ A1	✓ 1
Contract B: $40000 \times 1.045^{7}$ M1	✓ 1
= $54 434.47 A1	✓ 1
Part (b)
Contract A: $S_{10} = \frac{10}{2}(2\times 42000 + 9\times 1500) = $487\,500.00$ A1	✓ 1
Contract B: $S_{10} = \frac{40000(1.045^{10} - 1)}{0.045}$ M1	✓ 1
= $491 528.37 A1	✓ 1
Contract B gives greater total earnings (by about $4028) R1	✓ 1

Exam tip: For a comparison question, the reasoning mark depends on stating both totals and a clear conclusion in context. Give money answers to 2 decimal places.

+ Add to test✓ In your test195IBMI-030-03-r1-04Exponents and logarithms Paper 1Medium7 marks

The populations of two towns, A and B, are modelled t years after the start of 2025 by
$P_A = 12\,000(1.035)^t$ and $P_B = 18\,500(1.012)^t$.

(a)

Find the population of town A after 10 years.

(2)

(b)

Interpret the value 1.012 in the context of the model for town B.

(1)

(c)

Find the value of t at which the two populations are equal.

(3)

(d)

Suggest one reason why the model for town A may not be valid for large values of t.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
12000(1.035)¹⁰ M1	✓ 1
= 16 900 (16 927.2…) A1	✓ 1
Part (b)
The population of town B increases by 1.2% each year A1	✓ 1
Part (c)
12000(1.035)^t = 18500(1.012)^t M1	✓ 1
(1.035/1.012)^t = 18500/12000, so t = ln(1.5417)/ln(1.0227) or solved graphically M1	✓ 1
t = 19.3 years (19.26…) A1	✓ 1
Part (d)
E.g. a constant 3.5% growth rate cannot continue indefinitely because of limited housing, land or resources R1	✓ 1

Common error: Dividing the two growth factors is correct, but subtracting the percentages (3.5% − 1.2%) to form a single factor of 1.023 gives an inaccurate answer.

+ Add to test✓ In your test196IBMI-030-03-r2-03Exponents and logarithms Paper 1Medium7 marks

The acidity of a solution is measured by its pH, defined by
$\text{pH} = -\log_{10}[\text{H}^+]$,
where $[\text{H}^+]$ is the hydrogen ion concentration in mol dm⁻³.

(a)

A sample of lemon juice has $[\text{H}^+] = 6.3\times10^{-3}$ mol dm⁻³. Calculate its pH.

(2)

(b)

A sample of rainwater has a pH of 5.6. Find its hydrogen ion concentration.

(2)

(c)

Over several years, the pH of a mountain lake fell from 6.8 to 6.2. Find the factor by which the hydrogen ion concentration of the lake increased, and interpret your answer in context.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $-\log_{10}(6.3\times10^{-3})$	✓ 1
A1: pH = 2.20	✓ 1
Part (b)
M1: $[\text{H}^+] = 10^{-5.6}$	✓ 1
A1: $2.51\times10^{-6}$ mol dm⁻³	✓ 1
Part (c)
M1: forms the ratio $\dfrac{10^{-6.2}}{10^{-6.8}}$ or $10^{0.6}$	✓ 1
A1: factor = 3.98	✓ 1
R1: the hydrogen ion concentration of the lake water became about 4 times greater, even though the pH fell by only 0.6	✓ 1

Common error: A pH change of 0.6 does not mean the concentration changes by 0.6. On a logarithmic scale, a difference in pH corresponds to multiplying the concentration by 10 raised to that difference.

+ Add to test✓ In your test197IBMI-030-06-r2-05Financial mathematics, loans and annuities Paper 1Medium6 marks

Kenji wants to save 40 000 Canadian dollars (CAD) for a house deposit in 6 years. He makes equal deposits at the end of each month into an account that pays a nominal annual interest rate of 3.6%, compounded monthly.

(a)

Find the monthly deposit needed if the account starts with a balance of zero.

(3)

(b)

Kenji's grandmother gives him CAD 5000, which he puts into the account at the start. Find the new monthly deposit needed to reach CAD 40 000 in 6 years.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Uses 72 monthly periods at a monthly rate of 0.3% (e.g. N = 72, I% = 3.6, P/Y = C/Y = 12)	✓ 1
Correct set-up for end-of-month payments: PV = 0, FV = 40 000, or $\text{PMT}\times\frac{1.003^{72}-1}{0.003}=40000$	✓ 1
Monthly deposit = CAD 498.54	✓ 1
Part (b)
Includes the initial CAD 5000 with the same N and interest rate (e.g. PV = −5000, with PV and PMT both outflows), or finds that 5000 grows to $5000\times1.003^{72}\approx 6203.51$	✓ 1
Correct equation: $5000\times1.003^{72}+\text{PMT}\times\frac{1.003^{72}-1}{0.003}=40000$, or the equivalent TVM entry with FV = 40 000	✓ 1
Monthly deposit = CAD 421.23	✓ 1

Exam tip: In a TVM solver, money you pay into the account (the initial gift and the deposits) takes one sign, and the amount you receive at the end takes the opposite sign. The PMT shown will therefore be negative, so give the deposit as a positive amount.

+ Add to test✓ In your test198IBMI-030-03-r1-03Exponents and logarithms Paper 1Medium7 marks

A cup of tea is left on a table in a room. Its temperature, T °C, after t minutes is modelled by
$T(t) = 20 + 75e^{-kt}$, where k is a positive constant.
After 8 minutes the temperature of the tea is 60 °C.

(a)

Write down the initial temperature of the tea.

(1)

(b)

Find the value of k.

(2)

(c)

Find the time taken for the tea to cool to 35 °C.

(2)

(d)

Describe what happens to the temperature of the tea in the long term, justifying your answer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
T(0) = 20 + 75 = 95 °C A1	✓ 1
Part (b)
20 + 75e^−8k = 60 ⇒ e^−8k = 40/75 (or equivalent, including a correct GDC approach) M1	✓ 1
k = 0.0786 (0.078576…) A1	✓ 1
Part (c)
20 + 75e^−kt = 35 ⇒ e^−kt = 0.2 (or equivalent, including a correct GDC approach) M1	✓ 1
t = 20.5 minutes (20.48…) A1	✓ 1
Part (d)
Since k > 0, as t → ∞, e^−kt → 0 R1	✓ 1
so T decreases and approaches 20 °C (room temperature) from above, never falling below it A1	✓ 1

Exam tip: Store the unrounded value of k in your GDC and use it in part (c). Rounding a value early and then reusing it can introduce errors into later answers.

+ Add to test✓ In your test199IBMI-030-07-r2-03Arithmetic and geometric sequences in context Paper 1Medium7 marks

Lena starts a savings plan. She saves €50 in the first month. In each following month she saves €8 more than in the previous month.

(a)

Find the amount Lena saves in the 12th month.

(2)

(b)

Lena wants her total savings to reach at least €5000. Find the number of months it takes her to reach this target.

(4)

(c)

Find how much her total savings exceed €5000 at the end of that month.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $50+11\times8$	✓ 1
A1: €138	✓ 1
Part (b)
M1: use of the arithmetic series formula $\frac{n}{2}(2\times50+8(n-1))$	✓ 1
M1: sets this $\geq5000$ (or $=5000$), e.g. $4n^2+46n-5000=0$	✓ 1
A1: $n=30.07$ (from GDC or the quadratic formula)	✓ 1
A1: 31 months, since $S_{30}=4980$ euros, which is below the target	✓ 1
Part (c)
A1: $S_{31}=5270$, so she exceeds the target by €270	✓ 1

Exam tip: An answer of $n=30.07$ means the target is reached during month 31, not month 30. State a whole number of months and show why 30 months is not enough.

+ Add to test✓ In your test200IBMI-030-08-r1-03Exponents and logarithms Paper 1Medium7 marks

The sound level, $L$ decibels (dB), of a noise with intensity $I$ W m⁻² is given by $L = 10\log_{10}\left(\frac{I}{I_0}\right)$, where $I_0 = 1.0$×10⁻¹² W m⁻².
A workshop machine produces sound with intensity 4.5×10⁻⁵ W m⁻² at the operator's position.

(a)

Calculate the sound level of the machine at the operator's position.

(2)

(b)

Ear protection must be worn if the sound level reaches 95 dB. Find the intensity that corresponds to 95 dB.

(2)

(c)

A second, identical machine is switched on, so the intensity at the operator's position doubles. Find the increase in the sound level.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes correctly, L = 10 log₁₀(4.5×10⁻⁵ ÷ 1.0×10⁻¹²) = 10 log₁₀(4.5×10⁷)	✓ 1
A1: L = 76.5 dB (76.53…)	✓ 1
Part (b)
M1: rearranges to I = 1.0×10⁻¹² × 10^9.5	✓ 1
A1: I = 3.16×10⁻³ W m⁻²	✓ 1
Part (c)
M1: uses the doubled intensity 9.0×10⁻⁵ W m⁻² in the formula, OR writes the increase as 10 log₁₀(2I/I₀) − 10 log₁₀(I/I₀)	✓ 1
A1: new level is 79.5 dB (79.54…), OR simplifies the increase to 10 log₁₀2	✓ 1
A1: increase of 3.01 dB (accept 3.0 dB from subtracting the rounded levels)	✓ 1

Common error: Doubling the intensity does not double the decibel level. On a logarithmic scale it adds 10 log₁₀2 ≈ 3.01 dB. Keep unrounded values in your calculator when you subtract two levels.

+ Add to test✓ In your test201IBMI-030-01-r1-05Financial mathematics, loans and annuities Paper 1Medium7 marks

The Okafor family take out a $320 000 mortgage to buy a house. The loan is repaid with equal monthly payments over 25 years, at a nominal annual interest rate of 4.8%, compounded monthly.

(a)

Find the monthly payment.

(3)

(b)

Find the amount still owed immediately after the 60th payment.

(2)

(c)

Find the amount of interest paid during the first 5 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
N = 300 and I% = 4.8 with P/Y = C/Y = 12	✓ 1
PV = 320 000, FV = 0, solving for PMT	✓ 1
$1833.59	✓ 1
Part (b)
TVM with N = 60, PV = 320 000, PMT = −1833.59, solving for FV, or the present value of the remaining 240 payments	✓ 1
$282 544 (accept answers from $282 543 to $282 545, from rounded or unrounded PMT)	✓ 1
Part (c)
Principal repaid = 320 000 − 282 544 = $37 456, and total paid = 60 × 1833.59 = $110 015.40	✓ 1
Interest = $72 559.40 (accept answers from $72 558 to $72 561, follow through from their (b))	✓ 1

Exam tip: Interest paid over a period equals the total payments minus the reduction in the balance, not minus the full loan. In the first 5 years about two-thirds of each payment is interest.

+ Add to test✓ In your test202IBMI3S-1-01-05Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

Tomas takes out a car loan of $18 000. The loan has a nominal annual interest rate of 6.4%, compounded monthly. He repays the loan with equal payments at the end of each month over 5 years.

(a)

Find Tomas's monthly payment, to the nearest cent.

(3)

(b)

Find the total amount of interest Tomas pays over the 5 years. Give your answer to the nearest dollar.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of financial solver with N = 60, I% = 6.4, FV = 0	✓ 1
A1 PV = ±18 000, P/Y = C/Y = 12	✓ 1
A1 $351.35	✓ 1
Part (b)
M1 $60\times351.35$	✓ 1
A1 total repaid $21 081	✓ 1
A1 interest $3081	✓ 1

Exam tip: Enter the loan as positive and the payments as negative (or vice versa); a sign mismatch is the usual reason a solver returns an error.

+ Add to test✓ In your test203IBMI3S-1-01-07Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

A basketball is dropped from a height of 3.0 m onto a hard floor. After each bounce it rises to 70% of the height of the previous peak.

(a)

Find the height reached after the third bounce, to 3 significant figures.

(2)

(b)

Find the number of bounces after which the ball first rises to less than 2 cm.

(2)

(c)

Find the total distance the ball has travelled when it hits the floor for the sixth time. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $3.0\times0.7^{3}$	✓ 1
A1 1.03 m	✓ 1
Part (b)
M1 $3.0\times0.7^{n}<0.02$ (units consistent) and attempt to solve	✓ 1
A1 15 bounces	✓ 1
Part (c)
M1 recognising distance as $3.0+2\sum_{k=1}^{5}3.0(0.7)^k$	✓ 1
A1 correct sum of the five rebound heights	✓ 1
A1 14.6 m	✓ 1

Exam tip: After each bounce the ball travels up and down the same height, so every rebound height after the first drop is counted twice.

+ Add to test✓ In your test204IBMI3S-1-01-08Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

The sound intensity level $L$, in decibels (dB), of a sound with intensity $I$ watts per square metre is given by $L=10\log_{10}\left(\frac{I}{10^{-12}}\right)$.

(a)

A drill has intensity $3.2\times10^{-5}$ W m⁻². Calculate its sound intensity level, to 3 significant figures.

(2)

(b)

A concert speaker produces a level of 92 dB. Find its intensity, in the form $a\times10^{k}$ with $a$ to 3 significant figures.

(2)

(c)

A second identical speaker is switched on so that the intensity is multiplied by 4. Find the increase in sound intensity level, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $10\log_{10}\left(\frac{3.2\times10^{-5}}{10^{-12}}\right)$	✓ 1
A1 75.1 dB	✓ 1
Part (b)
M1 $\frac{I}{10^{-12}}=10^{9.2}$	✓ 1
A1 $1.58\times10^{-3}$ W m⁻²	✓ 1
Part (c)
M1 $10\log_{10}4$ (or difference of two levels)	✓ 1
A1 6.02 dB	✓ 1

Common error: Multiplying the intensity by a factor adds a fixed number of decibels; it does not multiply the decibel value by that factor.

+ Add to test✓ In your test205IBMI3S-1-01-10Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium5 marks

A scooter is bought for $3200. After 3 years its value is $1980. Its value decreases by a constant percentage each year.

(a)

Find the annual percentage rate of depreciation, to 3 significant figures.

(3)

(b)

Using the unrounded value of $r$, find the predicted value of the scooter after 6 years, to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $3200r^{3}=1980$	✓ 1
A1 $r=0.852$	✓ 1
A1 14.8 %	✓ 1
Part (b)
M1 $3200r^{6}$ (or $1980r^{3}$)	✓ 1
A1 $1225.13	✓ 1

Common error: The multiplier is $r\approx0.852$, so the depreciation rate is $1-r$, not $r$.

+ Add to test✓ In your test206IBMI3S-1-01-11Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The first row of seats in a stadium stand has 24 seats. Each row behind it has 3 more seats than the row in front.

(a)

Find the number of seats in row 20.

(2)

(b)

The stand must hold at least 2000 people. Find the least number of rows needed.

(4)

(c)

Find the number of spare seats when the stand has exactly this number of rows.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $24+19(3)$	✓ 1
A1 81	✓ 1
Part (b)
M1 use of $S_n=\frac{n}{2}\left(2(24)+(n-1)3\right)$	✓ 1
M1 setting $S_n\ge2000$ and solving (technology or quadratic)	✓ 1
A1 $n\ge29.8$	✓ 1
A1 30 rows	✓ 1
Part (c)
A1 $S_{30}-2000=2025-2000=25$ (ECF from (b))	✓ 1

Exam tip: A quadratic inequality from $S_n$ gives a non-integer boundary; always state the whole number of rows that satisfies the condition.

+ Add to test✓ In your test207IBMI3S-1-01b-05Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

A farm cooperative buys a combine harvester priced at €85 000. It pays a deposit of 20% of the price and borrows the rest. The loan has an annual interest rate of 6%, compounded monthly, and is repaid by equal payments at the end of each month for 5 years.

(a)

Find the amount borrowed.

(2)

(b)

Use a finance application to find the monthly payment.

(2)

(c)

Find the total interest paid over the 5 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 deposit = 0.2 × 85000 = 17000	✓ 1
A1 €68000	✓ 1
Part (b)
M1 N = 60, I% = 6, PV = 68000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €1314.63	✓ 1
Part (c)
M1 60 × monthly payment − 68000	✓ 1
A1 €10877.83 (accept €10877.80 if the rounded payment is used)	✓ 1

Common error: Enter the loan as a positive present value and the payments as negative (or the reverse) in the finance solver; mixing the signs gives an error or a wrong answer.

+ Add to test✓ In your test208IBMI3S-1-01b-06Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

An irrigation reservoir holds 12.5 megalitres (ML) at the start of week 1. During the dry season the volume at the start of each week is 8% less than the volume at the start of the previous week.

(a)

Write down the common ratio of the volumes at the start of successive weeks.

(1)

(b)

Find the volume of water in the reservoir at the start of week 7.

(2)

(c)

Show that the volume lost during week 1 is 1.00 ML.

(1)

(d)

The volumes lost in successive weeks form a geometric sequence. Find the total volume lost during the first 10 weeks.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.92	✓ 1
Part (b)
M1 12.5 × 0.92⁶	✓ 1
A1 7.58 ML	✓ 1
Part (c)
A1 12.5 × 0.08 = 1.00 ML	✓ 1
Part (d)
M1 S₁₀ = 1 × (1 − 0.92¹⁰) ÷ (1 − 0.92)	✓ 1
A1 7.07 ML	✓ 1

Exam tip: Week 7 begins after 6 weeks of loss, so the power of the ratio is 6, not 7.

+ Add to test✓ In your test209IBMI3S-1-01b-07Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

Atmospheric pressure P kilopascals (kPa) at height h metres above sea level is modelled by P = 101.3e^−0.000125h.

(a)

Find the pressure at a height of 3000 m.

(2)

(b)

An aircraft cruises at a height where the pressure is 25 kPa. Find this height, to the nearest 100 m.

(3)

(c)

Find the height at which the pressure is half of the sea-level pressure.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 101.3 × e^{−0.000125 × 3000}	✓ 1
A1 69.6 kPa	✓ 1
Part (b)
M1 25 = 101.3e^−0.000125h	✓ 1
M1 takes natural logarithms: ln(25 ÷ 101.3) = −0.000125h	✓ 1
A1 11200 m	✓ 1
Part (c)
M1 e^−0.000125h = 0.5	✓ 1
A1 h = ln 2 ÷ 0.000125 = 5545 m, so 5550 m (3 s.f.)	✓ 1

Exam tip: To undo an exponential, take natural logarithms of both sides after isolating the exponential term.

+ Add to test✓ In your test210IBMI3S-1-01b-08Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

A passenger aircraft is fuelled with 2.45 × 10⁴ litres of jet fuel. The density of the fuel is 0.80 kg per litre and each kilogram releases 4.3 × 10⁷ J of energy when burned.

(a)

Calculate the mass of fuel, in kg. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Calculate the energy released by this fuel, in joules, giving your answer in the form a × 10^k.

(2)

(c)

A planner estimates the energy by rounding each of the three given values to 1 significant figure first. Find the percentage error in the planner's estimate.

(2)

(d)

State whether the planner's estimate is an underestimate or an overestimate, and give a reason.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2.45 × 10⁴ × 0.80	✓ 1
A1 1.96 × 10⁴ kg	✓ 1
Part (b)
M1 1.96 × 10⁴ × 4.3 × 10⁷	✓ 1
A1 8.43 × 10¹¹ J	✓ 1
Part (c)
M1 estimate = 2 × 10⁴ × 0.8 × 4 × 10⁷ = 6.4 × 10¹¹ J	✓ 1
A1 (8.428 × 10¹¹ − 6.4 × 10¹¹) ÷ (8.428 × 10¹¹) × 100 = 24.1 %	✓ 1
Part (d)
R1 underestimate, because 2.45 × 10⁴ and 4.3 × 10⁷ were both rounded down (and 0.80 is unchanged)	✓ 1

Common error: Do not round the intermediate values before finding a percentage error: use the exact calculated value as the denominator.

+ Add to test✓ In your test211IBMI3S-1-01b-09Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The sound level L decibels (dB) of a sound of intensity I W m⁻² is given by L = 10 log₁₀(I ÷ 10⁻¹²).

(a)

A rehearsal room has a sound level of 85 dB. Find the intensity, giving your answer in the form a × 10^k.

(3)

(b)

Show that doubling the intensity of a sound increases its sound level by 3.01 dB, correct to 3 significant figures.

(2)

(c)

Four identical amplifiers, each producing a sound level of 82 dB on its own, play together. Assuming their intensities add, find the total sound level.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 85 = 10 log₁₀(I ÷ 10⁻¹²), so log₁₀(I ÷ 10⁻¹²) = 8.5	✓ 1
M1 I = 10⁻¹² × 10^8.5	✓ 1
A1 3.16 × 10⁻⁴ W m⁻²	✓ 1
Part (b)
M1 10 log₁₀(2I ÷ 10⁻¹²) − 10 log₁₀(I ÷ 10⁻¹²) = 10 log₁₀ 2	✓ 1
A1 10 log₁₀ 2 = 3.01 dB	✓ 1
Part (c)
M1 total intensity is 4 times the intensity of one, so level = 82 + 10 log₁₀ 4	✓ 1
A1 88.0 dB	✓ 1

Common error: Sound levels in decibels do not add directly: four 82 dB sources do not give 328 dB. Add the intensities, then convert.

+ Add to test✓ In your test212IBMI3S-1-01b-11Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

A flying club rents a hangar. The rent for year 1 is €18 000. The rent increases by 3.5% each year and is paid for the whole year.

(a)

Find the rent for year 8, to the nearest euro.

(2)

(b)

Find the total rent paid over the first 10 years, to the nearest euro.

(2)

(c)

Find the first year in which the rent exceeds €25 000.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18000 × 1.035⁷	✓ 1
A1 €22901	✓ 1
Part (b)
M1 S₁₀ = 18000 (1.035¹⁰ − 1) ÷ (1.035 − 1)	✓ 1
A1 €211165	✓ 1
Part (c)
M1 18000 × 1.035ⁿ⁻¹ > 25000, giving n > 10.5	✓ 1
A1 year 11	✓ 1

Common error: The rent in year n uses the power n − 1, because year 1 has no increase yet.

+ Add to test✓ In your test213IBMI3S-1-01r5-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

The concentration of chlorine in a swimming pool is modelled by C = 3.2 × 0.85^t, where C is in mg per litre and t is the time in hours after the chlorine is added.

(a)

Write down the initial concentration of chlorine.

(1)

(b)

Find the concentration after 3 hours. Give your answer to 3 significant figures.

(2)

(c)

The pool is safe to use when the concentration is at most 1.0 mg per litre. Find the time, to the nearest 0.01 hours, after which the pool is first safe to use.

(3)

(d)

State one limitation of using this model for a very long time.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 3.2 mg per litre	✓ 1
Part (b)
M1 substitution of t = 3 into the model	✓ 1
A1 1.97 mg per litre	✓ 1
Part (c)
M1 set up 3.2 × 0.85^t = 1.0	✓ 1
M1 use of logarithms (t = ln(1/3.2) ÷ ln 0.85) or a GDC solver	✓ 1
A1 t = 7.16 hours	✓ 1
Part (d)
R1 e.g. the concentration tends to zero but a pool is topped up / chlorine is added again, so the model no longer applies	✓ 1

Exam tip: Solving a·bt = k: divide by a first, then take logarithms of both sides.

+ Add to test✓ In your test214IBMI3S-1-01r5-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A town library borrows £40000 to refurbish its reading room. The loan has a nominal annual interest rate of 4.5%, compounded monthly. It is repaid by equal payments made at the end of each month for 8 years.

(a)

Use technology to find the monthly payment. Give your answer to the nearest penny.

(2)

(b)

Calculate the total amount of interest paid over the 8 years. Use the unrounded payment from (a) and give your answer to the nearest pound.

(3)

(c)

Find the amount still owed immediately after the 36th payment. Use the unrounded payment from (a) and give your answer to the nearest pound.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance solver with N = 96, I% = 4.5, PV = 40000, FV = 0, P/Y = 12, C/Y = 12	✓ 1
A1 £496.93 per month	✓ 1
Part (b)
M1 total repaid = 96 × payment	✓ 1
M1 interest = total repaid − 40000 = 47705.22 − 40000	✓ 1
A1 £7705	✓ 1
Part (c)
M1 finance solver with N = 60 (remaining payments) and the monthly payment, or balance = 40000(1.00375)³⁶ − payment × ((1.00375)³⁶ − 1)/0.00375	✓ 1
A1 £26655	✓ 1

Common error: Divide the nominal annual rate by 12 for the monthly rate; do not use 4.5% as the rate for each month.

+ Add to test✓ In your test215IBMI3S-1-01r5-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

The wind speed v m s⁻¹ at a height h metres above the ground at a wind farm site is modelled by v = 6.0 (h ÷ 10)^0.14, for h ≥ 10.

(a)

Calculate the wind speed at the hub height of 80 m. Give your answer to 3 significant figures.

(2)

(b)

Find the height at which the model gives a wind speed of 7.5 m s⁻¹. Give your answer to the nearest metre.

(3)

(c)

Explain why the model cannot be used for a height of 5 m.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitution of h = 80	✓ 1
A1 8.03 m s⁻¹	✓ 1
Part (b)
M1 set up 6.0(h/10)^0.14 = 7.5	✓ 1
M1 (h/10) = (7.5/6)^1/0.14 or GDC solver	✓ 1
A1 h = 49 m	✓ 1
Part (c)
R1 the model is only valid for h ≥ 10 (domain stated)	✓ 1

Exam tip: To undo a power of 0.14, raise both sides to the power 1/0.14 rather than dividing by 0.14.

+ Add to test✓ In your test216IBMI3S-1-01r5-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A swimmer increases the number of laps she swims by the same amount in each training session. She swims 14 laps in session 3 and 29 laps in session 8.

(a)

Find the common difference and the number of laps she swims in session 1.

(2)

(b)

Find the first session after which the total number of laps she has swum exceeds 500.

(3)

(c)

Find the total number of laps she swims in the first 20 sessions.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 d = (29 − 14) ÷ 5 = 3	✓ 1
A1 u₁ = 14 − 2 × 3 = 8	✓ 1
Part (b)
M1 S_n = (n/2)(2 × 8 + (n − 1) × 3) and S_n > 500	✓ 1
M1 solve the inequality or use a table on a GDC (S₁₆ = 488, S₁₇ = 544)	✓ 1
A1 session 17	✓ 1
Part (c)
M1 S₂₀ = 10 × (2 × 8 + 19 × 3)	✓ 1
A1 730 laps	✓ 1

Common error: Sn must be found for whole values of n; check the sessions either side of the crossing point.

+ Add to test✓ In your test217IBMI3S-1-01r5-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

Each year a library receives donated books. In 2021 it received 900 books. Each year after that the number received is 15% lower than the number received in the previous year.

(a)

Write down the common ratio of the sequence of yearly donations.

(1)

(b)

Find the number of books donated in 2026. Give your answer to the nearest whole number.

(2)

(c)

Find the total number of books donated from 2021 to 2030 inclusive. Give your answer to the nearest whole number.

(2)

(d)

The donations are assumed to continue in this pattern for ever. Find the greatest possible total number of books the library could receive.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.85	✓ 1
Part (b)
M1 u₆ = 900 × 0.85⁵	✓ 1
A1 399 books	✓ 1
Part (c)
M1 S₁₀ = 900(1 − 0.85¹⁰)/(1 − 0.85)	✓ 1
A1 4819 books	✓ 1
Part (d)
M1 S_∞ = u₁/(1 − r)	✓ 1
A1 900/0.15 = 6000 books	✓ 1

Exam tip: A sum to infinity exists only when |r| < 1; state this before using the formula.

+ Add to test✓ In your test218IBMI3S-1-01r5-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

The rotor of a wind turbine has a radius of 52 m. The area swept by the rotor is modelled as a circle.

(a)

Calculate the swept area. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a given to 3 significant figures.

(3)

(b)

A technician uses the approximation π ≈ 3 to estimate the area. Find the percentage error in her estimate of the area. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 A = π × 52²	✓ 1
A1 8494.9 m²	✓ 1
A1 8.49 × 10³ m²	✓ 1
Part (b)
M1 estimate = 3 × 52² = 8112	✓ 1
M1 percentage error = \|estimate − exact\| ÷ exact × 100, using the unrounded exact area	✓ 1
A1 4.51%	✓ 1

Common error: Use the unrounded value of the exact area in the percentage error formula, not the standard form value.

+ Add to test✓ In your test219IBMI3S-1-01r5-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A library had 18000 items in its catalogue at the start of 2010 and 31000 items at the start of 2024. The number of items is assumed to grow by the same percentage each year. Use unrounded values in later parts.

(a)

Find the annual percentage growth rate. Give your answer to 3 significant figures.

(3)

(b)

Using this model, find after how many years from the start of 2010 the catalogue first reaches 50000 items. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18000 × r¹⁴ = 31000	✓ 1
M1 r = (31000/18000)^1/14	✓ 1
A1 growth rate = 3.96%	✓ 1
Part (b)
M1 18000 × r^t = 50000	✓ 1
M1 t = ln(50000/18000) ÷ ln r (or GDC solver)	✓ 1
A1 26.3 years	✓ 1

Exam tip: Keep the unrounded growth factor in your calculator memory for the second part so rounding does not change the answer.

+ Add to test✓ In your test220IBMI3S-1-02r7-06Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

A marine research fund is built up by paying a fixed amount into an account at the end of each month. The account pays a nominal annual interest rate of 3%, compounded monthly.

(a)

A deposit of €150 is made each month for 5 years. Find the value of the fund at the end of the 5 years, correct to the nearest euro.

(3)

(b)

The fund needs to reach €12 000 after 5 years. Find the monthly deposit required, correct to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM solver with N = 60, I% = 3, PMT = −150, PV = 0, P/Y = C/Y = 12	✓ 1
A1 150 × (1.0025⁶⁰ − 1)/0.0025	✓ 1
A1 €9697	✓ 1
Part (b)
M1 solve for PMT with FV = 12000, N = 60	✓ 1
A1 12000 × 0.0025/(1.0025⁶⁰ − 1)	✓ 1
A1 €185.62	✓ 1

Exam tip: In a TVM solver, payments you make and the final value have opposite signs to the amount you receive.

+ Add to test✓ In your test221IBMI3S-1-02r7-07Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

A logistics company holds 5000 units of a product at the start of week 1. At the start of each following week, the stock is 8% lower than at the start of the previous week. The stock levels form a geometric sequence.

(a)

Write down the common ratio.

(1)

(b)

Find the first week in which the stock at the start of the week is below 2500 units.

(3)

(c)

Find the sum of the stock levels at the start of weeks 1 to 12, correct to the nearest unit.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.92	✓ 1
Part (b)
M1 5000 × 0.92ⁿ⁻¹ < 2500	✓ 1
A1 n − 1 > 8.31 (week 9 gives 2566, week 10 gives 2361)	✓ 1
A1 week 10	✓ 1
Part (c)
M1 S₁₂ = 5000(1 − 0.92¹²)/(1 − 0.92)	✓ 1
A1 39521 units	✓ 1

Exam tip: For the first term of a sequence beyond a threshold, test the terms either side of the boundary.

+ Add to test✓ In your test222IBMI3S-1-02r7-08Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The intensity of light in sea water at depth d metres is I = I₀ e^−kd, where I₀ is the intensity at the surface. At a depth of 10 m the intensity is 40% of the surface intensity.

(a)

Find the value of k, correct to 3 significant figures.

(2)

(b)

Find the intensity at a depth of 25 m as a percentage of the surface intensity. Use the unrounded value of k and give your answer correct to 3 significant figures.

(2)

(c)

Find the depth at which the intensity is 5% of the surface intensity. Use the unrounded value of k and give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 e^−10k = 0.4	✓ 1
A1 k = 0.0916	✓ 1
Part (b)
M1 e^−25k = 0.4^2.5	✓ 1
A1 10.1%	✓ 1
Part (c)
M1 e^−kd = 0.05	✓ 1
M1 d = ln 20 / k	✓ 1
A1 32.7 m	✓ 1

Exam tip: Keep the exact expression for the constant in your calculator rather than the rounded value.

+ Add to test✓ In your test223IBMI3S-1-02r7-09Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium5 marks

A logistics depot delivers an increasing number of packages each day. The numbers delivered on successive days form an arithmetic sequence. On day 5 the depot delivers 47 packages and on day 12 it delivers 96 packages.

(a)

Find the common difference and the number of packages delivered on day 1.

(3)

(b)

Find the total number of packages delivered over the first 25 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₂ − u₅ = 7d, so 96 − 47 = 7d	✓ 1
A1 d = 7	✓ 1
A1 u₁ = 47 − 4 × 7 = 19	✓ 1
Part (b)
M1 S₂₅ = (25/2)(2 × 19 + 24 × 7)	✓ 1
A1 2575	✓ 1

Exam tip: Subtracting two terms of an arithmetic sequence gives the difference multiplied by the gap between the term numbers.

+ Add to test✓ In your test224IBMI3S-1-02r7-10Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

An architect measures a rectangular atrium as 38.4 m long and 21.7 m wide.

(a)

Calculate the exact area of the atrium.

(2)

(b)

Write the area in part (a) in the form a × 10^k, where 1 ≤ a < 10, a correct to 3 significant figures and k an integer.

(1)

(c)

To check the result, the architect estimates the area by rounding each length to 1 significant figure and multiplying. Write down this estimate.

(1)

(d)

Calculate the percentage error of the estimate in part (c), taking the exact area as the true value. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 38.4 × 21.7	✓ 1
A1 833.28 m²	✓ 1
Part (b)
A1 8.33 × 10² m²	✓ 1
Part (c)
A1 40 × 20 = 800 m²	✓ 1
Part (d)
M1 \|800 − 833.28\| / 833.28 × 100	✓ 1
A1 3.99%	✓ 1

Common error: Rounding every number up or down before multiplying can compound errors; compare the estimate with the exact value.

+ Add to test✓ In your test225IBMI3S-1-02r10-06Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

A food manufacturer borrows £18 000 to buy a mixing machine. The loan has a nominal annual interest rate of 6%, compounded monthly. It is repaid by equal payments made at the end of each month for 5 years.

(a)

Calculate the monthly payment. Give your answer to the nearest penny.

(3)

(b)

Find the total amount repaid over the 5 years and hence the interest paid. Give your answers to the nearest penny, using the monthly payment in part (a).

(2)

(c)

Calculate the amount still owed immediately after the 24th payment. Give your answer to the nearest pound.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
TVM entries: N = 60, I% = 6, PV = 18 000, FV = 0, P/Y = C/Y = 12	✓ 1
Payments at end of period (or equivalent method)	✓ 1
£347.99	✓ 1
Part (b)
Total repaid = 60 × 347.99 = £20879.40	✓ 1
Interest = £2879.40	✓ 1
Part (c)
Use of TVM with N = 24 (or balance formula) with payment from part (a)	✓ 1
£11439	✓ 1

Exam tip: State the TVM entries you used (N, I%, PV, PMT, FV, P/Y, C/Y) so method marks can be awarded even if a number is mistyped.

+ Add to test✓ In your test226IBMI3S-1-02r10-07Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

The monthly data traffic on a mobile network is 450 terabytes (TB) in month 1. The traffic increases by 8% each month, so the traffic in month n forms a geometric sequence T_n.

(a)

Write down the common ratio of the sequence.

(1)

(b)

Calculate the traffic in month 12. Give your answer to the nearest TB.

(2)

(c)

Find the value of n for which T_n = 1500. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
1.08	✓ 1
Part (b)
T₁₂ = 450 × 1.08¹¹	✓ 1
1049 TB	✓ 1
Part (c)
450 × 1.08ⁿ⁻¹ = 1500	✓ 1
(n − 1) = log(1500/450) ÷ log 1.08 (or GDC solver)	✓ 1
n = 16.6	✓ 1

Exam tip: In a geometric sequence the n-th term uses the power n − 1; keep this in the equation before taking logarithms.

+ Add to test✓ In your test227IBMI3S-1-02r10-09Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The number of bacteria, N, in a sample of milk t hours after it was left out of a fridge is modelled by N = 500 × 2^t/4.

(a)

Calculate the number of bacteria after 10 hours. Give your answer to 3 significant figures.

(2)

(b)

Find the time taken for the number of bacteria to reach 20 000. Give your answer to 3 significant figures.

(3)

(c)

The model can also be written as N = 500e^kt. Find the value of k to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitution 500 × 2^2.5	✓ 1
2830	✓ 1
Part (b)
2^t/4 = 40	✓ 1
t/4 = log 40 ÷ log 2 (or GDC)	✓ 1
t = 21.3 hours	✓ 1
Part (c)
2^t/4 = e^kt so k = (ln 2) ÷ 4	✓ 1
k = 0.173	✓ 1

Exam tip: To change the base of an exponential, write the base as a power of e using a = e^(ln a).

+ Add to test✓ In your test228IBMI3S-1-02r10-10Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

In a telecommunications cable the loss in signal strength is measured in decibels (dB). If the power into the cable is P_in and the power out is P_out, the loss is 10 log₁₀(P_in ÷ P_out) dB.

(a)

A signal enters a cable with power 5.0 mW and leaves with power 0.32 mW. Calculate the loss in dB. Give your answer to 3 significant figures.

(2)

(b)

The cable loses 0.35 dB for every kilometre. Find the length of cable that gives a total loss of 30 dB. Give your answer to 3 significant figures.

(2)

(c)

A different cable has a total loss of 20 dB. A signal of power 5.0 mW enters it. Find the output power in mW.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
10 log₁₀(5.0 ÷ 0.32)	✓ 1
11.9 dB	✓ 1
Part (b)
30 ÷ 0.35	✓ 1
85.7 km	✓ 1
Part (c)
10 log₁₀(5 ÷ P_out) = 20 so 5 ÷ P_out = 10²	✓ 1
P_out = 0.05 mW	✓ 1

Common error: Every 10 dB of loss divides the power by 10; the decibel scale is logarithmic, so 20 dB is not simply twice the power loss of 10 dB.

+ Add to test✓ In your test229IBMI3S-1-02r10-14Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The atmospheric pressure, P kPa, at a height of h metres above sea level is modelled by P = 101.3e^−0.00013h.

(a)

Calculate the pressure at a base camp at a height of 5400 m. Give your answer to 3 significant figures.

(2)

(b)

Find the height at which the pressure is 34.0 kPa. Give your answer to the nearest 10 m.

(3)

(c)

Find the percentage by which the pressure decreases for each 1000 m increase in height. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitution of h = 5400	✓ 1
P = 50.2 kPa	✓ 1
Part (b)
101.3e^−0.00013h = 34.0	✓ 1
h = ln(34/101.3) ÷ (−0.00013) (use of logarithms or GDC)	✓ 1
h = 8400 m	✓ 1
Part (c)
Pressure is multiplied by e^−0.13 = 0.878… for each 1000 m	✓ 1
Decrease = 12.2%	✓ 1

Common error: The percentage decrease is 1 minus the multiplier, written as a percentage; the exponent −0.13 is not itself the percentage.

+ Add to test✓ In your test230IBMI3S-1-02r12-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium7 marks

A bridge authority invests €240 000 in a maintenance fund. The account pays 3.2% nominal annual interest, compounded quarterly.

(a)

Calculate the value of the fund after 8 years. Give your answer to the nearest cent.

(2)

(b)

Find the smallest whole number of quarters after which the fund first reaches €400 000.

(3)

(c)

Calculate the annual interest rate, compounded annually, that would give the same value after one year. Give your answer as a percentage to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 240000(1 + 0.032/4)³² (or TVM: N = 32, I% = 3.2, PV = −240000, P/Y = C/Y = 4)	✓ 1
A1 €309705.04	✓ 1
Part (b)
M1 240000(1.008)^q = 400000, or TVM with FV = 400000	✓ 1
A1 q = 64.1 (solution of the equation)	✓ 1
A1 65 quarters (the value after 64 quarters is below €400 000)	✓ 1
Part (c)
M1 (1.008)⁴ − 1	✓ 1
A1 3.24%	✓ 1

Exam tip: If a question asks for the smallest whole number of periods, solve first and then round up. Check by testing the whole number below your answer.

+ Add to test✓ In your test231IBMI3S-1-02r12-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium7 marks

A space probe is lowering its orbit around a planet by a series of manoeuvres. The period of its first orbit is 120 minutes and each orbit is 4 minutes shorter than the one before it. The probe completes exactly one orbit at each stage. The periods form an arithmetic sequence.

(a)

Find an expression for the period u_n of the nth orbit.

(2)

(b)

Find the number of the first orbit whose period is less than 70 minutes.

(2)

(c)

Show that the least number of orbits for which the total time first exceeds 1000 minutes is 10.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u_n = 120 + (n − 1)(−4)	✓ 1
A1 u_n = 124 − 4n	✓ 1
Part (b)
M1 124 − 4n < 70, so n > 13.5	✓ 1
A1 n = 14	✓ 1
Part (c)
M1 S₁₀ = 10/2 (120 + 84)	✓ 1
A1 S₁₀ = 1020 minutes, which exceeds 1000	✓ 1
A1 S₉ = 9/2 (120 + 88) = 936 minutes, which is below 1000, so 10 is the least number	✓ 1

Common error: Solving for n and stopping at a decimal. The number of orbits must be a whole number, so check the totals either side of your answer.

+ Add to test✓ In your test232IBMI3S-1-02r12-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium6 marks

Coffee beans leave a roaster at 200 °C and cool in a room at 22 °C. The temperature T °C of the beans t minutes after leaving the roaster is modelled by T = 22 + 178e^−0.30t.

(a)

Calculate the temperature of the beans after 5 minutes. Give your answer to 3 significant figures.

(2)

(b)

Find the time taken for the beans to cool to 30 °C. Give your answer to 3 significant figures.

(2)

(c)

The difference between the temperature of the beans and the room temperature halves at regular intervals. Find the length of this interval. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 22 + 178e^{−0.30 × 5}	✓ 1
A1 61.7 °C	✓ 1
Part (b)
M1 22 + 178e^−0.30t = 30, so e^−0.30t = 8/178	✓ 1
A1 t = 10.3 minutes	✓ 1
Part (c)
M1 e^−0.30t = 1/2, or ln 2 ÷ 0.30	✓ 1
A1 2.31 minutes	✓ 1

Exam tip: To isolate an exponential term, subtract the constant first and divide second, and only then take a natural logarithm.

+ Add to test✓ In your test233IBMI3S-1-02r12-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium7 marks

The number of stress cycles N that a steel bridge component survives is modelled by N = CS^−m, where S is the stress range in MPa and C and m are positive constants. A test at S = 80 gives N = 1.2 × 10⁶ and a test at S = 40 gives N = 9.6 × 10⁶.

(a)

Find the value of m.

(3)

(b)

Find the value of C. Give your answer in the form a × 10^k where 1 ≤ a < 10.

(2)

(c)

Calculate the number of cycles the component survives at S = 55. Give your answer in the form a × 10^k with a correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms two equations and divides: 9.6 × 10⁶ ÷ 1.2 × 10⁶ = (80 ÷ 40)^m	✓ 1
M1 2^m = 8, or m = ln 8 ÷ ln 2	✓ 1
A1 m = 3	✓ 1
Part (b)
M1 C = 1.2 × 10⁶ × 80³	✓ 1
A1 C = 6.144 × 10¹¹	✓ 1
Part (c)
M1 6.144 × 10¹¹ ÷ 55³	✓ 1
A1 3.69 × 10⁶	✓ 1

Common error: Treating the exponent as positive. A negative exponent means N falls as S rises, so check that your values show this.

+ Add to test✓ In your test234IBMI3S-1-02r12-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium6 marks

A bridge deck contains 4.8 × 10³ m³ of steel with density 7850 kg m⁻³.

(a)

Calculate the mass of the steel. Give your answer in kilograms in the form a × 10^k where 1 ≤ a < 10, with a correct to 3 significant figures.

(2)

(b)

Express the mass in tonnes (1 tonne = 1000 kg) in the form a × 10^k with a correct to 3 significant figures.

(1)

(c)

An engineer approximates the volume as 5 × 10³ m³ and the density as 8000 kg m⁻³. Calculate the percentage error in the engineer's estimate of the mass, taking 3.768 × 10⁷ kg as the exact value. Give your answer to 3 significant figures.

(2)

(d)

The deck must have a steel mass below 3.8 × 10⁷ kg. State whether the exact mass meets this limit, and whether the engineer's estimate would have suggested that it does.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 mass = 4.8 × 10³ × 7850 = 3.768 × 10⁷	✓ 1
A1 3.77 × 10⁷ kg	✓ 1
Part (b)
A1 3.77 × 10⁴ tonnes	✓ 1
Part (c)
M1 estimate = 5 × 10³ × 8000 = 4.0 × 10⁷ kg and percentage error = (4.0 × 10⁷ − 3.768 × 10⁷) ÷ 3.768 × 10⁷ × 100	✓ 1
A1 6.16%	✓ 1
Part (d)
R1 the exact mass 3.768 × 10⁷ kg is below the limit, so it meets it, but the estimate 4.0 × 10⁷ kg is above the limit, so it would wrongly suggest that it does not	✓ 1

Common error: Dividing by the estimated value in a percentage error calculation. Divide the difference by the exact value.

+ Add to test✓ In your test235IBMI3S-1-02r14-05Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A bridge maintenance fund of €250000 is invested at a nominal annual interest rate of 3.6%, compounded monthly. No money is added or withdrawn.

(a)

Find the value of the fund after 8 years, to the nearest euro.

(2)

(b)

Calculate the effective annual interest rate, to 3 significant figures.

(2)

(c)

Find the number of years, to 3 significant figures, until the value of the fund first doubles.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $250000\left(1+\frac{3.6}{1200}\right)^{96}$ (or TVM with N = 96, I% = 3.6, PV = −250000, P/Y = C/Y = 12)	✓ 1
A1 €333296	✓ 1
Part (b)
M1 $\left(1+\frac{0.036}{12}\right)^{12}-1$	✓ 1
A1 3.66%	✓ 1
Part (c)
M1 $(1.003)^{12t}=2$	✓ 1
A1 19.3 years	✓ 1

Exam tip: The nominal rate is divided by the number of compounding periods, but the effective annual rate is found by compounding that monthly rate 12 times.

+ Add to test✓ In your test236IBMI3S-1-02r14-06Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A café buys a coffee roaster with a loan of €48000. The loan is repaid with equal monthly payments made at the end of each month for 5 years. Interest is charged at a nominal annual rate of 6.2%, compounded monthly.

(a)

Calculate the monthly payment, giving your answer to the nearest cent.

(2)

(b)

Using the payment from part (a), find the total amount repaid and the total interest paid over the 5 years.

(2)

(c)

Using the payment from part (a), find the outstanding balance of the loan immediately after the 24th payment, to the nearest euro.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM with N = 60, I% = 6.2, PV = 48000, FV = 0, P/Y = C/Y = 12 (or amortisation formula)	✓ 1
A1 €932.44	✓ 1
Part (b)
A1 total repaid $60\times932.44$ = €55946.40	✓ 1
A1 interest = €7946.40	✓ 1
Part (c)
M1 balance = value of the loan after 24 months minus the value of the 24 payments (or TVM with N = 24 and the payment from (a), solving for FV)	✓ 1
M1 correct substitution, e.g. $48000(1+\frac{0.062}{12})^{24}-P\frac{(1+\frac{0.062}{12})^{24}-1}{\frac{0.062}{12}}$	✓ 1
A1 €30559	✓ 1

Exam tip: Interest is added monthly, so use I% = 6.2 with C/Y = 12 on the GDC, not 6.2 ÷ 12. State the payment as a negative cash flow if your GDC convention requires it.

+ Add to test✓ In your test237IBMI3S-1-02r14-07Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

Freshly roasted coffee beans are tipped into a cooling tray in a room at 22 °C. The temperature of the beans t minutes after being tipped in is modelled by $T=22+190e^{-0.085t}$, where T is in °C.

(a)

Write down the temperature of the beans when they are tipped in.

(1)

(b)

Find the temperature of the beans after 10 minutes, to 3 significant figures.

(2)

(c)

Find the time taken for the beans to cool to 40 °C, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 212 °C	✓ 1
Part (b)
M1 $22+190e^{-0.85}$	✓ 1
A1 103 °C	✓ 1
Part (c)
M1 $22+190e^{-0.085t}=40$ giving $e^{-0.085t}=\frac{18}{190}$	✓ 1
M1 take natural logarithms: $t=\frac{\ln(190/18)}{0.085}$	✓ 1
A1 27.7 minutes	✓ 1

Common error: Isolate the exponential term before taking logarithms: subtract 22 first, then divide by 190.

+ Add to test✓ In your test238IBMI3S-1-02r14-08Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

The number of stress cycles N that a bridge cable can withstand before fatigue failure is modelled by $N=4.0\times10^{12}\,S^{-3}$, where S is the stress range in MPa.

(a)

Find the number of cycles the cable can withstand when S = 75. Give your answer in the form a × 10^k with 1 ≤ a < 10 and a to 3 significant figures.

(2)

(b)

The relationship can be written as $\log_{10}N=p+q\log_{10}S$. Find the values of p and q.

(2)

(c)

Find the stress range for which the cable can withstand 1.0 × 10⁸ cycles. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\frac{4.0\times10^{12}}{75^{3}}$	✓ 1
A1 9.48 × 10⁶	✓ 1
Part (b)
M1 take $\log_{10}$ of both sides and use $\log(ab)=\log a+\log b$ and $\log S^{-3}=-3\log S$	✓ 1
A1 p = 12.6 (accept 12.602) and q = −3	✓ 1
Part (c)
M1 $4.0\times10^{12}S^{-3}=10^{8}$ giving $S^{3}=40000$ (or use of the log form)	✓ 1
A1 34.2 MPa	✓ 1

Common error: Taking logs of a product gives a sum: log(a·S⁻³) = log a + (−3) log S. The log of the constant is not multiplied by anything.

+ Add to test✓ In your test239IBMI3S-1-01r18-07Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium5 marks

A robot measures the distance to a wall by sending out a laser pulse and timing the round trip. The pulse travels at 3.00×10⁸ m s⁻¹ and the measured round-trip time is 2.4×10⁻⁸ s.

(a)

Calculate the distance to the wall as measured by the robot.

(2)

(b)

The actual distance to the wall is 3.52 m. Calculate the percentage error in the robot's measurement, to 3 significant figures.

(2)

(c)

The manufacturer claims that the sensor has an error of less than 2 %. State, with a reason, whether this reading supports the claim.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (3.00×10⁸ × 2.4×10⁻⁸) ÷ 2 (the pulse travels there and back)	✓ 1
A1 3.6 m	✓ 1
Part (b)
M1 \|3.6 − 3.52\| ÷ 3.52 × 100	✓ 1
A1 2.27 %	✓ 1
Part (c)
R1 It does not support the claim, because 2.27 % > 2 %	✓ 1

Common error: The time given is for the round trip, so the distance to the wall is half of speed × time.

+ Add to test✓ In your test240IBMI3S-1-01r18-08Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A fisher records the mass of fish caught each week. In week 1 the catch is 340 kg and in each later week the catch is 14 kg less than in the previous week, so the weekly catches form an arithmetic sequence.

(a)

Find the catch in week 15.

(2)

(b)

The fisher stops fishing in the first week in which the catch would be less than 100 kg. Find the week in which this occurs.

(3)

(c)

Find the total catch over the first 12 weeks.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 340 − 14 × 14 (or 340 + 14(−14))	✓ 1
A1 144 kg	✓ 1
Part (b)
M1 340 − 14(n − 1) < 100	✓ 1
A1 n − 1 > 17.1 (or n > 18.1)	✓ 1
A1 week 19	✓ 1
Part (c)
M1 S₁₂ = (12/2)(2 × 340 + 11 × (−14))	✓ 1
A1 3156 kg	✓ 1

Exam tip: A decreasing arithmetic sequence has a negative common difference; keep the sign when substituting into the formulas.

+ Add to test✓ In your test241IBMI3S-1-01r18-09Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A sewing machine bought for €10 000 loses value so that its value after t years is V = 10 000 × k^t, where k is a positive constant. After 2 years its value is €7225.

(a)

Find the value of k.

(2)

(b)

Find the value of the machine after 6 years, to the nearest euro.

(2)

(c)

The machine will be replaced when its value first falls below €1500. Find after how many years this happens. Give your answer to 4 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10000 × k² = 7225, so k² = 0.7225	✓ 1
A1 k = 0.85	✓ 1
Part (b)
M1 10000 × 0.85⁶	✓ 1
A1 €3771	✓ 1
Part (c)
M1 10000 × 0.85^t = 1500, so 0.85^t = 0.15	✓ 1
M1 t = ln 0.15 ÷ ln 0.85 (or use a solver/graph)	✓ 1
A1 11.67 years	✓ 1

Common error: When a logarithm of a number less than 1 is divided by the logarithm of the ratio (also less than 1), both are negative; check the final time is positive.

+ Add to test✓ In your test242IBMI3S-1-01r18-13Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A factory makes nylon fibre for fishing nets. One spool holds a fibre of circular cross-section with diameter 2.5×10⁻⁴ m and length 3.6×10³ m. The density of nylon is 1.14×10³ kg m⁻³.

(a)

Calculate the volume of fibre on the spool. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(2)

(b)

Use the unrounded volume to calculate the mass of fibre on the spool, in kg, to 3 significant figures.

(2)

(c)

A quality-control scale gives the mass of the spool's fibre as 0.215 kg. Taking the unrounded calculated mass as the exact value, find the percentage error in the scale reading, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 π × (1.25×10⁻⁴)² × 3.6×10³	✓ 1
A1 1.77×10⁻⁴ m³	✓ 1
Part (b)
M1 volume × 1.14×10³	✓ 1
A1 0.201 kg	✓ 1
Part (c)
M1 \|0.215 − exact mass\| ÷ exact mass × 100	✓ 1
A1 6.72 %	✓ 1

Exam tip: Keep unrounded values in your calculator for later steps, and only round when you write the final answer.

+ Add to test✓ In your test243IBMI3S-1-01r19-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

Starlight passing through a dust cloud is dimmed. The intensity I of the light after travelling x parsecs through the cloud is modelled by I = I₀ × 0.85^x, where I₀ is the intensity entering the cloud.

(a)

Find the fraction of the original intensity that remains after 3 parsecs. Give your answer to 3 significant figures.

(2)

(b)

Find the distance through the cloud at which the intensity has fallen to 40% of I₀. Give your answer to 3 significant figures.

(3)

(c)

The model can also be written as I = I₀e^−kx. Find the value of k, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 0.85³	✓ 1
A1 0.614	✓ 1
Part (b)
M1 set up 0.85^x = 0.4	✓ 1
M1 take logarithms: x = ln 0.4 / ln 0.85	✓ 1
A1 5.64 parsecs	✓ 1
Part (c)
M1 equate e^−k = 0.85 (or −k = ln 0.85)	✓ 1
A1 k = 0.163	✓ 1

Common error: When solving ax = c, take logarithms of both sides first; do not divide c by a.

+ Add to test✓ In your test244IBMI3S-1-01r19-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium5 marks

A club borrows €2400 to buy a set of training equipment. The loan has an interest rate of 6% per year, compounded monthly, and is repaid with 24 equal payments made at the end of each month.

(a)

Use your calculator's finance application to find the monthly payment, to the nearest cent.

(2)

(b)

Using your answer to (a), find the total amount of interest paid over the whole loan.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 correct TVM entries: N = 24, I% = 6, PV = 2400, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €106.37	✓ 1
Part (b)
M1 total repaid = 24 × 106.37 = €2552.88	✓ 1
A1 interest = 2552.88 − 2400	✓ 1
A1 €152.88	✓ 1

Exam tip: State the TVM settings you used (N, I%, PV, FV, P/Y, C/Y) so method marks can be awarded even if the final value is wrong.

+ Add to test✓ In your test245IBMI3S-1-01r19-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium5 marks

A sports science department invests €8000 in an account paying 3.5% per year, compounded quarterly.

(a)

Find the least number of complete quarters needed for the value of the investment to exceed €10 000.

(3)

(b)

Find the value of the investment after this many quarters, to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 set up 8000(1 + 0.035/4)ⁿ > 10 000 (or 1.00875ⁿ > 1.25)	✓ 1
A1 n > 25.6… (or table values: 25 quarters give €9946.69, 26 quarters give €10 033.73)	✓ 1
A1 26 quarters	✓ 1
Part (b)
M1 8000 × 1.00875²⁶	✓ 1
A1 €10 034	✓ 1

Common error: The number of compounding periods must be a whole number of quarters: round up, not to the nearest.

+ Add to test✓ In your test246IBMI3S-1-01r19-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium8 marks

A rail museum saves for a restoration project by paying €350 at the end of every month into an account that pays 4.8% per year, compounded monthly, for 5 years.

(a)

Find the value of the account at the end of the 5 years, to the nearest euro.

(3)

(b)

Find the amount of interest earned over the 5 years, using the unrounded value from (a), to the nearest euro.

(2)

(c)

The museum would prefer to reach exactly €25 000 after 5 years with equal monthly payments. Find the monthly payment required, to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM: N = 60, I% = 4.8, PV = 0, PMT = −350, P/Y = C/Y = 12	✓ 1
A1 FV = 23 681.06…	✓ 1
A1 €23 681	✓ 1
Part (b)
M1 interest = FV − 60 × 350 = FV − 21 000	✓ 1
A1 €2681	✓ 1
Part (c)
M1 TVM with FV = 25 000 and PMT unknown, same N, I%, P/Y, C/Y	✓ 1
A1 substitute values correctly	✓ 1
A1 €369.49	✓ 1

Exam tip: Interest earned is the final value minus the total of the payments made, not minus the first payment.

+ Add to test✓ In your test247IBMI3S-1-01r21-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

Calculator

In one season a hive collects 40 kg of nectar in week 1. Because flowers are fading, the nectar collected in each later week is 85% of that collected in the previous week. The weekly amounts form a geometric sequence.

(a)

Find the nectar collected in week 5. Give your answer to 3 significant figures.

(2)

(b)

The beekeeper stops counting once the weekly amount is below 2 kg. Find the first week in which the amount is below 2 kg.

(3)

(c)

Find the total amount of nectar the hive would collect if the pattern continued indefinitely. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_5=40\times0.85^{4}$	✓ 1
A1 20.9 kg	✓ 1
Part (b)
M1 $40\times0.85^{n-1}<2$	✓ 1
M1 $n-1>\frac{\ln0.05}{\ln0.85}$ ($=18.43$)	✓ 1
A1 week 20	✓ 1
Part (c)
M1 $S_\infty=\frac{40}{1-0.85}$	✓ 1
A1 267 kg	✓ 1

Exam tip: The sum to infinity formula can be used only because the common ratio 0.85 lies between $-1$ and $1$.

+ Add to test✓ In your test248IBMI3S-1-01r21-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

Calculator

A bacterial culture in a hospital laboratory is modelled by $N=800\,e^{kt}$, where $N$ is the number of bacteria and $t$ is the time in hours after the culture is started. After 4 hours there are 5200 bacteria.

(a)

State what the value 800 represents.

(1)

(b)

Find the value of $k$, correct to 3 significant figures.

(3)

(c)

Find the time at which the culture contains 50 000 bacteria. Use the unrounded value of $k$ and give your answer in hours to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the number of bacteria at the start ($t=0$)	✓ 1
Part (b)
M1 $5200=800e^{4k}$	✓ 1
M1 $4k=\ln 6.5$	✓ 1
A1 $k=0.468$	✓ 1
Part (c)
M1 $800e^{kt}=50000$, so $e^{kt}=62.5$	✓ 1
M1 $t=\frac{\ln62.5}{k}$	✓ 1
A1 $t=8.84$ hours	✓ 1

Common error: After substituting, divide both sides by 800 before taking the logarithm; taking ln of 5200 alone gives the wrong value of $k$.

+ Add to test✓ In your test249IBMI3S-1-01r21-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

Calculator

A Formula 1 car completes a lap in 88.20 seconds at an average speed of 83.47 m s⁻¹. Treat both values as exact.

(a)

Calculate the lap distance. Give your answer in the form $a\times10^{k}$ metres, where $1\le a<10$ and $k$ is an integer, with $a$ correct to 3 significant figures.

(2)

(b)

A fan estimates the distance by rounding each value to 2 significant figures and multiplying. Find the fan's estimate and its percentage error, correct to 3 significant figures. Use the unrounded distance 7362.054 m as the exact value.

(3)

(c)

A second fan rounds each value to 1 significant figure and multiplies. Find the percentage error of this estimate to 3 significant figures, using 7362.054 m as the exact value.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $83.47\times88.20=7362.054$	✓ 1
A1 $7.36\times10^{3}$ m	✓ 1
Part (b)
M1 estimate $=83\times88=7304$ m	✓ 1
M1 $\frac{7362.054-7304}{7362.054}\times100$	✓ 1
A1 $0.789\%$	✓ 1
Part (c)
M1 $\frac{7362.054-80\times90}{7362.054}\times100$	✓ 1
A1 $2.20\%$	✓ 1

Exam tip: Estimates made by rounding to 1 significant figure are quick checks for size, but they can carry a percentage error of several per cent.

+ Add to test✓ In your test250IBMI3S-1-01r21-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

Calculator

A patient is given 200 mg of a drug every 6 hours. Just before the next dose, 35% of the drug from the previous period is still in the body. Let $D_n$ mg be the amount of the drug in the body immediately after the $n$th dose, so $D_1=200$.

(a)

Show that $D_3=294.5$.

(2)

(b)

It can be shown that $D_n=\frac{200(1-0.35^{n})}{0.65}$. Find the smallest value of $n$ for which $D_n>300$.

(3)

(c)

The doctor changes the dose to $x$ mg so that the amount in the body can never exceed 350 mg in the long run. Find the value of $x$.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $D_2=200+0.35\times200=270$	✓ 1
A1 $D_3=200+0.35\times270=294.5$ (shown)	✓ 1
Part (b)
M1 $\frac{200(1-0.35^{n})}{0.65}>300$, giving $0.35^{n}<0.025$	✓ 1
M1 $n>\frac{\ln0.025}{\ln0.35}=3.51$	✓ 1
A1 $n=4$	✓ 1
Part (c)
M1 $\frac{x}{1-0.35}=350$	✓ 1
A1 $x=227.5$ mg	✓ 1

Exam tip: When dividing by $\ln 0.35$ (a negative number) the inequality sign reverses, so check the direction by testing $n=3$ and $n=4$.

+ Add to test✓ In your test251IBMI3S-1-02r21-06Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium6 marks

Calculator

A Formula 1 team's finance office exchanges £25 000 into euros (€) at the rate £1 = €1.1642. The bank charges a commission of 1.8% of the amount in pounds, taken before the exchange. Later the office exchanges all of the euros back into pounds at the rate €1 = £0.8551, with a commission of 1.8% of the amount in euros, taken before the exchange.

(a)

Calculate the number of euros received, to the nearest cent.

(2)

(b)

Using the amount from part (a), find the amount in pounds received after exchanging back, to the nearest penny.

(2)

(c)

Find the total loss, in pounds, caused by the two exchanges.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $25000\times0.982\times1.1642$	✓ 1
A1 €28 581.11	✓ 1
Part (b)
M1 $28581.11\times0.982\times0.8551$	✓ 1
A1 £23 999.79	✓ 1
Part (c)
M1 $25000-23999.79$	✓ 1
A1 £1000.21	✓ 1

Exam tip: Commission taken before the exchange reduces the amount first (multiply by $1-0.018=0.982$), then the rate is applied.

+ Add to test✓ In your test252IBMI3S-1-02r21-07Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium7 marks

Calculator

A hospital uses a radioactive tracer. The activity $A$ MBq of a dose, $t$ hours after it is prepared, is given by $A=400\times\left(\frac{1}{2}\right)^{t/6}$.

(a)

Find the activity after 15 hours, to 3 significant figures.

(2)

(b)

Find the time at which the activity first falls to 5% of its initial value. Give your answer in hours to 3 significant figures.

(3)

(c)

The model can also be written as $A=400e^{-kt}$. Find the value of $k$, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $400\times0.5^{15/6}$	✓ 1
A1 70.7 MBq	✓ 1
Part (b)
M1 $0.5^{t/6}=0.05$	✓ 1
M1 $\frac{t}{6}=\frac{\ln0.05}{\ln0.5}$	✓ 1
A1 25.9 hours	✓ 1
Part (c)
M1 $e^{-k}=0.5^{1/6}$ so $k=\frac{\ln2}{6}$	✓ 1
A1 0.116	✓ 1

Exam tip: For half-life models, the exponent $t/6$ means the activity halves every 6 hours; check your answer for $t=12$ gives 100 MBq.

+ Add to test✓ In your test253IBMI3S-1-02r21-09Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium7 marks

Calculator

The rows of seats in a Formula 1 grandstand increase in number by a constant amount from one row to the next. Row 3 has 23 seats and row 8 has 48 seats.

(a)

Find the number of seats in row 1 and the common difference.

(3)

(b)

The organisers want at least 3000 seats in total. Find the smallest number of rows needed.

(2)

(c)

Find the total number of seats in a grandstand with 25 rows.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $5d=48-23$	✓ 1
A1 $d=5$	✓ 1
A1 $u_1=13$ seats	✓ 1
Part (b)
M1 $\frac{n}{2}(26+5(n-1))\ge3000$ or tabulating $S_n$	✓ 1
A1 $n=33$ (since $S_{32}=2896$ and $S_{33}=3069$)	✓ 1
Part (c)
M1 $u_{25}=13+24\times5=133$ and $S_{25}=\frac{25}{2}(13+133)$	✓ 1
A1 1825 seats	✓ 1

Exam tip: When a total must be at least a target value, check both $S_{32}$ and $S_{33}$ so you can see which side of 3000 each lies.

+ Add to test✓ In your test254IBMI3S-1-02r21-10Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Medium7 marks

Calculator

A hospital foundation's endowment is worth €2.4 million today. The trustees want it to reach €3.5 million in exactly 7 years with no money added or withdrawn. Interest is compounded quarterly.

(a)

Find the nominal annual interest rate needed, as a percentage to 3 significant figures.

(3)

(b)

Find the effective annual rate, as a percentage to 3 significant figures.

(2)

(c)

Inflation is 2.1% per year. Using the unrounded effective annual rate, find the real annual growth rate, as a percentage to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance solver with $N=28$ (or $N=7$ with $C/Y=4$), $PV=-2.4$, $FV=3.5$, $PMT=0$, $P/Y=C/Y=4$	✓ 1
M1 solve for $I$	✓ 1
A1 5.43%	✓ 1
Part (b)
M1 $\left(\frac{3.5}{2.4}\right)^{1/7}-1$ or $\left(1+\frac{0.0543}{4}\right)^{4}-1$	✓ 1
A1 5.54%	✓ 1
Part (c)
M1 $\frac{1+0.055378}{1.021}-1$	✓ 1
A1 3.37%	✓ 1

Exam tip: The effective annual rate is always at least as large as the nominal rate when compounding is more often than yearly.

+ Add to test✓ In your test255IBMI3S-1-03r22-02Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A sports club borrows €18 000 to buy training equipment. The loan has a nominal annual interest rate of 6%, compounded monthly. It is repaid by equal payments made at the end of each month for 4 years.

(a)

Find the monthly payment, to the nearest cent.

(2)

(b)

Using the payment from (a) (€422.73), find the total amount repaid and the total interest paid.

(3)

(c)

Find the amount still owed immediately after the 24th payment. Use the unrounded payment from (a) and give your answer to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses a finance solver (or annuity formula) with N = 48, I% = 6, PV = 18000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €422.73	✓ 1
Part (b)
M1 48 × 422.73	✓ 1
A1 total repaid €20291.04	✓ 1
A1 interest = 20291.04 − 18000 = €2291.04	✓ 1
Part (c)
M1 balance equals present value of the 24 remaining payments (e.g. finance solver with N = 24, PMT = −payment, FV = 0), or equivalent method	✓ 1
A1 €9538.01	✓ 1

Exam tip: Write down every value you enter into a finance solver (N, I%, PV, PMT, FV, P/Y, C/Y); a wrong entry then still earns the method mark.

+ Add to test✓ In your test256IBMI3S-1-03r22-09Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium8 marks

The brightness of a comet is recorded each night. On night 1 its brightness is 125 units. Each night the brightness is 8% lower than on the previous night, so the nightly brightness values form a geometric sequence.

(a)

Write down the common ratio.

(1)

(b)

Find the brightness on night 10, to 3 significant figures.

(2)

(c)

The comet can no longer be seen once the brightness is below 40 units. Find the first night on which the brightness is below 40 units.

(3)

(d)

Find the total of the brightness values recorded on nights 1 to 15, to the nearest whole number.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.92	✓ 1
Part (b)
M1 $u_{10} = 125 \times 0.92^{9}$	✓ 1
A1 59.0 units	✓ 1
Part (c)
M1 $125 \times 0.92^{n-1} < 40$ (or comparison of terms)	✓ 1
A1 solves, e.g. $n - 1 > \dfrac{\ln(40/125)}{\ln 0.92}$, giving $n > 14.67$ (or night 14 gives 42.3, night 15 gives 38.9)	✓ 1
A1 night 15	✓ 1
Part (d)
M1 $S_{15} = \dfrac{125(1 - 0.92^{15})}{1 - 0.92}$	✓ 1
A1 1115 units	✓ 1

Exam tip: A percentage decrease of 8% gives a common ratio of 0.92; check that the ratio is below 1 for a decreasing sequence.

+ Add to test✓ In your test257IBMI3S-1-03r22-12Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium8 marks

The number of passengers per year, P, using a new rail link is modelled by $P = 18000 \times b^{t}$, where t is the number of years after the link opens. After 4 years the number of passengers per year is 23 150.

(a)

Find the value of b, to 4 significant figures.

(2)

(b)

Interpret the value of b in this context.

(1)

(c)

Use the unrounded value of b to estimate the number of passengers per year 7 years after the link opens. Give your answer to the nearest 100.

(2)

(d)

Use the unrounded value of b to find how many years it takes for the number of passengers per year to reach 40 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $18000b^4 = 23150$	✓ 1
A1 $b = \left(\dfrac{23150}{18000}\right)^{1/4}$ = 1.065	✓ 1
Part (b)
R1 the number of passengers per year increases by about 6.5% (6.5%) each year	✓ 1
Part (c)
M1 $18000 \times b^7$	✓ 1
A1 27958, so about 28000 passengers per year	✓ 1
Part (d)
M1 $18000 b^{t} = 40000$	✓ 1
M1 logarithms: $t = \dfrac{\ln(40000/18000)}{\ln b}$	✓ 1
A1 12.7 years	✓ 1

Exam tip: Solve an exponential equation by isolating the power first (divide by the starting value) and only then take logarithms.

+ Add to test✓ In your test258IBMI3S-1-03r22-13Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

The amount of caffeine, C mg, in the blood of an athlete t hours after drinking a sports supplement is modelled by $C = 200 \times 0.88^{t}$.

(a)

Find the amount of caffeine 3 hours after drinking the supplement, to 3 significant figures.

(2)

(b)

Find the time for the amount of caffeine to fall to half its initial value. Give your answer in hours to 3 significant figures.

(3)

(c)

The model can also be written as $C = 200e^{kt}$. Find the value of k, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $200 \times 0.88^{3}$	✓ 1
A1 136 mg	✓ 1
Part (b)
M1 $200 \times 0.88^{t} = 100$, so $0.88^{t} = 0.5$	✓ 1
M1 $t = \dfrac{\ln 0.5}{\ln 0.88}$	✓ 1
A1 5.42 hours	✓ 1
Part (c)
M1 $e^{k} = 0.88$ (equating $0.88^{t} = e^{kt}$)	✓ 1
A1 $k = \ln 0.88 = -0.128$	✓ 1

Common error: A negative value of k in $e^{kt}$ is correct for decay; do not drop the negative sign when writing the answer.

+ Add to test✓ In your test259IBMI3S-1-02r23-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 1Medium6 marks

The number of virus particles in a laboratory sample is modelled by N = 500e^kt, where t is the time in hours since the start of the experiment and k is a constant. After 4 hours there are 1350 virus particles.

(a)

Find the value of k, correct to 3 significant figures.

(3)

(b)

Find the time at which the model predicts there will be 10 000 virus particles. Use the unrounded value of k and give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1350 = 500e^4k	✓ 1
M1 k = ln(2.7)/4	✓ 1
A1 k = 0.248	✓ 1
Part (b)
M1 10 000 = 500e^kt so e^kt = 20	✓ 1
M1 t = ln 20 ÷ k	✓ 1
A1 12.1 hours	✓ 1

Exam tip: Store the unrounded value of k in your calculator rather than retyping 0.248.

+ Add to test✓ In your test260IBMI3S-1-02r23-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 1Medium7 marks

The sound level L, in decibels (dB), of a sound with intensity I W m⁻² is given by L = 10 log₁₀(I/10⁻¹²). A wind turbine produces a sound intensity of 3.0 × 10⁻⁷ W m⁻² at a nearby house.

(a)

Calculate the sound level at the house, to 3 significant figures.

(2)

(b)

A local limit is a sound level of 62 dB. Find the sound intensity corresponding to 62 dB, giving your answer in the form a × 10^k with a to 3 significant figures.

(3)

(c)

Show that doubling the sound intensity always increases the sound level by 3.01 dB, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(3.0 × 10⁻⁷/10⁻¹²)	✓ 1
A1 54.8 dB	✓ 1
Part (b)
M1 62 = 10 log₁₀(I/10⁻¹²)	✓ 1
M1 I = 10⁻¹² × 10^6.2	✓ 1
A1 1.58 × 10⁻⁶ W m⁻²	✓ 1
Part (c)
M1 increase = 10 log₁₀(2I/10⁻¹²) − 10 log₁₀(I/10⁻¹²) = 10 log₁₀ 2	✓ 1
A1 3.01 dB (shown, independent of I)	✓ 1

Common error: Sound level is a logarithmic scale: adding the same number of decibels multiplies the intensity by a fixed factor, so do not add intensities and decibels directly.

+ Add to test✓ In your test261IBMI3S-1-02r23-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 1Medium6 marks

A supermarket buys a refrigerated display unit for €4200. Its value falls by 12% at the end of each year.

(a)

Find the value of the unit after 3 years, to the nearest cent.

(2)

(b)

Find the smallest whole number of years after which the value is below €1500.

(2)

(c)

Find the value of the unit at the end of this time, to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4200 × 0.88³	✓ 1
A1 €2862.18	✓ 1
Part (b)
M1 4200 × 0.88ⁿ < 1500	✓ 1
A1 n = 9 (since 8.05… years of 12% falls are needed)	✓ 1
Part (c)
M1 4200 × 0.88⁹ (ECF from (b))	✓ 1
A1 €1329.21	✓ 1

Common error: A 12% fall each year multiplies by 0.88, not 0.12 and not 1.12.

+ Add to test✓ In your test262IBMI3S-1-02r23-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 1Medium5 marks

A community energy group plans to save €25 000 towards a small wind turbine. It makes equal deposits at the end of each month for 5 years into an account that pays a nominal annual interest rate of 3.6%, compounded monthly. The account starts with no money in it.

(a)

Find the monthly deposit needed so that the account contains exactly €25 000 after 5 years. Give your answer to the nearest cent.

(3)

(b)

Find the total interest earned over the 5 years. Use the unrounded monthly deposit and give your answer to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 N = 60, I% = 3.6, PV = 0, FV = 25 000, P/Y = C/Y = 12	✓ 1
M1 solves for PMT	✓ 1
A1 €380.91	✓ 1
Part (b)
M1 25 000 − 60 × 380.914…	✓ 1
A1 €2145.16	✓ 1

Exam tip: The total deposited is 60 × the monthly payment; the interest is whatever remains of the target.

+ Add to test✓ In your test263IBMI3S-1-03r23-05Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

The amount of data stored in a data centre is modelled by S(t) = 12e^kt petabytes (PB), where t is the time in months after the start of monitoring and k is a constant. After 6 months the data centre stores 19.8 PB.

(a)

Find the value of k. Give your answer to 3 significant figures.

(3)

(b)

Find the time, in months, for the stored data to reach 50 PB. Use the unrounded value of k. Give your answer to 3 significant figures.

(3)

(c)

Find the time, in months, for the stored data to double. Use the unrounded value of k. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes: 19.8 = 12e^6k	✓ 1
M1 takes natural logarithms: 6k = ln(19.8 / 12) = ln 1.65	✓ 1
A1 k = 0.0835	✓ 1
Part (b)
M1 sets 50 = 12e^kt, so e^kt = 50/12	✓ 1
M1 t = ln(50/12) / k	✓ 1
A1 17.1 months	✓ 1
Part (c)
M1 e^kt = 2, so t = ln 2 / k	✓ 1
A1 8.30 months	✓ 1

Common error: To undo an exponential with base e use the natural logarithm of both sides; do not divide the logarithm by the coefficient before dividing the numbers.

+ Add to test✓ In your test264IBMI3S-1-03r23-06Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

A bicycle wheel has a measured diameter of 0.668 m. A cyclist rides 42.2 km without slipping.

(a)

Calculate the circumference of the wheel in metres. Give your answer to 3 significant figures.

(2)

(b)

Calculate the number of times the wheel turns during the ride. Use the unrounded circumference. Write your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a given to 3 significant figures.

(3)

(c)

The cyclist uses 2.10 m as the circumference. Find the percentage error in this value, relative to the unrounded circumference from part (a). Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 circumference = π × 0.668 (= 2.0985…)	✓ 1
A1 2.10 m	✓ 1
Part (b)
M1 converts 42.2 km to 42 200 m	✓ 1
M1 divides 42 200 by π × 0.668 (= 20 108.8…)	✓ 1
A1 2.01 × 10⁴	✓ 1
Part (c)
M1 difference = \|2.10 − 2.098584…\| = 0.001416…	✓ 1
M1 divides by the exact value and multiplies by 100	✓ 1
A1 0.0675 %	✓ 1

Exam tip: Keep unrounded values in your calculator until the final step; write the unrounded value (to at least 4 s.f.) in your working so that later accuracy marks are secure.

+ Add to test✓ In your test265IBMI3S-1-03r23-09Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The sound level L, in decibels (dB), of a sound with intensity I W m⁻² is given by L = 10 log₁₀(I / 10⁻¹²). In a server room the sound intensity is 4.5 × 10⁻⁴ W m⁻².

(a)

Calculate the sound level in the server room. Give your answer to 3 significant figures.

(2)

(b)

A second identical server room is built next to the first and both operate together, so the total intensity doubles. Find the new sound level, to 3 significant figures.

(2)

(c)

A new regulation limits the sound level in a work area to 78 dB. Find the greatest permitted sound intensity. Give your answer in the form a × 10^k, with a to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes: 10 log₁₀(4.5 × 10⁻⁴ / 10⁻¹²) = 10 log₁₀(4.5 × 10⁸)	✓ 1
A1 86.5 dB	✓ 1
Part (b)
M1 new intensity = 9 × 10⁻⁴ W m⁻²	✓ 1
A1 89.5 dB	✓ 1
Part (c)
M1 78 = 10 log₁₀(I / 10⁻¹²)	✓ 1
A1 log₁₀(I / 10⁻¹²) = 7.8, so I = 10⁻¹² × 10^7.8	✓ 1
A1 6.31 × 10⁻⁵ W m⁻²	✓ 1

Common error: Doubling the intensity does not double the decibel level; the logarithm turns multiplication of intensity into addition of decibels.

+ Add to test✓ In your test266IBMI3S-1-01r24-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

The volume V m³ of water in an irrigation reservoir is modelled by V = 48 000 × 0.94^t, where t is the time in weeks after the start of the dry season.

(a)

Find the volume of water in the reservoir after 4 weeks. Give your answer to the nearest m³.

(2)

(b)

Find the time taken for the volume to fall to 20 000 m³. Give your answer in weeks to 3 significant figures.

(3)

(c)

Explain what the number 0.94 tells you about the reservoir.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 48 000 × 0.94⁴	✓ 1
A1 37 476 m³	✓ 1
Part (b)
M1 48 000 × 0.94^t = 20 000	✓ 1
M1 use of logarithms or a solver, e.g. t = ln(20 000 ÷ 48 000) ÷ ln 0.94	✓ 1
A1 14.1 weeks	✓ 1
Part (c)
R1 the volume each week is 94 % of the volume the week before, i.e. it falls by 6 % per week	✓ 1

Common error: A decay factor of 0.94 means a 6 % decrease, not a 94 % decrease; check which percentage the context asks for.

+ Add to test✓ In your test267IBMI3S-1-01r24-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

The sound level L decibels (dB) of a sound of intensity I W m⁻² is given by L = 10 log₁₀(I ÷ 10⁻¹²).

(a)

A guitar amplifier produces a sound level of 95 dB. Find the intensity of this sound. Give your answer in the form a × 10^k with 1 ≤ a < 10 and a correct to 3 significant figures.

(3)

(b)

Show that doubling the intensity of a sound increases its sound level by about 3 dB.

(2)

(c)

The intensities of identical speakers add. Each speaker alone produces a sound level of 82 dB. Find how many such speakers are needed to produce a total sound level of 92 dB.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 95 = 10 log₁₀(I ÷ 10⁻¹²)	✓ 1
M1 I = 10⁻¹² × 10^9.5	✓ 1
A1 3.16 × 10⁻³ W m⁻²	✓ 1
Part (b)
M1 10 log₁₀(2I ÷ 10⁻¹²) = 10 log₁₀2 + 10 log₁₀(I ÷ 10⁻¹²)	✓ 1
R1 10 log₁₀2 = 3.01… so the increase is about 3 dB	✓ 1
Part (c)
M1 intensity ratio: 10 log₁₀(N) = 92 − 82	✓ 1
M1 N = 10^{(92 − 82)/10}	✓ 1
A1 10 speakers	✓ 1

Common error: Sound levels in decibels do not add directly: two 82 dB speakers do not give 164 dB. Add the intensities, then convert back.

+ Add to test✓ In your test268IBMI3S-1-01r24-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium5 marks

An airliner cruises at a speed of 8.7 × 10² km h⁻¹ for 6.75 hours.

(a)

Calculate the distance travelled. Give your answer in the form a × 10^k where 1 ≤ a < 10 and a is correct to 3 significant figures.

(2)

(b)

A trainee estimates the distance by using 9 × 10² km h⁻¹ and 7 hours. Find the percentage error in the trainee's estimate, using the unrounded value 5872.5 km as the exact distance. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 870 × 6.75 (= 5872.5)	✓ 1
A1 5.87 × 10³ km	✓ 1
Part (b)
A1 estimate = 6300 km	✓ 1
M1 percentage error = \|6300 − 5872.5\| ÷ 5872.5 × 100	✓ 1
A1 7.28 %	✓ 1

Exam tip: Percentage error is measured against the exact value, not against the estimate.

+ Add to test✓ In your test269IBMI3S-1-01r24-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

A music school saves for a new concert piano. It deposits €250 into an account at the start of each month for 4 years. The account pays a nominal annual interest rate of 2.4 %, compounded monthly.

(a)

Find the value of the account at the end of the 4 years. Give your answer to the nearest euro.

(3)

(b)

Find the interest earned over the 4 years. Use the unrounded value from part (a) and give your answer to the nearest euro.

(2)

(c)

The piano costs €13 500. Find the monthly deposit, to the nearest cent, that would give exactly this amount after 4 years with the same account and the same deposit timing.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM solver with payments at the beginning of each period (or annuity-due formula) and monthly rate 0.2 %	✓ 1
A1 N = 48, I% = 2.4, PMT = −250, PV = 0, P/Y = C/Y = 12, payment at BEGIN	✓ 1
A1 €12 607	✓ 1
Part (b)
M1 12 606.855… − 48 × 250	✓ 1
A1 €607	✓ 1
Part (c)
M1 TVM solver with FV = 13 500 and PMT unknown (payments at BEGIN)	✓ 1
A1 correct setup: N = 48, I% = 2.4, P/Y = C/Y = 12	✓ 1
A1 €267.71	✓ 1

Common error: Check whether payments are made at the start or the end of each period; setting the solver to END gives a lower final value for the same deposits.

+ Add to test✓ In your test270IBMI3S-1-01r26-05Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A sample of food is contaminated with bacteria. The number of bacteria, N, at time t hours after contamination is modelled by N = 1200e^0.35t.

(a)

Find the number of bacteria after 4 hours. Give your answer to the nearest whole number.

(2)

(b)

Find the time at which the number of bacteria reaches 50 000. Give your answer in hours to 3 significant figures.

(3)

(c)

Find the time taken for the number of bacteria to double. Give your answer in hours to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 N = 1200e^{0.35 × 4}	✓ 1
A1 4866	✓ 1
Part (b)
M1 1200e^0.35t = 50000, so e^0.35t = 41.67…	✓ 1
M1 taking natural logarithms, 0.35t = ln 41.67…	✓ 1
A1 10.7 (hours)	✓ 1
Part (c)
M1 e^0.35t = 2	✓ 1
A1 t = ln 2 / 0.35 = 1.98 (hours)	✓ 1

Common error: To solve an exponential equation take logarithms of both sides after isolating the exponential; do not divide the exponent by the coefficient.

+ Add to test✓ In your test271IBMI3S-1-01r26-06Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A signal from a geostationary satellite travels at 3.00 × 10⁸ m s⁻¹. The satellite is at an altitude of 3.58 × 10⁷ m above a ground station.

(a)

Calculate the time taken for a signal to travel from the satellite to the ground station. Give your answer in seconds to 3 significant figures.

(2)

(b)

Hence find the time for a signal to travel from the ground station to the satellite and back. Use the unrounded value from part (a). Give your answer to 3 significant figures.

(2)

(c)

A student estimates the altitude as 3.6 × 10⁷ m. Find the percentage error in the student's estimate. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 time = distance ÷ speed = 3.58 × 10⁷ ÷ 3.00 × 10⁸	✓ 1
A1 0.119 (s)	✓ 1
Part (b)
M1 doubling the one-way time (2 × 0.11933…)	✓ 1
A1 0.239 (s)	✓ 1
Part (c)
M1 difference 3.6 × 10⁷ − 3.58 × 10⁷ = 2 × 10⁵ (m)	✓ 1
M1 dividing by the exact altitude 3.58 × 10⁷ and multiplying by 100	✓ 1
A1 0.559 (%)	✓ 1

Exam tip: Keep numbers in standard form throughout and combine the powers of ten separately from the decimal parts.

+ Add to test✓ In your test272IBMI3S-1-01r26-07Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A climber follows a ten-week training plan. In week 1 she walks 4.5 km. In each later week she walks 1.2 km further than in the previous week, so the weekly distances form an arithmetic sequence.

(a)

Find the distance she walks in week 10.

(2)

(b)

Find the total distance she walks over the ten weeks.

(2)

(c)

She continues to increase her distance in the same way beyond week 10. Show that her total distance first exceeds 200 km during week 16 and find her total distance after 16 weeks.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4.5 + 9 × 1.2	✓ 1
A1 15.3 (km)	✓ 1
Part (b)
M1 S₁₀ = (10/2)(4.5 + 15.3) (or equivalent)	✓ 1
A1 99 (km)	✓ 1
Part (c)
M1 S₁₅ = (15/2)(2 × 4.5 + 14 × 1.2) = 193.5 (km), which is less than 200	✓ 1
M1 S₁₆ = (16/2)(2 × 4.5 + 15 × 1.2)	✓ 1
A1 216 (km), which is greater than 200, so the total first exceeds 200 km in week 16	✓ 1

Exam tip: When a sum must first pass a target, calculate the sums either side of the target to justify the answer.

+ Add to test✓ In your test273IBMI3S-1-01r26-10Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

The power of a signal in an optical cable decreases with distance. At a distance d km from the transmitter the power is P = 80 × 10^−0.04d milliwatts.

(a)

Find the power at a distance of 15 km. Give your answer to 3 significant figures.

(2)

(b)

Find the distance at which the power falls to 5 mW. Give your answer to 3 significant figures.

(3)

(c)

Show that the power halves every 7.53 km, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 80 × 10^{−0.04 × 15}	✓ 1
A1 20.1 (mW)	✓ 1
Part (b)
M1 80 × 10^−0.04d = 5, so 10^−0.04d = 0.0625	✓ 1
M1 taking log₁₀, −0.04d = log₁₀ 0.0625 = −1.204…	✓ 1
A1 30.1 (km)	✓ 1
Part (c)
M1 10^−0.04x = 0.5 for the distance x that halves the power	✓ 1
A1 x = log₁₀ 2 / 0.04 = 7.53 (km) (AG)	✓ 1

Common error: log₁₀ of a number less than 1 is negative, so keep the minus signs when dividing by −0.04.

+ Add to test✓ In your test274IBMI3S-1-01r26-14Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A baker counts yeast cells in a sample of dough. The dough contains 3.2 × 10⁶ cells per gram and the sample has mass 0.45 g. Each cell has a volume of 6.5 × 10⁻¹¹ cm³.

(a)

Find the number of yeast cells in the sample. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

(2)

(b)

Hence find the total volume of the yeast cells in the sample. Give your answer in cm³ in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

(2)

(c)

A student measures the total volume as 9.0 × 10⁻⁵ cm³. Find the percentage error in the student's measurement. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3.2 × 10⁶ × 0.45	✓ 1
A1 1.44 × 10⁶	✓ 1
Part (b)
M1 1.44 × 10⁶ × 6.5 × 10⁻¹¹	✓ 1
A1 9.36 × 10⁻⁵ (cm³)	✓ 1
Part (c)
M1 \|9.0 × 10⁻⁵ − 9.36 × 10⁻⁵\| ÷ 9.36 × 10⁻⁵ × 100	✓ 1
A1 3.85 (%)	✓ 1

Exam tip: Multiply the decimal parts and add the powers of ten, then check that your answer is in the form required.

+ Add to test✓ In your test275IBMI3S-1-02r26-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A construction firm borrows €85 000 to buy bridge-inspection equipment. The loan has a nominal annual interest rate of 4.8 %, compounded monthly.

(a)

The loan is repaid with equal payments at the end of each month for 5 years. Find the monthly payment, to the nearest cent.

(2)

(b)

Find the total interest paid over the 5 years, using the unrounded payment. Give your answer to the nearest cent.

(2)

(c)

The firm decides instead to pay €1 650 at the end of each month. Use a financial solver to find the number of payments needed to repay the loan, and state the number of payments required as a whole number.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a financial solver with N = 60, I% = 4.8, PV = 85000, FV = 0, P/Y = 12, C/Y = 12	✓ 1
A1 €1596.28	✓ 1
Part (b)
M1 60 × 1596.2780… − 85000	✓ 1
A1 €10776.68	✓ 1
Part (c)
M1 solver with PV = 85000, PMT = −1650, FV = 0, I% = 4.8, P/Y = 12, C/Y = 12	✓ 1
A1 N = 57.8	✓ 1
A1 58 payments (the final payment is smaller)	✓ 1

Exam tip: A non-integer number of payments means the final payment is smaller than the others; round the count up to a whole number.

+ Add to test✓ In your test276IBMI3S-1-02r26-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

The power output P watts of a radioisotope generator on a space probe, t years after launch, is modelled by P = 300 × 2^−t/88.

(a)

Calculate the power output 20 years after launch. Give your answer to 3 significant figures.

(2)

(b)

The probe needs at least 180 W to operate all its instruments. Find the number of years after launch at which the power output falls to 180 W. Give your answer to 3 significant figures.

(3)

(c)

The model can also be written P = 300e^−kt. Find the value of k. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 300 × 2^−20/88	✓ 1
A1 256 (W)	✓ 1
Part (b)
M1 300 × 2^−t/88 = 180, so 2^−t/88 = 0.6	✓ 1
M1 taking logarithms, −t/88 × ln 2 = ln 0.6	✓ 1
A1 64.9 (years)	✓ 1
Part (c)
M1 e^−k = 2^−1/88, so k = (ln 2)/88	✓ 1
A1 0.00788	✓ 1

Exam tip: Change of base is easy using natural logarithms: 2x = ex lⁿ ².

+ Add to test✓ In your test277IBMI3S-1-02r26-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium5 marks

The span of a suspension bridge is exactly 1 247.6 m. Its deck uses steel of mass 3.6 × 10² kg per metre of span.

(a)

Write the span correct to 3 significant figures.

(1)

(b)

An engineer uses 1 200 m as a rough estimate of the span. Find the percentage error in this estimate. Give your answer to 3 significant figures.

(2)

(c)

Calculate the total mass of steel in the deck using the exact span. Give your answer in kg in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ, with a correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1250 (m)	✓ 1
Part (b)
M1 \|1200 − 1247.6\| ÷ 1247.6 × 100	✓ 1
A1 3.82 (%)	✓ 1
Part (c)
M1 360 × 1247.6 = 449136	✓ 1
A1 4.49 × 10⁵ (kg)	✓ 1

Common error: Do not round the span before multiplying; round only the final answer.

+ Add to test✓ In your test278IBMI3S-1-02r26-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

The number of members of a coffee-shop loyalty scheme grows each month by the same factor, so the monthly membership totals form a geometric sequence. In month 3 there are 72 members and in month 6 there are 576 members.

(a)

Find the common ratio of the sequence.

(2)

(b)

Find the number of members in month 1.

(2)

(c)

Show that month 8 is the first month in which the scheme has more than 2 000 members.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 r³ = 576 ÷ 72 = 8	✓ 1
A1 r = 2	✓ 1
Part (b)
M1 u₁ × 2² = 72	✓ 1
A1 18	✓ 1
Part (c)
M1 u₇ = 18 × 2⁶ = 1152, which is less than 2000	✓ 1
A1 u₈ = 18 × 2⁷ = 2304	✓ 1
R1 2304 > 2000 and the sequence is increasing, so month 8 is the first (AG)	✓ 1

Exam tip: When two terms of a geometric sequence are given, divide them to find a power of the common ratio.

+ Add to test✓ In your test279IBMI3S-1-02r26-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

The sound intensity level L decibels of traffic noise near a bridge is given by L = 10 log₁₀(I / 10⁻¹²), where I is the intensity in W m⁻².

(a)

Find the sound level when I = 3.2 × 10⁻⁶ W m⁻². Give your answer to 3 significant figures.

(2)

(b)

The sound level at a different time is 78 dB. Find the intensity at that time. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ, with a to 3 significant figures.

(2)

(c)

Two identical sources of noise are heard together, so the intensity doubles. Show that the sound level increases by 3.01 dB, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(3.2 × 10⁻⁶ / 10⁻¹²) = 10 log₁₀(3.2 × 10⁶)	✓ 1
A1 65.1 (dB)	✓ 1
Part (b)
M1 78 = 10 log₁₀(I / 10⁻¹²), so I = 10⁻¹² × 10^7.8	✓ 1
A1 6.31 × 10⁻⁵ (W m⁻²)	✓ 1
Part (c)
M1 increase = 10 log₁₀(2I / 10⁻¹²) − 10 log₁₀(I / 10⁻¹²) = 10 log₁₀ 2	✓ 1
A1 3.01 (dB) (AG)	✓ 1

Exam tip: Use the law log a − log b = log(a/b) to show that the increase does not depend on the starting intensity.

+ Add to test✓ In your test280IBMI3S-1-02r26-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A coffee importer invests €15 000 in an account that pays a nominal annual interest rate of r %, compounded quarterly. After 6 years the investment has grown to €18 600.

(a)

Find the value of r. Give your answer to 3 significant figures.

(3)

(b)

The same €15 000 is instead invested at 3.6 % per year compounded annually. Find its value after 6 years, to the nearest cent, and find how much less this is than €18 600.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18600 = 15000(1 + r/400)²⁴	✓ 1
M1 solving, (1 + r/400) = (1.24)^1/24 or use of a financial solver with N = 24 (or 6 years), PV = −15000, FV = 18600, P/Y = 4, C/Y = 4	✓ 1
A1 r = 3.60	✓ 1
Part (b)
M1 15000 × 1.036⁶	✓ 1
A1 €18545.98	✓ 1
A1 18600 − 18545.98 = €54.02 less	✓ 1

Common error: With quarterly compounding the number of periods is 4 × 6 = 24 and the rate per period is the nominal rate divided by 4.

+ Add to test✓ In your test281IBMI3S-1-02r26-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A space probe is travelling to Mars at a constant speed of 1.7 × 10⁴ m s⁻¹. The distance it must travel is 2.25 × 10¹¹ m.

(a)

Calculate the time taken for the journey in seconds. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ, with a to 3 significant figures.

(2)

(b)

Convert the unrounded time to days, and give your answer to 3 significant figures. (There are 86 400 seconds in a day.)

(2)

(c)

The mission plan states a journey time of 150 days. Use the unrounded time in days to find the percentage error in the plan's value. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2.25 × 10¹¹ ÷ 1.7 × 10⁴	✓ 1
A1 1.32 × 10⁷ (s)	✓ 1
Part (b)
M1 13235294… ÷ 86400	✓ 1
A1 153 (days)	✓ 1
Part (c)
M1 \|150 − 153.186…\| ÷ 153.186… × 100	✓ 1
A1 2.08 (%)	✓ 1

Common error: Rounding the time to 1.32 × 10⁷ s before converting to days gives a slightly different answer; keep the unrounded value.

+ Add to test✓ In your test282IBMI3S-1-02r28-01Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A sandstone block in a temple wall is a cuboid measuring 1.85 m by 0.72 m by 0.46 m. The density of the sandstone is 2600 kg m⁻³.

(a)

Calculate the volume of the block, in m³, to 3 significant figures.

(2)

(b)

Find the mass of the block, giving your answer in the form a × 10^k kg, where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(2)

(c)

The block is weighed and its true mass is 1.62 × 10³ kg. Using the unrounded mass 1593.072 kg from part (b), find the percentage error in the calculated mass, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1.85 × 0.72 × 0.46	✓ 1
A1 0.613 m³	✓ 1
Part (b)
M1 mass = density × volume	✓ 1
A1 1.59 × 10³ kg	✓ 1
Part (c)
M1 \|1593.072 − 1620\| / 1620 × 100	✓ 1
A1 1.66%	✓ 1

Common error: Percentage error is divided by the exact (true) value, not by the approximate one.

+ Add to test✓ In your test283IBMI3S-1-02r28-02Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A cyclist borrows €1800 to buy an electric bike. The loan has a nominal annual interest rate of 6%, compounded monthly, and is repaid by equal payments at the end of each month for 2 years.

(a)

Find the monthly payment, in €, correct to the nearest cent.

(3)

(b)

Using the payment from part (a) given to the nearest cent, find the total amount repaid.

(2)

(c)

Hence find the total interest paid on the loan.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 monthly rate 0.5% and N = 24	✓ 1
M1 use of the finance solver (PV = 1800, FV = 0, P/Y = C/Y = 12)	✓ 1
A1 € 79.78	✓ 1
Part (b)
M1 24 × 79.78	✓ 1
A1 € 1914.72	✓ 1
Part (c)
A1 1914.72 − 1800 = € 114.72 (ECF from (b))	✓ 1

Exam tip: When using a finance solver for a loan, enter the loan as a positive present value and the payments as negative (or vice versa) so the signs are opposite.

+ Add to test✓ In your test284IBMI3S-1-02r28-07Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A data centre adds 3.2 TB of storage in month 1, and in each later month adds 8% more than in the previous month.

(a)

Find the storage added in month 3, in TB to 3 significant figures.

(2)

(b)

Find the first month in which the storage added is more than 10 TB.

(3)

(c)

Find the total storage added in the first 12 months, in TB to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3.2 × 1.08²	✓ 1
A1 3.73 TB	✓ 1
Part (b)
M1 3.2 × 1.08^{n − 1} > 10	✓ 1
A1 n − 1 > 14.8…	✓ 1
A1 month 16	✓ 1
Part (c)
M1 S₁₂ = 3.2(1.08¹² − 1)/(1.08 − 1)	✓ 1
A1 60.7 TB	✓ 1

Exam tip: An increase of 8% per period means a common ratio of 1.08, not 0.08.

+ Add to test✓ In your test285IBMI3S-1-02r28-11Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

The population of a village is modelled by P = 1200 × 1.035^t, where t is the time in years since 2020.

(a)

Find the population predicted for 2030, to the nearest whole number.

(2)

(b)

State the annual percentage growth rate of the population.

(1)

(c)

Find the year in which the population is first predicted to exceed 2000.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 P = 1200 × 1.035¹⁰	✓ 1
A1 1693	✓ 1
Part (b)
A1 3.5%	✓ 1
Part (c)
M1 set 1200 × 1.035^t = 2000	✓ 1
M1 t = log(2000/1200) / log(1.035)	✓ 1
A1 t = 14.8…	✓ 1
A1 the year 2035	✓ 1

Exam tip: When solving for an exponent, first isolate the power, then take logarithms of both sides: t = log(b) / log(multiplier).

+ Add to test✓ In your test286IBMI3S-1-02r28-13Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium5 marks

A sealed isotope source used in a conservation laboratory has a half-life of 28.8 years. The mass remaining after t years is m = m₀ × 0.5^t/28.8.

(a)

Find the percentage of the original mass remaining after 50 years, to 3 significant figures.

(2)

(b)

Find the time taken for the mass to fall to 10% of its original value, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 0.5^50/28.8	✓ 1
A1 30.0%	✓ 1
Part (b)
M1 set 0.5^t/28.8 = 0.1	✓ 1
M1 t/28.8 = log 0.1 / log 0.5	✓ 1
A1 t = 95.7 years	✓ 1

Common error: Do not divide by 2 repeatedly with rounded values; write the decay as a power with exponent t/half-life and use logarithms.

+ Add to test✓ In your test287IBMI3S-1-02r28-14Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

The daily data traffic y TB of a data centre is modelled by y = a × b^x, where x is the number of months after opening. The traffic is 4.0 TB when x = 2 and 13.5 TB when x = 5.

(a)

Show that b = 1.5.

(3)

(b)

Find a, to 3 significant figures.

(2)

(c)

Use the model to predict the daily traffic, in TB to 3 significant figures, when x = 8.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4.0 = ab² and 13.5 = ab⁵	✓ 1
M1 divide: b³ = 13.5/4.0 = 3.375	✓ 1
A1 b = 1.5	✓ 1
Part (b)
M1 a = 4.0 / 1.5²	✓ 1
A1 a = 1.78	✓ 1
Part (c)
M1 y = (16/9) × 1.5⁸	✓ 1
A1 45.6 TB	✓ 1

Exam tip: To find two unknowns in y = a × bx from two points, divide the equations: a cancels and leaves a power of b.

+ Add to test✓ In your test288IBMI3S-1-01r29-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

The charge, C %, remaining in a robot's battery t hours after it is fully charged is modelled by C = 100e^−kt, where k is a positive constant. After 5 hours the charge remaining is 72%.

(a)

Find the value of k. Give your answer to 3 significant figures.

(2)

(b)

Find the time at which the charge remaining is 20%. Give your answer in hours to 3 significant figures. Use the unrounded value of k.

(3)

(c)

Find the percentage of its charge that the battery loses during each hour, as a percentage of the charge at the start of that hour. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 72 = 100e^−5k and attempt to take natural logarithms	✓ 1
A1 k = 0.0657	✓ 1
Part (b)
M1 20 = 100e^−kt	✓ 1
M1 t = ln 5 ÷ k or equivalent	✓ 1
A1 24.5 hours	✓ 1
Part (c)
M1 1 − e^−k	✓ 1
A1 6.36%	✓ 1

Exam tip: When a model is exponential with base e, take natural logarithms of both sides after isolating the exponential term.

+ Add to test✓ In your test289IBMI3S-1-01r29-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A fishing cooperative records its annual catch. In the first year the catch is 8.0 tonnes. The catches form a geometric sequence in which each year's catch is 15% greater than the previous year's catch.

(a)

Find the catch in the 10th year. Give your answer to 3 significant figures.

(2)

(b)

Find the first year in which the catch exceeds 40 tonnes.

(3)

(c)

Find the total catch over the first 13 years. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 8.0 × 1.15⁹	✓ 1
A1 28.1 tonnes	✓ 1
Part (b)
M1 8.0 × 1.15ⁿ⁻¹ > 40	✓ 1
M1 attempt to solve, for example n − 1 > ln 5 ÷ ln 1.15 = 11.5…	✓ 1
A1 year 13	✓ 1
Part (c)
M1 $\frac{8.0(1.15^{13}-1)}{1.15-1}$	✓ 1
A1 275 tonnes	✓ 1

Common error: '15% greater' gives a common ratio of 1.15, not 0.15.

+ Add to test✓ In your test290IBMI3S-1-01r29-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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Tomás takes out a loan of £15 000 to buy a fishing boat engine. The loan has a nominal annual interest rate of 5.4%, compounded monthly. He repays the loan with 48 equal payments made at the end of each month, so that the loan is fully repaid after 48 months.

(a)

Calculate the amount of each monthly payment. Give your answer to the nearest penny.

(3)

(b)

Calculate the total interest Tomás pays over the 48 months. Use the payment from part (a), rounded to the nearest penny.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of TVM solver with N = 48, I% = 5.4, PV = 15000, FV = 0, C/Y = 12	✓ 1
M1 correct signs for PV and FV, or $15000\times0.0045\div(1-1.0045^{-48})$	✓ 1
A1 £348.16	✓ 1
Part (b)
M1 48 × 348.16 − 15000	✓ 1
A1 £1,711.68 (ECF from (a))	✓ 1

Exam tip: Interest paid is total repaid minus the amount borrowed; do not multiply the interest rate by the number of years.

+ Add to test✓ In your test291IBMI3S-1-01r29-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A delivery robot has wheels of diameter 9.6 cm. A programmer calculates the distance travelled in 250 complete turns of a wheel using the circumference formula C = πd.

(a)

Calculate the exact distance travelled in 250 turns, using the π key on a calculator. Give your answer in centimetres to 4 significant figures.

(2)

(b)

The programmer instead uses π ≈ 3.14. Calculate the distance travelled in 250 turns using this value.

(1)

(c)

Write your answer to part (a) in metres in the form a × 10^k, where 1 ≤ a < 10 and k is an integer. Give a to 3 significant figures.

(1)

(d)

Calculate the percentage error in the distance found in part (b). Use the unrounded value of the exact distance. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 250 × π × 9.6	✓ 1
A1 7540 cm	✓ 1
Part (b)
A1 7536 cm	✓ 1
Part (c)
A1 7.54 × 10¹ m	✓ 1
Part (d)
M1 error = \|7539.82 − 7536\| ≈ 3.82	✓ 1
M1 divide by the exact distance and multiply by 100	✓ 1
A1 0.0507%	✓ 1

Common error: Percentage error uses the exact value as the denominator; using the approximate value gives a slightly different answer.

+ Add to test✓ In your test292IBMI3S-1-01r29-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A fashion show has seating arranged in rows. The first row has 14 seats and each row has 2 more seats than the row in front of it, so the numbers of seats in the rows form an arithmetic sequence.

(a)

Find the number of seats in the 10th row.

(2)

(b)

Show that the total number of seats in the first n rows is n(n + 13).

(2)

(c)

The total capacity of the seating is 420 seats. Find the number of rows.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 14 + 9 × 2	✓ 1
A1 32 seats	✓ 1
Part (b)
M1 $S_n=\frac{n}{2}(2\times14+(n-1)\times2)$	✓ 1
A1 $=\frac{n}{2}(2n+26)=n(n+13)$ (AG)	✓ 1
Part (c)
M1 n(n + 13) = 420	✓ 1
M1 attempt to solve, for example n² + 13n − 420 = 0 or use of GDC	✓ 1
A1 15 rows (reject the negative solution −28)	✓ 1

Exam tip: A quadratic from a sum formula has two roots; reject the one that is not a positive whole number of rows.

+ Add to test✓ In your test293IBMI3S-1-01r29-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium5 marks

A culture in a fish-processing plant contains 500 bacteria at time t = 0. The number of bacteria, N, at t hours is modelled by N = 500 × 2^t/3.5.

(a)

Calculate the number of bacteria after 10 hours. Give your answer to 3 significant figures.

(2)

(b)

Find the time taken for the number of bacteria to reach 20 000. Give your answer in hours to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 500 × 2^10/3.5	✓ 1
A1 3620	✓ 1
Part (b)
M1 500 × 2^t/3.5 = 20 000, so 2^t/3.5 = 40	✓ 1
M1 t/3.5 = log₂ 40 = ln 40 ÷ ln 2	✓ 1
A1 18.6 hours	✓ 1

Exam tip: Divide both sides by the initial value before taking logarithms, so you are solving a simple equation of the form 2x = 40.

+ Add to test✓ In your test294IBMI3S-1-01r29-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium5 marks

Anika saves for a new sewing machine by paying €250 into an account at the end of every month for 5 years. The account pays a nominal annual interest rate of 2.4%, compounded monthly.

(a)

Calculate the value of her savings immediately after the final payment. Give your answer to the nearest cent.

(3)

(b)

Calculate the total interest earned over the 5 years, using the value from part (a) rounded to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of TVM solver with N = 60, I% = 2.4, PV = 0, PMT = −250, C/Y = 12	✓ 1
M1 P/Y = 12 and correct FV, or $250\times\frac{1.002^{60}-1}{0.002}$	✓ 1
A1 €15,920.22	✓ 1
Part (b)
M1 15920.22 − 15000	✓ 1
A1 €920.22	✓ 1

Common error: The total paid in is 60 × €250 = €15 000; interest is the final value minus this amount, not minus €250.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

Elena invests €8000 in an account with a nominal annual interest rate of r%, compounded quarterly. After 6 years the value of her investment is €9350.

(a)

Find the value of r. Give your answer to 3 significant figures.

(4)

(b)

Using the unrounded value of r, find the value of the investment after 10 years, to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 9350 = 8000(1 + r/400)²⁴ or TVM solver with PV = −8000, FV = 9350, N = 24	✓ 1
M1 (9350/8000)^1/24 = 1 + r/400 or C/Y = 4, P/Y = 4 in solver	✓ 1
M1 attempt to solve for r	✓ 1
A1 r = 2.61	✓ 1
Part (b)
M1 8000(1 + r/400)⁴⁰	✓ 1
A1 €10,374	✓ 1

Exam tip: Check your rate by substituting it back into the compound interest formula; the result should be close to €9350.

+ Add to test✓ In your test296IBMI3S-1-02r29-06Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The weekly sales of a new jacket, in units, form a geometric sequence. In week 2, 96 jackets are sold and in week 5, 324 jackets are sold.

(a)

Find the common ratio and the number of jackets sold in week 1.

(3)

(b)

Find the first week in which more than 5000 jackets are sold.

(2)

(c)

Find the total number of jackets sold in the first 10 weeks. Give your answer to the nearest whole number.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁r = 96 and u₁r⁴ = 324, so r³ = 3.375	✓ 1
A1 r = 1.5	✓ 1
A1 u₁ = 64	✓ 1
Part (b)
M1 64 × 1.5ⁿ⁻¹ > 5000, then attempt to solve, for example n − 1 > ln 78.125 ÷ ln 1.5 = 10.7…	✓ 1
A1 week 12	✓ 1
Part (c)
M1 $\frac{64(1.5^{10}-1)}{1.5-1}$	✓ 1
A1 7,253	✓ 1

Common error: When solving 1.5ⁿ⁻¹ > 78.125, add 1 to the value found for n − 1 and then round up to a whole week.

+ Add to test✓ In your test297IBMI3S-1-02r29-07Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

Pavel has £95 000 in a retirement fund that pays a nominal annual interest rate of 3.9%, compounded monthly. He plans to withdraw the same amount at the end of each month for 15 years, so that the fund is exactly used up at the end of the 15 years.

(a)

Calculate the amount of each monthly withdrawal, to the nearest penny.

(3)

(b)

Find the total amount Pavel withdraws over the 15 years. Use the withdrawal from part (a).

(2)

(c)

Find the total interest the fund earns over the 15 years, using your answer to part (b).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of TVM solver with N = 180, I% = 3.9, PV = −95000, FV = 0, C/Y = 12	✓ 1
M1 correct signs, or $95000\times0.00325\div(1-1.00325^{-180})$	✓ 1
A1 £697.95	✓ 1
Part (b)
M1 180 × 697.95	✓ 1
A1 £125,631.00 (ECF from (a))	✓ 1
Part (c)
M1 125631.00 − 95000	✓ 1
A1 £30,631.00 (ECF from (b))	✓ 1

Exam tip: The interest earned is the difference between total withdrawn and the starting balance, because the fund ends at zero.

+ Add to test✓ In your test298IBMI3S-1-02r29-08Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium5 marks

The sound level, L decibels (dB), of a factory robot is given by L = 10 log₁₀(I/I₀), where I is the sound intensity in W m⁻² and I₀ = 1.0 × 10⁻¹² W m⁻².

(a)

The robot has an intensity of 3.2 × 10⁻⁵ W m⁻². Calculate its sound level, to 3 significant figures.

(2)

(b)

A quieter model has a sound level 6 dB lower than the original model. Find the number by which the intensity of the original model must be divided to give the intensity of the quieter model. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(3.2 × 10⁷)	✓ 1
A1 75.1 dB	✓ 1
Part (b)
M1 let the intensity ratio be k, so 6 = 10 log₁₀ k	✓ 1
M1 k = 10^0.6	✓ 1
A1 3.98	✓ 1

Common error: A change of 6 dB is not a change of intensity by a factor of 6; the decibel scale is logarithmic, so equal steps in dB are equal ratios of intensity.

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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium5 marks

A tailor buys 7.8 m of fabric at €12.45 per metre. To check the bill, the tailor estimates the cost by rounding each number to 1 significant figure.

(a)

Calculate the exact cost of the fabric.

(1)

(b)

Write the exact cost in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(1)

(c)

Find the tailor's estimate of the cost.

(1)

(d)

Calculate the percentage error in the estimate. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €97.11	✓ 1
Part (b)
A1 9.711 × 10¹	✓ 1
Part (c)
A1 10 × 8 = €80	✓ 1
Part (d)
M1 $\frac{\|97.11-80\|}{97.11}\times100$	✓ 1
A1 17.6%	✓ 1

Exam tip: Use the exact cost, not the estimate, as the denominator when finding a percentage error.

+ Add to test✓ In your test300IBMI3S-1-02r29-10Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

A hatchery feeds its fish 2.5 kg of feed on day 1. Each day it increases the amount of feed by 0.4 kg, so the daily amounts form an arithmetic sequence.

(a)

Find the amount of feed given on day 10.

(1)

(b)

Show that the total amount of feed given in the first n days is 0.2n² + 2.3n kg.

(2)

(c)

Find the smallest number of days after which the total feed given is at least 100 kg.

(2)

(d)

Find the total amount of feed given in these 18 days.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 6.1 kg	✓ 1
Part (b)
M1 $S_n=\frac{n}{2}(2\times2.5+(n-1)\times0.4)$	✓ 1
A1 $=\frac{n}{2}(0.4n+4.6)=0.2n^2+2.3n$ (AG)	✓ 1
Part (c)
M1 0.2n² + 2.3n ≥ 100 and attempt to solve, giving n ≥ 17.3…	✓ 1
A1 18 days	✓ 1
Part (d)
M1 0.2 × 18² + 2.3 × 18 or $\frac{18}{2}(5+17\times0.4)$	✓ 1
A1 106.2 kg	✓ 1

Common error: If the inequality gives n ≥ 17.3, the answer is 18 days, not 17; the total after 17 days is still below 100 kg.

+ Add to test✓ In your test301IBMI3S-1-02r29-11Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium5 marks

The number of fish, P, in a lake t years after monitoring begins is modelled by P = P₀e^kt, where P₀ and k are constants. There are 2400 fish after 3 years and 3900 fish after 8 years.

(a)

Find the value of k. Give your answer to 3 significant figures.

(3)

(b)

Find the value of P₀. Use the unrounded value of k and give your answer to the nearest whole number.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 P₀e^3k = 2400 and P₀e^8k = 3900	✓ 1
M1 e^5k = 3900/2400 and take natural logarithms	✓ 1
A1 k = 0.0971	✓ 1
Part (b)
M1 2400 = P₀e^3k	✓ 1
A1 P₀ = 1,793	✓ 1

Exam tip: Dividing the two equations removes P₀ and leaves a single equation in k.

+ Add to test✓ In your test302IBMI3S-1-01r30-04Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium5 marks

Wood from an archaeological dig contains radioactive carbon-14. The fraction of the original carbon-14 that remains after t years is modelled by $F = \left(\tfrac{1}{2}\right)^{t/5730}$.

(a)

Find the percentage of the original carbon-14 that remains after 2000 years. Give your answer to 3 significant figures.

(2)

(b)

A wooden beam is found to contain 62% of its original carbon-14. Find the age of the beam, to the nearest 10 years.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\left(\tfrac12\right)^{2000/5730}$	✓ 1
A1 78.5%	✓ 1
Part (b)
M1 $\left(\tfrac12\right)^{t/5730} = 0.62$	✓ 1
M1 use of logarithms or solver: $t = 5730 \times \frac{\ln 0.62}{\ln 0.5}$	✓ 1
A1 3951.75… so age = 3950 years	✓ 1

Exam tip: To bring an unknown exponent down, take logarithms of both sides; any base works as long as you use it on both sides.

+ Add to test✓ In your test303IBMI3S-1-01r30-07Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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A data-centre server costs €60 000 when new. Its value falls by 18% of its value at the start of each year.

(a)

Find the value of the server after 3 years. Give your answer to the nearest cent.

(2)

(b)

Find the least whole number of years after which the value of the server is below €15 000.

(3)

(c)

Find the value of the server at the end of the time found in (b). Give your answer to the nearest euro.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 60000 × 0.82³	✓ 1
A1 €33082.08	✓ 1
Part (b)
M1 60000 × 0.82ⁿ < 15000	✓ 1
M1 use of logarithms, table or graph: n > 6.99	✓ 1
A1 7 years	✓ 1
Part (c)
A1 60000 × 0.82⁷ = €14957	✓ 1
ECF from (b).	—

Common error: Writing the multiplier as 0.18 instead of 0.82. A fall of 18% leaves 82% of the value.

+ Add to test✓ In your test304IBMI3S-1-01r30-08Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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The daily data traffic at a data centre is modelled by $T = T_0 e^{kt}$, where T is in petabytes (PB), t is the time in years since monitoring began and T₀ and k are constants. At the start of monitoring T = 8.0 PB, and after 4 years T = 20.8 PB.

(a)

Find the value of k. Give your answer to 3 significant figures.

(3)

(b)

Using the unrounded value of k, find the time for the daily traffic to double. Give your answer to 3 significant figures.

(2)

(c)

Using the unrounded value of k, predict the daily traffic 10 years after monitoring began. Give your answer to 3 significant figures.

(2)

(d)

Suggest one reason why the model may be unreliable for large values of t.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 20.8 = 8.0e^4k	✓ 1
M1 $k = \frac{\ln 2.6}{4}$	✓ 1
A1 k = 0.239	✓ 1
Part (b)
M1 2 = e^kt, so $t = \frac{\ln 2}{k}$	✓ 1
A1 2.90 years	✓ 1
Part (c)
M1 8.0e^10k	✓ 1
A1 87.2 PB	✓ 1
Part (d)
R1 e.g. the traffic cannot grow without limit because of hardware, power or demand limits, so growth would level off	✓ 1

Exam tip: Keep the unrounded value of k in your calculator memory when later parts say to use it; rounding early changes the third significant figure.

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The energy E, in joules, released by an earthquake of magnitude M is modelled by $\log_{10} E = 4.8 + 1.5M$. A data-centre operator is assessing the risk from tremors near a site.

(a)

Find the energy released by an earthquake of magnitude 5.2. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(2)

(b)

Show that an increase of 1 in magnitude multiplies the energy released by about 31.6.

(2)

(c)

An earthquake releases 1.0 × 10¹⁵ J. Find its magnitude.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $E = 10^{4.8 + 1.5 \times 5.2} = 10^{12.6}$	✓ 1
A1 3.98 × 10¹² J	✓ 1
Part (b)
M1 $\log_{10}E_2 - \log_{10}E_1 = 1.5$, so $E_2/E_1 = 10^{1.5}$	✓ 1
A1 $10^{1.5} = 31.6$ (to 3 s.f.)	✓ 1
Part (c)
M1 $\log_{10}(10^{15}) = 15 = 4.8 + 1.5M$	✓ 1
A1 M = 6.8	✓ 1

Exam tip: If log₁₀ E = x then E = 10^x. Writing this step down earns a method mark even if the arithmetic that follows slips.

+ Add to test✓ In your test306IBMI3S-1-03r30-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

In a role-playing game the experience points (XP) needed to reach level n follow the model $X_n = a\times r^{n}$, where $a$ and $r$ are positive constants. Level 3 needs 864 XP and level 6 needs 2916 XP.

(a)

Show that $r=1.5$.

(2)

(b)

Find the value of $a$.

(2)

(c)

Find the XP needed to reach level 8.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $r^{3}=\frac{2916}{864}$ (or $ar^6\div ar^3$)	✓ 1
A1 $r^3=3.375$ so $r=1.5$ (AG)	✓ 1
Part (b)
M1 $a\times1.5^{3}=864$	✓ 1
A1 $a=256$	✓ 1
Part (c)
M1 $256\times1.5^{8}$	✓ 1
A1 6561 XP	✓ 1

Common error: Dividing the two XP values gives $r^3$, not $r$; take the cube root (the difference in levels), not the square root or a division by 3.

+ Add to test✓ In your test307IBMI3S-1-03r30-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

The concentration of a drug in a patient's blood is modelled by $C(t)=45e^{-0.23t}$, where $C$ is in mg per litre and $t$ is the time in hours after the injection, $t\ge0$.

(a)

Write down the initial concentration.

(1)

(b)

Find the concentration after 5 hours. Give your answer to 3 significant figures.

(2)

(c)

Find the time at which the concentration is 10 mg per litre. Give your answer to 3 significant figures.

(2)

(d)

The half-life is the time taken for the concentration to halve. Find the half-life of the drug in hours, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 45 mg per litre	✓ 1
Part (b)
M1 $45e^{-0.23\times5}$	✓ 1
A1 14.2 mg per litre	✓ 1
Part (c)
M1 $45e^{-0.23t}=10$ so $e^{-0.23t}=\frac{2}{9}$	✓ 1
A1 $t=6.54$ hours	✓ 1
Part (d)
M1 $e^{-0.23t}=\frac12$ (or $0.23t=\ln2$)	✓ 1
A1 3.01 hours (independent of the initial concentration)	✓ 1

Exam tip: The half-life of an exponential decay does not depend on the starting amount, so you do not need the 45 in part (d).

+ Add to test✓ In your test308IBMI3S-1-03r30-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A water company borrows €250 000 to replace pipework. The loan has a nominal annual interest rate of 4.2%, compounded monthly, and is repaid with equal payments at the end of each month for 15 years.

(a)

Calculate the monthly repayment. Give your answer to the nearest cent.

(3)

(b)

Using your repayment from part (a) given to the nearest cent, calculate the total interest paid over the 15 years.

(2)

(c)

Using the same repayment, find the amount still owed immediately after the 60th payment. Give your answer to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 identify $N=180$, $I\%=4.2$, $PV=250000$, $FV=0$, $P/Y=C/Y=12$	✓ 1
M1 use of TVM solver or annuity formula to find PMT	✓ 1
A1 €1874.38	✓ 1
Part (b)
M1 $180\times1874.38-250000$	✓ 1
A1 €87388.40	✓ 1
Part (c)
M1 TVM with $N=60$ (balance $FV$) (or equivalent)	✓ 1
A1 €183 405 (accept 183405.37)	✓ 1

Exam tip: When finding a balance part-way through a loan, set N to the number of payments already made and read off FV, then change its sign.

+ Add to test✓ In your test309IBMI3S-1-03r30-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

Water leaves a treatment plant through a pipe at a steady rate of 0.0425 m³ per second.

(a)

Calculate the volume of water, in m³, that leaves the plant in one day (24 hours).

(2)

(b)

Write your answer to part (a) correct to 3 significant figures, in the form $a\times10^{k}$ where $1\le a<10$ and $k$ is an integer.

(1)

(c)

A manager estimates the daily volume as 3700 m³. Find the percentage error in her estimate, correct to 3 significant figures.

(2)

(d)

A reservoir holds $2.4\times10^{5}$ m³. Using the exact daily volume from part (a), find how many days of the plant's output are needed to fill it. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $0.0425\times24\times60\times60$	✓ 1
A1 3672 m³	✓ 1
Part (b)
A1 $3.67\times10^{3}$ m³	✓ 1
Part (c)
M1 $\frac{3700-3672}{3672}\times100$	✓ 1
A1 0.763%	✓ 1
Part (d)
M1 $\frac{2.4\times10^{5}}{3672}$	✓ 1
A1 65.4 days	✓ 1

Common error: Convert the time to seconds before multiplying by a rate given per second; mixing hours and seconds is the usual slip.

+ Add to test✓ In your test310IBMI3S-1-03r30-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A mobile game gives a player a login reward each day. The reward is 20 coins on day 1, and each day's reward is 8 coins more than the previous day's. The rewards form an arithmetic sequence.

(a)

Find the reward on day 30.

(2)

(b)

The player wants a total of at least 5000 coins. Find the smallest number of consecutive days from day 1 that the player must log in.

(3)

(c)

Find the total number of coins the player has received at the end of the day found in part (b).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_{30}=20+29\times8$	✓ 1
A1 252 coins	✓ 1
Part (b)
M1 $S_n=\frac n2\left(40+8(n-1)\right)=4n^2+16n$	✓ 1
M1 solve $S_n\ge5000$ (inequality, table or graph), e.g. $S_{33}=4884$ and $S_{34}=5168$	✓ 1
A1 34 days	✓ 1
Part (c)
M1 $S_{34}=\frac{34}{2}(2\times20+33\times8)$	✓ 1
A1 5168 coins (ECF from (b))	✓ 1

Exam tip: For "at least" or "first exceeds" questions, check the terms either side of the crossing value so you give a whole number of days.

+ Add to test✓ In your test311IBMI3S-1-03r30-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A game file has a size of 48.6 GB. A player's internet connection downloads data at 12.4 megabits per second (Mbit/s). Use 1 GB = 8000 megabits (Mbit).

(a)

Write the size of the file in Mbit in the form $a\times10^{k}$, where $1\le a<10$ and $k$ is an integer. Give $a$ to 3 significant figures.

(2)

(b)

Find the time the download takes, in hours. Use the exact size of 388 800 Mbit and give your answer to 3 significant figures.

(2)

(c)

The player estimates the file size as 49 GB and the speed as 12 Mbit/s. Find the percentage error in the player's estimate of the download time. Use exact values for the true time and give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $48.6\times8000=388\,800$	✓ 1
A1 $3.89\times10^{5}$ Mbit	✓ 1
Part (b)
M1 $\frac{388800}{12.4\times3600}$	✓ 1
A1 8.71 hours	✓ 1
Part (c)
M1 estimated time $=\frac{49\times8000}{12}=32\,667$ s	✓ 1
M1 $\frac{\text{estimate}-\text{true}}{\text{true}}\times100$ with true time $=31\,355$ s	✓ 1
A1 4.18%	✓ 1

Common error: Megabits and gigabytes are different units; convert the file size to the same unit as the connection speed before dividing.

+ Add to test✓ In your test312IBMI3S-1-03r32-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

Mireille deposits €15 000 into an account that pays a nominal annual interest rate of 3.6%, compounded monthly. No further deposits or withdrawals are made.

(a)

Find the value of the account after 5 years. Give your answer to the nearest cent.

(2)

(b)

Find the interest earned over the 5 years.

(1)

(c)

Find how many years it takes for the account to first reach €21 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $15000\left(1+\dfrac{3.6}{100\times12}\right)^{60}$ or finance solver with N = 60, I% = 3.6, PV = −15000, PMT = 0, P/Y = C/Y = 12	✓ 1
A1 €17953.42	✓ 1
Part (b)
A1 €17953.42 − €15 000 = €2953.42 (ECF from (a))	✓ 1
Part (c)
M1 $15000\times1.003^{m} = 21000$ (or finance solver with FV = 21000)	✓ 1
M1 solve for $m$ using logarithms or a GDC: $m = 112.3$ months	✓ 1
A1 9.36 years	✓ 1

Common error: Using 3.6% as the monthly rate instead of 3.6 ÷ 12 = 0.3% per month overstates the balance enormously.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A data centre company borrows €240 000 to buy cooling equipment. The loan has a nominal annual interest rate of 4.8%, compounded monthly. It is repaid by equal payments made at the end of each month for 5 years.

(a)

Calculate the monthly repayment. Give your answer to the nearest cent.

(3)

(b)

Calculate the total amount of interest paid over the 5 years. Use the monthly repayment rounded to the nearest cent.

(2)

(c)

Find the amount still owed immediately after the 24th repayment. Give your answer to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance solver with N = 60, I% = 4.8, PV = 240000, FV = 0, P/Y = C/Y = 12	✓ 1
M1 attempt to find PMT (or use of $\dfrac{240000\,i}{1-(1+i)^{-60}}$ with $i = 0.004$)	✓ 1
A1 €4507.14	✓ 1
Part (b)
M1 $60\times 4507.14 - 240000$	✓ 1
A1 €30428.40	✓ 1
Part (c)
M1 finance solver with N = 24 (and the same I%, PV, PMT, P/Y, C/Y) to find FV, or balance formula	✓ 1
A1 €150835	✓ 1

Exam tip: State the values you enter into the finance solver (N, I%, PV, PMT, FV, P/Y, C/Y); a correct setup earns the method mark even if a value is mis-copied.

+ Add to test✓ In your test314IBMI3S-1-03r32-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium5 marks

The amount of data stored by a company doubles every 18 months. At the start of 2024 it stored 6 petabytes (PB). The amount stored after t years is modelled by $D = 6\times 2^{t/1.5}$ PB.

(a)

Find the amount of data stored 4 years after the start of 2024. Give your answer to 3 significant figures.

(2)

(b)

Find how many years it takes for the amount of stored data to reach 500 PB. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $D = 6\times2^{4/1.5}$	✓ 1
A1 38.1 PB	✓ 1
Part (b)
M1 $6\times2^{t/1.5} = 500$	✓ 1
M1 take logarithms (or use a GDC): $\dfrac{t}{1.5} = \dfrac{\ln(500/6)}{\ln 2}$	✓ 1
A1 9.57 years	✓ 1

Common error: Doubling every 18 months means the exponent is $t/1.5$ with $t$ in years; using $t/18$ mixes months and years.

+ Add to test✓ In your test315IBMI3S-1-03r32-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A cyclist rides a distance of 42.6 km in 1 hour 24 minutes.

(a)

Write 42.6 km in metres, in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(1)

(b)

Calculate her average speed in km h⁻¹. Give your answer to 3 significant figures.

(2)

(c)

Her bike computer displays her average speed as 31 km h⁻¹. Using the exact value of her speed, calculate the percentage error in the displayed speed. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4.26 × 10⁴ m	✓ 1
Part (b)
M1 $\dfrac{42.6}{1.4}$ (1 h 24 min = 1.4 h)	✓ 1
A1 30.4 km h⁻¹	✓ 1
Part (c)
M1 $\dfrac{\|31 - 30.4286\ldots\|}{30.4286\ldots}\times100$ (using the exact speed $\dfrac{42.6}{1.4}$)	✓ 1
A1 $\dfrac{0.5714\ldots}{30.4286\ldots}$	✓ 1
A1 1.88%	✓ 1

Common error: Convert 1 h 24 min to 1.4 h, not 1.24 h; minutes are out of 60, not 100.

+ Add to test✓ In your test316IBMI3S-1-03r32-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium5 marks

A data centre installs server racks each month, and the numbers installed in successive months form an arithmetic sequence. In month 3 it installs 40 racks and in month 8 it installs 75 racks.

(a)

Find the number of racks installed in month 1 and the common difference.

(3)

(b)

Find the total number of racks installed in the first 12 months.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_1 + 2d = 40$ and $u_1 + 7d = 75$	✓ 1
A1 $d = 7$	✓ 1
A1 $u_1 = 26$	✓ 1
Part (b)
M1 $S_{12} = \dfrac{12}{2}\,(2\times26 + 11\times7)$ (ECF from (a))	✓ 1
A1 774 racks	✓ 1

Exam tip: Two terms of an arithmetic sequence give two simultaneous equations in $u_1$ and $d$; subtract them to find $d$ first.

+ Add to test✓ In your test317IBMI3S-1-03r33-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A heritage trust deposits €45 000 into an account to fund the restoration of a historic building. The account pays a nominal annual interest rate of 3.2%, compounded quarterly.

(a)

Calculate the value of the investment after 6 years. Give your answer to the nearest euro.

(3)

(b)

Find the time, in years, for the investment to reach €60 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of compound interest with $k=4$, $r=3.2$ and $n=6$ (or $N=24$, $I\%=3.2$, PV = −45 000, P/Y = C/Y = 4)	✓ 1
A1 $45\,000\left(1+\frac{0.032}{4}\right)^{24}$ (or correct values entered in a finance solver)	✓ 1
A1 €54 484	✓ 1
Part (b)
M1 set up $45\,000(1.008)^{n}=60\,000$ where $n$ is the number of quarters	✓ 1
M1 solve using logarithms or a solver, obtaining $n=36.1$ quarters	✓ 1
A1 9.03 years (dividing the number of quarters by 4)	✓ 1

Exam tip: Compounding is quarterly, so the exponent counts quarters; convert to years only at the end.

+ Add to test✓ In your test318IBMI3S-1-03r33-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

In a coastal sea, the percentage of surface light that reaches a depth of d metres is modelled by $I=100e^{-0.19d}$.

(a)

Find the percentage of surface light that reaches a depth of 10 m. Give your answer to 3 significant figures.

(2)

(b)

Photosynthesis stops when less than 1% of surface light is available. Find the depth at which this happens. Give your answer to 3 significant figures.

(2)

(c)

Find the depth, in metres, through which the light intensity falls to half of its value. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute $d=10$: $100e^{-0.19\times10}$	✓ 1
A1 15.0%	✓ 1
Part (b)
M1 set up $100e^{-0.19d}=1$ (so $e^{-0.19d}=0.01$) and take natural logarithms	✓ 1
A1 24.2 m	✓ 1
Part (c)
M1 set up $e^{-0.19x}=0.5$ (accept any starting depth)	✓ 1
A1 3.65 m	✓ 1

Exam tip: Natural logarithms undo $e^{kx}$ directly; the minus sign in the exponent often cancels with a minus from $\ln$ of a fraction.

+ Add to test✓ In your test319IBMI3S-1-03r33-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

The floor of a circular atrium is measured to have a diameter of 18.4 m.

(a)

Calculate the area of the floor. Give your answer to 3 significant figures.

(2)

(b)

Write the area, in cm², in the form a × 10^k, where 1 ≤ a < 10 and k is an integer. Use your answer to (a).

(2)

(c)

In an early sketch, the architect uses a diameter of 18.0 m. Find the percentage error in the area calculated from this sketch value. Use unrounded areas and give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $\pi\times9.2^2$	✓ 1
A1 266 m²	✓ 1
Part (b)
M1 multiply by $10^4$ (since 1 m² = 10⁴ cm²)	✓ 1
A1 2.66 × 10⁶ cm²	✓ 1
Part (c)
M1 use of $\pi\times9^2$ for the sketch area (≈ 254.5 m²)	✓ 1
M1 use of percentage error $=\left\|\frac{v_A-v_E}{v_E}\right\|\times100$ with $v_E=\pi\times9.2^2$	✓ 1
A1 4.30%	✓ 1

Common error: Rounding the two areas before finding the percentage error can change the third significant figure; keep unrounded values until the end.

+ Add to test✓ In your test320IBMI3S-1-03r33-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

The value, €V, of a refrigerated delivery lorry t years after it is bought is modelled by $V=250\,000\times0.88^{t}$.

(a)

State the percentage by which the value of the lorry falls each year.

(1)

(b)

Find the value of the lorry after 4 years, to the nearest euro.

(2)

(c)

Find the time for the value to fall to €100 000. Give your answer in years, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 12%	✓ 1
Part (b)
M1 use of $250\,000\times0.88^4$	✓ 1
A1 €149 924	✓ 1
Part (c)
M1 set up $250\,000\times0.88^{t}=100\,000$ (so $0.88^t=0.4$)	✓ 1
M1 use of logarithms, e.g. $t=\frac{\ln0.4}{\ln0.88}$	✓ 1
A1 7.17 years	✓ 1

Exam tip: A multiplier of 0.88 means a fall of 12%: subtract from 1 to read the percentage change.

+ Add to test✓ In your test321IBMI3S-1-03r33-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A cargo vessel burns 135 tonnes of fuel each day. A voyage lasts 17.2 days. The cost of fuel is €620 per tonne.

(a)

Calculate the total mass of fuel burned on the voyage. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Calculate the total cost of the fuel for the voyage. Use the unrounded mass of fuel and give your answer in the form a × 10^k with a correct to 3 significant figures.

(2)

(c)

A planner estimates the cost by rounding each of 135, 17.2 and 620 to 1 significant figure before multiplying. Find the percentage error in this estimate, to 3 significant figures. Use the unrounded cost from (b).

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $135\times17.2\ (=2322)$	✓ 1
A1 2.32 × 10³ tonnes	✓ 1
Part (b)
M1 use of $2322\times620$	✓ 1
A1 1.44 × 10⁶ euros	✓ 1
Part (c)
M1 estimate $=100\times20\times600=1.2\times10^6$ euros	✓ 1
M1 use of percentage error $=\left\|\frac{v_A-v_E}{v_E}\right\|\times100$ with $v_E=1\,439\,640$	✓ 1
A1 16.6%	✓ 1

Exam tip: Percentage error compares the approximate value with the exact value; always keep the exact value unrounded while you work.

+ Add to test✓ In your test322IBMI3S-1-04r34-05Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A café-robotics company borrows €18 500 to buy a robot. The loan has a nominal annual interest rate of 6.2%, compounded monthly, and is repaid with equal payments at the end of each month for 4 years.

(a)

Calculate the monthly repayment, to the nearest cent.

(2)

(b)

Using your answer to part (a), find the total amount repaid over the 4 years.

(1)

(c)

Find the total interest paid.

(1)

(d)

Find the amount still owed immediately after the 24th payment, to the nearest euro. Use the unrounded monthly repayment from your calculator.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 correct use of a financial app or formula: N = 48, I% = 6.2, PV = 18500, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €436.17	✓ 1
Part (b)
A1 48 × 436.17 = €20 936.16	✓ 1
Part (c)
A1 20 936.16 − 18 500 = €2436.16 (ECF from (b))	✓ 1
Part (d)
M1 using the financial app with N = 24 and the same PV, I%, PMT (or the balance formula)	✓ 1
A1 €9821	✓ 1

Common error: In a loan question the present value and the payment have opposite signs on the financial app; entering both as positive gives an error or an incorrect result.

+ Add to test✓ In your test323IBMI3S-1-04r34-06Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A designer invests €6500 in an account paying a nominal annual interest rate of 3.6%, compounded quarterly.

(a)

Find the value of the investment after 5 years, giving your answer to the nearest cent.

(2)

(b)

Find the number of years it takes for the investment to reach €9000. Give your answer to 3 significant figures.

(3)

(c)

Find the effective annual interest rate, as a percentage to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 6500 × (1 + 0.036/4)²⁰	✓ 1
A1 €7775.65	✓ 1
Part (b)
M1 setting 6500 × (1.009)^4t = 9000	✓ 1
M1 solving using logarithms, a graph or the finance solver	✓ 1
A1 t = 9.08 years	✓ 1
Part (c)
M1 (1.009)⁴ − 1	✓ 1
A1 3.65 %	✓ 1

Exam tip: The effective annual rate is always slightly above the nominal rate when interest is compounded more than once a year.

+ Add to test✓ In your test324IBMI3S-1-04r34-07Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium5 marks

A fisheries technician estimates the volume of a cylindrical tank. The measured radius is 1.5 m and the measured height is 0.8 m. The true radius is 1.46 m and the true height is 0.78 m.

(a)

Calculate the volume using the measured values. Give your answer to 3 significant figures.

(2)

(b)

Calculate the true volume, to 3 significant figures.

(1)

(c)

Using the 3 significant figure volumes from parts (a) and (b), find the percentage error in the measured volume. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 π × 1.5² × 0.8	✓ 1
A1 5.65 m³	✓ 1
Part (b)
A1 5.22 m³	✓ 1
Part (c)
M1 (5.65 − 5.22) ÷ 5.22 × 100	✓ 1
A1 8.24 %	✓ 1

Common error: Small percentage errors in the radius are doubled in the volume formula because the radius is squared.

+ Add to test✓ In your test325IBMI3S-1-04r34-08Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A shop stacks rolls of fabric in rows. The first row has 4 rolls and each row has 3 more rolls than the row before it.

(a)

Find the number of rolls in the 10th row.

(2)

(b)

The shop has exactly 246 rolls. Find the number of complete rows that can be built.

(3)

(c)

A 13th row is added. Find the number of rolls needed for it.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₀ = 4 + 9 × 3	✓ 1
A1 31 rolls	✓ 1
Part (b)
M1 sum formula S_n = n/2 × (2 × 4 + (n − 1) × 3)	✓ 1
M1 setting S_n = 246 and solving (graph, table or equation)	✓ 1
A1 n = 12	✓ 1
Part (c)
A1 4 + 12 × 3 = 40 rolls	✓ 1

Exam tip: When solving Sn = a given total, a graph or table on your calculator is quicker than rearranging the quadratic by hand, and the positive integer solution is the one to keep.

+ Add to test✓ In your test326IBMI3S-1-04r34-09Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A regulator sets an annual quota for a fishery. The quota is 800 tonnes in year 1, and each year it is 6% lower than in the previous year.

(a)

Find the quota in year 10, to the nearest tonne.

(2)

(b)

Find the first year in which the quota is below 300 tonnes.

(3)

(c)

Find the total quota over the first 15 years, to the nearest tonne.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 800 × 0.94⁹	✓ 1
A1 458 tonnes	✓ 1
Part (b)
M1 setting 800 × 0.94^{n − 1} < 300	✓ 1
M1 solving or checking terms, n − 1 > 15.85	✓ 1
A1 year 17	✓ 1
Part (c)
M1 S₁₅ = 800(1 − 0.94¹⁵)/(1 − 0.94)	✓ 1
A1 8063 tonnes	✓ 1

Common error: The nth term of a geometric sequence uses the power n − 1, not n.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; 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The strength S of a radio signal received by an underwater robot at a distance d metres from its transmitter is modelled by S = 80e^−0.35d, where S is measured in arbitrary units.

(a)

Find the signal strength at d = 2.5, to 3 significant figures.

(2)

(b)

Show that d = ln(80/S) ÷ 0.35.

(2)

(c)

The robot loses contact when S falls to 20. Find the distance at which this happens, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 80e^{−0.35 × 2.5}	✓ 1
A1 33.3	✓ 1
Part (b)
M1 e^−0.35d = S/80 and taking natural logarithms, −0.35d = ln(S/80)	✓ 1
A1 d = −ln(S/80) ÷ 0.35 = ln(80/S) ÷ 0.35	✓ 1
Part (c)
M1 d = ln(80/20) ÷ 0.35 (using (b))	✓ 1
A1 3.96 m	✓ 1

Exam tip: Natural logarithm and the exponential function ex are inverses, so ln(ex) = x; use this to isolate the variable in the exponent.

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A trawler's net sweeps an area of 4.2 × 10⁴ m² each hour. The fishing ground has an area of 1.8 × 10⁹ m². In parts (b) and (c) use the unrounded value from the previous part.

(a)

Find the number of hours needed to sweep the whole fishing ground. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(2)

(b)

The trawler works 16 hours per day. Find the number of days needed, in the form a × 10^k with a to 3 significant figures.

(2)

(c)

Find the number of years needed, assuming the trawler works every day of a 365-day year. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1.8 × 10⁹ ÷ 4.2 × 10⁴	✓ 1
A1 4.29 × 10⁴ hours	✓ 1
Part (b)
M1 42 857.14… ÷ 16	✓ 1
A1 2.68 × 10³ days	✓ 1
Part (c)
M1 2678.57… ÷ 365	✓ 1
A1 7.34 years	✓ 1

Common error: Dividing powers of ten means subtracting the exponents; keep the unrounded value in your calculator between steps to avoid rounding drift.

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A supermarket chain reports annual online sales of €3 482 917.

(a)

Write this amount in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, giving a correct to 3 significant figures.

(2)

(b)

A journalist quotes the sales as €3 480 000. Calculate the percentage error in the journalist's figure. Give your answer to 3 significant figures.

(3)

(c)

State, with a reason, whether the journalist's figure is an over-estimate or an under-estimate.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for 3.48… × 10⁶ seen	✓ 1
A1 3.48 × 10⁶	✓ 1
Part (b)
M1 \|approximate − exact\| = \|3 480 000 − 3 482 917\| (= 2917)	✓ 1
M1 percentage error = \|v_A − v_E\| / \|v_E\| × 100 with v_E = 3 482 917	✓ 1
A1 0.0838%	✓ 1
Part (c)
R1 under-estimate, since 3 480 000 is less than 3 482 917	✓ 1

Common error: Divide by the exact value, not by the approximate value, when finding a percentage error.

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After a promotion ends, a retailer sells 2400 units of a product in week 1. Sales then fall by 8% each week, so weekly sales form a geometric sequence.

(a)

Find the number of units sold in week 9, to the nearest whole number.

(2)

(b)

Find the first week in which fewer than 1000 units are sold.

(3)

(c)

Find the total number of units sold during the first 10 weeks, to the nearest whole number.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2400 × 0.92⁸	✓ 1
A1 1232	✓ 1
Part (b)
M1 2400 × 0.92ⁿ⁻¹ < 1000 (or equation, or table/graph)	✓ 1
M1 takes logarithms or uses table/graph to find n − 1 > 10.5 (accept 10.499…)	✓ 1
A1 week 12	✓ 1
Part (c)
M1 S₁₀ = 2400(1 − 0.92¹⁰)/(1 − 0.92)	✓ 1
A1 16968	✓ 1

Exam tip: When a question asks for the first term that is below a value, finish by stating a whole number of weeks, not the decimal solution of the equation.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

The number of people infected in an outbreak is modelled by N = 25 × 1.35^t, where t is the time in days since the outbreak was first detected.

(a)

Find the number of people infected after 6 days, to the nearest whole number.

(2)

(b)

Interpret the value 1.35 in this model.

(1)

(c)

Find the value of t at which the model predicts that 1000 people are infected. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 25 × 1.35⁶	✓ 1
A1 151	✓ 1
Part (b)
R1 the number infected increases by 35% each day (accept: is multiplied by 1.35 each day)	✓ 1
Part (c)
M1 25 × 1.35^t = 1000 (or 1.35^t = 40)	✓ 1
M1 takes logarithms: t = ln 40 / ln 1.35 (or equivalent, or solver)	✓ 1
A1 12.3 days	✓ 1

Exam tip: If a model is exponential, taking logarithms of both sides after isolating the power is the quickest way to find the unknown exponent.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A solar installation firm fits 5 rooftop panels in week 1 of a project. In each following week it fits 3 more panels than in the previous week, so the numbers fitted each week form an arithmetic sequence.

(a)

Find the smallest number of weeks after which the total number of panels fitted first exceeds 200.

(4)

(b)

Find the number of panels fitted in week 20.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 S_n = n/2 (2 × 5 + (n − 1) × 3)	✓ 1
M1 sets S_n > 200 (or = 200) and solves (or uses a table/graph)	✓ 1
A1 S₁₀ = 185 and S₁₁ = 220 (or critical value 10.44…) seen	✓ 1
A1 11 weeks	✓ 1
Part (b)
M1 u₂₀ = 5 + 19 × 3	✓ 1
A1 62	✓ 1

Exam tip: After solving an inequality involving a sum, check the two whole numbers either side of the critical value to confirm which one first satisfies it.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Medium6 marks

The usable capacity, C, of a grid-storage battery after n charge cycles is modelled by C = 100 × 0.9992ⁿ, where C is a percentage of its original capacity.

(a)

Find the capacity of the battery after 300 cycles. Give your answer as a percentage to 3 significant figures.

(2)

(b)

Find the value of n for which the capacity is exactly 70%. Give your answer to 1 decimal place.

(3)

(c)

The battery is replaced as soon as its capacity falls below 70%. Explain why this happens after 446 cycles and not after 445 cycles.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 100 × 0.9992³⁰⁰	✓ 1
A1 78.7%	✓ 1
Part (b)
M1 100 × 0.9992ⁿ = 70	✓ 1
M1 n = ln 0.7 / ln 0.9992 (or solver)	✓ 1
A1 445.7	✓ 1
Part (c)
R1 the number of cycles is a whole number, and at 445 cycles the capacity is still above 70% (while at 446 cycles it is below 70%)	✓ 1

Common error: Rounding 445.6 down to 445 would give a capacity that is still above 70%, so the replacement point is the next whole cycle.

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Calculator

After an injection, the concentration of a drug in a patient's blood is modelled by C(t) = 45e^−0.18t, where C is in mg per litre and t is the time in hours since the injection.

(a)

Find the concentration of the drug 5 hours after the injection. Give your answer to three significant figures.

(2)

(b)

The patient needs a further injection when the concentration falls below 10 mg per litre. Find the time at which this happens, giving your answer in hours to three significant figures.

(3)

(c)

Find the half-life of the drug in this model, to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 5: 45e^−0.9	✓ 1
A1 18.3 mg per litre	✓ 1
Part (b)
M1 45e^−0.18t = 10	✓ 1
M1 correct use of logarithms (ln(10/45) = −0.18t) or a graphical/solver method	✓ 1
A1 8.36 hours	✓ 1
Part (c)
M1 e^−0.18t = 0.5	✓ 1
A1 3.85 hours	✓ 1

Exam tip: Isolate the exponential before taking ln: divide both sides by 45 first.

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Calculator

A designer creates a rectangular map for an online game. The map has length 3.45 km, given to three significant figures, and width 2.8 km, given to two significant figures.

(a)

Calculate the area of the map in m². Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Find the lower bound and the upper bound for the area of the map, in km², to three significant figures.

(3)

(c)

The game's promotional material states that the area is 9.5 km². Taking 9.66 km² as the exact area, find the percentage error in the stated value, to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3450 × 2800	✓ 1
A1 9.66 × 10⁶ m²	✓ 1
Part (b)
M1 lower bounds 3.445 and 2.75, upper bounds 3.455 and 2.85 used	✓ 1
A1 lower bound 9.47 km²	✓ 1
A1 upper bound 9.85 km²	✓ 1
Part (c)
M1 \|9.5 − 9.66\| ÷ 9.66 × 100	✓ 1
A1 1.66%	✓ 1

Exam tip: Lower bound of a product uses both lower bounds; the upper bound uses both upper bounds.

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Calculator

A game studio invests £15 000 in an account that pays a nominal annual interest rate of 4.2%, compounded quarterly. No further money is added or withdrawn.

(a)

Find the value of the investment after 4 years. Give your answer to the nearest penny.

(2)

(b)

Find how many years it takes for the investment to reach £25 000. Give your answer to three significant figures.

(3)

(c)

Find the effective annual interest rate of the account, as a percentage to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 15 000 × (1 + 0.042/4)²⁴ (or solver with N = 6, I% = 4.2, PV = −15 000, P/Y = 1, C/Y = 4)	✓ 1
A1 £19 273.60	✓ 1
Part (b)
M1 15 000 × (1.0105)^4t = 25 000 (or solver set up with FV = 25 000)	✓ 1
M1 use of logarithms or solver to find t	✓ 1
A1 12.2 years	✓ 1
Part (c)
M1 (1 + 0.042/4)⁴ − 1	✓ 1
A1 4.27%	✓ 1

Exam tip: The effective annual rate compares accounts that compound at different frequencies.

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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

Calculator

A swimming club invests €8000 in an account that pays a nominal annual interest rate of 3.6%, compounded quarterly. No money is added or withdrawn.

(a)

Find the smallest number of complete years after which the value of the investment exceeds €12 000.

(3)

(b)

Find the value of the investment after 7 years. Give your answer to the nearest cent.

(2)

(c)

Find the interest earned in the first 7 years. Use the unrounded value from part (b) and give your answer to the nearest cent.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $8000\left(1+\frac{3.6}{4\times100}\right)^{4n}=12000$ or use of a finance solver (N, I%=3.6, PV=−8000, FV=12 000, P/Y=C/Y=4)	✓ 1
M1 solving for $n$ ($n=11.31\ldots$)	✓ 1
A1 12 years	✓ 1
Part (b)
M1 $8000\times1.009^{28}$	✓ 1
A1 €10 281.17	✓ 1
Part (c)
A1 €2281.17	✓ 1

Common error: Quarterly compounding means 4 periods per year at a quarter of the annual rate; using 7 periods or the full 3.6% each quarter gives a much larger value.

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Calculator

The efficiency, $E$ per cent, of a wind turbine gearbox $t$ years after installation is modelled by $E=95\,\mathrm{e}^{-0.021t}$, for $t\ge0$.

(a)

Find the efficiency of the gearbox after 10 years. Give your answer to three significant figures.

(2)

(b)

Find the time at which the efficiency first falls to 80%. Give your answer to three significant figures.

(2)

(c)

The efficiency can also be written as $E=95\times r^t$. Find the percentage by which the efficiency decreases each year. Give your answer to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $95\mathrm{e}^{-0.21}$	✓ 1
A1 $77.0\%$	✓ 1
Part (b)
M1 $95\mathrm{e}^{-0.021t}=80$ and attempt to take natural logarithms: $t=\frac{1}{0.021}\ln\frac{95}{80}$	✓ 1
A1 $t=8.18$ years	✓ 1
Part (c)
M1 $r=\mathrm{e}^{-0.021}=0.97922\ldots$ and decrease $=1-r$	✓ 1
A1 $2.08\%$	✓ 1

Exam tip: Keep unrounded values in your calculator until the final line; rounding $\mathrm{e}^{-0.021}$ to 0.979 early would give a percentage decrease of 2.1% instead of 2.08%.

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Calculator

A town council borrows €15 000 to refurbish a library reading room. The loan has a nominal annual interest rate of 5.4%, compounded monthly. It is repaid by equal payments made at the end of each month for 4 years.

(a)

Find the monthly payment. Give your answer to the nearest cent.

(3)

(b)

Using your answer to (a), find the total amount repaid over the 4 years.

(2)

(c)

Hence find the total interest paid over the 4 years.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of finance solver or annuity formula with $n=48$, monthly rate $0.0045$ (N=48, I%=5.4, PV=15 000, FV=0, P/Y=C/Y=12)	✓ 1
M1 correct set-up with FV = 0 (balance cleared after 48 payments)	✓ 1
A1 €348.16	✓ 1
Part (b)
M1 $48\times348.16$	✓ 1
A1 €16 711.68	✓ 1
Part (c)
A1 €1711.68 (ECF from (b))	✓ 1

Exam tip: Set the future value to zero for a loan that is fully repaid, and make sure payments per year and compounding periods per year both read 12 on your calculator.

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A swimmer increases the distance she swims each week by a constant amount. In week 1 she swims 1.5 km. Over the first 10 weeks she swims a total of 37.5 km. The weekly distances form an arithmetic sequence with common difference $d$ km.

(a)

Find the value of $d$.

(3)

(b)

Find the first week in which she swims at least 5.2 km.

(2)

(c)

Find the distance, in km, that she swims in week 10.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $S_{10}=\frac{10}{2}(2\times1.5+9d)$	✓ 1
M1 $5(3+9d)=37.5$	✓ 1
A1 $d=0.5$	✓ 1
Part (b)
M1 $1.5+0.5(n-1)\ge5.2$ (or equality, or table)	✓ 1
A1 week 9 ($n\ge8.4$)	✓ 1
Part (c)
M1 $1.5+9\times0.5$	✓ 1
A1 6 km	✓ 1

Common error: The sum formula uses the number of terms $n$, whereas the $n$th term formula uses $(n-1)$ common differences; mixing the two is the usual source of a wrong $d$.

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Calculator

A new library gains 400 new members in its first year. In each later year the number of new members is 8% fewer than in the previous year. The numbers of new members each year form a geometric sequence.

(a)

Find the number of new members in year 6. Give your answer to the nearest whole number.

(2)

(b)

Find the total number of new members gained in the first 6 years. Give your answer to the nearest whole number.

(2)

(c)

Show that, however many years pass, the total number of members gained can never reach 5000, and state the value the total approaches.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $400\times0.92^5$	✓ 1
A1 264	✓ 1
Part (b)
M1 $S_6=\frac{400(1-0.92^6)}{1-0.92}$	✓ 1
A1 1968	✓ 1
Part (c)
M1 $\|r\|<1$ so the sum to infinity exists, $S_\infty=\frac{400}{1-0.92}$	✓ 1
A1 $S_\infty=5000$, so the total approaches but never reaches 5000	✓ 1

Exam tip: A sum to infinity only exists when $|r|<1$; state this condition before using $S_\infty=\frac{u_1}{1-r}$.

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Calculator

Wind speed $v$ m s⁻¹ at a height $h$ metres above ground is modelled by $v=5.2\left(\frac{h}{10}\right)^{\alpha}$, where $\alpha$ is a constant. At a height of 80 m the wind speed is measured as 7.9 m s⁻¹.

(a)

Find the value of $\alpha$. Give your answer to three significant figures.

(2)

(b)

Using $\alpha=0.201$, find the predicted wind speed at a height of 120 m. Give your answer to three significant figures.

(2)

(c)

Using $\alpha=0.201$, find the height at which the predicted wind speed is 10 m s⁻¹. Give your answer to the nearest metre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $7.9=5.2\times8^{\alpha}$ and attempt to use logarithms: $\alpha=\frac{\ln(7.9/5.2)}{\ln8}$	✓ 1
A1 $\alpha=0.201$	✓ 1
Part (b)
M1 $5.2\times12^{0.201}$	✓ 1
A1 $8.57$ m s⁻¹	✓ 1
Part (c)
M1 $10=5.2\left(\frac{h}{10}\right)^{0.201}$ leading to $h=10\times\left(\frac{10}{5.2}\right)^{1/0.201}$	✓ 1
A1 259 m	✓ 1

Common error: To undo a power, raise both sides to the reciprocal power, here $1/0.201$; taking a logarithm of one side only, or dividing by 0.201, gives an answer that is far too small.

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Calculator

A library building fund receives a deposit of €300 at the end of every month. The fund earns a nominal annual interest rate of 4.2%, compounded monthly. Deposits continue for 6 years and the account starts at €0.

(a)

Find the value of the fund at the end of the 6 years. Give your answer to the nearest euro.

(3)

(b)

Find the total interest earned by the fund over the 6 years. Use the unrounded value from part (a) and give your answer to the nearest euro.

(2)

(c)

The library wants the fund to reach exactly €25 000 after 6 years, using equal monthly deposits and the same interest rate. Find the monthly deposit required. Give your answer to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of finance solver or annuity formula with $n=72$ and monthly rate $0.0035$	✓ 1
A1 correct set-up (N=72, I%=4.2, PV=0, PMT=−300, P/Y=C/Y=12)	✓ 1
A1 €24 517	✓ 1
Part (b)
M1 total deposited $=300\times72=21\,600$	✓ 1
A1 €2917	✓ 1
Part (c)
M1 set FV = 25 000 and solve for PMT	✓ 1
A1 €305.91	✓ 1

Exam tip: Check the cash-flow signs on your calculator: deposits are entered as negative values (money paid out) and the future value is positive.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium6 marks

A library trust invests €85 000 in an account paying a nominal annual interest rate of 2.8%, compounded monthly. No further deposits or withdrawals are made.

(a)

Calculate the value of the investment after 6 years, giving your answer to the nearest cent.

(3)

(b)

The trust needs the investment to reach €120 000 to fund an extension. Find the number of years, to 3 significant figures, for this to happen.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of compound interest with n = 72 (or TVM: N = 72, I% = 2.8, PV = −85 000, P/Y = C/Y = 12)	✓ 1
A1 correct substitution: 85 000(1 + 0.028/12)⁷²	✓ 1
A1 €100 529.94	✓ 1
Part (b)
M1 85 000(1 + 0.028/12)^12t = 120 000 (or TVM with FV = 120 000)	✓ 1
A1 147.96… months (N = 148)	✓ 1
A1 12.3 years	✓ 1

Common error: With monthly compounding, enter the nominal rate (2.8%) and set the compounding periods per year to 12; do not divide the rate by 12 and also use 12 periods per year.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A company installs wind turbines in a new farm. It installs 4 turbines in year 1 and then 3 more turbines each year than in the previous year, so the numbers installed each year form an arithmetic sequence.

(a)

Find the number of turbines installed in year 15.

(2)

(b)

Find the smallest number of years after which the total number of turbines installed is at least 150.

(3)

(c)

Find the total number of turbines installed in the first 12 years.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4 + 14 × 3	✓ 1
A1 46	✓ 1
Part (b)
M1 S_n = n/2 (8 + 3(n − 1)) ≥ 150 (or a GDC table)	✓ 1
A1 S₉ = 144 and S₁₀ = 175	✓ 1
A1 10 years	✓ 1
Part (c)
M1 S₁₂ = 12/2 (8 + 11 × 3)	✓ 1
A1 246	✓ 1

Common error: Check the integer on either side of the solution: the sum must be at least 150, so test both n = 9 and n = 10.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium7 marks

After the heater in a hot tub is switched off, the water temperature T °C after t minutes is modelled by T = 24 + 14e^−0.06t.

(a)

Find the temperature of the water when the heater is switched off.

(1)

(b)

Find the temperature after 10 minutes, to 3 significant figures.

(2)

(c)

Write down the value that the temperature approaches as t becomes very large.

(1)

(d)

Find the time at which the temperature falls to 30 °C, in minutes to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 38 °C	✓ 1
Part (b)
M1 24 + 14e^−0.6	✓ 1
A1 31.7 °C	✓ 1
Part (c)
A1 24 °C	✓ 1
Part (d)
M1 24 + 14e^−0.06t = 30, so e^−0.06t = 6/14	✓ 1
M1 take natural logs: −0.06t = ln(6/14)	✓ 1
A1 14.1 minutes	✓ 1

Exam tip: The constant term in a model of the form a + be⁻kt is the long-term value; here it is the room temperature.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium5 marks

A library pays €18 500 a year for access to a journal database. The price is expected to rise by 3.1% each year.

(a)

Find the price of the database after 6 years, to the nearest euro.

(2)

(b)

After how many years will the price have doubled? Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18 500 × 1.031⁶	✓ 1
A1 €22 219	✓ 1
Part (b)
M1 1.031ⁿ = 2 (or 18 500 × 1.031ⁿ = 37 000)	✓ 1
M1 solve using logs or GDC: n = ln 2 / ln 1.031	✓ 1
A1 22.7 years	✓ 1

Exam tip: Doubling time is independent of the starting price: solve (1 + rate)ⁿ = 2.

+ Add to test✓ In your test348IBMI3S-1-02r39-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 1Medium7 marks

A swimmer swims 4.0 km in week 1 of a plan. Each week she swims 8% further than in the previous week.

(a)

Find the distance she swims in week 12, to 3 significant figures.

(2)

(b)

Find the first week in which she swims more than 12 km.

(3)

(c)

Find the total distance she swims in weeks 1 to 16, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4.0 × 1.08¹¹	✓ 1
A1 9.33 km	✓ 1
Part (b)
M1 4.0 × 1.08ⁿ⁻¹ > 12	✓ 1
A1 1.08¹⁴ = 2.94 and 1.08¹⁵ = 3.17 (or GDC table)	✓ 1
A1 week 16	✓ 1
Part (c)
M1 S₁₆ = 4.0(1.08¹⁶ − 1)/(1.08 − 1)	✓ 1
A1 121 km	✓ 1

Common error: In a geometric sequence the exponent in the nth term is n − 1, but in the sum formula it is n.

+ Add to test✓ In your test349IBMI3S-1-03r40-07Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

A grain merchant estimates the number of wheat grains in a sack. A single grain has mass 4.5 × 10⁻² g and the sack contains 25 kg of grain.

(a)

Estimate the number of grains in the sack. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, correct to 3 significant figures.

(2)

(b)

The grains in the sack are later counted exactly, and there are 572 316. Calculate the percentage error in the estimate. Use the unrounded estimate from (a) and give your answer to 3 significant figures.

(2)

(c)

Seed is sold at €2.40 per 10 000 grains. Using the exact count of 572 316 grains, calculate the cost of the grains in the sack, to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 25 × 1000 ÷ 0.045 (convert kg to g)	✓ 1
A1 5.56 × 10⁵	✓ 1
Part (b)
M1 \|555555.6 − 572316\| ÷ 572316 × 100	✓ 1
A1 2.93%	✓ 1
Part (c)
M1 572316 ÷ 10000 × 2.40	✓ 1
A1 €137.36	✓ 1

Common error: Percentage error is measured relative to the exact value, not the estimate, so divide by 572 316.

+ Add to test✓ In your test350IBMI3S-1-03r40-08Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

The sound intensity level, L decibels (dB), of a sound with intensity I W m⁻² is given by L = 10 log₁₀(I ÷ I₀), where I₀ = 1.0 × 10⁻¹² W m⁻². A loudspeaker at a concert produces an intensity of 3.2 × 10⁻³ W m⁻² at a listener's seat.

(a)

Calculate the sound intensity level at the seat. Give your answer to 3 significant figures.

(2)

(b)

A sound level of 85 dB is the limit for long exposure. Find the intensity at which L = 85. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, to 3 significant figures.

(3)

(c)

A second identical loudspeaker is added next to the first, so the intensity at the seat doubles. Calculate the new sound intensity level, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(3.2 × 10⁻³ ÷ 10⁻¹²)	✓ 1
A1 95.1 dB	✓ 1
Part (b)
M1 85 = 10 log₁₀(I ÷ 10⁻¹²), so I ÷ 10⁻¹² = 10^8.5	✓ 1
M1 I = 10⁻¹² × 10^8.5	✓ 1
A1 3.16 × 10⁻⁴ W m⁻²	✓ 1
Part (c)
M1 10 log₁₀(2 × 3.2 × 10⁻³ ÷ 10⁻¹²) (or 10 log₁₀ 2 added to the answer to (a))	✓ 1
A1 98.1 dB	✓ 1

Common error: Doubling the intensity does not double the decibel value: it adds 10 log₁₀ 2 ≈ 3.01 dB to the level.

+ Add to test✓ In your test351IBMI3S-1-03r40-10Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

An aircraft descends from a cruising altitude of 10 500 m. In the first minute of the descent it loses 1 200 m of altitude. In each later minute it loses 60 m less altitude than in the minute before, so the altitude lost each minute forms an arithmetic sequence.

(a)

Find the altitude lost in the 8th minute.

(2)

(b)

Find the total altitude lost in the first 8 minutes.

(2)

(c)

Find the altitude of the aircraft at the end of the 10th minute.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1200 + 7 × (−60)	✓ 1
A1 780 m	✓ 1
Part (b)
M1 S₈ = 8/2 (1200 + 780)	✓ 1
A1 7920 m	✓ 1
Part (c)
M1 S₁₀ = 10/2 (2 × 1200 + 9 × (−60))	✓ 1
A1 9300 m lost	✓ 1
A1 altitude 1200 m	✓ 1

Common error: The common difference is negative (−60); using +60 gives altitude lost increasing each minute.

+ Add to test✓ In your test352IBMI3S-1-03r40-14Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

A regional airline buys a turboprop aircraft for €4.5 million. Its value falls by 12% of its value at the start of each year.

(a)

Find the value of the aircraft after 5 years. Give your answer in millions of euros to 3 significant figures.

(2)

(b)

Find the number of complete years after which the value of the aircraft is first below €1 million.

(3)

(c)

Find the amount by which the value falls during the 5th year. Give your answer in millions of euros to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4.5 × 0.88⁵	✓ 1
A1 €2.37 million	✓ 1
Part (b)
M1 4.5 × 0.88ⁿ < 1	✓ 1
M1 n > ln(1/4.5) ÷ ln 0.88	✓ 1
A1 n > 11.77, so 12 years	✓ 1
Part (c)
M1 4.5 × 0.88⁴ × 0.12 (or 4.5 × 0.88⁴ − 4.5 × 0.88⁵)	✓ 1
A1 €0.324 million	✓ 1

Common error: A fall of 12% each year multiplies by 0.88 each year; it does not remove a fixed 12% of the original price each year.

+ Add to test✓ In your test353IBMI3S-1-01r41-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A hospital borrows €250 000 to buy a scanner. The loan has a nominal annual interest rate of 4.8%, compounded monthly. It is repaid by equal payments at the end of each month for 5 years.

(a)

Find the monthly payment. Give your answer to the nearest cent.

(3)

(b)

Using the unrounded monthly payment, find the total interest paid over the 5 years. Give your answer to the nearest euro.

(2)

(c)

Find the amount of interest included in the first monthly payment.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a TVM solver with $N=60$, $I\%=4.8$, $PV=250000$, $FV=0$, $P/Y=C/Y=12$	✓ 1
M1 attempt to solve for PMT	✓ 1
A1 €4694.94	✓ 1
Part (b)
M1 $60\times 4694.935\ldots - 250000$	✓ 1
A1 €31 696	✓ 1
Part (c)
M1 $250000\times\frac{0.048}{12}$	✓ 1
A1 €1000	✓ 1

Common error: Enter the loan and the payment with opposite signs in the TVM solver, otherwise the GDC reports an error or a negative value.

+ Add to test✓ In your test354IBMI3S-1-01r41-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium8 marks

A Formula 1 tyre has a slow puncture. Its pressure, P bar, is modelled by $P = 2.4e^{-0.035t}$, where t is the time in hours since the puncture began.

(a)

Find the pressure after 10 hours. Give your answer to 3 significant figures.

(2)

(b)

The team must replace the tyre when the pressure falls to 1.5 bar. Find the value of t at which this happens. Give your answer to 3 significant figures.

(3)

(c)

Find the time, in hours, for the pressure to fall to half of its initial value. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $2.4e^{-0.035\times10}$	✓ 1
A1 1.69 bar	✓ 1
Part (b)
M1 $2.4e^{-0.035t}=1.5$	✓ 1
M1 correct use of $\ln$ or a GDC solver	✓ 1
A1 13.4 hours	✓ 1
Part (c)
M1 $e^{-0.035t} = 0.5$	✓ 1
M1 $t=\frac{\ln 0.5}{-0.035}$	✓ 1
A1 19.8 hours	✓ 1

Exam tip: When you solve an exponential equation with logarithms, isolate the exponential first, then take $\ln$ of both sides.

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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A patient is given 200 mg of a drug every 6 hours. Just before the next dose, 20% of the drug present remains in the patient's body; the rest has been removed. Let $S_n$ mg be the amount of the drug in the body immediately after the nth dose.

(a)

Find $S_3$.

(2)

(b)

Show that $S_n = 250(1-0.2^n)$.

(3)

(c)

Find the greatest amount of the drug that the body approaches in the long term.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $S_2 = 200 + 0.2\times200 = 240$ or $200+40+8$	✓ 1
A1 248 mg	✓ 1
Part (b)
M1 recognise a geometric series with first term 200 and common ratio 0.2	✓ 1
M1 $S_n = \frac{200(1-0.2^n)}{1-0.2}$	✓ 1
A1 $\frac{200}{0.8}=250$, leading to the given result (AG)	✓ 1
Part (c)
A1 250 mg (as $n\to\infty$, $0.2^n\to0$)	✓ 1

Exam tip: For a 'show that' question, every step of the working must be visible and the final line must match the result given.

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A beekeeper saves for new hives by paying €150 into an account at the end of every quarter (3 months). The account pays a nominal annual interest rate of 3.2%, compounded quarterly.

(a)

Find the value of the account immediately after the 32nd payment. Give your answer to the nearest cent.

(3)

(b)

The beekeeper needs €6000. Find the number of quarters, to 3 significant figures, for the account to reach €6000, and hence state the number of complete payments needed.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM solver with $N=32$, $I\%=3.2$, $PV=0$, $PMT=-150$, $P/Y=C/Y=4$	✓ 1
M1 attempt to solve for FV	✓ 1
A1 €5445.71	✓ 1
Part (b)
M1 solve for N with $FV=6000$ and the same other entries	✓ 1
A1 $N=34.8$	✓ 1
A1 35 payments (since the value after 34 payments is below €6000)	✓ 1
R1 reference to rounding up because payments are whole and the value after 35 payments is above €6000	✓ 1

Common error: Rounding the number of payments down gives a total that is still below the target.

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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A hospital laboratory grows a bacterial culture containing $3.2\times10^{7}$ bacteria per mL. A sample of 0.85 mL is taken. Each bacterium has mass $9.5\times10^{-13}$ g.

(a)

Find the number of bacteria in the sample. Give your answer in the form $a\times10^{k}$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(2)

(b)

Find the total mass of the bacteria in the sample, in grams. Give your answer in the form $a\times10^{k}$ with $a$ correct to 3 significant figures.

(2)

(c)

A technician estimates the mass as $2.6\times10^{-5}$ g. Taking $2.584\times10^{-5}$ g as the exact value, find the percentage error. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $3.2\times10^{7}\times0.85$	✓ 1
A1 $2.72\times10^{7}$	✓ 1
Part (b)
M1 $2.72\times10^{7}\times9.5\times10^{-13}$	✓ 1
A1 $2.58\times10^{-5}$ g	✓ 1
Part (c)
M1 $\left\|\frac{2.6\times10^{-5}-2.584\times10^{-5}}{2.584\times10^{-5}}\right\|\times100$	✓ 1
A1 0.619 %	✓ 1

Common error: When multiplying numbers in standard form, multiply the decimals and add the powers of 10; then check that the first number is between 1 and 10.

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Calculator

Dev buys a games console for €1200 using a loan. The loan has a nominal annual interest rate of 7.2%, compounded monthly. It is repaid with equal payments made at the end of each month for 2 years, after which nothing is owed.

(a)

Calculate the monthly payment. Give your answer to the nearest cent.

(2)

(b)

Using the payment rounded to the nearest cent, find the total amount Dev repays.

(2)

(c)

Find the total interest Dev pays on the loan.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM entries: N = 24, I% = 7.2, PV = 1200, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €53.84	✓ 1
Part (b)
M1 24 × 53.84	✓ 1
A1 €1292.16	✓ 1
Part (c)
M1 1292.16 − 1200 (ECF from (b))	✓ 1
A1 €92.16	✓ 1

Exam tip: Write down each TVM input (including FV = 0 and the payments per year) so method marks are secure even if a key is mistyped.

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Calculator

After an injection, the concentration of a drug in a patient's blood is modelled by C(t) = C₀ e^−kt mg/L, where t is the time in hours after injection and k is a positive constant. Immediately after injection C = 80 mg/L. After 3 hours C = 50 mg/L.

(a)

Find the value of k, correct to 3 significant figures.

(3)

(b)

The drug is no longer effective when the concentration falls below 10 mg/L. Use the unrounded value of k to find the time, in hours, for the concentration to fall to 10 mg/L. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 C₀ = 80	✓ 1
M1 50 = 80e^−3k	✓ 1
A1 k = 0.157	✓ 1
Part (b)
M1 10 = 80e^−kt	✓ 1
M1 t = ln 8 / k (or equivalent use of logarithms, or a graph)	✓ 1
A1 13.3 hours	✓ 1

Common error: Store the unrounded value of k on your calculator; using 0.157 can change the third significant figure of later answers.

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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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Calculator

A rectangular reservoir has length 1.2 × 10³ m, width 3.8 × 10² m and a uniform depth of 8.0 m. One cubic metre holds 1000 litres.

(a)

Calculate the volume of the reservoir in litres. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(3)

(b)

A planner estimates the volume by first rounding each of the three dimensions to 1 significant figure. Use the exact volume 3.648 × 10⁶ m³ to find the percentage error in the planner's estimate. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1200 × 380 × 8 = 3648000 m³	✓ 1
M1 multiplies by 1000	✓ 1
A1 3.65 × 10⁹ litres	✓ 1
Part (b)
A1 estimate = 1000 × 400 × 8 = 3.2 × 10⁶ m³	✓ 1
M1 \|3.2 × 10⁶ − 3.648 × 10⁶\| / 3.648 × 10⁶ × 100	✓ 1
A1 12.3 %	✓ 1

Common error: In standard form, check that the first number lies in [1, 10): 3648000 000 is 3.648 × 10⁹, not 36.48 × 10⁸.

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Calculator

In an online game, a player earns 40 coins on day 1. On each later day the player earns 7% more coins than on the previous day. Treat all quantities as exact (do not round the daily amounts).

(a)

Find the number of coins earned on day 14, to the nearest whole coin.

(2)

(b)

Find the smallest number of days after which the total number of coins earned first exceeds 2000.

(3)

(c)

Find the total number of coins earned over the number of days found in part (b), to the nearest whole coin.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 40 × 1.07¹³	✓ 1
A1 96 coins	✓ 1
Part (b)
M1 40(1.07ⁿ − 1)/0.07 > 2000 (or table/graph of the sum)	✓ 1
A1 n > 22.2… (or sums for 22 and 23 days found: 1960 and 2137)	✓ 1
A1 23 days	✓ 1
Part (c)
A1 2137 coins (ECF from (b))	✓ 1

Exam tip: For a sum of a geometric sequence, check your answer by evaluating the total for n and n − 1 days to confirm the first day it passes the target.

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Calculator

A water company invests €5000 in a fund that pays a nominal annual interest rate of 3.6%, compounded quarterly. Interest is added at the end of each quarter.

(a)

Find the smallest number of complete quarters after which the value of the fund is at least €6500.

(4)

(b)

Find the effective annual interest rate of the fund. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 quarterly rate 0.036/4 = 0.9%	✓ 1
M1 5000(1.009)ⁿ ≥ 6500 (or TVM solver for N)	✓ 1
A1 n = 29.3… quarters (accept 29.28)	✓ 1
A1 30 quarters	✓ 1
Part (b)
M1 (1.009)⁴ − 1	✓ 1
A1 3.65 %	✓ 1

Common error: The nominal rate must be divided by the number of compounding periods per year before it is used in the formula.

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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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Calculator

A bacterial culture in a water tank initially contains 2.0 × 10³ cells. The population doubles every 3.5 hours, so the number of cells after t hours is N = 2000 × 2^t/3.5.

(a)

Find the number of cells after 12 hours. Give your answer correct to 3 significant figures.

(2)

(b)

Find the time, in hours, for the population to reach 1.0 × 10⁶ cells. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2000 × 2^12/3.5	✓ 1
A1 2.15 × 10⁴ (21534…)	✓ 1
Part (b)
M1 2^t/3.5 = 500	✓ 1
M1 t = 3.5 log 500 / log 2 (or equivalent use of logarithms, or a graph)	✓ 1
A1 31.4 hours	✓ 1

Exam tip: To solve for an unknown exponent, isolate the power first, then take logarithms of both sides.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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Calculator

A vaccine vial contains 5.5 mL of liquid. The concentration is 3.2 × 10⁹ particles per mL. Each dose uses 0.35 mL.

(a)

Calculate the total number of particles in the vial. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

Find the greatest number of complete doses that can be taken from one vial.

(2)

(c)

A technician estimates the total number of particles by using the concentration rounded to 1 significant figure (3 × 10⁹ particles per mL). Find the percentage error in her estimate of the total.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3.2 × 10⁹ × 5.5	✓ 1
A1 1.76 × 10¹⁰	✓ 1
Part (b)
M1 5.5 ÷ 0.35 = 15.7…	✓ 1
A1 15 doses (a partial dose is not complete)	✓ 1
Part (c)
A1 estimate = 3 × 10⁹ × 5.5 = 1.65 × 10¹⁰	✓ 1
M1 (1.76 × 10¹⁰ − 1.65 × 10¹⁰) / 1.76 × 10¹⁰ × 100	✓ 1
A1 6.25 %	✓ 1

Common error: Rounding 15.7 doses to 16 is wrong: a dose that cannot be filled completely is not counted, so round down.

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A logistics company invests €85 000 in an account that pays a nominal annual interest rate of 3.6 %, compounded quarterly. No further money is added or withdrawn.

(a)

Calculate the value of the investment after 6 years. Give your answer to the nearest cent.

(3)

(b)

Find the least whole number of years after which the value of the investment is at least €170 000.

(3)

(c)

Using your answer to part (a), find the interest earned during the first 6 years.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to use the compound interest formula or a finance solver	✓ 1
A1 correct values, e.g. N = 24 (or 6 × 4), I% = 3.6, PV = 85 000, P/Y = C/Y = 4, or 85 000 × (1 + 0.036/4)²⁴	✓ 1
A1 €105,391.82	✓ 1
Part (b)
M1 85 000 × (1.009)⁴ⁿ = 170 000 (or ≥), or equivalent with a solver	✓ 1
A1 n = 19.34 (4n = 77.4)	✓ 1
A1 20 years (after 19 years the value is €167,937.45 which is below €170 000)	✓ 1
Part (c)
A1 €20,391.82	✓ 1

Common error: With quarterly compounding the rate per quarter is 3.6 ÷ 4 = 0.9 % and there are 4 compounding periods in each year, so N = 4 × the number of years.

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A forklift truck is bought for €48 000. Its value falls by 14 % each year, so its value after n years is 48 000 × 0.86ⁿ euros.

(a)

Find the number of complete years after which the value of the forklift first falls below €15 000.

(3)

(b)

Find the value of the forklift after 5 years. Give your answer to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 48 000 × 0.86ⁿ = 15 000 (or < 15 000)	✓ 1
A1 n = 7.71 (or table of values showing the change of side)	✓ 1
A1 8 years (after 7 years the value is €16,700.54, after 8 years €14,362.46)	✓ 1
Part (b)
M1 48 000 × 0.86⁵	✓ 1
A1 €22,580.50	✓ 1

Common error: A fall of 14 % each year means multiplying by 0.86, not by 0.14 and not by 1.14.

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A freight company pays €1 200 at the end of every month into a savings account. The account pays a nominal annual interest rate of 2.4 %, compounded monthly. The account is empty before the first payment.

(a)

Find the number of months needed for the balance to reach at least €150 000.

(3)

(b)

Find the balance in the account after 8 years. Give your answer to the nearest cent.

(2)

(c)

Find the total interest earned during the 8 years. Give your answer to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 setting up 1 200 × ((1.002ⁿ − 1)/0.002) = 150 000, or using a finance solver with PV = 0, PMT = −1 200, FV = 150 000, I% = 2.4, P/Y = C/Y = 12	✓ 1
A1 n = 111.7	✓ 1
A1 112 months (balance after 111 payments is €148,976.76, after 112 payments €150,474.71)	✓ 1
Part (b)
M1 attempt to use a finance solver or annuity formula with N = 96	✓ 1
A1 €126,862.92	✓ 1
Part (c)
M1 total paid in is 96 × 1 200 = €115 200 and interest = balance − 115 200	✓ 1
A1 €11,663	✓ 1

Common error: Payments at the end of each month earn interest from the following month, so PMT mode must be set to END, not BEGIN, in a finance solver.

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The sound level L, in decibels (dB), of a sound of intensity I W m⁻² is given by L = 10 log₁₀(I/I₀), where I₀ = 1.0 × 10⁻¹² W m⁻².

(a)

A cleaning robot produces a sound of intensity 3.2 × 10⁻⁶ W m⁻² at a certain distance. Calculate the sound level. Give your answer to 3 significant figures.

(2)

(b)

A second robot produces a sound level of 72 dB. Find its sound intensity. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(2)

(c)

Show that doubling the intensity of any sound increases the sound level by 3.01 dB, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(3.2 × 10⁻⁶ / 10⁻¹²) = 10 log₁₀(3.2 × 10⁶)	✓ 1
A1 65.1 dB (65.05… before rounding)	✓ 1
Part (b)
M1 72 = 10 log₁₀(I/10⁻¹²), so I = 10⁻¹² × 10^7.2	✓ 1
A1 1.58 × 10⁻⁵ W m⁻² (1.5849… × 10⁻⁵)	✓ 1
Part (c)
M1 difference of levels = 10 log₁₀(2I/I₀) − 10 log₁₀(I/I₀) = 10 log₁₀ 2 (using the law for logarithms of a quotient)	✓ 1
A1 10 log₁₀ 2 = 3.01 dB (3.0103… before rounding), independent of I	✓ 1

Exam tip: For a 'show that' question, write the exact expression first (here 10 log₁₀ 2) and only then give the rounded value stated in the question.

+ Add to test✓ In your test369IBMI3S-1-02r43-11Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium7 marks

A fishery stores sonar data. The size of its archive is modelled by D = 3.2 × 2^t/1.5 terabytes (TB), where t is the time in years since the archive was created.

(a)

Find the size of the archive after 4.5 years.

(2)

(b)

The storage system can hold 1000 TB. Find the time at which the model predicts the archive reaches this size. Give your answer to 3 significant figures.

(3)

(c)

The model can also be written as D = 3.2e^kt. Find the value of k to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 3.2 × 2^4.5/1.5 = 3.2 × 2³	✓ 1
A1 25.6 TB	✓ 1
Part (b)
M1 3.2 × 2^t/1.5 = 1000	✓ 1
M1 correct method to solve (logarithms, graph or solver)	✓ 1
A1 t = 12.4 years (12.43… before rounding)	✓ 1
Part (c)
M1 2^t/1.5 = e^{(ln 2)t/1.5}, so k = (ln 2)/1.5	✓ 1
A1 k = 0.462 (0.46209… before rounding)	✓ 1

Common error: Write the exponent as a fraction of t (here t/1.5) and keep the brackets when you enter it into the calculator.

+ Add to test✓ In your test370IBMI3S-1-04r44-05Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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A telecoms company invests €85 000 in an account that pays a nominal annual interest rate of 3.6%, compounded quarterly.

(a)

Calculate the value of the investment after 6 years. Give your answer to the nearest euro.

(3)

(b)

The company wants the investment to be worth €110 000 after 6 years. Find the nominal annual interest rate, compounded quarterly, that is needed. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 85 000 × (1 + 3.6/(4 × 100))^{4 × 6}	✓ 1
A1 = 85 000 × 1.009²⁴	✓ 1
A1 €105 392	✓ 1
Part (b)
M1 85 000 × (1 + r/400)²⁴ = 110 000 (or use of a finance solver with N = 24 or N = 6, C/Y = 4)	✓ 1
A1 (1 + r/400) = (110/85)^1/24 = 1.0108…	✓ 1
A1 r = 4.32%	✓ 1

Exam tip: With quarterly compounding the rate per period is the nominal rate divided by 4 and the number of periods is the number of years multiplied by 4.

+ Add to test✓ In your test371IBMI3S-1-04r44-06Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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A mobile network company borrows €18 000 to buy equipment. The loan is repaid with equal payments at the end of each month for 5 years. The nominal annual interest rate is 5.4%, compounded monthly.

(a)

Find the monthly payment. Give your answer to the nearest cent.

(3)

(b)

Find the total amount of interest paid over the 5 years. Use the unrounded payment and give your answer to the nearest euro.

(2)

(c)

Express the total interest as a percentage of the amount borrowed. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a finance solver with N = 60, I% = 5.4, PV = 18 000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 correct values entered (rate per month 0.45%)	✓ 1
A1 €342.99	✓ 1
Part (b)
M1 total repaid = 60 × 342.99… = 20 579.4…	✓ 1
A1 interest = €2579	✓ 1
Part (c)
M1 2579.4…/18 000 × 100	✓ 1
A1 14.3%	✓ 1

Common error: The monthly rate is the nominal annual rate divided by 12, and there are 12 × 5 = 60 payments, not 5.

+ Add to test✓ In your test372IBMI3S-1-04r44-07Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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In food science the acidity of a liquid is measured by pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol dm⁻³. A fruit juice has [H⁺] = 3.9 × 10⁻⁴.

(a)

Calculate the pH of the juice. Give your answer to 3 significant figures.

(2)

(b)

After fermentation the pH of the juice is 3.2. Find the hydrogen ion concentration, giving your answer in the form a × 10^k with a to 3 significant figures.

(2)

(c)

Find how many times greater the hydrogen ion concentration is after fermentation than before. Use the unrounded value from (b) and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 −log₁₀(3.9 × 10⁻⁴)	✓ 1
A1 3.41	✓ 1
Part (b)
M1 [H⁺] = 10^−3.2	✓ 1
A1 6.31 × 10⁻⁴ mol dm⁻³	✓ 1
Part (c)
M1 10^−3.2/(3.9 × 10⁻⁴)	✓ 1
A1 1.62	✓ 1

Common error: A lower pH means a higher concentration of H⁺; each unit of pH changes the concentration by a factor of 10.

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The power of a radio signal in a cable is modelled by P(d) = 40 × 0.88^d milliwatts, where d is the distance in km from the transmitter.

(a)

Find the power of the signal at a distance of 3 km. Give your answer to 3 significant figures.

(2)

(b)

A repeater is needed when the power falls to 5 mW. Find the distance at which this happens. Give your answer to 3 significant figures.

(3)

(c)

Find the distance over which the power of the signal halves. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 40 × 0.88³	✓ 1
A1 27.3 mW	✓ 1
Part (b)
M1 40 × 0.88^d = 5	✓ 1
M1 use of logarithms or a solver: d = ln(1/8)/ln 0.88	✓ 1
A1 16.3 km	✓ 1
Part (c)
M1 0.88^d = 0.5	✓ 1
A1 5.42 km	✓ 1

Exam tip: To solve for an exponent, isolate the exponential term first, then take logarithms of both sides.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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A mountaineer follows a training plan over 12 days. On the first day she climbs 360 m. Each day she climbs the same extra height more than on the previous day, so the daily heights form an arithmetic sequence. The total height she climbs over the 12 days is 6960 m.

(a)

Find the extra height, in metres, that she climbs each day compared with the previous day.

(3)

(b)

Find the height she climbs on day 12.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 S₁₂ = (12/2)(2 × 360 + 11d)	✓ 1
M1 6960 = 6(720 + 11d)	✓ 1
A1 d = 40 m	✓ 1
Part (b)
M1 u₁₂ = 360 + 11 × 40	✓ 1
A1 800 m (ECF from (a))	✓ 1

Exam tip: Both sum formulas work: use Sn = (n/2)(2u₁ + (n − 1)d) when the last term is not yet known.

+ Add to test✓ In your test375IBMI3S-1-02r45-03Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

The concentration of a drug in a patient's blood is modelled by C(t) = 240 × 0.82^t mg per litre, where t is the time in hours after the drug is given.

(a)

Write down the initial concentration.

(1)

(b)

Find the concentration after 3 hours. Give your answer correct to 3 significant figures.

(2)

(c)

Find the time after which the concentration first falls below 50 mg per litre. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 240 mg per litre	✓ 1
Part (b)
M1 240 × 0.82³	✓ 1
A1 132 mg per litre	✓ 1
Part (c)
M1 240 × 0.82^t = 50	✓ 1
M1 use of logarithms: t = log(50/240) ÷ log(0.82)	✓ 1
A1 t = 7.90 hours	✓ 1

Common error: When solving an exponential equation with logarithms, divide by log of the base: the exponent does not come out by dividing 50 by 240 alone.

+ Add to test✓ In your test376IBMI3S-1-02r45-04Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A Formula 1 circuit has a lap length of 5412 m. A driver completes a lap in 91.8 seconds.

(a)

Calculate the total distance for 52 laps. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(2)

(b)

Calculate the driver's mean speed for the lap in m s⁻¹, correct to 3 significant figures.

(2)

(c)

A commentator says that the mean speed was 60 m s⁻¹. Taking 5412/91.8 as the exact speed, find the percentage error in the commentator's value. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 5412 × 52 = 281 424	✓ 1
A1 2.81 × 10⁵ m	✓ 1
Part (b)
M1 5412 ÷ 91.8 = 58.954…	✓ 1
A1 59.0 m s⁻¹	✓ 1
Part (c)
M1 \|60 − 58.954…\| ÷ 58.954… × 100	✓ 1
A1 1.77%	✓ 1

Exam tip: Percentage error is measured against the exact value; keep the unrounded speed in your calculator for part (c).

+ Add to test✓ In your test377IBMI3S-1-02r45-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

A hospital charity invests €25 000 in an account that pays a nominal annual interest rate of 3.2%, compounded quarterly. No money is added or withdrawn.

(a)

Find the value of the investment after 6 years, to the nearest cent.

(3)

(b)

The charity needs the investment to reach €38 000. Find the number of years this takes. Give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 identify n = 24 quarters and quarterly rate 0.8%	✓ 1
M1 25 000 × (1 + 0.032/4)²⁴ (or TVM solver: N = 24, I% = 3.2, PV = −25 000, P/Y = C/Y = 4)	✓ 1
A1 €30 268.63	✓ 1
Part (b)
M1 25 000 × 1.008^4t = 38 000	✓ 1
M1 solve using logarithms or TVM solver (4t = 52.5…)	✓ 1
A1 13.1 years	✓ 1

Common error: In compound interest the number of periods is the number of years multiplied by the compounding frequency; divide the nominal rate by the frequency too.

+ Add to test✓ In your test378IBMI3S-1-02r45-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium7 marks

A grandstand at a motor-racing circuit has rows of seats. The first row has 28 seats and each row has 3 more seats than the row in front of it.

(a)

Find the number of seats in row 15.

(2)

(b)

Find the total number of seats in the first 15 rows.

(2)

(c)

The grandstand is to hold exactly 1500 seats. Find the number of rows needed.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 28 + 14 × 3	✓ 1
A1 70	✓ 1
Part (b)
M1 (15/2)(28 + 70)	✓ 1
A1 735	✓ 1
Part (c)
M1 (n/2)(2 × 28 + (n − 1) × 3) = 1500, so 3n² + 53n − 3000 = 0	✓ 1
M1 solve the quadratic (GDC or formula) giving n = 24 or n = −41.7	✓ 1
A1 24 rows (negative solution rejected)	✓ 1

Exam tip: When solving for the number of terms, reject any negative or non-integer solution and say why.

+ Add to test✓ In your test379IBMI3S-1-02r45-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Medium6 marks

Sound level L in decibels (dB) is related to sound intensity I in W m⁻² by L = 10 log₁₀(I/10⁻¹²).

(a)

A mechanic is advised to wear ear protection at 85 dB. Find the intensity at this level, giving your answer in the form a × 10^k with a correct to 3 significant figures.

(2)

(b)

The intensity in a quiet hospital ward at night is 2.5 × 10⁻⁹ W m⁻². Find the sound level, correct to 3 significant figures.

(2)

(c)

A Formula 1 car produces a sound level of 130 dB close to the track. Find how many times greater its intensity is than the intensity at 85 dB. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 85 = 10 log₁₀(I/10⁻¹²), so I = 10^8.5 × 10⁻¹²	✓ 1
A1 3.16 × 10⁻⁴ W m⁻²	✓ 1
Part (b)
M1 10 log₁₀(2.5 × 10⁻⁹/10⁻¹²)	✓ 1
A1 34.0 dB	✓ 1
Part (c)
M1 ratio = 10^{(130 − 85)/10}	✓ 1
A1 3.16 × 10⁴	✓ 1

Common error: Each increase of 10 dB multiplies the intensity by 10; it is not an increase of 10 in the intensity itself.

+ Add to test✓ In your test380IBMI3S-1-01r47-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

The number of bees in a hive, P, is modelled by P = P₀e^kt, where t is the time in weeks since the start of the season and P₀ and k are constants. At t = 3 there are 24 600 bees and at t = 8 there are 36 400 bees.
In all parts, use unrounded values from earlier parts in later calculations.

(a)

Show that k = ln(36 400 ÷ 24 600) ÷ 5, and find the value of k correct to 3 significant figures.

(3)

(b)

Find P₀, correct to the nearest hundred.

(2)

(c)

According to the model, find the time at which the number of bees reaches 50 000. Give your answer in weeks, correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms the equation 36 400 = 24 600e^5k (dividing the two equations)	✓ 1
A1 e^5k = 36 400/24 600 and takes logarithms to give k = ln(36 400 ÷ 24 600) ÷ 5	✓ 1
A1 k = 0.0784	✓ 1
Part (b)
M1 substitutes into 24 600 = P₀e^3k	✓ 1
A1 19 400	✓ 1
Part (c)
M1 sets up 50 000 = P₀e^kt with their values	✓ 1
M1 takes natural logarithms (or uses solver)	✓ 1
A1 12.1 weeks	✓ 1

Common error: Substituting a rounded value of k too early can change the third significant figure of later answers; store values in the calculator memory.

+ Add to test✓ In your test381IBMI3S-1-01r47-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

A patient has 4.8 × 10⁶ red blood cells in each microlitre of blood, and a total blood volume of 5.1 litres. There are 10⁶ microlitres in one litre.

(a)

Calculate the total number of red blood cells in the patient's blood. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(3)

(b)

A nurse estimates the total as 5 × 10⁶ × 5 × 10⁶. Taking 2.448 × 10¹³ as the exact value, find the percentage error in the nurse's estimate. Give your answer to 3 significant figures.

(2)

(c)

Each red blood cell has diameter 7.5 × 10⁻⁶ m. If all the cells were placed in a straight line, touching end to end, find the length of the line in kilometres. Give your answer in the form a × 10^k with a correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 4.8 × 10⁶ × 5.1 × 10⁶	✓ 1
A1 2.448 × 10¹³ seen or implied	✓ 1
A1 2.45 × 10¹³	✓ 1
Part (b)
M1 use of \|estimate − exact\| ÷ exact × 100	✓ 1
A1 2.12 %	✓ 1
Part (c)
M1 2.448 × 10¹³ × 7.5 × 10⁻⁶ = 1.836 × 10⁸ m	✓ 1
A1 converts metres to kilometres (÷ 1000)	✓ 1
A1 1.84 × 10⁵ km	✓ 1

Common error: Convert units before multiplying powers of ten, and check that the coefficient a in standard form lies between 1 and 10.

+ Add to test✓ In your test382IBMI3S-1-01r47-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

Pit-crew trainees at a racing team practise tyre changes. The numbers of practice stops completed in successive sessions form an arithmetic sequence. In session 4 the trainees complete 31 stops and in session 9 they complete 61 stops.

(a)

Find the number of stops in session 1 and the common difference.

(3)

(b)

Find the smallest number of sessions after which the total number of stops completed exceeds 1000.

(3)

(c)

Find the total number of stops completed in these sessions.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms u₁ + 3d = 31 and u₁ + 8d = 61	✓ 1
A1 d = 6	✓ 1
A1 u₁ = 13	✓ 1
Part (b)
M1 uses S_n = (n/2)(26 + 6(n − 1)) > 1000, that is n(3n + 10) > 1000	✓ 1
A1 S₁₆ = 928 and S₁₇ = 1037 (or solves n > 16.x by solver)	✓ 1
A1 17 sessions	✓ 1
Part (c)
M1 S₁₇ = (17/2)(2 × 13 + 16 × 6)	✓ 1
A1 1037 stops	✓ 1

Common error: With two terms given, write two equations in u₁ and d and solve them; do not assume the first term equals the common difference.

+ Add to test✓ In your test383IBMI3S-1-01r47-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium8 marks

A Formula 1 team puts €30 000 into an investment account with a nominal annual interest rate of r%, compounded quarterly. After 12 years the value of the investment is €60 000.

(a)

Find the value of r, correct to 3 significant figures.

(3)

(b)

Find the effective annual interest rate, as a percentage correct to 3 significant figures. Use the unrounded value of r.

(2)

(c)

The team leaves the money invested at the same rate. Find the total number of years, from the start, for the investment to reach €90 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 sets up 30 000(1 + r/400)⁴⁸ = 60 000	✓ 1
M1 solves, for example 1 + r/400 = 2^1/48	✓ 1
A1 r = 5.82	✓ 1
Part (b)
M1 (1 + r/400)⁴ − 1 with unrounded r	✓ 1
A1 5.95 %	✓ 1
Part (c)
M1 sets up 30 000(1.01454…)^4t = 90 000 (unrounded quarterly rate)	✓ 1
M1 solves using logarithms or solver	✓ 1
A1 19.0 years	✓ 1

Common error: The quarterly rate is r/4 per cent, so there are 4 compounding periods per year; do not use r as a yearly multiplier when interest is compounded quarterly.

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arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

Calculator

A cycling club invests €2500 in an account that pays a nominal annual interest rate of 3.6%, compounded monthly, towards the cost of a new team trailer.

(a)

Find the value of the investment after 5 years. Give your answer to the nearest cent.

(3)

(b)

The club instead finds an account that compounds quarterly. Find the nominal annual interest rate this account needs to pay for €2500 to grow to €3400 in exactly 8 years. Give your answer as a percentage to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 monthly rate $\frac{3.6}{12}=0.3\%$ and 60 compounding periods	✓ 1
M1 $2500\left(1+\frac{0.036}{12}\right)^{60}$ or TVM solver with N=60, I%=3.6, PV=−2500, P/Y=C/Y=12	✓ 1
A1 €2,992.24	✓ 1
Part (b)
M1 $2500\left(1+\frac{r}{400}\right)^{32}=3400$	✓ 1
M1 correct method to solve, e.g. $r=400\left(\left(\frac{3400}{2500}\right)^{1/32}-1\right)$	✓ 1
A1 3.86%	✓ 1

Common error: For quarterly compounding over 8 years use $N=32$ and a rate of $r/4$ per period; do not use the annual rate with 32 periods.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

Calculator

A cyclist increases her weekly training distance by the same amount each week. In week 3 she rides 110 km and in week 10 she rides 187 km.

(a)

Find the common difference and the distance ridden in week 1.

(3)

(b)

Find the total distance she rides in the first 12 weeks.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_1+2d=110$ and $u_1+9d=187$ (or $7d=187-110$)	✓ 1
A1 $d=11$ km	✓ 1
A1 $u_1=88$ km	✓ 1
Part (b)
M1 use of $S_n=\frac{n}{2}(2u_1+(n-1)d)$	✓ 1
M1 $\frac{12}{2}(2\times88+11\times11)$	✓ 1
A1 1782 km	✓ 1

Exam tip: Two terms of an arithmetic sequence give two simultaneous equations in $u_1$ and $d$; subtracting them removes $u_1$.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

Calculator

A data centre processes requests in batches. The numbers of thousands of requests processed in successive hours form a geometric sequence with $u_2=36$ and $u_5=121.5$. All terms are positive.

(a)

Find the common ratio.

(2)

(b)

Find $u_1$.

(2)

(c)

Find the total number of thousands of requests processed in the first 10 hours. Give your answer to the nearest whole number.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\frac{u_5}{u_2}=r^3$, i.e. $r^3=\frac{121.5}{36}$	✓ 1
A1 $r=1.5$	✓ 1
Part (b)
M1 $u_1=\frac{36}{1.5}$ or $u_1\times1.5=36$	✓ 1
A1 $u_1=24$	✓ 1
Part (c)
M1 $S_{10}=\frac{24(1.5^{10}-1)}{1.5-1}$	✓ 1
A1 2720	✓ 1

Exam tip: Divide two terms of a geometric sequence to isolate a power of $r$: $u_5\div u_2=r^3$.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium5 marks

Calculator

The strength of a signal on a data centre link is measured on a logarithmic scale. A signal of power $P$ watts has level $D=10\log_{10}\left(\frac{P}{P_0}\right)$ decibels, where $P_0$ is a fixed reference power.

(a)

A signal has power $P=25\,000\,P_0$. Find its level $D$, correct to 3 significant figures.

(2)

(b)

The level of a second signal is 6 decibels higher than the level of the first signal. Find the value of $\frac{P_2}{P_1}$, the ratio of the powers of the two signals. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $10\log_{10}(25\,000)$	✓ 1
A1 44.0	✓ 1
Part (b)
M1 $D_2-D_1=10\log_{10}\left(\frac{P_2}{P_1}\right)=6$	✓ 1
M1 $\log_{10}\left(\frac{P_2}{P_1}\right)=0.6$	✓ 1
A1 $10^{0.6}=3.98$	✓ 1

Common error: A change in level on a logarithmic scale corresponds to a ratio of powers, not a difference: $\log a-\log b=\log\frac{a}{b}$.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

Calculator

An archaeologist measures a rectangular stone block as 2.35 m long, 1.20 m wide and 0.85 m high.

(a)

Calculate the volume of the block in m³.

(1)

(b)

The stone has a density of 2650 kg m⁻³. Calculate the mass of the block, giving your answer in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$, with $a$ correct to 3 significant figures.

(2)

(c)

A colleague estimates the dimensions as 2.4 m, 1.2 m and 0.9 m and multiplies them to estimate the volume. Calculate the percentage error in her estimate of the volume, correct to 3 significant figures. Use the exact volume from part (a).

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 2.397 m³	✓ 1
Part (b)
M1 $2650\times2.397$ (6,352.0 kg)	✓ 1
A1 $6.35\times10^3$ kg	✓ 1
Part (c)
M1 colleague's volume $2.4\times1.2\times0.9=2.592$	✓ 1
M1 $\frac{2.592-2.397}{2.397}\times100$	✓ 1
A1 8.14%	✓ 1

Exam tip: Show the estimated and exact values before substituting into the percentage error formula so that method marks are secure.

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arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Medium6 marks

Calculator

A velodrome has a seating block with 18 rows. The first row has 24 seats, and each row has 3 more seats than the row in front of it.

(a)

Find the number of seats in the last row.

(2)

(b)

Find the total number of seats in the block.

(2)

(c)

Each seat costs €85 to install. Find the total installation cost in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$, with $a$ correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $24+17\times3$	✓ 1
A1 75 seats	✓ 1
Part (b)
M1 $\frac{18}{2}(24+75)$	✓ 1
A1 891 seats	✓ 1
Part (c)
M1 $891\times85$ (€75,735)	✓ 1
A1 $7.57\times10^4$ €	✓ 1

Exam tip: In standard form the number before the power of ten must lie in $1\le a<10$; 75 735 becomes $7.57\times10^4$, not $75.7\times10^3$.

+ Add to test✓ In your test390IBMI-030-05-10Approximation, percentage error and standard form Paper 1Hard6 marks

A courier estimates the mass of a parcel as 4.2 kg. The percentage error in this estimate is 12.5%.

(a)

Find the two possible values of the actual mass of the parcel.

(3)

(b)

The actual mass, rounded to the nearest kilogram, is 5 kg. State the actual mass of the parcel, giving a reason.

(2)

(c)

The courier charges $3.15 per kg of actual mass. Calculate the charge for this parcel.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for an equation such as $\frac{\|4.2-A\|}{A}=0.125$	✓ 1
A1 A = 4.8 kg	✓ 1
A1 A = 3.73 kg (3.7333…)	✓ 1
Part (b)
A1 4.8 kg	✓ 1
R1 because 3.73 rounds to 4 kg, not 5 kg, while 4.8 rounds to 5 kg	✓ 1
Part (c)
A1 $15.12	✓ 1

Exam tip: The percentage error is measured relative to the actual value, which is unknown. Do not just work out 4.2 × 1.125 and 4.2 × 0.875. Instead, solve 4.2 = A(1 ± 0.125).

+ Add to test✓ In your test391IBMI3S-1-01-14Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Hard7 marks

A patient is given 200 mg of a drug every 8 hours. Between doses, 60% of the drug in the body is eliminated, so 40% remains. Let $u_n$ be the amount of drug in the body, in mg, immediately after the $n$th dose, so that $u_1=200$.

(a)

Find $u_3$.

(2)

(b)

Show that $u_n=\frac{1000}{3}\left(1-0.4^{n}\right)$.

(2)

(c)

The treatment is safe to continue only while $u_n$ stays below 330 mg. Find the least value of $n$ for which $u_n>330$.

(2)

(d)

Find the amount of drug in the body immediately before the 7th dose. Give your answer to 3 significant figures.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_2=200+0.4(200)=280$ or $200+200(0.4)+200(0.4)^2$	✓ 1
A1 312 mg	✓ 1
Part (b)
M1 recognising $u_n=200\left(1+0.4+\dots+0.4^{n-1}\right)$ as a geometric sum with $r=0.4$	✓ 1
A1 $u_n=\frac{200(1-0.4^{n})}{1-0.4}=\frac{1000}{3}(1-0.4^{n})$ (AG)	✓ 1
Part (c)
M1 solving $\frac{1000}{3}(1-0.4^{n})>330$ (technology or logarithms)	✓ 1
A1 $n=6$	✓ 1
Part (d)
A1 $0.4\times u_6=0.4\times331.97\ldots=133$ mg (ECF from (c))	✓ 1

Common error: The amount just before a dose is 40% of the amount just after the previous dose; do not add the next 200 mg until the dose is given.

+ Add to test✓ In your test392IBMI3S-1-02r12-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 1Hard7 marks

A space probe's power supply produces a power P watts that halves every 87.7 years. At launch the power is 470 W, so P = 470 × (1/2)^t/87.7, where t is the time in years after launch.

(a)

Show that the model can be written as P = 470e^−kt and find the value of k to 3 significant figures.

(2)

(b)

The probe's instruments need at least 120 W. Find the time after launch at which the power first falls to 120 W. Give your answer to 3 significant figures.

(2)

(c)

A second probe has power P = 520 × (1/2)^t/64. Find the time after launch at which the two probes have equal power. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (1/2)^t/87.7 = e^{−(ln 2)t/87.7}, so k = ln 2 ÷ 87.7	✓ 1
A1 k = 0.00790	✓ 1
Part (b)
M1 470e^−0.00790t = 120, so t = ln(470/120) ÷ 0.00790 (accept use of unrounded k)	✓ 1
A1 t = 173 years	✓ 1
Part (c)
M1 470 × (1/2)^t/87.7 = 520 × (1/2)^t/64	✓ 1
M1 takes logarithms: ln(520/470) = t ln 2 (1/64 − 1/87.7), or equivalent, or graph intersection	✓ 1
A1 t = 34.5 years	✓ 1

Exam tip: When two exponential models are set equal, take logarithms of both sides and collect the terms in t before dividing.

+ Add to test✓ In your test393IBMI3S-1-02r14-09Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Hard7 marks

A space agency wants a mission fund to be worth £500000 exactly 10 years from now. It will pay the same amount into the fund at the start of every quarter (40 payments in all). The fund earns interest at a nominal annual rate of 3.2%, compounded quarterly.

(a)

Write down the quarterly interest rate as a percentage.

(1)

(b)

Find the quarterly payment required, to the nearest pound.

(3)

(c)

Using your answer to part (b), and assuming the fund reaches exactly £500000, find the total interest earned over the 10 years, to the nearest £1000.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.8%	✓ 1
Part (b)
M1 TVM with N = 40, I% = 3.2, FV = 500000, P/Y = C/Y = 4, payments at the beginning of each period (or the formula for the sum of a geometric series with first term $P(1.008)$ and ratio 1.008)	✓ 1
M1 correct equation, e.g. $P\times1.008\times\frac{1.008^{40}-1}{0.008}=500000$	✓ 1
A1 £10571	✓ 1
Part (c)
M1 total paid in = $40\times$ quarterly payment	✓ 1
M1 interest = 500000 − total paid in	✓ 1
A1 £77000	✓ 1

Common error: Payments at the start of each period earn interest for one extra period. On the GDC set the payment mode to BEGIN, not END.

+ Add to test✓ In your test394IBMI3S-1-01r29-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Hard6 marks

The numbers of followers of two fashion accounts, A and B, t years after 1 January 2024 are modelled by A(t) = 2000 × 1.06^t and B(t) = 500 × 1.14^t.

(a)

Find the value of t for which the two accounts have the same number of followers. Give your answer to 3 significant figures.

(4)

(b)

Find the number of followers each account has at this time. Use the unrounded value of t and give your answer to the nearest 10 followers.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2000 × 1.06^t = 500 × 1.14^t	✓ 1
M1 rearrange to (1.14 ÷ 1.06)^t = 4	✓ 1
M1 t = ln 4 ÷ ln(1.14 ÷ 1.06)	✓ 1
A1 t = 19.1	✓ 1
Part (b)
M1 substitute into A(t) or B(t)	✓ 1
A1 6070 followers	✓ 1

Exam tip: Collect the two exponential terms on one side first; (1.14 ÷ 1.06)t is far easier to solve than taking logarithms of each side separately.

+ Add to test✓ In your test395IBMI3S-1-02r29-14Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 1Hard6 marks

The popularity scores of two clothing designs, P and Q, t months after their launch are modelled by P = 800 × 1.09^t and Q = 1500 × 0.95^t.

(a)

Find the value of t at which the two designs have equal popularity scores. Give your answer to 3 significant figures.

(4)

(b)

Find the popularity score at this time. Use the unrounded value of t and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 800 × 1.09^t = 1500 × 0.95^t	✓ 1
M1 (1.09 ÷ 0.95)^t = 1.875	✓ 1
M1 t = ln 1.875 ÷ ln(1.09 ÷ 0.95)	✓ 1
A1 t = 4.57 months	✓ 1
Part (b)
M1 substitute into P or Q	✓ 1
A1 1190	✓ 1

Exam tip: Both models must give the same score at the intersection, so substituting into either one is a free check on your answer.

+ Add to test✓ In your test396IBMI3S-1-03r32-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 1Hard8 marks

The daily network traffic, in terabytes (TB), passing through two data centres t days after the start of a monitoring period is modelled by
Centre A: $T_A = 120\times1.35^{t}$
Centre B: $T_B = 450\times1.12^{t}$

(a)

Find the value of t for which the two centres carry the same daily traffic. Give your answer to 3 significant figures.

(3)

(b)

Find the daily traffic, in TB, at this time. Use the unrounded value of t and give your answer to 3 significant figures.

(2)

(c)

A technician plots log₁₀ T_A against t and finds that the points lie on a straight line. Find the gradient of this line. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $120\times1.35^{t} = 450\times1.12^{t}$ (or GDC intersection of the two graphs)	✓ 1
M1 rearrange to $\left(\dfrac{1.35}{1.12}\right)^{t} = 3.75$ and take logarithms	✓ 1
A1 $t = 7.08$	✓ 1
Part (b)
M1 substitute $t$ into either model	✓ 1
A1 1000 TB to 3 s.f. (unrounded 1003.5; both models give the same value)	✓ 1
Part (c)
M1 $\log_{10}T_A = \log_{10}120 + t\log_{10}1.35$	✓ 1
M1 identify gradient as $\log_{10}1.35$	✓ 1
A1 0.130	✓ 1

Exam tip: Taking logarithms of an exponential model $y = ab^{t}$ gives $\log y = \log a + t\log b$, a straight line whose gradient is $\log b$.

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Calculator

The concentration, C mg/L, of a pollutant in a water supply t years after a clean-up begins is modelled by C = A × b^t + 15, where A and b are positive constants and 0 < b < 1. At the start of the clean-up, C = 135. After 3 years, C = 63.

(a)

Find the value of A.

(1)

(b)

Find the value of b, correct to 3 significant figures.

(3)

(c)

Water is safe to use when the concentration first falls to 25 mg/L. Find the time after the start of the clean-up at which this happens. Give your answer in years, correct to 3 significant figures (using either the unrounded or rounded value of b gives the same answer).

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 135 = A + 15, so A = 120	✓ 1
Part (b)
M1 63 = 120b³ + 15	✓ 1
M1 b³ = 0.4, so b = 0.4^1/3	✓ 1
A1 b = 0.737	✓ 1
Part (c)
M1 25 = 120b^t + 15, so b^t = 1/12	✓ 1
M1 t = ln(1/12) / ln b (or equivalent use of logarithms, or a graph)	✓ 1
A1 8.14 years	✓ 1

Common error: Subtract the constant 15 before dividing by A and taking logarithms; taking logs of the whole expression gives a wrong equation.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Hard7 marks

A public aquarium has a cylindrical tank. A trainee measures the radius of the tank as 3.2 m and the depth of the water as 1.5 m. The accurate values are radius 3.4 m and depth 1.4 m. The volume of a cylinder is V = πr²h.

(a)

Calculate the volume of water in the tank using the trainee’s measurements. Give your answer to 3 significant figures.

(2)

(b)

Calculate the volume of water using the accurate values. Give your answer to 3 significant figures.

(1)

(c)

Using the unrounded volumes, find the percentage error in the volume calculated from the trainee’s measurements. Give your answer to 3 significant figures.

(2)

(d)

Seawater has a density of 1 025 kg m⁻³. The water in the tank is only 90 % of the volume found in part (b). Find the mass of water in the tank. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k ∈ ℤ, with a given to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 π × 3.2² × 1.5	✓ 1
A1 48.3 m³	✓ 1
Part (b)
A1 50.8 m³	✓ 1
Part (c)
M1 \|48.25 − 50.84\| / 50.84 × 100 (accurate volume in the denominator)	✓ 1
A1 5.09 %	✓ 1
Part (d)
M1 0.9 × V_accurate × 1 025 (using the accurate volume)	✓ 1
A1 4.69 × 10⁴ kg	✓ 1

Common error: Percentage error is measured against the accurate value, so divide by the accurate volume, not by the volume from the measurements.

+ Add to test✓ In your test399IBMI3S-1-01r42-13Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Hard7 marks

A marine research trust invests €20 000 in an account with a nominal annual interest rate of r %, compounded monthly. After 7 years the investment is worth €26 000. No money is added or withdrawn.

(a)

Find the value of r. Give your answer to 3 significant figures.

(3)

(b)

Find the effective annual interest rate of this account, as a percentage to 3 significant figures. Use the unrounded value of r.

(2)

(c)

A second account pays a nominal annual rate of 3.9 %, compounded annually. By comparing effective annual rates, state which account gives the greater value after 7 years and justify your answer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to use a finance solver or 20 000(1 + r/1 200)⁸⁴ = 26 000	✓ 1
A1 correct values, e.g. N = 84, PV = −20 000, FV = 26 000, PMT = 0, P/Y = C/Y = 12	✓ 1
A1 r = 3.75	✓ 1
Part (b)
M1 (26 000/20 000)^1/7 − 1, or (1 + r/1 200)¹² − 1	✓ 1
A1 3.82 % (accept 3.82 % if the rounded value of r is used)	✓ 1
Part (c)
M1 effective annual rate of the second account is 3.9 %, compared with 3.82 %	✓ 1
A1 the second account gives the greater value, because its effective annual rate is higher	✓ 1

Common error: A nominal rate compounded more often than once a year gives an effective rate larger than the nominal rate, so compare effective rates, not nominal rates.

+ Add to test✓ In your test400IBMI3S-1-02r45-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 1Hard7 marks

A bee charity invests €20 000 for 10 years in an account paying a nominal annual interest rate of 4.2%, compounded monthly. Over the same period, prices rise by 2.3% per year.

(a)

Find the value of the investment after 10 years, to the nearest cent.

(2)

(b)

Find the factor by which prices rise over the 10 years, correct to 4 decimal places.

(1)

(c)

Find the value of the investment after 10 years in terms of today's prices, to the nearest euro. Use the unrounded values.

(2)

(d)

Find the equivalent real annual growth rate of the investment, as a percentage correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 20 000 × (1 + 0.042/12)¹²⁰	✓ 1
A1 €30 416.92	✓ 1
Part (b)
A1 1.023¹⁰ = 1.2553	✓ 1
Part (c)
M1 30 416.92… ÷ 1.023¹⁰	✓ 1
A1 €24 230	✓ 1
Part (d)
M1 (1 + 0.042/12)¹² ÷ 1.023 (annual growth factor 1.0194…)	✓ 1
A1 1.94%	✓ 1

Exam tip: Compare money at different dates only after adjusting for price changes, and divide by the price factor (do not subtract percentages).

+ Add to test✓ In your test401IBMI-030-00-07Approximation, percentage error and standard form Paper 2Medium11 marks

In a simple model, the Earth and Mars orbit the Sun in circles in the same plane. The orbital radius of the Earth is 1.50×10⁸ km and that of Mars is 2.28×10⁸ km. Radio signals travel at 3.00×10⁵ km s⁻¹.
Mars may be modelled as a sphere of diameter 6.79×10³ km.

(a)

Find the minimum distance between the Earth and Mars according to this model. Give your answer in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(2)

(b)

Calculate the time, in minutes, for a radio signal to travel from the Earth to Mars at this minimum distance.

(2)

(c)

Calculate the time, in minutes, for a radio signal to travel between the planets when they are on opposite sides of the Sun.

(2)

(d)

Calculate the volume of Mars in km³, giving your answer in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(3)

(e)

The volume of the Earth is 1.08×10¹² km³. Find how many times as large the volume of the Earth is as the volume of Mars.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for 2.28×10⁸ − 1.50×10⁸	✓ 1
A1 7.80×10⁷ km	✓ 1
Part (b)
M1 for $\frac{7.80\times10^{7}}{3.00\times10^{5}}=260$ s	✓ 1
A1 $\frac{260}{60}=4.33$ minutes	✓ 1
Part (c)
M1 for a distance of 1.50×10⁸ + 2.28×10⁸ = 3.78×10⁸ km and $\frac{3.78\times10^{8}}{3.00\times10^{5}}=1260$ s	✓ 1
A1 $\frac{1260}{60}=21.0$ minutes	✓ 1
Part (d)
M1 for radius = 3.395×10³ km, seen or implied (for example by the use of $\frac{\pi d^3}{6}$)	✓ 1
M1 for $\frac{4}{3}\pi(3.395\times10^{3})^3$ or $\frac{\pi(6.79\times10^{3})^3}{6}$	✓ 1
A1 1.64×10¹¹ km³	✓ 1
Part (e)
M1 for $\frac{1.08\times10^{12}}{1.639...\times10^{11}}$	✓ 1
A1 6.59 (accept 6.58 to 6.59)	✓ 1

Common error: The volume formula needs the radius, not the diameter. Halve 6.79×10³ before cubing, or use $\frac{\pi d^3}{6}$.

+ Add to test✓ In your test402IBMI-030-05-12Approximation, percentage error and standard form Paper 2Medium10 marks

In one year, global carbon dioxide emissions were 3.71×10¹⁰ tonnes. Country A emitted 5.06×10⁹ tonnes and had a population of 3.34×10⁸. The mass of Earth's atmosphere is 5.15×10¹⁸ kg (1 tonne = 1000 kg).

(a)

Calculate the percentage of global emissions produced by Country A.

(2)

(b)

Calculate the emissions per person in Country A, in tonnes.

(2)

(c)

Write the global emissions in kg. Hence find the ratio of the mass of the atmosphere to the mass of one year's global emissions. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$.

(3)

(d)

A journalist rounds global emissions to 4×10¹⁰ tonnes and Country A's emissions to 5×10⁹ tonnes. The journalist then reports Country A's share as 12.5%. Calculate the percentage error in the journalist's figure, compared with your unrounded answer to part (a).

(2)

(e)

The journalist changed Country A's figure by about 1.2% and the global figure by about 7.8%. Explain why the percentage error in the reported share is larger than either of these changes.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for $\frac{5.06\times10^9}{3.71\times10^{10}} \times 100$	✓ 1
A1 13.6% (13.638…)	✓ 1
Part (b)
M1 for 5.06×10⁹ ÷ 3.34×10⁸	✓ 1
A1 15.1 tonnes per person (15.149…)	✓ 1
Part (c)
A1 3.71×10¹³ kg	✓ 1
M1 for 5.15×10¹⁸ ÷ 3.71×10¹³	✓ 1
A1 1.39×10⁵	✓ 1
Part (d)
M1 for $\frac{\|12.5-13.638\ldots\|}{13.638\ldots} \times 100\%$	✓ 1
A1 8.35%	✓ 1
Part (e)
R1 the numerator was rounded down and the denominator was rounded up, so both roundings make the quotient smaller and their effects combine rather than cancel	✓ 1

Exam tip: In part (d), use the unrounded value 13.638… from your calculator as the exact value. Using 13.6 gives 8.09%, which is not accurate enough.

+ Add to test✓ In your test403IBMI-030-08-r2-03Exponents and logarithms Paper 2Medium10 marks

An invasive weed is spreading across the surface of a pond. The area covered, A m², t days after it was first measured is modelled by
$A(t) = 2.4 \times b^{t}$
where $b$ is a constant. Three days after the first measurement, the weed covers 5.9 m².

(a)

Find the value of b.

(2)

(b)

Hence write down the percentage increase in the area covered each day.

(1)

(c)

Find the time it takes for the area covered to double.

(2)

(d)

The surface area of the pond is 500 m². Use the model to find the time at which the weed covers the whole surface.

(3)

(e)

Explain why this model is unlikely to be accurate close to the time found in part (d).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: forms the equation $2.4b^{3} = 5.9$	✓ 1
A1: $b = 1.35$ (1.34963…)	✓ 1
Part (b)
A1: 35.0% (accept 35%)	✓ 1
Part (c)
M1: forms the equation $2.4b^{t} = 4.8$ or $b^{t} = 2$	✓ 1
A1: $t = 2.31$ days	✓ 1
Part (d)
M1: forms the equation $2.4b^{t} = 500$	✓ 1
M1: uses logarithms, $t = \frac{\ln(500/2.4)}{\ln b}$, or solves with GDC	✓ 1
A1: $t = 17.8$ days	✓ 1
Part (e)
R1: the pond surface and resources such as light and nutrients are limited, so exponential growth cannot continue	✓ 1
R1: the rate of spread will slow down as the area covered approaches 500 m², so the model overestimates the area or underestimates the time	✓ 1

Exam tip: Keep the unrounded value of b in your GDC for parts (c) and (d). Using $b = 1.35$ in part (d) gives a time of about 17.8 days here, but early rounding often changes the third significant figure.

+ Add to test✓ In your test404IBMI-030-03-r2-04Exponents and logarithms Paper 2Medium11 marks

A cup of coffee contains 180 mg of caffeine. The amount of caffeine, $C$ mg, remaining in an adult's body $t$ hours after drinking the coffee is modelled by
$C(t) = 180\,e^{-kt}$, $t\ge0$,
where $k$ is a positive constant. The half-life of caffeine in this adult's body is 5.5 hours.

(a)

Show that $k = 0.126$, correct to three significant figures.

(2)

(b)

Find the amount of caffeine remaining 8 hours after the coffee is drunk.

(2)

(c)

Find the time taken for the amount of caffeine to fall to 20 mg.

(3)

(d)

The adult drinks one cup of this coffee at 07:00 and a second, identical cup at 12:00. Assuming the model applies to each cup independently, find the total amount of caffeine in the adult's body at 15:00.

(3)

(e)

State one limitation of the model in this context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: sets up the equation $90 = 180e^{-5.5k}$, or $e^{-5.5k} = 0.5$	✓ 1
A1: $k = \dfrac{\ln 2}{5.5} = 0.126027\ldots$, which gives 0.126	✓ 1
Part (b)
M1: substitutes $t=8$, $180e^{-0.126027\times8}$	✓ 1
A1: 65.7 mg	✓ 1
Part (c)
M1: sets up the equation $180e^{-kt} = 20$	✓ 1
M1: rearranges with logarithms, $t = \dfrac{\ln 9}{k}$, or uses a GDC	✓ 1
A1: $t = 17.4$ hours	✓ 1
Part (d)
M1: uses $t=8$ for the first cup and $t=3$ for the second cup	✓ 1
M1: adds the two amounts, $180e^{-8k} + 180e^{-3k}$ (65.68 + 123.33)	✓ 1
A1: 189 mg	✓ 1
Part (e)
R1: any valid limitation, e.g. it assumes all the caffeine is absorbed instantly when the coffee is drunk; the rate of removal varies between people and with factors such as food or medication; it predicts that caffeine is never completely removed	✓ 1

Exam tip: In a 'show that' question you must show the unrounded value, such as 0.126027…, before rounding it. In later parts, use the unrounded value of k so that rounding errors do not build up.

+ Add to test✓ In your test405IBMI-030-07-r1-05Arithmetic and geometric sequences in context Paper 2Medium13 marks

Priya is offered two jobs.
Offer A: a salary of $42 000 in year 1, rising by $1800 each year.
Offer B: a salary of $38 000 in year 1, rising by 6% each year.

(a)

Find Priya's salary in year 5 under each offer.

(3)

(b)

Find the total amount Priya would earn over the first 10 years under each offer.

(4)

(c)

Priya plans to stay in the job for exactly 10 years. Using your answers to the previous part, state which offer earns her more money, giving a reason.

(1)

(d)

Find the first year in which the annual salary under Offer B is greater than under Offer A.

(2)

(e)

Find the least number of complete years after which the total earned under Offer B is greater than the total earned under Offer A.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Offer A: $42000 + 4(1800) = $49\,200$ A1	✓ 1
Offer B: $38000 \times 1.06^{4}$ M1	✓ 1
$= $47\,974.12$ A1	✓ 1
Part (b)
Offer A: $\frac{10}{2}(2\times42000 + 9\times1800)$ M1	✓ 1
$= $501\,000$ A1	✓ 1
Offer B: $\frac{38000(1.06^{10}-1)}{0.06}$ M1	✓ 1
$= $500\,870.21$ A1	✓ 1
Part (c)
Offer A, because its 10-year total ($501 000) is about $130 more than Offer B's ($500 870.21) R1	✓ 1
Part (d)
Compares $38000(1.06)^{n-1}$ with $42000 + 1800(n-1)$ by table, graph or solver, e.g. year 6: B $50 852.57 < A $51 000; year 7: B $53 903.73 > A $52 800 M1	✓ 1
Year 7 A1	✓ 1
Part (e)
Sets up $\frac{38000(1.06^{n}-1)}{0.06} > \frac{n}{2}(84000 + 1800(n-1))$ or equivalent M1	✓ 1
Evaluates both totals for $n = 10$ and $n = 11$: $n=10$: B $500 870 < A $501 000; $n=11$: B $568 922 > A $561 000 (or uses a graph or solver to find the crossing at $n \approx 10.02$) M1	✓ 1
11 years A1	✓ 1

Exam tip: Check whether the question compares the salary in a single year (terms) or the amount earned so far (sums). The two cross over in different years.

+ Add to test✓ In your test406IBMI-030-02-r1-05Arithmetic and geometric sequences in context Paper 2Medium11 marks

A ball is dropped from a height of 2.5 m onto a hard floor. After each bounce it rises to 70% of the height it reached before that bounce. Assume the ball moves only vertically.

(a)

Find the height the ball reaches after the 4th bounce.

(2)

(b)

Find the greatest number of bounces for which the ball still rises to a height of at least 1 cm after the bounce.

(3)

(c)

Find the total vertical distance the ball has travelled when it hits the floor for the 5th time.

(3)

(d)

Using the model, find the total vertical distance the ball would travel if it bounced indefinitely.

(2)

(e)

Comment on the validity, in context, of your answer to the previous part.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Uses $2.5\times0.7^{4}$ M1 ; 0.600 m (0.60025) A1	✓ 2
Part (b)
Forms $2.5\times0.7^{n}\ge0.01$ (or an equivalent with consistent units) M1 ; $n\le15.48$, or bounce 15 gives about 1.19 cm and bounce 16 gives about 0.831 cm A1 ; 15 bounces A1	✓ 3
Part (c)
Recognises that the initial 2.5 m is travelled once and each of the first 4 bounce heights is travelled twice M1 ; uses $2.5+2\times\frac{1.75(1-0.7^{4})}{0.3}$ or $2.5+2(1.75+1.225+0.8575+0.60025)$ M1 ; 11.4 m (11.3655) A1	✓ 3
Part (d)
Uses $2.5+2\times\frac{1.75}{1-0.7}$ M1 ; 14.2 m (14.1667) A1	✓ 2
Part (e)
The model assumes infinitely many bounces, but a real ball comes to rest after a finite number of bounces, for example because of energy losses. The answer is therefore an upper estimate, although the later bounces add very little distance R1	✓ 1

Common error: The first drop is travelled only once. Every height after a bounce is travelled twice, once up and once down. Add the 2.5 m separately rather than doubling it.

+ Add to test✓ In your test407IBMI-030-06-r2-03Financial mathematics, loans and annuities Paper 2Medium10 marks

Marguerite retires with 250 000 Swiss francs (CHF) in an annuity account. The account pays a nominal annual interest rate of 4.2%, compounded quarterly. She plans to withdraw an equal amount at the end of every quarter for 20 years, so that the account is empty after the final withdrawal.

(a)

Find the amount of each quarterly withdrawal.

(3)

(b)

Find the total interest earned by the account over the 20 years.

(2)

(c)

Marguerite instead withdraws CHF 5000 at the end of each quarter. Find the number of full CHF 5000 withdrawals she can make.

(3)

(d)

Find the amount left in the account immediately after the last full CHF 5000 withdrawal.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
N = 80 and I% = 4.2 with P/Y = C/Y = 4, or quarterly rate r = 0.0105	✓ 1
PV = 250 000, FV = 0	✓ 1
Withdrawal = CHF 4634.58	✓ 1
Part (b)
80 × 4634.58 = CHF 370 766.40	✓ 1
Interest = 370 766.40 − 250 000 = CHF 120 766.40 (accept CHF 120 766 from an unrounded withdrawal)	✓ 1
Part (c)
TVM with PV = 250 000, PMT = −5000, FV = 0, I% = 4.2, P/Y = C/Y = 4	✓ 1
N = 71.27…	✓ 1
71 full withdrawals, because the balance is not enough for a 72nd full one	✓ 1
Part (d)
$250000(1.0105)^{71}-5000\times\frac{(1.0105)^{71}-1}{0.0105}$, or TVM with N = 71 solving for FV	✓ 1
= CHF 1344.37	✓ 1

Exam tip: For an annuity paid from a lump sum, the lump sum is PV and the final balance is FV = 0. Give PV and PMT opposite signs on the calculator.

+ Add to test✓ In your test408IBMI-030-01-r2-03Financial mathematics, loans and annuities Paper 2Medium11 marks

Marta takes out a mortgage of 240 000 pounds to buy a flat. The loan is to be repaid over 25 years by equal monthly payments made at the end of each month. The nominal annual interest rate is 4.8%, compounded monthly.

(a)

Find Marta's monthly payment.

(3)

(b)

Find the total interest Marta would pay over the 25 years.

(2)

(c)

Find the amount Marta still owes immediately after her 120th payment.

(3)

(d)

After the 120th payment the interest rate rises to 6.0% per annum, compounded monthly. Marta keeps the original end date of the mortgage. Find her new monthly payment.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
N = 300 and I% = 4.8 with C/Y = 12, or $i = 0.004$ per month M1	✓ 1
PV = 240 000, FV = 0, or $\frac{240000 \times 0.004}{1-1.004^{-300}}$ M1	✓ 1
1375.19 pounds A1	✓ 1
Part (b)
$300 \times 1375.19 - 240000$ M1	✓ 1
172 557 pounds (accept 172 557 to 172 558) A1	✓ 1
Part (c)
Uses N = 120 with PMT = 1375.19 and finds FV, or uses the present value of the remaining 180 payments M1	✓ 1
Correct set-up, e.g. $1375.19 \times \frac{1-1.004^{-180}}{0.004}$ A1	✓ 1
176 213 pounds (accept 176 212 to 176 213) A1	✓ 1
Part (d)
N = 180, I% = 6, PV = 176 213 (their (c)), FV = 0 M1	✓ 1
Correct method, e.g. $\frac{176213 \times 0.005}{1-1.005^{-180}}$ A1	✓ 1
1486.99 pounds (accept 1486.98 to 1486.99) A1	✓ 1

Exam tip: Store the unrounded payment and balance in your calculator and carry them forward. Rounding to the nearest pound in an early part can push a later answer outside the accepted range.

+ Add to test✓ In your test409IBMI-030-02-r2-05Arithmetic and geometric sequences in context Paper 2Medium11 marks

Marco is training for a long-distance event. He runs 3.2 km on day 1 of his training plan and increases his daily distance by 0.4 km each day.
His friend Lena runs 2.5 km on day 1 and increases her daily distance by 6% each day.

(a)

Find the distance Marco runs on day 15.

(2)

(b)

Find the total distance Marco runs in the first 15 days.

(2)

(c)

Find the first day on which Lena runs further than Marco.

(3)

(d)

Marco's coach sets him a target of running a total of at least 200 km. Find the minimum number of days needed to reach this target.

(3)

(e)

Comment on the suitability of Lena's model for predicting her daily distance after many months of training.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$u_{15} = 3.2 + 14 \times 0.4$ M1	✓ 1
= 8.8 km A1	✓ 1
Part (b)
$S_{15} = \frac{15}{2}(3.2 + 8.8)$ M1	✓ 1
= 90 km A1	✓ 1
Part (c)
Marco: $3.2 + 0.4(n-1)$; Lena: $2.5 \times 1.06^{n-1}$ M1	✓ 1
Compares values using a GDC table or graph, e.g. day 32: Marco 15.6 km, Lena 15.2 km; day 33: Marco 16.0 km, Lena 16.1 km M1	✓ 1
Day 33 A1	✓ 1
Part (d)
$\frac{n}{2}(2\times 3.2 + 0.4(n-1)) \geq 200$ M1	✓ 1
$0.2n^2 + 3n - 200 = 0$ solved, giving $n = 25$ (or $-40$) A1	✓ 1
25 days. The total after 25 days is exactly 200 km, so the target is met. The negative root is rejected. A1	✓ 1
Part (e)
Not suitable: exponential growth is unbounded. For example, by day 60 the model predicts about 78 km per day, which is unrealistic for a runner. R1	✓ 1

Exam tip: When you compare an arithmetic model with a geometric one, a GDC table of both sequences is the most reliable method. Quote the values on both sides of the change to show where it happens.

+ Add to test✓ In your test410IBMI-030-06-r1-05Financial mathematics, loans and annuities Paper 2Medium10 marks

A small bakery needs to borrow £40 000 for new ovens. It is offered two loans. Each loan is repaid over 6 years with payments at the end of each period.
Option A: nominal annual rate 7.2%, compounded monthly, with monthly payments.
Option B: nominal annual rate 6.9%, compounded quarterly, with quarterly payments. A one-off arrangement fee of £500 is paid when the loan starts.

(a)

Find the monthly payment for Option A.

(2)

(b)

Find the quarterly payment for Option B.

(2)

(c)

Calculate the effective annual interest rate for each option, giving your answers as percentages to 3 significant figures.

(2)

(d)

Taking the fee into account, determine which option costs the bakery less in total, and by how much.

(3)

(e)

Explain why Option B costs more in total even though its effective annual rate is lower.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 N = 72, I% = 7.2, PV = 40 000, FV = 0, C/Y = P/Y = 12	✓ 1
A1 £685.81	✓ 1
Part (b)
M1 N = 24, I% = 6.9, PV = 40 000, FV = 0, C/Y = P/Y = 4	✓ 1
A1 £2049.54	✓ 1
Part (c)
A1 Option A: (1 + 0.072/12)¹² − 1 = 7.44%	✓ 1
A1 Option B: (1 + 0.069/4)⁴ − 1 = 7.08%	✓ 1
Part (d)
M1 total = payment × number of payments, with the £500 fee added for B	✓ 1
A1 A: 72 × 685.81 = £49 378.32; B: 24 × 2049.54 + 500 = £49 688.96	✓ 1
A1 Option A is cheaper, by £310.64 (accept £310.69 from unrounded payments, or £311)	✓ 1
Part (e)
R1 Option B charges about £190 less in interest, but the £500 fee more than cancels this saving	✓ 1

Exam tip: A lower effective annual rate does not always mean a cheaper loan. Fees paid up front are part of the true cost, so include them before you choose.

+ Add to test✓ In your test411IBMI-030-01-r2-05Financial mathematics, loans and annuities Paper 2Medium12 marks

Lena is saving for a house deposit. She compares two savings accounts.
Account A: 5.1% per annum, compounded annually.
Account B: 5.0% per annum, compounded monthly.

(a)

By finding the effective annual interest rate of Account B, determine which account gives the better return. Justify your answer.

(4)

(b)

Lena chooses Account B. She deposits 350 euros at the end of every month for 5 years. Find the value of her savings at the end of the 5 years.

(3)

(c)

Inflation is expected to be 2.4% per year over the 5 years. Find the value of Lena's savings at the end of the 5 years in terms of today's money.

(2)

(d)

Lena wants exactly 30 000 euros in Account B after 5 years of equal monthly deposits, each made at the end of a month. Find the monthly deposit she needs to make.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$\left(1+\frac{0.05}{12}\right)^{12} - 1$ or equivalent GDC method M1	✓ 1
5.116% (accept 5.12%; a value rounded to 5.1% does not allow a valid comparison) A1	✓ 1
Compares this explicitly with the 5.1% effective rate of Account A, e.g. 5.116% > 5.1% R1	✓ 1
Account B gives the better return A1	✓ 1
Part (b)
Attempts a TVM calculation or the annuity formula with 60 monthly periods M1	✓ 1
Correct values: either N = 60, I% = 5, C/Y = P/Y = 12, PMT = ±350, PV = 0, or $350 \times \frac{(1+0.05/12)^{60}-1}{0.05/12}$ A1	✓ 1
23 802.13 euros (accept 23 800) A1	✓ 1
Part (c)
$\frac{23802.13}{1.024^5}$ M1	✓ 1
21 140.54 euros (accept 21 100) A1	✓ 1
Part (d)
Attempts to solve for the payment with 60 monthly periods M1	✓ 1
Correct values: either N = 60, I% = 5, C/Y = P/Y = 12, FV = 30 000, PV = 0, or $\frac{30000 \times 0.05/12}{(1+0.05/12)^{60}-1}$ A1	✓ 1
441.14 euros A1	✓ 1

Exam tip: When a question asks you to justify a choice, state both effective rates to enough decimal places to tell them apart and compare them explicitly. Naming the better account without a comparison does not earn the reasoning mark.

+ Add to test✓ In your test412IBMI3S-1-01-06Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium12 marks

Sara saves $250 at the end of every month in an account that pays a nominal annual interest rate of 4.8%, compounded monthly.

(a)

Find the value of Sara's savings after 10 years, to the nearest cent.

(3)

(b)

Find the total interest earned over the 10 years, to the nearest dollar.

(2)

(c)

Sara's brother deposits a single lump sum into an identical account instead, making no further deposits. Find the lump sum needed for it to be worth $38 407.99 after 10 years. Give your answer to the nearest cent.

(2)

(d)

Find the least number of complete months of saving after which Sara's balance exceeds $50 000.

(3)

(e)

Find the effective annual rate of interest, as a percentage to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of financial solver or annuity formula with N = 120	✓ 1
A1 I% = 4.8, PMT = ∓250, PV = 0, P/Y = C/Y = 12	✓ 1
A1 $38 407.99	✓ 1
Part (b)
M1 total deposited $=250\times120=30\,000$	✓ 1
A1 interest $8408	✓ 1
Part (c)
M1 solver with N = 120, I% = 4.8, PMT = 0, FV = ∓38 407.99	✓ 1
A1 $23 788.99	✓ 1
Part (d)
M1 solver with FV = ∓50 000 and PMT = ∓250	✓ 1
A1 $n=147.2$	✓ 1
A1 148 months	✓ 1
Part (e)
M1 $\left(1+\frac{0.048}{12}\right)^{12}-1$	✓ 1
A1 4.91 %	✓ 1

Common error: When the balance must exceed a target after a whole number of payments, round the solver's N up to the next integer, not to the nearest integer.

+ Add to test✓ In your test413IBMI3S-1-01b-10Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium12 marks

An open-air concert area has 30 rows of seats. Row 1 has 24 seats and each row has 3 more seats than the row in front of it.

(a)

Find the number of seats in row 20.

(2)

(b)

Show that the total number of seats in the concert area is 2025.

(2)

(c)

Tickets for rows 1 to 10 cost €85 each and tickets for rows 11 to 30 cost €60 each. Find the total income if every seat is sold.

(3)

(d)

The organisers need an income of at least €120 000. They assume the same percentage of seats is sold at each price. Find the least percentage of seats that must be sold.

(2)

(e)

A different design has 25 rows with 30 seats in row 1 and a constant increase of d seats per row, giving a total of 2100 seats. Find d and explain why this design cannot be built exactly.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₂₀ = 24 + 19 × 3	✓ 1
A1 81 seats	✓ 1
Part (b)
M1 S₃₀ = 30/2 (2 × 24 + 29 × 3)	✓ 1
A1 15 × 135 = 2025	✓ 1
Part (c)
M1 S₁₀ = 10/2 (48 + 9 × 3) = 375 seats in rows 1 to 10	✓ 1
M1 2025 − 375 = 1650 seats in rows 11 to 30	✓ 1
A1 375 × 85 + 1650 × 60 = €130875	✓ 1
Part (d)
M1 120000 ÷ 130875	✓ 1
A1 91.7 %	✓ 1
Part (e)
M1 25/2 (2 × 30 + 24d) = 2100	✓ 1
A1 d = 4.5	✓ 1
R1 row 2 would have 34.5 seats, which is not a whole number	✓ 1

Exam tip: For a sum of an arithmetic series use Sn = n/2 (2u₁ + (n − 1)d) and check that n is the number of terms, not the last term number you want.

+ Add to test✓ In your test414IBMI3S-1-02r14-10Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium13 marks

The vertical hangers on one side of a suspension bridge are numbered 1 to 16 starting from the centre of the span. Hanger 1 is 3.2 m long, and each hanger is 1.45 m longer than the previous one, so the lengths form an arithmetic sequence.

(a)

Find the length of hanger 16.

(2)

(b)

Find the total length of the 16 hangers.

(2)

(c)

The steel cable for the hangers costs €24.60 per metre. Find the cost of the cable for these 16 hangers.

(2)

(d)

The bridge has four identical sets of 16 hangers. Find the total cost of the cable, to the nearest euro.

(1)

(e)

Corrosion reduces the diameter of a hanger cable. The diameter after n years is modelled by $D_n=64\times0.996^{\,n}$ millimetres. Find the diameter after 15 years, to 3 significant figures.

(2)

(f)

A hanger must be replaced when its diameter falls below 58 mm. Find, to 3 significant figures, the number of years after which the diameter reaches 58 mm.

(3)

(g)

Give one limitation of using this model for the diameter over a long period.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $3.2+15\times1.45$	✓ 1
A1 24.95 m	✓ 1
Part (b)
M1 $\frac{16}{2}(3.2+24.95)$ or $\frac{16}{2}(2\times3.2+15\times1.45)$	✓ 1
A1 225.2 m	✓ 1
Part (c)
M1 $225.2\times24.60$	✓ 1
A1 €5539.92	✓ 1
Part (d)
A1 $4\times5539.92$ = €22160	✓ 1
Part (e)
M1 $64\times0.996^{15}$	✓ 1
A1 60.3 mm	✓ 1
Part (f)
M1 $64\times0.996^{n}=58$	✓ 1
M1 take logarithms: $n=\frac{\ln(58/64)}{\ln0.996}$	✓ 1
A1 24.6 years	✓ 1
Part (g)
R1 e.g. corrosion rates are unlikely to stay at a constant 0.4% per year (they depend on weather, coating, load); the model may not represent real corrosion	✓ 1

Exam tip: For the sum of an arithmetic sequence, use whichever of the two formulas matches what you know: first and last term, or first term and common difference.

+ Add to test✓ In your test415IBMI3S-1-02r14-11Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium13 marks

A space probe is 4.30 × 10¹² m from Earth. Radio signals travel at 2.998 × 10⁸ m s⁻¹.

(a)

Calculate the time for a signal to travel from the probe to Earth. Give your answer in seconds in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(2)

(b)

Using the unrounded travel time, express it in hours and minutes, to the nearest minute.

(2)

(c)

A student uses 3.00 × 10⁸ m s⁻¹ for the speed of the signal. Find the percentage error in the travel time that results. Give your answer to 3 significant figures.

(2)

(d)

The power of the received signal is inversely proportional to the square of the distance. At 4.30 × 10¹² m the received power is 2.4 × 10⁻¹⁹ W. Find the received power when the probe is 1.50 × 10¹² m from Earth. Give your answer to 3 significant figures.

(3)

(e)

The change in signal level between two powers P₁ and P₂ is $10\log_{10}\left(\frac{P_2}{P_1}\right)$ decibels. Find the increase in signal level when the probe moves from 4.30 × 10¹² m to 1.50 × 10¹² m. Give your answer to 3 significant figures.

(2)

(f)

The receiver needs a signal power of at least 1.0 × 10⁻¹⁹ W. Find the greatest distance from Earth at which this is possible. Give your answer in units of 10¹² m, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\frac{4.30\times10^{12}}{2.998\times10^{8}}$	✓ 1
A1 1.43 × 10⁴ s	✓ 1
Part (b)
M1 $\frac{14343}{60}=239$ minutes (to the nearest minute) or dividing by 3600 for hours	✓ 1
A1 3 hours 59 minutes	✓ 1
Part (c)
M1 $\frac{\|\text{approximate}-\text{exact}\|}{\text{exact}}\times100$ with times 14333 s and 14343 s (or equivalent using speeds)	✓ 1
A1 0.0667%	✓ 1
Part (d)
M1 $P\,d^{2}$ constant, or $P\propto\frac{1}{d^{2}}$	✓ 1
M1 $P=2.4\times10^{-19}\times\left(\frac{4.30}{1.50}\right)^{2}$	✓ 1
A1 1.97 × 10⁻¹⁸ W	✓ 1
Part (e)
M1 $10\log_{10}(8.22)$ or $10\log_{10}\left(\frac{4.30}{1.50}\right)^{2}$	✓ 1
A1 9.15 dB	✓ 1
Part (f)
M1 $d^{2}=4.30^{2}\times\frac{2.4\times10^{-19}}{1.0\times10^{-19}}$ (in units of $10^{12}$ m)	✓ 1
A1 6.66	✓ 1

Common error: "Inversely proportional to the square" means doubling the distance divides the power by 4. Squaring the distance ratio is easy to forget.

+ Add to test✓ In your test416IBMI3S-1-02r14-12Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium13 marks

In a coffee laboratory, the moisture content of roasting beans is modelled by $W=12e^{-kt}$, where W is the moisture content as a percentage of the bean mass, t is the time in minutes since the roast began and k is a positive constant. After 5 minutes the moisture content is 7.2%. In parts (b) and (c), use the unrounded value of k.

(a)

Find the value of k, to 3 significant figures.

(2)

(b)

Find the moisture content after 9 minutes, to 3 significant figures.

(2)

(c)

Find the time at which the moisture content first falls to 3%. Give your answer to 3 significant figures.

(3)

(d)

The acidity of a brewed coffee is measured by its pH, where $\text{pH}=-\log_{10}[\text{H}^+]$ and [H⁺] is the hydrogen ion concentration in mol dm⁻³. A coffee has pH 5.2. Find its hydrogen ion concentration, in the form a × 10^k with a to 3 significant figures.

(2)

(e)

A second coffee has a hydrogen ion concentration of 2.5 × 10⁻⁵ mol dm⁻³. Find its pH, to 3 significant figures.

(2)

(f)

Find how many times greater the hydrogen ion concentration of the second coffee is than that of the first. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $12e^{-5k}=7.2$ giving $e^{-5k}=0.6$	✓ 1
A1 $k=\frac{\ln(12/7.2)}{5}=$ 0.102	✓ 1
Part (b)
M1 $12e^{-9k}$	✓ 1
A1 4.78%	✓ 1
Part (c)
M1 $12e^{-kt}=3$ giving $e^{-kt}=0.25$	✓ 1
M1 take natural logarithms: $t=\frac{\ln4}{k}$	✓ 1
A1 13.6 minutes	✓ 1
Part (d)
M1 $[\text{H}^+]=10^{-5.2}$	✓ 1
A1 6.31 × 10⁻⁶ mol dm⁻³	✓ 1
Part (e)
M1 $-\log_{10}(2.5\times10^{-5})$	✓ 1
A1 4.60	✓ 1
Part (f)
M1 $\frac{2.5\times10^{-5}}{10^{-5.2}}$	✓ 1
A1 3.96	✓ 1

Exam tip: Keep unrounded values in your calculator memory for later parts. Rounding k to 3 significant figures too early can change the third significant figure of the next answer.

+ Add to test✓ In your test417IBMI3S-1-01r18-05Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium10 marks

A fishing cooperative borrows €42 000 to buy a new boat. The loan is repaid over 8 years by equal payments made at the end of every month. The annual interest rate is 4.5 %, compounded monthly.

(a)

Calculate the monthly payment. Give your answer to the nearest cent.

(3)

(b)

Use the unrounded monthly payment to calculate the total amount repaid over the 8 years. Give your answer to the nearest cent.

(2)

(c)

Hence find the total interest paid on the loan, to the nearest cent.

(2)

(d)

Find the amount still owed immediately after the 36th payment. Give your answer to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to use the loan (TVM) formula or solver	✓ 1
A1 N = 96, I% = 4.5, PV = 42000, FV = 0, P/Y = C/Y = 12 (monthly rate 0.375 %)	✓ 1
A1 €521.78	✓ 1
Part (b)
M1 96 × (monthly payment)	✓ 1
A1 €50090.48	✓ 1
Part (c)
M1 total repaid − 42000	✓ 1
A1 €8090.48	✓ 1
Part (d)
M1 attempt to find balance, e.g. TVM with N = 36 or 60 remaining payments	✓ 1
A1 correct setup (e.g. N = 60, PMT = monthly payment, FV = 0, I% = 4.5)	✓ 1
A1 €27987.73	✓ 1

Common error: Using the annual rate 4.5 % for each monthly period instead of 0.375 % gives a payment that is far too large.

+ Add to test✓ In your test418IBMI3S-1-01r18-06Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium9 marks

A fashion label has 12 000 followers at the end of month 1. Its follower count then increases by 6 % each month, so the follower counts at the end of each month form a geometric sequence.

(a)

Write down the common ratio of the sequence.

(1)

(b)

Find the number of followers at the end of month 10. Give your answer to the nearest whole number.

(2)

(c)

Find the first month at the end of which the label has more than 25 000 followers.

(3)

(d)

A sponsor pays the label €0.002 for each follower counted at the end of each of the first 12 months. Find the total payment, giving your answer to the nearest euro.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.06	✓ 1
Part (b)
M1 12000 × 1.06⁹	✓ 1
A1 20274	✓ 1
Part (c)
M1 12000 × 1.06ⁿ⁻¹ > 25000	✓ 1
A1 n − 1 > 12.6 (or n > 13.6)	✓ 1
A1 month 14	✓ 1
Part (d)
M1 S₁₂ = 12000(1.06¹² − 1)/(1.06 − 1)	✓ 1
A1 202439 followers counted in total (accept 202439)	✓ 1
A1 €405	✓ 1

Exam tip: "First month" questions need a whole-number answer: solve the inequality, then round up to the next integer.

+ Add to test✓ In your test419IBMI3S-1-01r18-12Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium10 marks

The sound level L, in decibels (dB), of a noise of intensity I W m⁻² is given by L = 10 log₁₀(I/I₀), where I₀ = 1.0×10⁻¹² W m⁻². A packing robot in a fashion warehouse produces a noise of intensity 3.3×10⁻⁵ W m⁻².

(a)

Calculate the sound level produced by one robot, to 3 significant figures.

(2)

(b)

Show that doubling the intensity of a noise increases the sound level by approximately 3 dB.

(2)

(c)

The warehouse must keep the total sound level at or below 85 dB. Each running robot produces the same intensity, and the intensities add. Find the greatest number of robots that can run at the same time.

(4)

(d)

Find the sound level when this number of robots is running. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(3.3×10⁻⁵ ÷ 1.0×10⁻¹²)	✓ 1
A1 75.2 dB	✓ 1
Part (b)
M1 10 log₁₀(2I/I₀) = 10 log₁₀ 2 + 10 log₁₀(I/I₀)	✓ 1
A1 10 log₁₀ 2 = 3.01 dB, which is approximately 3 dB (AG)	✓ 1
Part (c)
M1 10 log₁₀(I/10⁻¹²) = 85, so I = 10^8.5 × 10⁻¹²	✓ 1
A1 3.16×10⁻⁴ W m⁻² maximum total intensity	✓ 1
M1 3.16×10⁻⁴ ÷ 3.3×10⁻⁵ = 9.58	✓ 1
A1 9 robots	✓ 1
Part (d)
M1 10 log₁₀(9 × 3.3×10⁻⁵ ÷ 10⁻¹²)	✓ 1
A1 84.7 dB	✓ 1

Exam tip: Intensities add, but decibel levels do not: combine the intensities first and then convert to decibels.

+ Add to test✓ In your test420IBMI3S-1-01r19-08Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Medium9 marks

After a sprint, the skin temperature T °C of an athlete's calf is modelled, t minutes after she steps into a water bath at 22 °C, by T = 22 + 14e^−0.12t.

(a)

Find the temperature at the moment she enters the bath.

(2)

(b)

Write down the temperature the model predicts in the long term, and give a reason.

(2)

(c)

Find the temperature after 10 minutes, to 3 significant figures.

(2)

(d)

Find the time taken for the temperature to fall to 25 °C. Give your answer in minutes to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 0	✓ 1
A1 36 °C	✓ 1
Part (b)
A1 22 °C	✓ 1
R1 as t increases e^−0.12t tends to 0 (so T approaches 22)	✓ 1
Part (c)
M1 22 + 14e^−1.2	✓ 1
A1 26.2 °C	✓ 1
Part (d)
M1 set up 22 + 14e^−0.12t = 25, so e^−0.12t = 3/14	✓ 1
M1 take natural logarithms: −0.12t = ln(3/14)	✓ 1
A1 12.8 minutes	✓ 1

Exam tip: To solve for an exponent, isolate the exponential term first and only then take logarithms.

+ Add to test✓ In your test421IBMI3S-1-01r19-09Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Medium9 marks

The mean distance from the Earth to the Sun is 1.496 × 10⁸ km. The speed of light is 3.00 × 10⁸ m s⁻¹.

(a)

Calculate the time light takes to travel from the Sun to the Earth. Give your answer in seconds, to 3 significant figures.

(2)

(b)

A student approximates the distance as 1.5 × 10⁸ km and the speed as 3 × 10⁸ m s⁻¹, giving a time of 500 s. Taking the exact time as 498.666… s, find the percentage error in the student's value, to 3 significant figures.

(3)

(c)

One light-year is the distance light travels in 365 days. Show that 1 light-year is approximately 9.46 × 10¹⁵ m, and write the exact calculated value in the form a × 10^k.

(2)

(d)

A nearby star is 4.0 × 10¹⁶ m from the Earth. Using 1 light-year = 9.46 × 10¹⁵ m, find its distance in light-years, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 convert: 1.496 × 10¹¹ m ÷ 3.00 × 10⁸ m s⁻¹	✓ 1
A1 499 s	✓ 1
Part (b)
M1 correct percentage error formula	✓ 1
M1 substitute 500 and 498.666… (unrounded)	✓ 1
A1 0.267%	✓ 1
Part (c)
M1 3.00 × 10⁸ × 365 × 24 × 3600	✓ 1
A1 9.4608 × 10¹⁵ m, which rounds to 9.46 × 10¹⁵ m	✓ 1
Part (d)
M1 4.0 × 10¹⁶ ÷ 9.46 × 10¹⁵	✓ 1
A1 4.23 light-years	✓ 1

Common error: Convert km to m before using a speed given in m s⁻¹; a factor of 1000 is the usual slip.

+ Add to test✓ In your test422IBMI3S-1-02r21-11Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 2Medium14 marks

Calculator

A hospital needs €480 000 for an MRI scanner and compares two ten-year loans. Option A: 4.8% per year, compounded monthly, repaid by equal payments at the end of each month. Option B: 4.5% per year, compounded annually, repaid by equal payments at the end of each year. Give money answers to the nearest cent.

(a)

Find the monthly payment under Option A.

(3)

(b)

Using the monthly payment of €5044.35, find the total amount repaid under Option A.

(2)

(c)

Find the effective annual interest rate for Option A, as a percentage to 3 significant figures.

(2)

(d)

Find the annual payment under Option B.

(2)

(e)

Using the annual payment of €60 661.83, find how much more is repaid in total under Option B than under Option A.

(2)

(f)

The finance director says: "Option B has the lower interest rate, so it must cost less to repay in total." Use your answers to comment on this statement.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance solver with $N=120$, $PV=480000$, $FV=0$	✓ 1
M1 $I=4.8$, $P/Y=C/Y=12$	✓ 1
A1 €5044.35	✓ 1
Part (b)
M1 $120\times5044.35$	✓ 1
A1 €605 322.00	✓ 1
Part (c)
M1 $\left(1+\frac{0.048}{12}\right)^{12}-1$	✓ 1
A1 4.91%	✓ 1
Part (d)
M1 finance solver with $N=10$, $I=4.5$, $PV=480000$, $FV=0$, $P/Y=C/Y=1$	✓ 1
A1 €60 661.83	✓ 1
Part (e)
M1 $10\times60661.83=606618.30$	✓ 1
A1 €1296.30	✓ 1
Part (f)
R1 the effective rate for B (4.5%) is lower than for A (4.91%)	✓ 1
R1 but the total repaid under B is higher, so the statement about total cost is incorrect	✓ 1
R1 payments under B are made only once a year, so the balance stays higher for longer and more interest is paid	✓ 1

Common error: The nominal rate 4.8% compounded monthly is not comparable with 4.5% compounded annually; convert to effective annual rates first.

+ Add to test✓ In your test423IBMI3S-1-02r21-14Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 2Medium12 marks

Calculator

A Formula 1 race consists of 57 laps. Each lap is 5.793 km long. A car's fuel allowance for the race is 110 kg, and the fuel releases 4.2 × 10⁷ J of energy per kilogram. Treat all given values as exact.

(a)

Calculate the race distance in metres. Give your answer in the form $a\times10^{k}$ where $1\le a<10$ and $k$ is an integer, with $a$ correct to 3 significant figures.

(2)

(b)

Calculate the total energy available from the fuel allowance. Give your answer in the form $a\times10^{k}$ joules.

(2)

(c)

A commentator estimates the race distance by rounding the number of laps and the lap length to 1 significant figure. Find the percentage error in this estimate, correct to 3 significant figures, using the exact distance 330.201 km.

(3)

(d)

A car completes every lap in 91.46 s. Find the total race time in hours, minutes and seconds, with the seconds to the nearest whole second.

(3)

(e)

Find the car's average speed for the race in km h⁻¹, correct to 3 significant figures. Use the exact distance 330.201 km and the unrounded race time.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $57\times5.793=330.201$ km and convert to metres	✓ 1
A1 $3.30\times10^{5}$ m	✓ 1
Part (b)
M1 $110\times4.2\times10^{7}$	✓ 1
A1 $4.62\times10^{9}$ J	✓ 1
Part (c)
M1 estimate $=60\times6=360$ km	✓ 1
M1 $\frac{360-330.201}{330.201}\times100$	✓ 1
A1 9.02%	✓ 1
Part (d)
M1 $57\times91.46=5213.22$ s	✓ 1
M1 $5213.22=86\times60+53.22$	✓ 1
A1 1 h 26 min 53 s	✓ 1
Part (e)
M1 $\frac{330.201}{5213.22/3600}$	✓ 1
A1 228 km h⁻¹	✓ 1

Exam tip: Convert the time to hours before using km h⁻¹: divide seconds by 3600, not 60.

+ Add to test✓ In your test424IBMI3S-1-03r22-06Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium11 marks

A groundsman measures a rectangular training pitch. The length is 105 m and the width is 68 m, each measured to the nearest metre.

(a)

Write down the lower and upper bounds for the length and for the width.

(2)

(b)

Calculate the area of the pitch using the measured values.

(1)

(c)

Find the lower and upper bounds for the area of the pitch.

(3)

(d)

The club later measures the pitch accurately with a laser: the length is 104.7 m and the width is 67.9 m. Find the true area and state whether it lies within the bounds found in (c).

(2)

(e)

Find the percentage error in the area 7140 m² from (b), compared with the true area. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 length: 104.5 m ≤ L < 105.5 m	✓ 1
A1 width: 67.5 m ≤ W < 68.5 m	✓ 1
Part (b)
A1 105 × 68 = 7140 m²	✓ 1
Part (c)
M1 uses bounds for both dimensions, 104.5 × 67.5 and 105.5 × 68.5	✓ 1
A1 lower bound 7053.75 m²	✓ 1
A1 upper bound 7226.75 m²	✓ 1
Part (d)
M1 104.7 × 67.9 = 7109.13 m²	✓ 1
R1 7109.13 lies between 7053.75 and 7226.75, so it is within the bounds	✓ 1
Part (e)
M1 absolute error \|7140 − 7109.13\| = 30.87	✓ 1
M1 divides by the true area and multiplies by 100	✓ 1
A1 0.434 %	✓ 1

Common error: Lower and upper bounds of a product use the lower bounds of both factors together (for the lower bound) and the upper bounds of both together (for the upper bound).

+ Add to test✓ In your test425IBMI3S-1-03r22-11Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium9 marks

The seating in a sports academy's lecture theatre is arranged in rows. The first row has 24 seats and each row has the same number of extra seats more than the row in front of it, so the numbers of seats form an arithmetic sequence. The first 20 rows contain 1050 seats altogether.

(a)

Find the number of extra seats in each row.

(2)

(b)

Find the number of seats in row 20.

(2)

(c)

The academy wants a theatre with at least 2000 seats. Find the least number of rows needed.

(3)

(d)

Find the total number of seats in a theatre with this number of rows.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\dfrac{20}{2}(2 \times 24 + 19d) = 1050$	✓ 1
A1 d = 3	✓ 1
Part (b)
M1 $24 + 19 \times 3$	✓ 1
A1 81	✓ 1
Part (c)
M1 $\dfrac{n}{2}(48 + 3(n-1)) \ge 2000$ (or equation / table)	✓ 1
A1 solves, e.g. $3n^2 + 45n - 4000 \ge 0$ giving $n \ge 29.78$	✓ 1
A1 30 rows (29 rows give 1914 seats)	✓ 1
Part (d)
M1 $S_{30} = \dfrac{30}{2}(2 \times 24 + 29 \times 3)$	✓ 1
A1 2025	✓ 1

Common error: In 'at least' questions, check the whole numbers either side of your solution; the least valid integer is the next one up from the non-integer root.

+ Add to test✓ In your test426IBMI3S-1-03r23-08Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium9 marks

A cycling team travelling from the UK to France exchanges £12 500 into euros (€). The exchange rate is £1 = €1.1642. The bank charges a commission of 1.5% of the amount being exchanged, taken before the exchange is made.

(a)

Calculate the number of euros the team receives. Give your answer to the nearest cent.

(3)

(b)

The team spends €9800 and exchanges the remaining euros back into pounds. The rate is €1 = £0.8576 and the bank again charges 1.5% commission on the amount being exchanged. Use your answer to part (a). Calculate the number of pounds received, to the nearest penny.

(3)

(c)

Suppose that no money had been spent and all of the euros had been exchanged back into pounds at the same rates. Find the percentage of the original £12 500 lost to commission and rate differences. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 deducts commission: 12 500 × (1 − 0.015) = £12 312.50	✓ 1
A1 multiplies by 1.1642: 12 312.50 × 1.1642	✓ 1
A1 €14 334.21	✓ 1
Part (b)
M1 remaining euros = 14 334.21 − 9800 = €4534.21	✓ 1
M1 (4534.21 × 0.985) × 0.8576	✓ 1
A1 £3830.21	✓ 1
Part (c)
M1 round-trip factor 0.985 × 1.1642 × 0.985 × 0.8576	✓ 1
A1 factor = 0.96869…	✓ 1
A1 loss = 3.13 %	✓ 1

Exam tip: With two exchanges, multiply the rates and commission factors in the order they occur; then the net effect is a single multiplier you can interpret as a percentage.

+ Add to test✓ In your test427IBMI3S-1-03r23-13Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium9 marks

A data centre's nightly backup files grow in size so that the daily sizes form a geometric sequence. The backup on day 2 is 18.0 GB and the backup on day 5 is 60.75 GB.

(a)

Find the common ratio and the size of the backup on day 1.

(4)

(b)

Find the first day on which the backup is larger than 500 GB.

(3)

(c)

Find the total size of the backups over the first 14 days. Give your answer to 3 significant figures, in GB.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₅ / u₂ = r³ = 60.75 / 18	✓ 1
A1 r = 1.5	✓ 1
M1 u₁ = 18 / 1.5	✓ 1
A1 12 GB	✓ 1
Part (b)
M1 solves 12 × 1.5ⁿ⁻¹ > 500	✓ 1
A1 day 10 gives 461.3 GB and day 11 gives 692.0 GB (or n − 1 > 9.2…)	✓ 1
A1 day 11	✓ 1
Part (c)
M1 S₁₄ = 12(1.5¹⁴ − 1) / (1.5 − 1)	✓ 1
A1 6980 GB (6982.3… to 3 s.f.)	✓ 1

Common error: The ratio between terms that are three steps apart is r³, not 3r; take a cube root to find r.

+ Add to test✓ In your test428IBMI3S-1-01r30-03Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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A data centre is being fitted out with server racks. In week 1 the installers fit 12 racks. In each following week they fit 5 more racks than in the previous week, so the numbers of racks fitted each week form an arithmetic sequence.

(a)

Find the number of racks fitted in week 10.

(2)

(b)

Find the total number of racks fitted in the first 15 weeks.

(3)

(c)

The data centre needs 1500 racks in total. Find the least number of complete weeks after which at least 1500 racks would have been fitted if the pattern continued.

(3)

(d)

In week 23 the installers stop as soon as exactly 1500 racks have been fitted. Find the number of racks fitted in week 23.

(2)

(e)

Each rack costs €2350 to fit. Calculate the fitting cost of the racks installed in the first 15 weeks. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u_n = 12 + (n − 1) × 5	✓ 1
A1 57 racks	✓ 1
Part (b)
M1 use of S_n = (n/2)(2u₁ + (n − 1)d)	✓ 1
A1 S₁₅ = (15/2)(2 × 12 + 14 × 5)	✓ 1
A1 705 racks	✓ 1
Part (c)
M1 S_n = (n/2)(24 + 5(n − 1)) ≥ 1500	✓ 1
A1 S₂₂ = 1419 and S₂₃ = 1541 (or solver / graph giving n ≥ 22.…)	✓ 1
A1 23 weeks	✓ 1
Part (d)
M1 1500 − S₂₂ = 1500 − 1419	✓ 1
A1 81 racks	✓ 1
Part (e)
M1 705 × 2350 = 1656750	✓ 1
A1 1.66 × 10⁶ €	✓ 1
ECF from (b).	—

Common error: Using the formula for the n-th term when the total is wanted. Decide whether the question asks about one week or about all weeks up to that week.

+ Add to test✓ In your test429IBMI3S-1-01r30-06Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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A cyclist plans a training programme lasting 12 weeks. In Plan A, the distance cycled in week 1 is 20 km and the distance increases by 4% each week, so the weekly distances form a geometric sequence.

(a)

Find the distance cycled in week 12 under Plan A. Give your answer to 3 significant figures.

(2)

(b)

Find the total distance cycled in the 12 weeks under Plan A. Give your answer to 3 significant figures.

(2)

(c)

Find the first week in which the distance cycled under Plan A exceeds 50 km.

(3)

(d)

In Plan B the distance in week 1 is 20 km and the distance increases by 1.5 km each week. For Plan B, find (i) the distance cycled in week 12; (ii) the total distance cycled in the 12 weeks.

(3)

(e)

Show that Plan A has a longer distance than Plan B in week 32 but not in week 31, if both plans were continued beyond 12 weeks.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 20 × 1.04¹¹	✓ 1
A1 30.8 km	✓ 1
Part (b)
M1 $\frac{20(1.04^{12}-1)}{1.04-1}$	✓ 1
A1 301 km	✓ 1
Part (c)
M1 20 × 1.04ⁿ⁻¹ > 50	✓ 1
M1 use of logarithms, table or graph: n − 1 > 23.36…	✓ 1
A1 week 25	✓ 1
Part (d)
(i) A1 20 + 11 × 1.5 = 36.5 km	✓ 1
(ii) M1 (12/2)(2 × 20 + 11 × 1.5)	✓ 1
A1 339 km	✓ 1
Part (e)
M1 week 31: A = 20 × 1.04³⁰ = 64.9 (km), B = 20 + 30 × 1.5 = 65 (km)	✓ 1
A1 week 32: A = 20 × 1.04³¹ = 67.5 (km), B = 66.5 (km)	✓ 1
R1 A < B in week 31 and A > B in week 32,	✓ 1

Exam tip: When you compare a geometric and an arithmetic model, a table of values on your calculator around the crossover is the quickest check.

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Layers of sediment at an archaeological site were laid down one above another. The thickness of the first (lowest) layer is 24 cm, and each layer above it is 85% as thick as the layer below it.

(a)

Find the thickness of the third layer. Give your answer to 3 significant figures.

(2)

(b)

Find the total thickness of the first 8 layers. Give your answer to 3 significant figures.

(2)

(c)

Find the least number of layers needed for the total thickness to exceed 130 cm.

(3)

(d)

Find the number of the first layer that is thinner than 2 cm.

(3)

(e)

Each layer is estimated to have taken 35 years to form. Find the time taken for the first 11 layers to form, in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 24 × 0.85²	✓ 1
A1 17.3 cm	✓ 1
Part (b)
M1 $\frac{24(1-0.85^{8})}{1-0.85}$	✓ 1
A1 116 cm	✓ 1
Part (c)
M1 $\frac{24(1-0.85^{n})}{0.15} > 130$	✓ 1
A1 n > 10.3… (or S₁₀ = 128.5 and S₁₁ = 133.2)	✓ 1
A1 11 layers	✓ 1
Part (d)
M1 24 × 0.85ⁿ⁻¹ < 2	✓ 1
A1 n − 1 > 15.3… (or u₁₆ = 2.10 and u₁₇ = 1.78)	✓ 1
A1 layer 17	✓ 1
Part (e)
M1 11 × 35 = 385	✓ 1
A1 3.85 × 10² years	✓ 1
ECF from (c).	—

Common error: Using n instead of n − 1 as the exponent when finding the n-th term of a geometric sequence. The first term has exponent 0.

+ Add to test✓ In your test431IBMI3S-1-03r32-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Medium11 marks

A cyclist buys a road bike for €3200. Its value decreases by 15% each year.

(a)

Find the value of the bike after 4 years. Give your answer to the nearest euro.

(2)

(b)

Find the number of complete years after which the value of the bike first falls below €1000.

(3)

(c)

The cyclist decides to save for a replacement bike. She pays €50 into an account at the end of every month for 5 years. The account pays a nominal annual interest rate of 2.4%, compounded monthly. Find the amount in the account after 5 years. Give your answer to the nearest cent.

(2)

(d)

A new bike of the same model costs €3200 now, and its price increases by 3% each year. Find its price after 5 years. Give your answer to the nearest cent.

(2)

(e)

Find the extra amount the cyclist would need, in addition to the savings from part (c), to buy the new bike after 5 years. Use the answers to (c) and (d) as given to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $3200\times0.85^{4}$	✓ 1
A1 €1670	✓ 1
Part (b)
M1 $3200\times0.85^{n} < 1000$ (or equals sign)	✓ 1
M1 solve: $n > \dfrac{\ln(1000/3200)}{\ln0.85}$ or use a GDC table/graph	✓ 1
A1 7.16... so 8 years	✓ 1
Part (c)
M1 finance solver with N = 60, I% = 2.4, PV = 0, PMT = −50, P/Y = C/Y = 12 (or sum of a geometric series)	✓ 1
A1 €3184.04	✓ 1
Part (d)
M1 $3200\times1.03^{5}$	✓ 1
A1 €3709.68	✓ 1
Part (e)
M1 $ 3709.68 - 3184.04$ (ECF from (c) and (d))	✓ 1
A1 €525.64	✓ 1

Common error: A price that increases by 3% is multiplied by 1.03; using 0.03 or 0.97 gives the wrong direction or size of change.

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A small shop owner borrows €17 500 to buy refrigeration equipment. The loan is repaid by equal payments at the end of each month for 5 years. The nominal annual interest rate is 6.6%, compounded monthly.

(a)

Calculate the monthly repayment. Give your answer to the nearest cent.

(3)

(b)

Find the total amount repaid over the 5 years. Use the payment from (a), rounded to the nearest cent.

(2)

(c)

Hence write down the total interest paid.

(1)

(d)

The owner decides to repay the remaining loan in full immediately after the 24th payment. Find the amount owed at that moment, to the nearest euro.

(3)

(e)

Find the amount of interest saved by repaying early, compared with making all 60 payments. Give your answer to the nearest euro.

(2)

(f)

Calculate the effective annual interest rate. Give your answer as a percentage to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 N = 60, I% = 6.6, PV = 17 500, FV = 0, P/Y = 12, C/Y = 12	✓ 1
M1 payments at the end of each period, solving for PMT	✓ 1
A1 €343.23	✓ 1
Part (b)
M1 60 × 343.23	✓ 1
A1 €20593.80	✓ 1
Part (c)
A1 €3093.80 (ECF from (b))	✓ 1
Part (d)
M1 N = 36 remaining payments (or N = 24 with PV = 17 500 and FV unknown)	✓ 1
M1 uses PMT from (a) with I% = 6.6, C/Y = 12 to find PV (or FV)	✓ 1
A1 €11182	✓ 1
Part (e)
M1 36 × 343.23 − 11182 (or 36 × payment − balance)	✓ 1
A1 €1174	✓ 1
Part (f)
M1 (1 + 0.066/12)¹² − 1	✓ 1
A1 6.80%	✓ 1

Common error: The balance owed is the present value of the payments still to be made, not the amount borrowed minus the payments already made.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Medium12 marks

Solar farm A produces 4.80 GWh of electricity in year 1. Because the panels age, its output each year is 0.5% lower than in the previous year.

(a)

Write down the common ratio of the geometric sequence of annual outputs.

(1)

(b)

Find the output of farm A in year 10. Give your answer to 3 significant figures.

(2)

(c)

Find the total output of farm A over the first 25 years. Give your answer to 1 decimal place.

(2)

(d)

Find the first year in which the output of farm A is less than 4.2 GWh.

(3)

(e)

Farm B has an output of 5.10 GWh in year 1, and its output falls by 0.045 GWh each year. Find the total output of farm B over 25 years, and hence the amount by which this exceeds the total output of farm A. Use the unrounded total for farm A. Give your final answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.995	✓ 1
Part (b)
M1 4.8 × 0.995⁹	✓ 1
A1 4.59 GWh	✓ 1
Part (c)
M1 S₂₅ = 4.8(1 − 0.995²⁵)/(1 − 0.995)	✓ 1
A1 113.1 GWh	✓ 1
Part (d)
M1 4.8 × 0.995ⁿ⁻¹ < 4.2 (or equation, table or graph)	✓ 1
M1 n − 1 > ln(4.2/4.8)/ln 0.995 = 26.6… (accept 26.64)	✓ 1
A1 year 28	✓ 1
Part (e)
M1 S₂₅ = 25/2 × (2 × 5.1 + 24 × (−0.045))	✓ 1
A1 114 GWh (accept 114.0)	✓ 1
M1 114 − 113.1… (difference from the farm A total in (c), unrounded)	✓ 1
A1 0.931 GWh	✓ 1

Common error: A fall of 0.5% each year is a common ratio of 0.995, not 0.5 and not −0.005.

+ Add to test✓ In your test434IBMI3S-1-01r37-08Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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Calculator

A patient takes a 150 mg tablet at the same time every morning. Each day, 65% of the drug present in the body is eliminated, so 35% remains 24 hours later. The amount of drug in the body is measured just after each tablet is taken. Assume the tablet is absorbed immediately.

(a)

Find the amount of drug from the first tablet that is still in the body just before the fourth tablet is taken. Give your answer to three significant figures.

(2)

(b)

Show that the total amount of drug in the body just after the fourth tablet is 227 mg, to three significant figures.

(3)

(c)

If the patient continues for a very long time, the amount just after a tablet approaches a maximum value. Find this value to three significant figures.

(2)

(d)

Find the smallest number of tablets after which the amount in the body exceeds 99% of the maximum value from part (c). Use the unrounded maximum value.

(3)

(e)

Find the amount of drug in the body just after this tablet. Give your answer to the nearest 0.1 mg.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 150 × 0.35³	✓ 1
A1 6.43 mg	✓ 1
Part (b)
M1 recognise geometric series with first term 150 and common ratio 0.35	✓ 1
M1 S₄ = 150(1 − 0.35⁴)/(1 − 0.35)	✓ 1
A1 227.3… so 227 mg	✓ 1
Part (c)
M1 S_∞ = 150/(1 − 0.35)	✓ 1
A1 231 mg	✓ 1
Part (d)
M1 150(1 − 0.35ⁿ)/0.65 > 0.99 × 150/0.65, so 0.35ⁿ < 0.01	✓ 1
M1 solve, e.g. n > ln 0.01/ln 0.35 = 4.39 (or table of values)	✓ 1
A1 n = 5	✓ 1
Part (e)
M1 150(1 − 0.35⁵)/(1 − 0.35)	✓ 1
A1 229.6 mg (ECF from (d))	✓ 1

Exam tip: In a geometric series the sum to infinity only exists when |r| < 1; state the ratio before you use the formula.

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Calculator

A nurse prepares a dose of a drug for a patient. The correct dose is 0.075 mg per kg of body mass. The patient's body mass is measured as 64.8 kg, correct to the nearest 0.1 kg.

(a)

Using a mass of 64.8 kg, calculate the dose in mg, and write the dose in grams in the form a × 10^k, where 1 ≤ a < 10 and k is an integer.

(2)

(b)

The nurse's syringe delivers 4.9 mg. Taking 4.86 mg as the exact dose, find the percentage error in the delivered amount, to three significant figures.

(2)

(c)

Find the lower bound and the upper bound for the correct dose, in mg. Give exact values.

(3)

(d)

State, with a reason, whether a delivered dose of 4.9 mg lies within the range of possible correct doses.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4.86 mg	✓ 1
A1 4.86 × 10⁻³ g	✓ 1
Part (b)
M1 \|4.9 − 4.86\| ÷ 4.86 × 100	✓ 1
A1 0.823%	✓ 1
Part (c)
M1 mass bounds 64.75 kg and 64.85 kg identified	✓ 1
A1 lower bound 4.85625 mg	✓ 1
A1 upper bound 4.86375 mg	✓ 1
Part (d)
A1 No	✓ 1
R1 4.9 > 4.86375, so it is above the upper bound	✓ 1

Exam tip: Mass given to the nearest 0.1 kg is accurate to ±0.05 kg.

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Calculator

Sam is saving to buy a gaming PC costing £2400. She makes equal deposits at the end of each month into an account paying a nominal annual interest rate of 3.6%, compounded monthly.

(a)

Sam wants to have £2400 after 2 years. Find the monthly deposit she needs to make, to the nearest penny.

(3)

(b)

Using a deposit of £96.59, find the total amount Sam deposits in the 2 years, and the difference between £2400 and this total.

(2)

(c)

Sam instead deposits £90 each month. Find the smallest whole number of months after which she has at least £2400.

(2)

(d)

Find the amount in Sam's account after the number of months found in part (c). Give your answer to the nearest penny.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance solver set up with N = 24, I% = 3.6, PV = 0, P/Y = C/Y = 12	✓ 1
M1 FV = ±2400 and PMT solved for	✓ 1
A1 £96.59	✓ 1
Part (b)
A1 96.59 × 24 = £2318.16	✓ 1
A1 £81.84 (accept £81.77 to £81.84 from unrounded payment)	✓ 1
Part (c)
M1 solver with PMT = ±90 and FV = ±2400 giving N = 25.7…	✓ 1
A1 26 months	✓ 1
Part (d)
M1 N = 26, PMT = ±90, solve for FV	✓ 1
A1 £2429.89	✓ 1

Exam tip: Interest compounded monthly at a nominal annual rate uses a monthly rate of I/12 percent: set P/Y = C/Y = 12.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 2Medium9 marks

The radius r of the circle swept by a wind-turbine blade is measured as 52.4 m, correct to the nearest 0.1 m. The swept area is A = πr².

(a)

Calculate A using r = 52.4, to the nearest m².

(2)

(b)

Write down the lower bound and the upper bound for r.

(2)

(c)

Calculate the lower and upper bounds for A and hence show that A is 8600 m² when written to 2 significant figures.

(3)

(d)

The true radius is later found to be 52.43 m. Calculate the percentage error in the area found in (a), using the area for 52.43 m as the exact value. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 π × 52.4²	✓ 1
A1 8626 m²	✓ 1
Part (b)
A1 52.35 m	✓ 1
A1 52.45 m	✓ 1
Part (c)
M1 π × 52.35² = 8610 (8609.6…) and π × 52.45² = 8643 (8642.5…)	✓ 1
A1 both bounds round to 8600 to 2 s.f.	✓ 1
R1 so 2 s.f. is the greatest accuracy that can be justified (3 s.f. would be 8610 or 8640)	✓ 1
Part (d)
M1 π × 52.43² = 8635.9… and \|8626.06… − 8635.9…\| / 8635.9… × 100	✓ 1
A1 0.114%	✓ 1

Exam tip: The bounds of a number rounded to the nearest 0.1 are ±0.05 either side; the bounds of a calculation come from substituting the bounds, not by rounding the answer.

+ Add to test✓ In your test438IBMI3S-1-03r40-13Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium12 marks

A cylindrical grain silo has radius 3.2 m and height 9.5 m, each measured to the nearest 0.1 m. The volume of a cylinder is V = πr²h. Wheat stored in the silo has density 780 kg m⁻³.

(a)

Calculate the volume of the silo, in m³, using the measured values. Give your answer to 3 significant figures.

(2)

(b)

Calculate the mass of wheat in a full silo. Use the unrounded volume. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, to 3 significant figures.

(2)

(c)

The farmer says the silo holds about 300 m³. Calculate the percentage error in this estimate, taking the volume found in (a), unrounded, as the exact value. Give your answer to 3 significant figures.

(2)

(d)

Find the lower and upper bounds for the volume of the silo, each to 4 significant figures.

(3)

(e)

Use your bounds from (d) to find the greatest number of significant figures to which the volume of the silo can be stated with certainty. Justify your answer.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 π × 3.2² × 9.5	✓ 1
A1 306 m³	✓ 1
Part (b)
M1 305.614 × 780	✓ 1
A1 2.38 × 10⁵ kg	✓ 1
Part (c)
M1 \|300 − 305.614\| ÷ 305.614 × 100	✓ 1
A1 1.84%	✓ 1
Part (d)
M1 using radius bounds 3.15 and 3.25 and height bounds 9.45 and 9.55	✓ 1
A1 lower bound π × 3.15² × 9.45 = 294.6 m³	✓ 1
A1 upper bound π × 3.25² × 9.55 = 316.9 m³	✓ 1
Part (e)
M1 round the bounds to 2 significant figures: 290 and 320	✓ 1
R1 the two roundings differ, so 2 significant figures is not certain	✓ 1
A1 to 1 significant figure both bounds round to 300 m³, so the answer is 1 significant figure	✓ 1

Exam tip: To decide how many significant figures are justified, round both bounds: the answer may only keep digits on which the lower and upper bounds agree.

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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium13 marks

Two Formula 1 teams plan their annual wind-tunnel spending. Team X spends €8.0 million in year 1 and increases its spending by 15% each year (a geometric sequence). Team Y spends €8.0 million in year 1 and increases its spending by €1.2 million each year (an arithmetic sequence).

(a)

Find the amount spent by Team Y in year 6.

(2)

(b)

Find the amount spent by Team X in year 6. Give your answer to 3 significant figures.

(2)

(c)

Find the first year in which Team X spends more in that year than Team Y.

(3)

(d)

Find the first year in which the total spent by Team X since year 1 exceeds €100 million.

(3)

(e)

Find how much more Team X spends than Team Y in total over the first 10 years. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $8+5\times1.2$	✓ 1
A1 €14.0 million	✓ 1
Part (b)
M1 $8\times1.15^5$	✓ 1
A1 €16.1 million	✓ 1
Part (c)
M1 compare $8\times1.15^{n-1}$ with $8+1.2(n-1)$, using a GDC graph or table	✓ 1
A1 year 2 values equal (9.2 and 9.2), or year 3 values 10.58 and 10.4	✓ 1
A1 year 3	✓ 1
Part (d)
M1 $\frac{8(1.15^n-1)}{0.15}>100$ or a table of $S_n$	✓ 1
A1 $S_7=88.5$ and $S_8=109.8$	✓ 1
A1 year 8	✓ 1
Part (e)
M1 $S_{10}$ for Team X: $\frac{8(1.15^{10}-1)}{0.15}=162.43\ldots$	✓ 1
M1 $S_{10}$ for Team Y: $\frac{10}{2}(2\times8+9\times1.2)=134$	✓ 1
A1 €28.4 million	✓ 1

Common error: 'Spends more in that year' compares single terms, while 'total spent' compares sums of terms; do not mix the two.

+ Add to test✓ In your test440IBMI3S-1-01r42-10Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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An architect designs a tower in which the floor area decreases from one floor to the next. The ground floor (floor 1) has an area of 640 m² and floor 2 has an area of 588.8 m². The floor areas form a geometric sequence.

(a)

Show that the common ratio of the sequence is 0.92.

(2)

(b)

Find the area of floor 8. Give your answer to the nearest square metre.

(2)

(c)

Find the total floor area of the first 12 floors. Give your answer to the nearest square metre.

(3)

(d)

Floors are built only while the floor area is at least 300 m². Find the number of the highest floor that can be built.

(3)

(e)

The total floor area of the tower must not exceed 6 000 m². Find the greatest number of floors the tower can have.

(3)

(f)

Floor covering costs €85 per square metre. Using the unrounded total area of the first 12 floors from part (c), find the cost of covering all 12 floors. Give your answer to the nearest euro.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 588.8 / 640	✓ 1
A1 = 0.92 (shown)	✓ 1
Part (b)
M1 640 × 0.92⁷	✓ 1
A1 357 m² (357.0)	✓ 1
Part (c)
M1 attempt to use the sum formula S_n = 640(1 − 0.92ⁿ)/(1 − 0.92)	✓ 1
A1 correct substitution with n = 12	✓ 1
A1 5059 m² (5058.7)	✓ 1
Part (d)
M1 640 × 0.92ⁿ⁻¹ = 300 (or ≥ 300), or a table of values	✓ 1
A1 n − 1 = 9.09, so n = 10.09	✓ 1
A1 floor 10 (floor 10 has area 302.2 m², floor 11 has area 278.0 m²)	✓ 1
Part (e)
M1 640(1 − 0.92ⁿ)/0.08 ≤ 6 000, leading to 0.92ⁿ ≥ 0.25	✓ 1
A1 n ≤ 16.63 (from logarithms or a table)	✓ 1
A1 16 floors (total area of 16 floors is 5,893 m², of 17 floors 6,061 m²)	✓ 1
Part (f)
M1 85 × total area of 12 floors (ECF from (c))	✓ 1
A1 €429,987	✓ 1

Common error: The nth term of a geometric sequence is u₁rⁿ⁻¹, so floor 8 uses the power 7, not 8.

+ Add to test✓ In your test441IBMI3S-1-01r42-11Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium12 marks

A port handles 1 200 containers in its first year of operation (year 1). Two expansion plans are compared.
Plan A: the number of containers handled increases by 150 each year, so the yearly totals form an arithmetic sequence.
Plan B: the number of containers handled increases by 9 % each year, so the yearly totals form a geometric sequence.

(a)

Find the number of containers handled in year 10 under Plan A.

(2)

(b)

Find the number of containers handled in year 10 under Plan B. Give your answer to the nearest whole number.

(2)

(c)

Find the first year in which Plan B handles more containers than Plan A.

(3)

(d)

Find the total number of containers handled over the first 10 years under Plan A.

(2)

(e)

Find the total number of containers handled over the first 10 years under Plan B. Give your answer to the nearest whole number.

(2)

(f)

Using the unrounded total for Plan B, find how many more containers Plan A handles than Plan B over the 10 years. Give your answer to the nearest whole number.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1 200 + 9 × 150	✓ 1
A1 2550 containers	✓ 1
Part (b)
M1 1 200 × 1.09⁹	✓ 1
A1 2606 containers	✓ 1
Part (c)
M1 comparing 1 200 + 150(n − 1) with 1 200 × 1.09ⁿ⁻¹, using a table or graph	✓ 1
A1 year 9: Plan A 2400, Plan B 2391 (Plan B is still lower)	✓ 1
A1 year 10	✓ 1
Part (d)
M1 (10/2)(2 × 1 200 + 9 × 150)	✓ 1
A1 18750 containers	✓ 1
Part (e)
M1 1 200(1.09¹⁰ − 1)/(1.09 − 1)	✓ 1
A1 18232 containers	✓ 1
Part (f)
A1 518 containers (ECF)	✓ 1

Exam tip: When two sequences are compared, a table of both sequences side by side shows the crossover year and the margin at a glance.

+ Add to test✓ In your test442IBMI3S-1-02r43-04Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium13 marks

A fashion boutique borrows €24 000 to buy cutting equipment. The loan has a nominal annual interest rate of 6.6%, compounded monthly, and is repaid by equal payments at the end of each month for 5 years.

(a)

Calculate the monthly payment. Give your answer to the nearest cent.

(3)

(b)

Calculate the total interest paid over the 5 years. Use the unrounded monthly payment.

(2)

(c)

The boutique decides instead to pay €500 each month. Find the least number of monthly payments needed to repay the loan.

(3)

(d)

With the original monthly payment, calculate the balance of the loan immediately after the 24th payment. Use the unrounded monthly payment and give your answer to the nearest cent.

(3)

(e)

Find the balance in (d) as a percentage of the original loan. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM solver (or annuity formula) with N = 60, I = 6.6, PV = 24 000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 correct inputs (monthly rate 0.55%)	✓ 1
A1 €470.71 (470.7125… before rounding)	✓ 1
Part (b)
M1 60 × 470.7125… − 24 000	✓ 1
A1 €4242.75	✓ 1
Part (c)
M1 solving for N with PMT = −500 (TVM solver or formula)	✓ 1
A1 N = 55.88…	✓ 1
A1 56 payments (the final payment is smaller than €500)	✓ 1
Part (d)
M1 TVM with N = 24 (or 24 000 × 1.0055²⁴ − payment × (1.0055²⁴ − 1)/0.0055)	✓ 1
A1 correct inputs	✓ 1
A1 €15 335.38	✓ 1
Part (e)
M1 15335.38 / 24000 × 100 (ECF from (d))	✓ 1
A1 63.9%	✓ 1

Exam tip: State the TVM inputs you used (N, I%, PV, PMT, FV, P/Y, C/Y); the signs of PV and PMT must be opposite. Keep the unrounded payment in your calculator for later parts.

+ Add to test✓ In your test443IBMI3S-1-02r43-14Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium12 marks

A solar-powered rover uses a power of 72 W for 4.2 hours. Both values are given correct to 2 significant figures.

(a)

Write down the lower and upper bounds of the power and of the time.

(2)

(b)

Calculate the energy used, in watt-hours (Wh), using the given values. Give your answer to 3 significant figures.

(2)

(c)

Convert the energy in (b), using the unrounded value, to joules, given that 1 Wh = 3600 J. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(2)

(d)

Find the lower and upper bounds for the energy used, giving each to 3 significant figures.

(3)

(e)

The value 302.4 Wh is used as an estimate of the true energy. Find the largest possible percentage error in this estimate. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 power: 71.5 ≤ P < 72.5 W	✓ 1
A1 time: 4.15 ≤ t < 4.25 hours	✓ 1
Part (b)
M1 72 × 4.2	✓ 1
A1 302 Wh (302.4 before rounding)	✓ 1
Part (c)
M1 302.4 × 3600 = 1 088 640	✓ 1
A1 1.09 × 10⁶ J	✓ 1
Part (d)
M1 lower bound = 71.5 × 4.15 (or upper = 72.5 × 4.25)	✓ 1
A1 lower bound 297 Wh (296.725)	✓ 1
A1 upper bound 308 Wh (308.125)	✓ 1
Part (e)
M1 percentage error = \|302.4 − true\| / true × 100, evaluated at the lower bound (296.725)	✓ 1
M1 comparison with the upper bound: (308.125 − 302.4)/308.125 × 100 = 1.86%	✓ 1
A1 largest percentage error 1.91% (1.9126… before rounding)	✓ 1

Common error: To find the greatest possible error, test both ends of the interval: dividing by the smaller true value gives the larger percentage error.

+ Add to test✓ In your test444IBMI3S-1-04r44-10Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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A telecoms engineer saves by paying €250 into an account at the end of every month for 8 years. The account pays a nominal annual interest rate of 3.0%, compounded monthly.

(a)

Find the value of the account after 8 years. Give your answer to the nearest euro.

(3)

(b)

Find the total interest earned over the 8 years. Give your answer to the nearest euro. Use the unrounded value from (a).

(2)

(c)

Find the minimum number of whole months of payments needed for the account to reach at least €30 000.

(3)

(d)

Find the single amount that would need to be invested at the start for 8 years, at the same rate, to give the same value as in (a). Give your answer to the nearest euro. Use the unrounded value from (a).

(3)

(e)

The engineer wants the account to reach exactly €30 000 after 8 years using equal monthly payments. Find the monthly payment required. Give your answer to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a finance solver with N = 96, I% = 3, PV = 0, PMT = −250, P/Y = C/Y = 12	✓ 1
A1 correct values entered (rate per month 0.25%)	✓ 1
A1 €27 087	✓ 1
Part (b)
M1 total paid in = 250 × 96 = €24 000	✓ 1
A1 interest = €3087	✓ 1
Part (c)
M1 250((1.0025)ⁿ − 1)/0.0025 = 30 000 (or solver with FV = 30 000)	✓ 1
M1 solve for n: n = 105.08…	✓ 1
A1 106 months	✓ 1
Part (d)
M1 PV = 27 086.8…/1.0025⁹⁶ (or solver with N = 96, FV = 27 086.8…, PMT = 0)	✓ 1
A1 correct substitution	✓ 1
A1 €21 314	✓ 1
Part (e)
M1 payment = 250 × 30 000/27 086.8… (or solver with FV = 30 000)	✓ 1
A1 €276.89	✓ 1

Exam tip: Check the sign convention on your calculator's finance solver: payments you make are entered as negative values, and the future value is then positive.

+ Add to test✓ In your test445IBMI3S-1-02r45-06Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Medium12 marks

A Formula 1 team takes out a loan of €450 000 to buy equipment. The loan has a nominal annual interest rate of 6%, compounded monthly. The team repays the loan with equal payments at the end of every month for 5 years.

(a)

Find the monthly payment, to the nearest cent.

(3)

(b)

Using your answer to (a), find the total amount repaid over the 5 years.

(2)

(c)

Hence find the total interest paid.

(1)

(d)

Find the amount still owed immediately after the 24th payment. Use the payment from (a) and give your answer to the nearest euro.

(3)

(e)

Immediately after the 24th payment, the team makes an extra one-off payment of €100 000, and the remaining 36 monthly payments are recalculated so that the loan is repaid on time. Using your rounded answer to (d), find the new monthly payment, to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 identify N = 60 and monthly rate 0.5%	✓ 1
M1 use of TVM solver (PV = 450 000, FV = 0, P/Y = C/Y = 12) or the annuity formula	✓ 1
A1 €8699.76	✓ 1
Part (b)
M1 60 × 8699.76	✓ 1
A1 €521 985.60	✓ 1
Part (c)
A1 €71 985.60 (ECF from (b))	✓ 1
Part (d)
M1 TVM solver with N = 24 (or 450 000 × 1.005²⁴ − 8699.76 × (1.005²⁴ − 1)/0.005)	✓ 1
A1 €285 969.99	✓ 1
A1 €285 970	✓ 1
Part (e)
M1 285 970 − 100 000 = 185 970	✓ 1
M1 TVM solver with N = 36, PV = 185 970, FV = 0	✓ 1
A1 €5657.57	✓ 1

Exam tip: A repayment question is solved by setting the final value to zero; show the values you enter into the finance solver.

+ Add to test✓ In your test446IBMI3S-1-01r47-05Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium9 marks

After a race, the temperature of a Formula 1 tyre, T °C, t minutes after the car stops is modelled by
T = 25 + 75e^−0.08t, for t ≥ 0.

(a)

Write down the temperature of the tyre when the car stops.

(2)

(b)

Find the temperature of the tyre 10 minutes after the car stops. Give your answer to 3 significant figures.

(2)

(c)

Find the time taken for the tyre to cool to 40 °C. Give your answer in minutes to 3 significant figures.

(3)

(d)

State the temperature that the tyre approaches in the long term, and give its meaning in this context.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 0 into the model	✓ 1
A1 100 °C	✓ 1
Part (b)
M1 substitutes t = 10	✓ 1
A1 58.7 °C	✓ 1
Part (c)
M1 sets 25 + 75e^−0.08t = 40	✓ 1
M1 isolates the exponential e^−0.08t = 0.2 (or uses a graph or solver)	✓ 1
A1 20.1 minutes	✓ 1
Part (d)
A1 25 °C	✓ 1
R1 the temperature of the surroundings (air temperature)	✓ 1

Exam tip: To undo ekt take natural logarithms of both sides, but first isolate the exponential term. A graphing check on the calculator confirms the answer.

+ Add to test✓ In your test447IBMI3S-1-01r47-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium12 marks

Two beekeepers, A and B, each harvested 80 kg of honey in their first year. Beekeeper A's yearly harvest increases by a fixed amount, d kg, every year, so the yearly harvests form an arithmetic sequence. Over the first 10 years, A harvests a total of 1250 kg.
Beekeeper B's yearly harvest increases by 9% every year.

(a)

Find the value of d.

(3)

(b)

Find A's harvest in year 10.

(1)

(c)

Find the first year after year 1 in which B's harvest is greater than A's.

(3)

(d)

Find the total amount of honey harvested by B in the first 10 years. Give your answer to the nearest 0.1 kg.

(3)

(e)

Beekeeper B says: 'My yearly harvest is now larger than A's, so I must have harvested more in total.' Use your answers to comment on this claim.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of S₁₀ = 5(2 × 80 + 9d) (or equivalent)	✓ 1
M1 sets 5(160 + 9d) = 1250	✓ 1
A1 d = 10	✓ 1
Part (b)
A1 170 kg (ECF from (a), 80 + 9d)	✓ 1
Part (c)
M1 B's harvest in year n is 80 × 1.09ⁿ⁻¹ and A's is 80 + 10(n − 1) (ECF)	✓ 1
M1 compares by table, graph or inequality	✓ 1
A1 year 10 (B = 173.75… kg, A = 170 kg; in year 9 B = 159.4… kg, A = 160 kg)	✓ 1
Part (d)
M1 use of S_n = u₁(rⁿ − 1)/(r − 1) with u₁ = 80, r = 1.09, n = 10	✓ 1
A1 correct substitution	✓ 1
A1 1215.4 kg	✓ 1
Part (e)
R1 compares totals: A harvested 1250 kg and B harvested 1215.4 kg	✓ 1
R1 the claim is incorrect, because B's harvest was smaller than A's in earlier years, so the total is still smaller	✓ 1

Common error: The nth term of a geometric sequence is u₁rⁿ⁻¹, not u₁rⁿ. Check by testing n = 1.

+ Add to test✓ In your test448IBMI3S-1-01r48-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Medium12 marks

Calculator

A company borrows €250 000 to buy cooling equipment for a data centre. The loan has a nominal annual interest rate of 6%, compounded monthly. It is repaid by equal monthly payments made at the end of each month for 5 years.

(a)

Find the amount of each monthly payment, correct to the nearest cent.

(3)

(b)

Find the total amount repaid over the 5 years. Use the payment rounded to the nearest cent from part (a).

(2)

(c)

Find the total interest paid over the 5 years.

(1)

(d)

Find the amount still owed immediately after the 24th payment. Give your answer to the nearest euro.

(3)

(e)

Express the total interest paid as a percentage of the total amount repaid. Use your answers to parts (b) and (c). Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 N=60, I%=6, PV=250 000, FV=0, P/Y=C/Y=12 (TVM solver) or use of annuity formula	✓ 1
M1 correct equation in the payment, e.g. $250\,000=\text{PMT}\times\frac{1-1.005^{-60}}{0.005}$	✓ 1
A1 €4,833.20	✓ 1
Part (b)
M1 $60\times4,833.20$	✓ 1
A1 €289,992.00 (ECF from (a))	✓ 1
Part (c)
A1 €39,992.00 (ECF from (b))	✓ 1
Part (d)
M1 TVM with N=36 remaining payments (or N=24 with FV unknown)	✓ 1
M1 correct calculation of balance, e.g. $250\,000\times1.005^{24}-\text{PMT}\times\frac{1.005^{24}-1}{0.005}$	✓ 1
A1 €158,872	✓ 1
Part (e)
M1 $\frac{\text{interest}}{\text{total repaid}}\times100$	✓ 1
M1 correct substitution, e.g. $\frac{39,992.00}{289,992.00}\times100$	✓ 1
A1 13.8% (ECF from (b) and (c))	✓ 1

Common error: With monthly payments the rate per period is $\frac{6}{12}=0.5\%$ and $N=60$; entering $N=5$ or I%=0.5 with P/Y=1 gives the wrong payment.

+ Add to test✓ In your test449IBMI-030-00-11Approximation, percentage error and standard form Paper 2Hard13 marks

A city has a population of 2.4×10⁶, correct to two significant figures. Each person uses on average 145 litres of water per day, correct to three significant figures. 1 m³ = 1000 litres.
The city's reservoir holds exactly 4.2×10⁷ m³ of water.

(a)

Using the given values, calculate the total daily water use of the city in litres and in m³. Give both answers in the form $a\times10^k$ where $1\le a<10$ and $k\in\mathbb{Z}$.

(3)

(b)

Write down the lower and upper bounds of the population and of the daily use per person. Hence find the upper bound of the city's daily water use in m³.

(4)

(c)

Assuming no rainfall and no other supply, find the number of complete days for which the reservoir is guaranteed to supply the city.

(2)

(d)

A planner calculates the supply time using the daily water use in m³ found in the first part of this question. Calculate the percentage error of the planner's value compared with the guaranteed supply time of 117.8... days found in the previous part. Comment on the planner's method in context.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 for 2.4×10⁶ × 145	✓ 1
A1 3.48×10⁸ litres	✓ 1
A1 3.48×10⁵ m³	✓ 1
Part (b)
A1 population: 2.35×10⁶ and 2.45×10⁶	✓ 1
A1 use per person: 144.5 and 145.5 litres	✓ 1
M1 for 2.45×10⁶ × 145.5 ÷ 1000	✓ 1
A1 3.56×10⁵ m³ (3.56475×10⁵)	✓ 1
Part (c)
M1 for $\frac{4.2\times10^{7}}{3.56475\times10^{5}}=117.8...$	✓ 1
A1 117 complete days	✓ 1
Part (d)
M1 for $\frac{4.2\times10^{7}}{3.48\times10^{5}}=120.7$ days	✓ 1
M1 for $\left\|\frac{120.69...-117.82...}{117.82...}\right\|\times100$	✓ 1
A1 2.44 %	✓ 1
R1 the planner overestimates how long the water will last, so plans based on it could leave the city short; the worst-case (upper bound) usage should be used	✓ 1

Exam tip: For a guaranteed number of days, divide by the upper bound of usage and round down: 117.8 days gives only 117 complete days.

+ Add to test✓ In your test450IBMI-030-06-r1-06Financial mathematics, loans and annuities Paper 2Hard12 marks

Aisha borrows £30 000 to pay for a postgraduate course. She repays the loan with payments of £438.26 at the end of each month for 7 years. Interest is charged at a nominal annual rate of r%, compounded monthly.

(a)

Show that r = 6.00, correct to 3 significant figures.

(2)

(b)

Using r = 6, find the amount Aisha still owes immediately after her 36th payment. Give your answer to the nearest pound.

(3)

(c)

Immediately after her 36th payment, Aisha pays a lump sum of £7500 off the loan. She continues paying £438.26 per month, with a smaller final payment if needed.
Find the number of further payments she makes, and hence state how many months earlier the loan is repaid than originally planned.

(4)

(d)

Instead, Aisha decides to keep the original end date after paying the £7500 lump sum. Find her new monthly payment.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM entry: N = 84, PV = 30 000, PMT = −438.26, FV = 0, C/Y = P/Y = 12, solving for I%	✓ 1
A1 I% = 6.0004… so r = 6.00	✓ 1
Part (b)
M1 N = 36 solving for FV, or the present value of the 48 remaining payments	✓ 1
M1 correct values entered: PV = 30 000, PMT = −438.26, I% = 6	✓ 1
A1 £18 661	✓ 1
Part (c)
A1 new balance = 18 661 − 7500 = £11 161	✓ 1
M1 PV = 11 161, PMT = −438.26, I% = 6, FV = 0, solving for N	✓ 1
A1 N = 27.3…, so 28 further payments (27 full payments and one smaller payment)	✓ 1
A1 48 − 28 = 20 months earlier	✓ 1
Part (d)
M1 PV = 11 161 (their balance after the lump sum), N = 48	✓ 1
M1 I% = 6, FV = 0, solving for PMT	✓ 1
A1 £262.12	✓ 1

Common error: A non-integer N, such as 27.3, means a final part payment. That still counts as a payment, so the loan takes 28 months, not 27.

+ Add to test✓ In your test451IBMI-030-02-r1-06Arithmetic and geometric sequences in context Paper 2Hard12 marks

A patient takes a 250 mg dose of a medication every 8 hours. By the time the next dose is due, 60% of the medication in the body has been eliminated, so 40% remains. Let $A_n$ be the amount of medication in the body, in mg, immediately after the nth dose.

(a)

Show that $A_3=390$ mg.

(2)

(b)

Find $A_6$.

(2)

(c)

The medication is effective only once the amount immediately after a dose is at least 400 mg. Find how many hours after the first dose this first happens.

(3)

(d)

Find the long-term amount immediately after a dose, according to the model.

(2)

(e)

Instead, the doctor gives a larger first dose of L mg, followed by 250 mg every 8 hours. Find the value of L for which the amount immediately after every dose is the same.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$A_3=250+250(0.4)+250(0.4)^2$ or $A_2=350$, $A_3=0.4\times350+250$ M1 ; $250+100+40=390$ A1	✓ 2
Part (b)
Uses $\frac{250(1-0.4^{6})}{1-0.4}$ M1 ; 415 mg (414.96) A1	✓ 2
Part (c)
Forms $\frac{250(1-0.4^{n})}{0.6}\ge400$ or tests values: $A_3=390$, $A_4=406$ M1 ; $n=4$ A1 ; 24 hours after the first dose A1	✓ 3
Part (d)
Uses the sum to infinity $\frac{250}{1-0.4}$ M1 ; 417 mg (416.67) A1	✓ 2
Part (e)
Forms $0.4L+250=L$ M1 ; rearranges to $0.6L=250$ M1 ; $L=417$ mg (416.67) A1	✓ 3

Exam tip: In part (c) the answer needed is a time, not a number of doses. The 4th dose is taken after 3 intervals of 8 hours, so it is 24 hours after the first dose, not 32 hours.

+ Add to test✓ In your test452IBMI-030-07-r1-06Arithmetic and geometric sequences in context Paper 2Hard13 marks

A fish farm has 8000 fish in a lake at the start. At the end of each year, 10% of the fish in the lake are harvested, and then 500 young fish are added. Let $F_n$ be the number of fish after $n$ years, so $F_0 = 8000$ and $F_n = 0.9F_{n-1} + 500$. Assume no fish die or are born naturally.

(a)

Find $F_1$ and $F_2$.

(2)

(b)

Show that $F_n = 8000(0.9)^n + 500\left(1 + 0.9 + 0.9^2 + \dots + 0.9^{n-1}\right)$, and hence that $F_n = 5000 + 3000(0.9)^n$.

(3)

(c)

Find the number of fish in the lake after 10 years, to the nearest whole fish.

(2)

(d)

Find the least number of years after which there are fewer than 5200 fish in the lake.

(3)

(e)

Describe what happens to the number of fish in the long term. The farmer wants the long-term population to be 6000 fish, still harvesting 10% each year. Find the number of young fish that should be added each year.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$F_1 = 0.9(8000) + 500 = 7700$ A1	✓ 1
$F_2 = 0.9(7700) + 500 = 7430$ A1	✓ 1
Part (b)
Expands the rule, e.g. $F_2 = 0.9^2(8000) + 500(0.9) + 500$, and generalises the pattern R1	✓ 1
Sums the geometric series: $500 \times \frac{1-0.9^{n}}{1-0.9} = 5000(1-0.9^{n})$ M1	✓ 1
$F_n = 8000(0.9)^n + 5000 - 5000(0.9)^n = 5000 + 3000(0.9)^n$ A1	✓ 1
Part (c)
$5000 + 3000(0.9)^{10}$ M1	✓ 1
$= 6046$ fish A1	✓ 1
Part (d)
$5000 + 3000(0.9)^n < 5200 \Rightarrow 0.9^{n} < \frac{1}{15}$ M1	✓ 1
$n > 25.7$ (by logarithms or GDC) A1	✓ 1
26 years A1	✓ 1
Part (e)
$0.9^{n} \to 0$, so the population decreases towards 5000 fish (the harvest of 10% then balances the 500 added) R1	✓ 1
In the long term, $10\%$ of the population equals the number added: $0.1 \times 6000 = J$, or $\frac{J}{0.1} = 6000$ M1	✓ 1
$J = 600$ fish per year A1	✓ 1

Exam tip: A 'show that' answer must reach the printed result through visible algebra. Write the geometric sum formula and simplify it step by step; do not just check a value of $n$.

+ Add to test✓ In your test453IBMI-030-02-r2-06Arithmetic and geometric sequences in context Paper 2Hard14 marks

Two factories are reducing their carbon dioxide emissions.
Factory A emits 48 000 tonnes in year 1, and its annual emission decreases by 8% each year.
Factory B emits 40 000 tonnes in year 1, and its annual emission decreases by 1500 tonnes each year.

(a)

Find the annual emission of Factory A in year 5.

(2)

(b)

Find the total emission of Factory A over the first 10 years.

(2)

(c)

Find the first year in which the annual emission of Factory A is less than that of Factory B.

(3)

(d)

Find the number of years needed for the total emission of Factory A to first exceed 450 000 tonnes.

(3)

(e)

Find the total emission of Factory B over the first 20 years, and explain why this model cannot be used to find the total emission of Factory B over the first 30 years.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$48000 \times 0.92^{4}$ M1	✓ 1
= 34 400 tonnes (34 386.9) A1	✓ 1
Part (b)
$S_{10} = \frac{48000(1 - 0.92^{10})}{1 - 0.92}$ M1	✓ 1
= 339 000 tonnes (339 367) A1	✓ 1
Part (c)
Sets up $48000 \times 0.92^{n-1} < 40000 - 1500(n-1)$ M1	✓ 1
Year 5: A = 34 387, B = 34 000; year 6: A = 31 636, B = 32 500 A1	✓ 1
Year 6 A1	✓ 1
Part (d)
$\frac{48000(1 - 0.92^{n})}{0.08} > 450000$ M1	✓ 1
$0.92^{n} < 0.25$, giving $n > 16.6$ A1	✓ 1
17 years A1	✓ 1
Part (e)
$S_{20} = \frac{20}{2}(2\times 40000 - 19\times 1500)$ M1	✓ 1
= 515 000 tonnes A1	✓ 1
$40000 - 1500(n-1) < 0$ when $n > 27.7$ M1	✓ 1
So the model predicts negative annual emissions from year 28, which is impossible R1	✓ 1

Common error: Giving 16.6 years as the answer. The total only exceeds 450 000 tonnes once a whole year is complete, so the result must be rounded up to 17 years.

+ Add to test✓ In your test454IBMI-030-08-r1-06Exponents and logarithms Paper 2Hard14 marks

A cup of black coffee is left on a table in a room. Its temperature, $T$ °C, $t$ minutes after it is poured is modelled by $T = 21 + 69e^{-kt}$, where $k$ is a positive constant.
Five minutes after it is poured, the temperature of the coffee is 78 °C.

(a)

Write down the temperature of the coffee when it is poured.

(1)

(b)

Find the value of $k$.

(3)

(c)

Find the temperature of the coffee 20 minutes after it is poured.

(2)

(d)

The coffee is pleasant to drink once its temperature falls to 60 °C. Find how long after pouring this happens.

(3)

(e)

State what happens to the temperature of the coffee in the long term, and interpret this value in context.

(2)

(f)

At the same moment, a cup of coffee with milk is poured at 75 °C. Its temperature is modelled by $T = 21 + 54e^{-0.031t}$. Using your value of $k$, find the time at which the two cups of coffee are at the same temperature.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: 90 °C	✓ 1
Part (b)
M1: substitutes t = 5 and T = 78: 78 = 21 + 69e^−5k	✓ 1
M1: rearranges to e^−5k = 57/69 and takes ln, or uses a GDC solver	✓ 1
A1: k = 0.0382 (0.038211…)	✓ 1
Part (c)
M1: substitutes t = 20 with their k	✓ 1
A1: T = 53.1 °C	✓ 1
Part (d)
M1: sets up 21 + 69e^−kt = 60	✓ 1
M1: rearranges to e^−kt = 39/69 and takes ln, or uses a GDC solver	✓ 1
A1: t = 14.9 minutes	✓ 1
Part (e)
A1: T approaches 21 °C, because e^−kt → 0 as t increases	✓ 1
R1: 21 °C is the temperature of the room, so the coffee cools to room temperature	✓ 1
Part (f)
M1: sets the models equal: 69e^−kt = 54e^−0.031t	✓ 1
M1: takes ln or uses a GDC to obtain (k − 0.031)t = ln(69/54)	✓ 1
A1: t = 34.0 minutes	✓ 1

Exam tip: Store the unrounded value of k in your GDC and use it in later parts. If you use a rounded value, accuracy marks can be lost through rounding errors.

+ Add to test✓ In your test455IBMI-030-07-r2-06Arithmetic and geometric sequences in context Paper 2Hard14 marks

A lake is stocked with 5000 fish. Each year the fish population grows by 8%, and then at the end of the year 600 fish are removed by licensed anglers. Let $P_n$ be the number of fish at the end of year $n$, after the fish have been removed, with $P_0=5000$.

(a)

Show that $P_1=4800$.

(2)

(b)

Find $P_3$.

(2)

(c)

Show that $P_n=5000(1.08)^n-600\left(1+1.08+1.08^2+\dots+1.08^{n-1}\right)$, and hence that $P_n=7500-2500(1.08)^n$.

(4)

(d)

Using the formula from part (c), find the year in which the fish population would be wiped out.

(3)

(e)

The lake authority wants the population never to fall below 5000. Find the largest number of fish, $R$, that can be removed each year, and justify your answer.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $5000\times1.08-600$	✓ 1
A1: $5400-600=4800$ (AG)	✓ 1
Part (b)
M1: repeated use of $P_{n+1}=1.08P_n-600$, e.g. $P_2=4584$	✓ 1
A1: $P_3=4350.72$, so about 4351 fish (accept 4350.72, 4351 or 4350)	✓ 1
Part (c)
M1: expands at least two steps, e.g. $P_2=5000(1.08)^2-600(1.08)-600$	✓ 1
A1: generalises correctly to the given expression	✓ 1
M1: uses the sum of a geometric series, $\frac{1.08^n-1}{0.08}$, so that 600 times the bracket becomes $7500(1.08^n-1)$	✓ 1
A1: $5000(1.08)^n-7500(1.08)^n+7500=7500-2500(1.08)^n$ (AG)	✓ 1
Part (d)
M1: sets $7500-2500(1.08)^n=0$, i.e. $1.08^n=3$	✓ 1
A1: $n=\frac{\ln3}{\ln1.08}=14.3$	✓ 1
A1: during year 15, since $P_{14}\approx157>0$ and $P_{15}\approx-430<0$	✓ 1
Part (e)
M1: requires the yearly growth to be at least the number removed, $0.08\times5000\geq R$ (or $P_1=5400-R\geq5000$)	✓ 1
A1: $R=400$	✓ 1
R1: with $R=400$, each year's growth of 400 exactly replaces the fish removed, so the population stays at 5000. Any larger $R$ makes $P_1<5000$, and the population then decreases every year	✓ 1

Common error: Treating the removal as a percentage, or subtracting the 600 fish before applying the 8% growth. Follow the order stated in the context: first grow, then remove.

+ Add to test✓ In your test456IBMI-030-03-r1-06Exponents and logarithms Paper 2Hard14 marks

A patient is given doses of a medicine. After a single dose of D mg, the amount of medicine remaining in the body t hours later is modelled by $A(t) = De^{-kt}$. The half-life of the medicine is 6 hours.

(a)

Show that k ≈ 0.116.

(2)

(b)

A single dose of 400 mg is given. Find the amount remaining after 10 hours.

(2)

(c)

Instead, the patient receives a 400 mg dose every 6 hours. Find the amount of medicine in the body immediately after the 4th dose.

(2)

(d)

Show that the amount immediately after the nth dose is $800(1 - 0.5^n)$ mg, and state the long-term value this amount approaches.

(2)

(e)

For safety, the amount immediately after any dose must not exceed 780 mg. Find the maximum number of doses that can be given on this schedule.

(3)

(f)

A doctor proposes giving 400 mg doses every T hours indefinitely, so that the amount immediately after a dose never exceeds 600 mg. Find the minimum value of T.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
e^−6k = 0.5 M1	✓ 1
k = ln 2 / 6 = 0.11552… ≈ 0.116 A1	✓ 1
Part (b)
400e^−10k (= 400 × 2^−10/6) M1	✓ 1
= 126 mg (125.99…); accept 125 mg from k = 0.116 A1	✓ 1
Part (c)
400 + 200 + 100 + 50, since each earlier dose halves every 6-hour interval M1	✓ 1
= 750 mg A1	✓ 1
Part (d)
Geometric series with u₁ = 400 and r = 0.5 (the fraction left after one half-life): S_n = 400(1 − 0.5ⁿ)/(1 − 0.5) = 800(1 − 0.5ⁿ) M1	✓ 1
As n → ∞, 0.5ⁿ → 0, so the amount approaches 800 mg A1	✓ 1
Part (e)
800(1 − 0.5ⁿ) ≤ 780 M1	✓ 1
0.5ⁿ ≥ 0.025 ⇒ n ≤ 5.32 M1	✓ 1
Maximum of 5 doses A1	✓ 1
Part (f)
The amount after each dose increases towards the sum to infinity 400/(1 − e^−kT), with r = e^−kT M1	✓ 1
400/(1 − e^−kT) ≤ 600 ⇒ e^−kT ≤ 1/3 M1	✓ 1
Minimum T = 6 ln 3 / ln 2 = 9.51 hours; accept 9.47 hours from k = 0.116 A1	✓ 1

Exam tip: When doses repeat at equal intervals, the amounts form a geometric series whose common ratio is the fraction remaining over one interval; the sum to infinity gives the long-term level.

+ Add to test✓ In your test457IBMI-030-01-r1-06Financial mathematics, loans and annuities Paper 2Hard14 marks

Ravi is 40 years old. He wants to retire at 65 and then receive $2500 at the end of every month for 25 years. His pension fund earns a nominal annual interest rate of 4.5%, compounded monthly, both before and after retirement. He already has $40 000 in the fund. He will add an equal deposit at the end of every month for the 25 years until he retires.

(a)

Find the amount the fund must contain when Ravi retires.

(3)

(b)

Find the monthly deposit Ravi must make until he retires.

(4)

(c)

Find the total amount Ravi pays into the fund, including the initial $40 000. Hence find the total interest earned over the 50 years.

(3)

(d)

Inflation is expected to average 2% per year. Find the value, in today's money, of a $2500 payment made 25 years from now.

(2)

(e)

Comment on whether Ravi's plan meets his needs, referring to your answer to the previous part.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Recognises a present value of an annuity with PMT = 2500	✓ 1
N = 300, I% = 4.5, FV = 0, P/Y = C/Y = 12	✓ 1
$449 775.80	✓ 1
Part (b)
Value of the existing $40 000 after 300 months: $40000(1.00375)^{300}$	✓ 1
= $122 949.70	✓ 1
TVM with N = 300, PV = −40 000, FV = 449 775.80, solving for PMT; or the remaining $326 826.10 equated to the future value of the deposits	✓ 1
$591.01	✓ 1
Part (c)
Total paid in = 300 × 591.01 + 40 000	✓ 1
= $217 303	✓ 1
Total received = 300 × 2500 = $750 000, so interest ≈ $532 697	✓ 1
Part (d)
$\frac{2500}{1.02^{25}}$	✓ 1
$1523.83	✓ 1
Part (e)
A $2500 payment at 65 buys only about as much as $1524 does today	✓ 1
Payments later in retirement are worth even less in real terms, so Ravi may need payments that rise with inflation, or higher deposits	✓ 1

Common error: The $40 000 already in the fund also earns interest for 25 years. Leaving out its growth makes the required monthly deposit far too large.

+ Add to test✓ In your test458IBMI3S-1-01-13Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

The area covered by algae on a pond is modelled by $A=A_0b^{t}$ square metres, where $t$ is the time in days since the start of observation. The area was 9.0 m² after 2 days and 30.375 m² after 5 days.

(a)

Show that $b=1.5$ and $A_0=4$.

(4)

(b)

Find the time at which the area first reaches 200 m². Give your answer in days to 3 significant figures.

(3)

(c)

The model can be written as $A=4e^{kt}$. Find the value of $k$, to 3 significant figures.

(2)

(d)

The pond has a surface area of 5000 m². The algae become hazardous when they cover 10% of the surface. Find the time, in days to 3 significant figures, at which this happens.

(2)

(e)

State one reason why the model is unlikely to remain valid for large values of $t$.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 dividing: $b^{3}=\frac{30.375}{9.0}$	✓ 1
A1 $b^{3}=3.375$ so $b=1.5$	✓ 1
M1 $A_0(1.5)^2=9$	✓ 1
A1 $A_0=4$ (AG)	✓ 1
Part (b)
M1 $4(1.5)^{t}=200$	✓ 1
M1 taking logarithms: $t=\frac{\ln50}{\ln1.5}$	✓ 1
A1 9.65 days	✓ 1
Part (c)
M1 $e^{k}=1.5$ or $k=\ln1.5$	✓ 1
A1 0.405	✓ 1
Part (d)
M1 $4(1.5)^{t}=500$	✓ 1
A1 11.9 days	✓ 1
Part (e)
R1 e.g. the area cannot exceed the surface area of the pond / growth is limited by nutrients and space	✓ 1

Exam tip: In a 'show that' question every line of working must be written down, and the final line must match the stated result exactly.

+ Add to test✓ In your test459IBMI3S-1-01b-14Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard14 marks

On a guitar the vibrating length of a string is shortened by pressing it against a fret. The vibrating length L_n mm when the string is pressed at fret n is given by L_n = 650 × 2^−n/12, with L₀ = 650 (the open string).

(a)

Show that L₁, L₂, L₃, … form a geometric sequence and write down its common ratio.

(2)

(b)

Find the vibrating length when the string is pressed at fret 7.

(2)

(c)

Find the distance from fret 0 (the nut) to fret 7.

(2)

(d)

The gap between fret n − 1 and fret n is g_n = L_n₋₁ − L_n. Show that g₁, g₂, … is geometric with first term 36.5 mm, correct to 3 significant figures, and the same common ratio as in (a).

(3)

(e)

Find the first fret for which the gap between it and the previous fret is less than 15 mm.

(3)

(f)

The guitar has 22 frets. Use the formula for the sum of a geometric series to find the distance from the nut to fret 22.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 L_n ÷ L_n₋₁ = 2^−n/12 ÷ 2^{−(n−1)/12}	✓ 1
A1 = 2^−1/12, which is constant, so the sequence is geometric with ratio 0.944 (3 s.f.)	✓ 1
Part (b)
M1 650 × 2^−7/12	✓ 1
A1 434 mm	✓ 1
Part (c)
M1 650 − 433.82…	✓ 1
A1 216 mm	✓ 1
Part (d)
M1 g_n = L_n₋₁(1 − 2^−1/12)	✓ 1
R1 L_n₋₁ is geometric with ratio 2^−1/12, so g_n is geometric with the same ratio	✓ 1
A1 g₁ = 650(1 − 2^−1/12) = 36.5 mm	✓ 1
Part (e)
M1 36.48 × 0.9439ⁿ⁻¹ < 15	✓ 1
A1 n − 1 > 15.4	✓ 1
A1 fret 17	✓ 1
Part (f)
M1 S₂₂ = 36.48 × (1 − 0.9439²²) ÷ (1 − 0.9439)	✓ 1
A1 468 mm	✓ 1

Exam tip: The sum of the first 22 gaps is the distance from the nut to fret 22, so you can check your sum by evaluating 650 − L₂₂ directly.

+ Add to test✓ In your test460IBMI3S-1-01r5-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard12 marks

A wind farm operator sets up a fund to pay for the decommissioning of its turbines. The fund earns interest at 3.2% per year, compounded annually. The operator pays €12000 into the fund at the start of each year for 15 years, the first payment being made at the start of year 1.

(a)

Find the value at the end of year 15 of the first payment of €12000. Give your answer to the nearest euro.

(2)

(b)

Show that the total value of the fund at the end of year 15 is approximately €233735.

(3)

(c)

The operator needs the fund to reach €250000 at the end of year 15. Find the yearly payment, to the nearest euro, that would achieve this.

(3)

(d)

With the original payments of €12000, no further payments are made after year 15. Find the number of additional years, to 3 significant figures, until the fund first reaches €250000.

(3)

(e)

State one assumption made in this model.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 12000 × 1.032¹⁵	✓ 1
A1 €19248	✓ 1
Part (b)
M1 recognises a geometric series with first term 12000 × 1.032 and common ratio 1.032	✓ 1
M1 S₁₅ = 12000 × 1.032 × (1.032¹⁵ − 1) ÷ (1.032 − 1)	✓ 1
A1 233735 (accept 233735.3), shown to at least 4 significant figures	✓ 1
Part (c)
M1 set up R × 1.032 × (1.032¹⁵ − 1) ÷ 0.032 = 250000	✓ 1
M1 rearrange for R or use a GDC finance solver (annuity due)	✓ 1
A1 €12835	✓ 1
Part (d)
M1 233735 × 1.032^t = 250000 (ECF from (b))	✓ 1
M1 t = ln(250000/value) ÷ ln 1.032 or GDC solver	✓ 1
A1 t = 2.14 years	✓ 1
Part (e)
R1 e.g. the interest rate stays constant at 3.2% for the whole period	✓ 1

Common error: Payments at the start of each year earn interest for one extra year compared with payments at the end; this is why the first term is 12000 × 1.032.

+ Add to test✓ In your test461IBMI3S-1-01r5-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

Two friends, Anna and Ben, save for a swimming club trip. Anna saves €110 in month 1 and increases her saving by €12 each month, so her monthly savings form an arithmetic sequence. Ben saves €90 in month 1 and increases his saving by 8% each month, so his monthly savings form a geometric sequence.

(a)

Find the amount Anna saves in month 12.

(2)

(b)

Find the amount Ben saves in month 12. Give your answer to the nearest cent.

(2)

(c)

Find the first month in which the amount Ben saves is greater than the amount Anna saves in the same month.

(3)

(d)

Find the first month in which the total amount Ben has saved is greater than the total amount Anna has saved.

(4)

(e)

Find, to the nearest cent, how much more Ben has saved than Anna at the end of that month.

(2)

(f)

Explain why Ben's total can pass Anna's even though Anna saves more than Ben in the early months.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁₂ = 110 + 11 × 12	✓ 1
A1 €242	✓ 1
Part (b)
M1 u₁₂ = 90 × 1.08¹¹	✓ 1
A1 €209.85	✓ 1
Part (c)
M1 set up 90 × 1.08ⁿ⁻¹ > 110 + 12(n − 1)	✓ 1
M1 use of a GDC table or graph intersection	✓ 1
A1 month 17 (month 16: €285.50 vs €290; month 17: €308.33 vs €302)	✓ 1
Part (d)
M1 S_n for Anna: (n/2)(2 × 110 + (n − 1) × 12)	✓ 1
M1 S_n for Ben: 90(1.08ⁿ − 1) ÷ 0.08	✓ 1
M1 compare using a GDC table or graph	✓ 1
A1 month 24	✓ 1
Part (e)
M1 difference of the two totals at the month found in (d) (ECF from (d))	✓ 1
A1 €56.83	✓ 1
Part (f)
R1 Ben's monthly saving grows by a fixed percentage, so it increases by larger and larger amounts, while Anna's increases by a fixed amount each month	✓ 1

Exam tip: When two totals are compared, check the months either side of the crossing point to show that the first month is correct.

+ Add to test✓ In your test462IBMI3S-1-01r5-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

The power output P kW of a wind turbine at wind speeds between 6 m s⁻¹ and 12 m s⁻¹ is modelled by P = av^b, where v is the wind speed in m s⁻¹ and a and b are constants. The turbine produces 260 kW at 6 m s⁻¹ and 1000 kW at 10 m s⁻¹. The rated (maximum) power of the turbine is 1500 kW. In parts (b) to (g) use unrounded values from earlier parts.

(a)

Show that ln P = ln a + b ln v.

(2)

(b)

Find the value of b. Give your answer to 3 significant figures.

(3)

(c)

Find the value of a. Give your answer to 3 significant figures.

(2)

(d)

Calculate the power predicted by the model at 12 m s⁻¹. Give your answer to the nearest kW.

(2)

(e)

Comment on whether this prediction is realistic for this turbine.

(1)

(f)

State what the gradient of a graph of ln P against ln v represents.

(1)

(g)

Find the wind speed at which the model first gives the rated power of 1500 kW. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 take natural logarithms of both sides: ln P = ln(a v^b)	✓ 1
A1 ln a + ln(v^b) = ln a + b ln v, using the laws of logarithms	✓ 1
Part (b)
M1 two equations: ln 260 = ln a + b ln 6 and ln 1000 = ln a + b ln 10 (or ratio 1000/260 = (10/6)^b)	✓ 1
M1 subtract / take logarithms of the ratio: b = ln(1000/260) ÷ ln(10/6)	✓ 1
A1 b = 2.64	✓ 1
Part (c)
M1 a = 260 ÷ 6^b	✓ 1
A1 a = 2.31	✓ 1
Part (d)
M1 P = a × 12^b	✓ 1
A1 1617 kW (ECF from (b) and (c))	✓ 1
Part (e)
R1 the prediction is greater than the rated power of 1500 kW, so the model is not valid up to 12 m s⁻¹	✓ 1
Part (f)
A1 the gradient equals the constant b (the power of v in the model)	✓ 1
Part (g)
M1 set a v^b = 1500	✓ 1
M1 v = (1500 ÷ a)^1/b or GDC solver	✓ 1
A1 v = 11.66 m s⁻¹	✓ 1

Exam tip: To find an unknown exponent from two data points, divide the two equations first; the constant a then cancels.

+ Add to test✓ In your test463IBMI3S-1-02r7-14Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard13 marks

The sound level, L decibels (dB), at a distance r metres from a ship’s engines is modelled by L = 172 − 20 log₁₀ r, where r ≥ 1. The sound intensity corresponding to a level L is proportional to 10^L/10.

(a)

Find the sound level at a distance of 200 m, correct to 1 decimal place.

(2)

(b)

Find the distance at which the sound level is 100 dB, correct to 3 significant figures.

(3)

(c)

Two identical ships are at the same distance from a listener. Show that the sound level is greater than that from one ship by 3.0 dB, correct to 1 decimal place.

(3)

(d)

One ship is 500 m from a hydrophone and a second identical ship is 800 m from it. Find the combined sound level at the hydrophone, correct to 1 decimal place. Use unrounded values of the two individual levels.

(4)

(e)

Suggest one reason why the model may be unreliable for very large distances.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 172 − 20 log₁₀ 200	✓ 1
A1 126.0 dB	✓ 1
Part (b)
M1 20 log₁₀ r = 72	✓ 1
M1 r = 10^3.6	✓ 1
A1 3980 m	✓ 1
Part (c)
M1 the total intensity is twice that of one ship	✓ 1
M1 increase in level = 10 log₁₀ 2	✓ 1
A1 3.01 dB, which is 3.0 dB to 1 d.p. (shown)	✓ 1
Part (d)
M1 individual levels 118.02 dB and 113.94 dB	✓ 1
M1 add the intensities: 10^L₁/10 + 10^L₂/10	✓ 1
M1 combined level = 10 log₁₀(sum)	✓ 1
A1 119.5 dB	✓ 1
Part (e)
R1 sound is absorbed and scattered by the water, so the level falls faster than the model predicts (or the model ignores absorption)	✓ 1

Common error: Sound levels in decibels cannot be added directly; convert to intensities, add, then convert back.

+ Add to test✓ In your test464IBMI3S-1-02r10-11Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

A mountaineering club saves for a new mountain hut. It makes a deposit of €4000 into an account on day one, and then deposits €350 at the end of each month. The account pays a nominal annual interest rate of 2.4%, compounded monthly. The €4000 is deposited at the start of month 1 and the first €350 deposit is made at the end of month 1.
Give all money answers to the nearest euro, unless told otherwise.

(a)

Calculate the value after 6 years of the monthly deposits of €350 (not including the initial €4000).

(3)

(b)

Calculate the value after 6 years of the initial €4000 deposit.

(2)

(c)

Hence find the total value of the account after 6 years.

(1)

(d)

Find the total interest earned in the 6 years.

(2)

(e)

The club needs €40 000 for the hut. Find the number of months, n, after which the account value is exactly €40 000. Give your answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
TVM or formula with N = 72, I% = 2.4, PMT = −350, PV = 0, P/Y = C/Y = 12	✓ 1
Correct use of monthly rate 0.2% (0.002) with 72 months	✓ 1
€27 076	✓ 1
Part (b)
4000 × 1.002⁷²	✓ 1
€4 619	✓ 1
Part (c)
€31 695 (sum of parts (a) and (b))	✓ 1
Part (d)
Total paid in = 4000 + 72 × 350 = €29 200	✓ 1
Interest = €2 495	✓ 1
Part (e)
4000 × 1.002ⁿ + 350 × (1.002ⁿ − 1) ÷ 0.002 = 40 000	✓ 1
179 000 × 1.002ⁿ = 215 000 (or equivalent form; or GDC solver)	✓ 1
n = log(215 000/179 000) ÷ log 1.002 (use of logarithms)	✓ 1
n = 91.7 months	✓ 1

Exam tip: With regular deposits and an initial lump sum, find each part's future value separately and add them; this is easier to check than entering everything into one TVM line.

+ Add to test✓ In your test465IBMI3S-1-02r10-12Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard14 marks

A climber plans to ascend 4900 m from base camp to a summit. In Plan A she climbs 900 m on day 1 and, on each later day, 85% of the distance climbed on the previous day. In Plan B she climbs 700 m on day 1 and, on each later day, 60 m less than on the previous day, until the distance climbed would no longer be positive.

(a)

In Plan A, find the distance climbed on day 4 and the total distance climbed in the first 8 days. Give your answers to the nearest metre.

(4)

(b)

Find the sum to infinity of the distances in Plan A, and hence explain whether the summit can be reached with Plan A.

(2)

(c)

In Plan A, find the value of n for which the total distance climbed in n days is exactly 4900 m. Give your answer to 3 significant figures, and state the least number of whole days needed.

(3)

(d)

In Plan B, find the distance climbed on the last day on which the distance is positive.

(2)

(e)

Find the total distance climbed in Plan B, and state whether the summit can be reached with Plan B.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Day 4: 900 × 0.85³	✓ 1
553 m	✓ 1
Use of S_n = u₁(1 − rⁿ) ÷ (1 − r) with n = 8	✓ 1
4365 m	✓ 1
Part (b)
S_∞ = 900 ÷ (1 − 0.85) = 6000 m	✓ 1
6000 > 4900, so the summit can be reached (in theory, as the total gets close to 6000)	✓ 1
Part (c)
900(1 − 0.85ⁿ) ÷ 0.15 = 4900	✓ 1
0.85ⁿ = 0.18333… and use of logarithms	✓ 1
n = 10.4, so 11 days	✓ 1
Part (d)
700 − 60(n − 1) > 0 gives n < 12.67…, so the last day is day 12	✓ 1
Distance = 700 − 60 × 11 = 40 m	✓ 1
Part (e)
Use of S₁₂ = 12/2 (700 + 40) (ECF from part (d))	✓ 1
4440 m	✓ 1
4440 < 4900, so the summit cannot be reached with Plan B	✓ 1

Common error: A sum to infinity only exists when |r| < 1, and the total of a geometric sequence never exceeds it; check that this value is above the target before looking for n.

+ Add to test✓ In your test466IBMI3S-1-02r10-13Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard14 marks

A cylindrical food tin has a measured diameter of 15.6 cm and a measured height of 8.2 cm. Each measurement is correct to the nearest 0.1 cm. The tin is filled to 95% of its volume with soup.

(a)

Use the measured values to calculate the volume of the tin. Give your answer to the nearest cm³.

(3)

(b)

Calculate the lower bound and the upper bound for the volume of the tin. Give your answers to the nearest cm³.

(4)

(c)

Find the percentage error in the volume from part (a) if the actual volume is the upper bound. Give your answer to 3 significant figures.

(2)

(d)

Calculate the volume of soup, in cm³, when the tin has the volume in part (a). Give your answer to the nearest cm³.

(2)

(e)

The label says that each tin contains at least 1.45 litres of soup. (1 litre = 1000 cm³.) Use the lower bound of the volume to decide whether the label is justified. Give the volume of soup in litres to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Radius = 7.8 cm	✓ 1
Volume = π × 7.8² × 8.2	✓ 1
1567 cm³	✓ 1
Part (b)
Use of lower bounds: diameter 15.55 cm (radius 7.775 cm) and height 8.15 cm	✓ 1
Lower bound = 1548 cm³	✓ 1
Use of upper bounds: diameter 15.65 cm (radius 7.825 cm) and height 8.25 cm	✓ 1
Upper bound = 1587 cm³	✓ 1
Part (c)
Error = upper bound − volume from part (a)	✓ 1
Percentage error = error ÷ upper bound × 100 = 1.24%	✓ 1
Part (d)
0.95 × volume from part (a)	✓ 1
1489 cm³	✓ 1
Part (e)
Soup volume at lower bound = 0.95 × lower bound ÷ 1000	✓ 1
1.47 litres	✓ 1
1.47 > 1.45, so the label is justified	✓ 1

Exam tip: To guarantee a minimum, use the lower bounds of every measurement; for a maximum, use the upper bounds.

+ Add to test✓ In your test467IBMI3S-1-02r12-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 2Hard12 marks

A coffee roastery plans its weekly output of roasted coffee. In week 1 the output is 120 kg. Under Plan A, the output increases by 6% each week, so the weekly outputs form a geometric sequence.

(a)

Find the least number of weeks for which the total output under Plan A first exceeds 5000 kg.

(4)

(b)

Under Plan B, the weekly outputs form an arithmetic sequence with first term 120 kg and common difference d kg. The total output over the first 22 weeks is exactly 5200 kg. Find d to 3 significant figures.

(3)

(c)

Using d = 11.1, find the number of the first week in which the weekly output under Plan A is greater than the weekly output under Plan B.

(3)

(d)

Calculate the weekly output under Plan A in week 10. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 S_n = 120(1.06ⁿ − 1) ÷ 0.06	✓ 1
M1 sets S_n > 5000, so 1.06ⁿ > 3.5 (solve or use a table)	✓ 1
A1 S₂₁ = 4799 kg is below 5000 kg	✓ 1
A1 S₂₂ = 5207 kg exceeds 5000 kg, so n = 22	✓ 1
Part (b)
M1 S₂₂ = 22/2 (2 × 120 + 21d) = 5200	✓ 1
M1 2640 + 231d = 5200	✓ 1
A1 d = 11.1	✓ 1
Part (c)
M1 120(1.06)ⁿ⁻¹ > 120 + 11.1(n − 1)	✓ 1
M1 solves using a graph or table	✓ 1
A1 n = 16 (week 15 gives 271.3 kg for A and 275.4 kg for B; week 16 gives 287.6 kg and 286.5 kg)	✓ 1
Part (d)
M1 120 × 1.06⁹	✓ 1
A1 203 kg	✓ 1

Common error: Using un = u₁rⁿ for a geometric sequence. The exponent is n − 1.

+ Add to test✓ In your test468IBMI3S-1-02r12-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 2Hard12 marks

A coffee roastery buys a roasting machine for €48 000. The machine loses 14% of its value each year, so after n years its value is 48000 × 0.86ⁿ euros. The cost of a new machine of the same type rises by 2.8% each year, so after n years it is 48000 × 1.028ⁿ euros.

(a)

Calculate the value of the machine after 3 years. Give your answer to the nearest euro.

(2)

(b)

Calculate the cost of a new machine after 3 years. Give your answer to the nearest euro.

(1)

(c)

The roastery will replace the machine when the cost of a new machine exceeds the value of the old machine by more than €45 000. Find the least whole number of years after which this happens.

(4)

(d)

The yearly decrease of 14% is equivalent to a constant percentage decrease each month. Find this monthly percentage decrease. Give your answer to 3 significant figures.

(3)

(e)

Calculate the total percentage decrease in the value of the machine over its first 3 years. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 48000 × 0.86³	✓ 1
A1 €30531	✓ 1
Part (b)
A1 €52146	✓ 1
Part (c)
M1 48000(1.028ⁿ − 0.86ⁿ) > 45000	✓ 1
M1 solves using a graph or table	✓ 1
A1 for n = 7 the difference is €41536, which is below €45 000	✓ 1
A1 for n = 8 the difference is €45504, which is above €45 000, so n = 8	✓ 1
Part (d)
M1 (1 − r)¹² = 0.86	✓ 1
M1 1 − r = 0.86^1/12 = 0.9875	✓ 1
A1 r = 1.25%	✓ 1
Part (e)
M1 (1 − 0.86³) × 100	✓ 1
A1 36.4%	✓ 1

Common error: Multiplying the yearly percentage by 3 for a 3-year change. Each year's decrease applies to a smaller value, so use the multiplier 0.86³.

+ Add to test✓ In your test469IBMI3S-1-02r14-13Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard13 marks

A council borrows €2400000 to build a footbridge. The loan is to be repaid over 20 years by equal monthly payments made at the end of each month. Interest is charged at a nominal annual rate of 4.8%, compounded monthly.

(a)

Calculate the monthly payment, to the nearest cent.

(2)

(b)

Using your answer to part (a), find the total amount repaid over the 20 years and the total interest paid, to the nearest euro.

(2)

(c)

Using your answer to part (a), find the outstanding balance immediately after the 60th payment, to the nearest euro.

(3)

(d)

Immediately after the 60th payment, the council pays an extra lump sum of €300000, using the balance from part (c) rounded to the nearest euro. The remaining balance is then repaid by equal monthly payments over the remaining 180 months, at the same interest rate. Find the new monthly payment, to the nearest cent.

(3)

(e)

Using the monthly payments from parts (a) and (d), each rounded to the nearest cent, find the total amount of interest saved over the remaining 180 months by paying the lump sum. Give your answer to the nearest euro.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM with N = 240, I% = 4.8, PV = 2400000, FV = 0, P/Y = C/Y = 12 (or amortisation formula)	✓ 1
A1 €15574.98	✓ 1
Part (b)
A1 total repaid $240\times15574.98$ = €3737995	✓ 1
A1 interest = €1337995	✓ 1
Part (c)
M1 balance = value of the loan after 60 months minus the value of the 60 payments (or TVM with N = 60, solving for FV; accept N = 180 and solving for PV)	✓ 1
M1 correct substitution, e.g. $2400000(1.004)^{60}-P\frac{(1.004)^{60}-1}{0.004}$	✓ 1
A1 €1995732	✓ 1
Part (d)
A1 remaining balance €1695732	✓ 1
M1 TVM with N = 180, I% = 4.8, PV = remaining balance, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €13233.74	✓ 1
Part (e)
M1 original remaining payments $180\times15574.98$; new total paid after month 60 $=300000+180\times13233.74$	✓ 1
M1 difference of the two totals	✓ 1
A1 €121423	✓ 1

Common error: The lump sum itself is not interest saved. Compare the total of all money paid after month 60 under each plan; the lump sum counts as money paid under the new plan.

+ Add to test✓ In your test470IBMI3S-1-02r14-14Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard13 marks

A space probe carries 120 kg of fuel for course-correction burns. The first burn uses 9.0 kg of fuel and each later burn uses 85% of the fuel used in the previous burn, so the amounts of fuel used form a geometric sequence.

(a)

Find the mass of fuel used in the 6th burn. Give your answer to 3 significant figures.

(2)

(b)

Find the total mass of fuel used in the first 10 burns. Give your answer to 3 significant figures.

(2)

(c)

At least 75 kg of fuel must be kept in reserve for the rest of the mission, so no more than 45 kg may be used for burns. Find the greatest number of burns that can be made.

(3)

(d)

Find the mass of fuel remaining after the number of burns found in part (c), to the nearest kilogram.

(2)

(e)

The time between burn n and burn n + 1 is u_n days, where u₁ = 12 and each interval is 3 days longer than the previous one. Find the time between burn 7 and burn 8.

(2)

(f)

Find the total time from the first burn to the last burn when 8 burns are made.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $9.0\times0.85^{5}$	✓ 1
A1 3.99 kg	✓ 1
Part (b)
M1 $\frac{9.0(1-0.85^{10})}{1-0.85}$	✓ 1
A1 48.2 kg	✓ 1
Part (c)
M1 $\frac{9.0(1-0.85^{n})}{0.15}\le45$ giving $0.85^{n}\ge0.25$	✓ 1
M1 $n\le\frac{\ln0.25}{\ln0.85}=8.53$ (or testing values: 8 burns use 43.65 kg, 9 burns use 46.1 kg)	✓ 1
A1 8 burns	✓ 1
Part (d)
M1 $120-\frac{9.0(1-0.85^{8})}{0.15}$ (or $120-43.65$)	✓ 1
A1 76 kg	✓ 1
Part (e)
M1 $12+6\times3$	✓ 1
A1 30 days	✓ 1
Part (f)
M1 $\frac{7}{2}(2\times12+6\times3)$ (sum of the first 7 intervals)	✓ 1
A1 147 days	✓ 1

Common error: With 8 burns there are only 7 intervals between them. Check how many terms a sum needs before using the sum formula.

+ Add to test✓ In your test471IBMI3S-1-01r18-11Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

A fisheries authority sets a quota for the mass of fish that may be landed each year. In year 1 the quota is 5000 tonnes. In each later year the quota is 5 % lower than in the previous year, so the yearly quotas form a geometric sequence.

(a)

Find the quota in year 6. Give your answer to the nearest tonne.

(2)

(b)

Find the total quota over the first 10 years. Give your answer to the nearest tonne.

(3)

(c)

Find the first year in which the quota is less than 3200 tonnes.

(3)

(d)

Find the smallest number of years for which the total quota over those years exceeds 45 000 tonnes.

(3)

(e)

The price of fish is €1200 per tonne in year 1 and rises by €40 per tonne each year (an arithmetic sequence). Assuming the full quota is landed and sold in year 4, calculate the total value of the fish. Give your answer to the nearest euro.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 5000 × 0.95⁵	✓ 1
A1 3869 tonnes	✓ 1
Part (b)
M1 S₁₀ = 5000(1 − 0.95¹⁰)/(1 − 0.95)	✓ 1
A1 correct substitution with n = 10, r = 0.95	✓ 1
A1 40126 tonnes	✓ 1
Part (c)
M1 5000 × 0.95ⁿ⁻¹ < 3200	✓ 1
A1 n − 1 > 8.7 (or n > 9.7)	✓ 1
A1 year 10	✓ 1
Part (d)
M1 5000(1 − 0.95ⁿ)/0.05 > 45000	✓ 1
A1 n > 11.7 (accept value from a solver or table)	✓ 1
A1 12 years	✓ 1
Part (e)
M1 quota in year 4 = 5000 × 0.95³ = 4286.875 tonnes	✓ 1
M1 price in year 4 = 1200 + 3 × 40 = €1320 per tonne	✓ 1
A1 €5658675	✓ 1

Common error: The nth term of a geometric sequence uses rⁿ⁻¹, but the sum of the first n terms uses rⁿ; mixing these up gives an answer one year out.

+ Add to test✓ In your test472IBMI3S-1-01r18-14Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard12 marks

A boat-building firm borrows €30 000. The annual interest rate is 5.4 %, compounded monthly. The firm repays the loan by making a payment of €650 at the end of each month. Every payment is €650, except that the last payment is smaller than €650 and clears the loan exactly.

(a)

Show that the firm needs more than 51 payments but fewer than 53 payments. Give the value of N you find to 3 significant figures.

(3)

(b)

Calculate the amount still owed immediately after the 51st payment. Give your answer to the nearest cent.

(3)

(c)

Find the amount of the 52nd (final) payment. Use the unrounded answer to part (b) and give your answer to the nearest cent.

(2)

(d)

Find the total amount repaid by the firm, to the nearest cent.

(2)

(e)

Find the total interest paid as a percentage of the amount borrowed. Use the unrounded final payment and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to use the TVM solver with I% = 5.4, PV = 30000, PMT = −650, FV = 0, P/Y = C/Y = 12	✓ 1
A1 N = 51.9	✓ 1
R1 so 52 payments are needed in total (51 full payments and one smaller final payment)	✓ 1
Part (b)
M1 attempt to find the balance with N = 51 (or 30000 × 1.0045⁵¹ − 650 × (1.0045⁵¹ − 1)/0.0045)	✓ 1
A1 correct substitution of monthly rate 0.0045	✓ 1
A1 €550.77	✓ 1
Part (c)
M1 balance × 1.0045	✓ 1
A1 €553.25 (ECF from (b))	✓ 1
Part (d)
M1 51 × 650 + final payment	✓ 1
A1 €33703.25 (ECF from (c))	✓ 1
Part (e)
M1 (total repaid − 30000) ÷ 30000 × 100	✓ 1
A1 12.3 % (ECF)	✓ 1

Common error: Rounding the number of payments to 52 and then assuming all 52 payments are €650 overstates the cost: the final payment is only the remaining balance plus one month of interest.

+ Add to test✓ In your test473IBMI3S-1-01r19-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard12 marks

A sports club takes out a loan of €18 000 to buy gym equipment. The loan has an interest rate of 5.4% per year, compounded monthly, and is repaid by equal payments at the end of each month over 4 years.

(a)

Find the monthly payment, to the nearest cent.

(3)

(b)

Find the amount still owed immediately after the 18th payment, to the nearest cent. Use the unrounded monthly payment.

(3)

(c)

Find the total interest paid if the loan runs for the full 4 years, to the nearest euro. Use the unrounded monthly payment.

(2)

(d)

Immediately after the 18th payment the interest rate changes to 6.6% per year, still compounded monthly, and the club keeps the remaining 30 monthly payments. Using the balance from (b) to the nearest cent, find the new monthly payment, to the nearest cent.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM: N = 48, I% = 5.4, PV = 18 000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 correct entries and method	✓ 1
A1 €417.80	✓ 1
Part (b)
M1 TVM with N = 18 (same PV, PMT, I%, P/Y, C/Y)	✓ 1
A1 correct entries	✓ 1
A1 €11 700.11	✓ 1
Part (c)
M1 48 × 417.7967… − 18 000	✓ 1
A1 €2054	✓ 1
Part (d)
M1 PV = 11 700.11, N = 30, I% = 6.6, FV = 0	✓ 1
A1 correct TVM entries	✓ 1
M1 solve for PMT (or 11 700.11 × 0.0055/(1 − 1.0055⁻³⁰))	✓ 1
A1 €424.13	✓ 1

Exam tip: After a rate change, start a new TVM calculation using the outstanding balance as the present value.

+ Add to test✓ In your test474IBMI3S-1-01r21-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

Calculator

A hospital charity fund starts with €250 000. It earns interest at 3.6% per year, compounded quarterly. At the end of each quarter, immediately after the interest is added, the charity withdraws €6000.

(a)

Find the balance of the fund immediately after the 12th withdrawal. Give your answer to the nearest cent.

(3)

(b)

Find the effective annual interest rate. Give your answer as a percentage to 3 significant figures.

(2)

(c)

Find how many full withdrawals of €6000 the fund can pay.

(3)

(d)

The trustees instead choose an equal withdrawal at the end of each quarter so that the fund is exactly used up after 15 years. Find the size of this withdrawal, to the nearest cent.

(2)

(e)

Using the withdrawal of €5410.70 from part (d), find the total interest earned by the fund over the 15 years.

(2)

(f)

Find the largest quarterly withdrawal that would allow the fund to last indefinitely without the balance falling.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance solver with $N=12$, $I=3.6$, $PV=250000$, $PMT=-6000$, $P/Y=C/Y=4$	✓ 1
M1 $FV$ unknown	✓ 1
A1 €202 704.30	✓ 1
Part (b)
M1 $\left(1+\frac{0.036}{4}\right)^{4}-1$	✓ 1
A1 $3.65\%$	✓ 1
Part (c)
M1 finance solver with $PV=250000$, $PMT=-6000$, $FV=0$ and $N$ unknown	✓ 1
A1 $N=52.46\ldots$	✓ 1
A1 52 full withdrawals	✓ 1
Part (d)
M1 $N=60$, $I=3.6$, $PV=250000$, $FV=0$, $P/Y=C/Y=4$	✓ 1
A1 €5410.70	✓ 1
Part (e)
M1 total withdrawn minus initial amount: $60\times5410.70-250000$	✓ 1
A1 €74 642.00	✓ 1
Part (f)
M1 withdrawal equals interest on €250 000 in one quarter: $250000\times0.009$	✓ 1
A1 €2250	✓ 1

Common error: The quarterly rate is 3.6% ÷ 4 = 0.9%; using 3.6% for each quarter makes every answer far too large.

+ Add to test✓ In your test475IBMI3S-1-01r21-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

Calculator

The number of bees $P$ in a new colony $t$ weeks after it is established is modelled by $P=P_0e^{kt}$, where $P_0$ and $k$ are constants. The colony contains 14 500 bees at $t=2$ and 22 300 bees at $t=5$.

(a)

Find the value of $k$, correct to 3 significant figures.

(3)

(b)

Using the unrounded value of $k$, find $P_0$ to the nearest 100.

(2)

(c)

Find the time taken for the colony to double in size. Give your answer in weeks to 3 significant figures.

(2)

(d)

Find the time at which the model predicts 60 000 bees. Use the unrounded values of $k$ and $P_0$ and give your answer in weeks to 3 significant figures.

(3)

(e)

A graph is drawn of $\ln P$ against $t$. State the gradient of the straight line obtained.

(1)

(f)

Using the unrounded values, find the intercept of this line on the $\ln P$ axis, correct to 3 significant figures.

(2)

(g)

Give one reason why the model cannot be valid for very large values of $t$.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\frac{P(5)}{P(2)}=e^{3k}$, that is $\frac{22300}{14500}=e^{3k}$	✓ 1
M1 $k=\frac{1}{3}\ln\frac{22300}{14500}$	✓ 1
A1 $k=0.143$	✓ 1
Part (b)
M1 $P_0=14500e^{-2k}$	✓ 1
A1 10 900	✓ 1
Part (c)
M1 $e^{kt}=2$	✓ 1
A1 4.83 weeks	✓ 1
Part (d)
M1 $60000=P_0e^{kt}$	✓ 1
M1 $t=\frac{1}{k}\ln\frac{60000}{P_0}$	✓ 1
A1 11.9 weeks	✓ 1
Part (e)
A1 gradient $=k=0.143$ (3 s.f.)	✓ 1
Part (f)
M1 intercept $=\ln P_0=\ln14500-2k$	✓ 1
A1 9.29	✓ 1
Part (g)
R1 the colony cannot grow without limit because of limited space or food (accept any valid reason)	✓ 1

Exam tip: Taking natural logarithms turns $P=P_0e^{kt}$ into $\ln P=\ln P_0+kt$, which is why a straight-line graph confirms exponential growth.

+ Add to test✓ In your test476IBMI3S-1-02r21-12Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 2Hard14 marks

Calculator

The temperature $T$ °C of a Formula 1 brake disc, $t$ seconds after the car leaves a high-speed corner, is modelled by $T=30+Ae^{-kt}$, where $A$ and $k$ are positive constants. At $t=0$ the temperature is 850 °C, and at $t=20$ the temperature is 520 °C.

(a)

Find the value of $A$.

(2)

(b)

Find the value of $k$, correct to 3 significant figures.

(3)

(c)

Using the unrounded value of $k$, find the temperature at $t=40$, to the nearest degree.

(2)

(d)

Find the time when the temperature first falls to 200 °C. Use the unrounded value of $k$ and give your answer to 3 significant figures.

(3)

(e)

Write down the equation of the horizontal asymptote of the graph of $T$ against $t$, and explain what it represents in this context.

(2)

(f)

Find the average rate of change of temperature between $t=0$ and $t=20$, including its units.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $850=30+Ae^{0}$	✓ 1
A1 $A=820$	✓ 1
Part (b)
M1 $520=30+820e^{-20k}$, so $e^{-20k}=\frac{490}{820}$	✓ 1
M1 $k=-\frac{1}{20}\ln\frac{490}{820}$	✓ 1
A1 $k=0.0257$	✓ 1
Part (c)
M1 $T=30+820e^{-40k}$	✓ 1
A1 323 °C	✓ 1
Part (d)
M1 $200=30+820e^{-kt}$, so $e^{-kt}=\frac{170}{820}$	✓ 1
M1 $t=\frac{1}{k}\ln\frac{820}{170}$	✓ 1
A1 61.1 s	✓ 1
Part (e)
A1 $T=30$	✓ 1
R1 the temperature of the surroundings: the disc cools towards but never goes below it	✓ 1
Part (f)
M1 $\frac{520-850}{20}$	✓ 1
A1 $-16.5$ °C per second	✓ 1

Exam tip: In models of the form $T=c+Ae^{-kt}$, subtract the constant $c$ before taking logarithms.

+ Add to test✓ In your test477IBMI3S-1-02r21-13Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms Paper 2Hard13 marks

Calculator

The number of bacteria $B$ per square centimetre on a hospital surface, $t$ hours after cleaning stops, is modelled by $\ln B=4.2+0.34t$.

(a)

Find the number of bacteria per square centimetre at $t=0$, correct to 3 significant figures.

(2)

(b)

Show that the model can be written as $B=a\times b^{t}$ and find the values of $a$ and $b$, each correct to 3 significant figures.

(3)

(c)

Interpret the value of $b$ in context.

(1)

(d)

Find the number of bacteria per square centimetre after 12 hours, to the nearest 10.

(2)

(e)

Find the time at which $B$ reaches 100 000. Give your answer in hours to 3 significant figures.

(3)

(f)

Find the time taken for the number of bacteria to double. Give your answer in hours to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $B=e^{4.2}$	✓ 1
A1 66.7	✓ 1
Part (b)
M1 $B=e^{4.2+0.34t}=e^{4.2}\times\left(e^{0.34}\right)^{t}$	✓ 1
A1 $a=66.7$	✓ 1
A1 $b=1.40$	✓ 1
Part (c)
R1 the number of bacteria is multiplied by 1.40 (about a 40% increase) each hour	✓ 1
Part (d)
M1 $\ln B=4.2+0.34\times12=8.28$	✓ 1
A1 3940	✓ 1
Part (e)
M1 $\ln100000=4.2+0.34t$	✓ 1
M1 $t=\frac{\ln100000-4.2}{0.34}$	✓ 1
A1 21.5 hours	✓ 1
Part (f)
M1 $e^{0.34t}=2$ so $t=\frac{\ln2}{0.34}$	✓ 1
A1 2.04 hours	✓ 1

Exam tip: To undo a natural logarithm, raise $e$ to the power of both sides of the equation.

+ Add to test✓ In your test478IBMI3S-1-03r22-10Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard15 marks

Two crews lay track on a new rail line. Crew A lays 1.8 km of track in week 1 and then 0.2 km more each week than in the previous week. Crew B lays 1.2 km of track in week 1 and then 12% more each week than in the previous week. The distances laid by crew A form an arithmetic sequence and the distances laid by crew B form a geometric sequence.

(a)

Find the length of track laid by crew A in week 10.

(2)

(b)

Find the length of track laid by crew B in week 10, to 3 significant figures.

(2)

(c)

Find the total length laid by crew A in the first 10 weeks.

(2)

(d)

Find the total length laid by crew B in the first 10 weeks, to 3 significant figures.

(2)

(e)

Find the first week in which crew B lays more track than crew A.

(4)

(f)

The new line is 62 km long. Find the total length laid by the two crews together in the first 12 weeks, to 3 significant figures, and state whether the line is complete.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_{10} = 1.8 + 9 \times 0.2$	✓ 1
A1 3.6 km	✓ 1
Part (b)
M1 $u_{10} = 1.2 \times 1.12^{9}$	✓ 1
A1 3.33 km	✓ 1
Part (c)
M1 $S_{10} = \dfrac{10}{2}(2 \times 1.8 + 9 \times 0.2)$	✓ 1
A1 27.0 km	✓ 1
Part (d)
M1 $S_{10} = \dfrac{1.2(1.12^{10} - 1)}{0.12}$	✓ 1
A1 21.1 km	✓ 1
Part (e)
M1 compares $1.2 \times 1.12^{n-1}$ with $1.8 + 0.2(n-1)$ (table, graph or solver)	✓ 1
A1 week 11: A lays 3.8 km and B lays 3.73 km, so B is not yet ahead	✓ 1
A1 week 12: A lays 4.0 km and B lays 4.17 km	✓ 1
A1 week 12	✓ 1
Part (f)
M1 $S_{12}$ for A: 34.8 km and $S_{12}$ for B: 1.2(1.12¹² − 1)/0.12 = 29.0 km	✓ 1
A1 total 63.76 km (≈ 63.8 km)	✓ 1
R1 63.8 > 62, so the line is complete	✓ 1

Exam tip: For a question such as 'first week when B exceeds A' state the values in the two consecutive weeks either side of the change to justify your answer.

+ Add to test✓ In your test479IBMI3S-1-03r22-14Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard12 marks

The apparent magnitude m of a star is related to the energy flux F received from it, in W m⁻², by $m = -2.5\log_{10}\left(\dfrac{F}{F_0}\right)$, where $F_0 = 2.5 \times 10^{-8}$ W m⁻².

(a)

A star has flux $F = 4.0 \times 10^{-10}$ W m⁻². Find its apparent magnitude, to 3 significant figures.

(2)

(b)

A second star has apparent magnitude 1.20. Find the flux received from it. Give your answer in the form a × 10^k where 1 ≤ a < 10 and a is given to 3 significant figures.

(3)

(c)

Two stars have flux values $F_1$ and $F_2$ and apparent magnitudes $m_1$ and $m_2$ with $m_1 - m_2 = 5$. Show that $F_2 = 100F_1$.

(3)

(d)

The second star has luminosity $L = 3.5 \times 10^{27}$ W, and $F = \dfrac{L}{4\pi d^2}$ where d is its distance from the Earth. Use the unrounded flux from (b) to find d in light-years. Take 1 light-year = 9.46 × 10¹⁵ m. Give your answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $m = -2.5\log_{10}(4.0 \times 10^{-10} / 2.5 \times 10^{-8})$	✓ 1
A1 4.49	✓ 1
Part (b)
M1 rearranges: $\log_{10}(F/F_0) = -1.20/2.5 = -0.48$	✓ 1
M1 $F = 2.5 \times 10^{-8} \times 10^{-0.48}$	✓ 1
A1 8.28 × 10⁻⁹ W m⁻²	✓ 1
Part (c)
M1 $m_1 - m_2 = -2.5\log_{10}(F_1/F_0) + 2.5\log_{10}(F_2/F_0) = 5$	✓ 1
M1 uses the laws of logarithms: $2.5\log_{10}(F_2/F_1) = 5$, so $\log_{10}(F_2/F_1) = 2$	✓ 1
A1 $F_2/F_1 = 10^2 = 100$, so $F_2 = 100F_1$ (AG)	✓ 1
Part (d)
M1 $d = \sqrt{L/(4\pi F)}$	✓ 1
A1 $d = 1.83 \times 10^{17}$ m	✓ 1
M1 divides by 9.46 × 10¹⁵	✓ 1
A1 19.4 light-years	✓ 1

Exam tip: In 'show that' questions write every step of the argument, because the final result is already given.

+ Add to test✓ In your test480IBMI3S-1-02r23-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 2Hard14 marks

A retailer borrows €60 000 to refit a shop. The loan has a nominal annual interest rate of 5.4%, compounded monthly. The retailer plans to repay it with payments of €1250 at the end of each month, until the loan is repaid. The final payment may be smaller than €1250.

(a)

Use a financial solver to show that the number of months needed to repay the loan is approximately 54.2.

(3)

(b)

Find the balance owed immediately after the 54th payment of €1250. Give your answer to the nearest cent.

(3)

(c)

The final payment is made one month after the 54th. Find the amount of the final payment, to the nearest cent. Use the unrounded balance from (b).

(2)

(d)

Find the total amount paid and the total interest paid over the life of the loan. Use the unrounded final payment.

(2)

(e)

The retailer is offered an alternative of equal monthly payments over exactly 48 months at the same interest rate. Find the monthly payment, to the nearest cent.

(2)

(f)

Find how much less interest is paid with the 48-month plan than with the original plan. Use the unrounded monthly payment for the 48-month plan, and give your answer to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 sets N unknown with I% = 5.4, PV = 60 000, PMT = −1250, FV = 0, P/Y = C/Y = 12	✓ 1
A1 correct entries for all values	✓ 1
A1 N = 54.2 (shown)	✓ 1
Part (b)
M1 N = 54 with the same inputs, solving for FV	✓ 1
A1 magnitude of FV	✓ 1
A1 €247.49	✓ 1
Part (c)
M1 247.49… × (1 + 0.054/12)	✓ 1
A1 €248.61	✓ 1
Part (d)
M1 54 × 1250 + 248.607…	✓ 1
A1 total paid €67 748.61 and interest paid €7748.61	✓ 1
Part (e)
M1 solves for PMT with N = 48	✓ 1
A1 €1392.66	✓ 1
Part (f)
M1 48 × 1392.655… − 60 000 = 6847.48 for the alternative plan	✓ 1
A1 €901.13 (ECF from (d))	✓ 1

Exam tip: When a loan is not an exact number of payments, the last payment is the leftover balance plus one month’s interest.

+ Add to test✓ In your test481IBMI3S-1-02r23-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 2Hard12 marks

At the peak of an outbreak, a town records 1500 new cases in a week; this is week 1. After the peak, the number of new cases each week is 80% of the number in the previous week, so the weekly numbers form a geometric sequence.

(a)

Find the number of new cases in week 5, to the nearest whole number.

(2)

(b)

Find the first week in which there are fewer than 50 new cases.

(3)

(c)

Find the total number of cases the model predicts if the outbreak continued indefinitely.

(2)

(d)

Find the smallest number of weeks after which the cumulative total of cases first exceeds 95% of the total in (c).

(3)

(e)

Find the cumulative total number of cases after this number of weeks, to the nearest whole number.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1500 × 0.8⁴	✓ 1
A1 614	✓ 1
Part (b)
M1 1500 × 0.8ⁿ⁻¹ < 50	✓ 1
M1 solves using logarithms or a GDC: n − 1 > 15.2	✓ 1
A1 week 17	✓ 1
Part (c)
M1 1500 ÷ (1 − 0.8)	✓ 1
A1 7500	✓ 1
Part (d)
M1 1500(1 − 0.8ⁿ)/(1 − 0.8) > 0.95 × 7500	✓ 1
M1 0.8ⁿ < 0.05 so n > 13.4	✓ 1
A1 14 weeks	✓ 1
Part (e)
M1 7500(1 − 0.8¹⁴) (ECF from (d))	✓ 1
A1 7170	✓ 1

Common error: The sum to infinity formula only applies when the common ratio satisfies |r| < 1; here r = 0.8.

+ Add to test✓ In your test482IBMI3S-1-02r23-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form Paper 2Hard12 marks

Sunlight reaches the top of the Earth’s atmosphere with an intensity of 1.36 × 10³ W m⁻². The Earth intercepts this energy over a circular disc of radius 6.37 × 10⁶ m. Power = intensity × area.

(a)

Calculate the power intercepted by the Earth. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures.

(3)

(b)

Calculate the energy intercepted in one year of 365 days, in joules. Use the unrounded power and give your answer to 3 significant figures in the form a × 10^k.

(2)

(c)

The world’s annual energy use is 6.0 × 10²⁰ J. Find how many times greater the energy intercepted in a year is than the annual energy use, to 3 significant figures.

(2)

(d)

A student estimates the power by rounding each given value to 1 significant figure and using π ≈ 3. Calculate the student’s estimate of the power.

(2)

(e)

Find the percentage error in the student’s estimate, taking the unrounded answer to (a) as the exact value. Give your answer to 3 significant figures.

(2)

(f)

Suggest one reason why the percentage error is so large.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 area = π × (6.37 × 10⁶)² = 1.2747… × 10¹⁴ m²	✓ 1
M1 power = 1.36 × 10³ × area	✓ 1
A1 1.73 × 10¹⁷ W	✓ 1
Part (b)
M1 power × 365 × 24 × 3600	✓ 1
A1 5.47 × 10²⁴ J	✓ 1
Part (c)
M1 5.467… × 10²⁴ ÷ (6.0 × 10²⁰)	✓ 1
A1 9.11 × 10³ times	✓ 1
Part (d)
M1 1 × 10³ × 3 × (6 × 10⁶)²	✓ 1
A1 1.08 × 10¹⁷ W	✓ 1
Part (e)
M1 (1.7336… × 10¹⁷ − 1.08 × 10¹⁷) ÷ 1.7336… × 10¹⁷ × 100	✓ 1
A1 37.7%	✓ 1
Part (f)
R1 any one of: the radius is squared, so rounding 6.37 down to 6 reduces the area by about 11%; π ≈ 3 underestimates by about 4.5%; rounding 1.36 to 1 reduces the intensity by about 26% (errors compound when values are multiplied)	✓ 1

Exam tip: Rounding several quantities in the same direction before multiplying makes the errors compound, so round only at the end.

+ Add to test✓ In your test483IBMI3S-1-03r23-10Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

Radiocarbon dating uses the fact that the fraction of carbon-14 remaining in organic material after t years is N = (1/2)^t/5730 times the original amount. An archaeologist studies a wooden beam from a settlement.

(a)

Find the percentage of the original carbon-14 remaining after 2000 years. Give your answer to 3 significant figures.

(2)

(b)

The beam contains 31.4% of its original carbon-14. Find the age of the beam, giving your answer to 3 significant figures.

(3)

(c)

The measured percentage is only accurate to within ± 0.5 percentage points, so the true percentage lies between 30.9% and 31.9%. Find the corresponding range of possible ages, giving each limit to 3 significant figures.

(4)

(d)

The decay can also be modelled by N = e^−kt. Find the value of k correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes: (1/2)^2000/5730	✓ 1
A1 78.5 %	✓ 1
Part (b)
M1 sets (1/2)^t/5730 = 0.314	✓ 1
M1 takes logarithms: t = 5730 × ln 0.314 / ln 0.5 (= 9575.8…)	✓ 1
A1 9580 years	✓ 1
Part (c)
M1 uses 0.309 in the model: t = 5730 × ln 0.309 / ln 0.5	✓ 1
A1 9710 years (upper limit of age)	✓ 1
M1 uses 0.319 in the model	✓ 1
A1 9450 years (lower limit of age)	✓ 1
Part (d)
M1 equates (1/2)^t/5730 = e^−kt	✓ 1
M1 takes logarithms: −(t/5730) ln 2 = −kt, or equivalent	✓ 1
A1 k = ln 2 / 5730 = 1.21 × 10⁻⁴ per year	✓ 1

Exam tip: A lower percentage of carbon-14 means an older sample, so the smaller percentage gives the upper age limit; sketching the decay shape helps you assign the limits correctly.

+ Add to test✓ In your test484IBMI3S-1-03r23-11Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

Maya is a cyclist who is saving for retirement. She pays €400 at the end of every month into an account for 15 years. The account pays a nominal annual interest rate of 4.2%, compounded monthly.

(a)

Find the amount in the account at the end of the 15 years. Give your answer to the nearest cent.

(3)

(b)

Maya then stops paying in. She leaves the money in a different account with a nominal annual interest rate of 3.0%, compounded monthly, and withdraws equal amounts at the end of every month for 20 years, after which the balance is zero. Use your answer to part (a). Find the monthly withdrawal, to the nearest cent.

(4)

(c)

Calculate the total amount Maya withdraws over the 20 years. Use the withdrawal from part (b).

(2)

(d)

Maya paid in 180 × €400 = €72 000 in total. Find the total amount withdrawn as a percentage gain on the amount she paid in. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses a finance solver or annuity formula with PMT = −400, I% = 4.2, C/Y = P/Y = 12	✓ 1
A1 N = 180	✓ 1
A1 €100 062.45	✓ 1
Part (b)
M1 uses a finance solver with PV = 100 062.45, FV = 0	✓ 1
A1 N = 240, I% = 3.0	✓ 1
M1 solves for PMT	✓ 1
A1 €554.94	✓ 1
Part (c)
M1 240 × 554.94	✓ 1
A1 €133 185.60	✓ 1
Part (d)
M1 gain = 133 185.60 − 72 000 = €61 185.60	✓ 1
M1 divides by 72 000 and multiplies by 100	✓ 1
A1 85.0 %	✓ 1

Common error: When the interest rate changes, the new calculation starts from the balance at that moment as the present value; do not carry the old rate or the old payments forward.

+ Add to test✓ In your test485IBMI3S-1-03r23-14Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

Two cyclists, Ana and Ben, follow different training plans. On day 1 each cycles 20 km. Ana increases her distance by 3 km each day. Ben increases his distance by 10% each day.

(a)

Find the distance Ana cycles on day 12.

(2)

(b)

Find the distance Ben cycles on day 12. Give your answer to 3 significant figures.

(2)

(c)

Find the first day on which Ben cycles further than Ana.

(3)

(d)

Find how much further Ben has cycled than Ana in total over the first 16 days. Give your answer to 3 significant figures.

(5)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 arithmetic sequence u₁₂ = 20 + 11 × 3	✓ 1
A1 53 km	✓ 1
Part (b)
M1 geometric sequence u₁₂ = 20 × 1.1¹¹	✓ 1
A1 57.1 km	✓ 1
Part (c)
M1 compares 20 × 1.1ⁿ⁻¹ with 20 + 3(n − 1) using a table or graph	✓ 1
A1 day 9: Ben 42.87 km, Ana 44 km	✓ 1
A1 day 10: Ben 47.16 km, Ana 47 km, so day 10	✓ 1
Part (d)
M1 Ana: S₁₆ = (16/2)(2 × 20 + 15 × 3)	✓ 1
A1 680 km	✓ 1
M1 Ben: S₁₆ = 20(1.1¹⁶ − 1) / 0.1	✓ 1
A1 719 km (718.99…)	✓ 1
A1 39.0 km	✓ 1

Common error: Arithmetic growth adds the same amount each day, while geometric growth multiplies; geometric growth starts slower here but eventually overtakes.

+ Add to test✓ In your test486IBMI3S-1-01r24-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

The concentration of nitrate in a field drain, N mg per litre, is modelled by N = A × k^t, where t is the time in days after fertiliser is spread, and A and k are constants. When t = 1, N = 72. When t = 4, N = 30.375.

(a)

Find the value of k and the value of A.

(4)

(b)

Find the time at which the concentration first falls to 10 mg per litre. Give your answer in days to 3 significant figures.

(3)

(c)

The model can also be written as N = 96e^−bt. Find the value of b to 3 significant figures.

(2)

(d)

Find the time taken for the concentration to halve. Give your answer in days to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 dividing the two equations: k³ = 30.375 ÷ 72	✓ 1
A1 k = 0.75	✓ 1
M1 A = 72 ÷ 0.75	✓ 1
A1 A = 96	✓ 1
Part (b)
M1 96 × 0.75^t = 10	✓ 1
M1 use of logarithms or a solver, e.g. t = ln(10 ÷ 96) ÷ ln 0.75	✓ 1
A1 t = 7.86 days	✓ 1
Part (c)
M1 e^−b = 0.75 or b = −ln 0.75	✓ 1
A1 b = 0.288	✓ 1
Part (d)
M1 0.75^t = 0.5 (or e^−0.288t = 0.5)	✓ 1
A1 t = ln 0.5 ÷ ln 0.75	✓ 1
A1 2.41 days	✓ 1

Exam tip: When two data points fit an exponential model, divide the equations to eliminate A before solving for k.

+ Add to test✓ In your test487IBMI3S-1-01r24-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard14 marks

A dairy cooperative invests €60 000 in an account that pays a nominal annual interest rate of 3.8 %, compounded quarterly. The annual rate of inflation is constant at 2.1 %.

(a)

Find the value of the investment after 6 years. Give your answer to the nearest euro.

(3)

(b)

Show that the effective annual interest rate is 3.85 %, correct to 3 significant figures.

(2)

(c)

Find the value, after 6 years, of the investment in terms of the purchasing power of today's euros. Use the unrounded value from part (a) and give your answer to the nearest euro.

(3)

(d)

Find the real annual growth rate of the investment, as a percentage to 3 significant figures. Use the unrounded effective annual rate from part (b).

(3)

(e)

Using the unrounded real growth rate, find the number of years for the value of the investment, in today's euros, to double. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 60 000 × (1 + 0.038 ÷ 4)²⁴	✓ 1
A1 correct substitution of quarterly rate 0.95 % and 24 periods	✓ 1
A1 €75 284	✓ 1
Part (b)
M1 (1 + 0.038 ÷ 4)⁴ − 1	✓ 1
A1 0.038545… = 3.85 % (3 s.f.)	✓ 1
Part (c)
M1 divide by inflation factor 1.021⁶	✓ 1
A1 75 284.056… ÷ 1.021⁶	✓ 1
A1 €66 458	✓ 1
Part (d)
M1 (1.038545… ÷ 1.021) − 1	✓ 1
A1 0.0171841…	✓ 1
A1 1.72 %	✓ 1
Part (e)
M1 (1.017184…)^t = 2	✓ 1
M1 use of logarithms or a solver, e.g. t = ln 2 ÷ ln(1.017184…)	✓ 1
A1 40.7 years	✓ 1

Exam tip: Compare real values by dividing by the inflation factor over the same period; subtracting 2.1 % from the interest rate is only a rough approximation.

+ Add to test✓ In your test488IBMI3S-1-01r24-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard11 marks

A guitar student plans to increase practice time over 12 weeks. In week 1 the student practises for 20 minutes.
Plan A: each week's practice time is 10 % more than the previous week, so the weekly times form a geometric sequence.
Plan B: each week's practice time is a fixed number of minutes more than the previous week, so the weekly times form an arithmetic sequence.

(a)

Find the practice time in week 5 under Plan A. Give your answer to 3 significant figures.

(2)

(b)

Find the total practice time over the 12 weeks under Plan A. Give your answer to the nearest minute.

(3)

(c)

Under Plan B the total practice time over the 12 weeks is the same as under Plan A. Using the unrounded total from part (b), find the weekly increase d in minutes, to 3 significant figures.

(3)

(d)

Find how many more minutes the student practises in week 12 under Plan A than under Plan B. Use unrounded values and give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 20 × 1.1⁴	✓ 1
A1 29.3 minutes	✓ 1
Part (b)
M1 use of S₁₂ = 20(1.1¹² − 1) ÷ (1.1 − 1)	✓ 1
A1 correct substitution	✓ 1
A1 428 minutes	✓ 1
Part (c)
M1 (12/2)(2 × 20 + 11d) = 427.6856…	✓ 1
M1 correct rearrangement: 40 + 11d = 71.2809…	✓ 1
A1 d = 2.84 minutes	✓ 1
Part (d)
M1 Plan A: 20 × 1.1¹¹ (= 57.0623…)	✓ 1
M1 Plan B: 20 + 11 × 2.8437… (= 51.2809…)	✓ 1
A1 5.78 minutes	✓ 1

Exam tip: For geometric sequences use un = u₁rⁿ⁻¹; the exponent is n − 1, so week 12 uses 1.1¹¹, not 1.1¹².

+ Add to test✓ In your test489IBMI3S-1-01r26-11Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard12 marks

A mountaineer is saving for an expedition. She pays €250 into a savings account at the end of every month for 3 years. The account pays a nominal annual interest rate of 4.2 %, compounded monthly. The expedition currently costs €9 800 and its cost increases by 3 % each year.

(a)

Calculate the value of her savings at the end of the 3 years. Give your answer to the nearest cent.

(3)

(b)

Find the cost of the expedition after 3 years. Give your answer to the nearest cent.

(2)

(c)

Determine whether her savings from part (a) are enough to pay for the expedition after 3 years, and state the difference.

(1)

(d)

Find the smallest monthly payment, to the nearest cent, that would give enough savings to pay the cost found in part (b) after 3 years. Use the unrounded cost.

(3)

(e)

Instead of monthly payments, she could deposit a single sum into the same account now. Find the single sum needed to reach the cost found in part (b) after 3 years. Use the unrounded cost and give your answer to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a financial solver with N = 36, I% = 4.2, PMT = −250, PV = 0, P/Y = 12, C/Y = 12	✓ 1
A1 correct set-up (end-of-month payments)	✓ 1
A1 €9573.76	✓ 1
Part (b)
M1 9800 × 1.03³	✓ 1
A1 €10708.72	✓ 1
Part (c)
A1 not enough, as 9573.76 < 10708.72, shortfall €1134.96 (ECF from (a) and (b))	✓ 1
Part (d)
M1 FV = 10708.7246… with N = 36, I% = 4.2, PV = 0, P/Y = 12, C/Y = 12	✓ 1
M1 solving for PMT	✓ 1
A1 €279.64	✓ 1
Part (e)
M1 FV = 10708.7246…, N = 36, PMT = 0, I% = 4.2, P/Y = 12, C/Y = 12	✓ 1
M1 solving for PV, or PV = FV ÷ (1 + 0.042/12)³⁶	✓ 1
A1 €9443.05	✓ 1

Exam tip: Check the sign convention on your calculator: money paid in and money received must have opposite signs.

+ Add to test✓ In your test490IBMI3S-1-01r26-13Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard12 marks

A tray of custard is removed from an oven at time t = 0 and left in a kitchen at constant temperature 20 °C. Its temperature, T °C, after t minutes is modelled by T = 20 + Ae^−kt, where A and k are positive constants. The initial temperature of the custard is 95 °C. After 10 minutes the temperature is 52 °C.

(a)

Find the value of A.

(1)

(b)

Find the value of k. Give your answer to 3 significant figures.

(3)

(c)

Use k = 0.0852 to find the time taken for the custard to cool to 35 °C. Give your answer in minutes to 3 significant figures.

(3)

(d)

Use k = 0.0852 to find the temperature of the custard after 60 minutes. Give your answer to 3 significant figures.

(2)

(e)

Explain why this model predicts that the temperature of the custard never falls below 20 °C.

(1)

(f)

Show that the time taken for the difference between the custard temperature and 20 °C to halve is the same at every stage, and find this time. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 A = 75	✓ 1
Part (b)
M1 52 = 20 + 75e^−10k, so e^−10k = 32/75	✓ 1
M1 taking natural logarithms, −10k = ln(32/75)	✓ 1
A1 k = 0.0852	✓ 1
Part (c)
M1 35 = 20 + 75e^−0.0852t, so e^−0.0852t = 0.2	✓ 1
M1 t = ln 5 / 0.0852	✓ 1
A1 18.9 (minutes)	✓ 1
Part (d)
M1 20 + 75e^{−0.0852 × 60}	✓ 1
A1 20.5 (°C)	✓ 1
Part (e)
R1 as t increases the term 75e^−0.0852t tends to 0 but is always positive, so T approaches 20 from above	✓ 1
Part (f)
M1 75e^{−k(t + h)} = ½ × 75e^−kt gives e^−kh = ½, independent of t	✓ 1
A1 h = ln 2 / 0.0852 = 8.14 (minutes)	✓ 1

Exam tip: Isolate the exponential term before taking logarithms; subtract the constant (here 20) first.

+ Add to test✓ In your test491IBMI3S-1-02r26-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard13 marks

During roasting, the temperature T °C of a coffee bean at time t minutes after it enters the roaster is modelled by T = 215 − 190e^−0.21t.

(a)

Find the temperature of the bean after 4 minutes. Give your answer to 3 significant figures.

(2)

(b)

Find the time at which the bean reaches 196 °C. Give your answer to 3 significant figures.

(3)

(c)

State the initial temperature of the bean and explain why the model predicts that the temperature never reaches 215 °C.

(2)

(d)

A second batch follows the model T = 215 − 190e^−kt, and is 105 °C after 3 minutes. Find the value of k. Give your answer to 3 significant figures.

(3)

(e)

Use k = 0.182 to find the time for the second batch to reach 170 °C, and compare it with the time for the first batch to reach 170 °C. State which batch heats up more quickly. Give times in minutes to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 215 − 190e^−0.84	✓ 1
A1 133 (°C)	✓ 1
Part (b)
M1 190e^−0.21t = 19, so e^−0.21t = 0.1	✓ 1
M1 taking natural logarithms, t = ln 10 / 0.21	✓ 1
A1 11.0 (minutes)	✓ 1
Part (c)
A1 initial temperature 25 °C	✓ 1
R1 e^−0.21t is always positive, so 190e^−0.21t > 0 and T < 215	✓ 1
Part (d)
M1 105 = 215 − 190e^−3k, so e^−3k = 110/190	✓ 1
M1 taking natural logarithms, k = ln(190/110) / 3	✓ 1
A1 0.182	✓ 1
Part (e)
M1 second batch: 190e^−0.182t = 45, t = ln(190/45) / 0.182	✓ 1
A1 7.91 (minutes)	✓ 1
A1 first batch: ln(190/45) / 0.21 = 6.86 (minutes), so the first batch heats up more quickly	✓ 1

Common error: Subtract the temperature from the asymptote first: 215 − T = 190e⁻kt, and only then take logarithms.

+ Add to test✓ In your test492IBMI3S-1-02r26-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

A council borrows €2 400 000 to repair a bridge. Lender X charges a nominal annual interest rate of 3.9 %, compounded annually. The loan is repaid in equal payments at the end of each year for 15 years.

(a)

Calculate the annual payment. Give your answer to the nearest cent.

(3)

(b)

Find the total amount repaid and the total interest paid. Use the unrounded annual payment and give your answers to the nearest cent.

(2)

(c)

Find the amount still owed immediately after the fifth payment. Use the unrounded annual payment and give your answer to the nearest cent.

(3)

(d)

Lender Y offers a nominal annual interest rate of 3.7 %, compounded quarterly. Find the equivalent effective annual rate of lender Y, correct to 3 significant figures, and state which lender's rate is lower.

(3)

(e)

With lender Y the council repays the €2 400 000 in equal payments at the end of each quarter for 15 years. Find the saving in total interest compared with lender X. Use unrounded values and give your answer to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a financial solver with N = 15, I% = 3.9, PV = 2400000, FV = 0, P/Y = 1, C/Y = 1	✓ 1
A1 correct set-up of end-of-year payments	✓ 1
A1 €214352.00	✓ 1
Part (b)
A1 total repaid = 15 × 214351.9961… = €3215279.94	✓ 1
A1 interest = €815279.94	✓ 1
Part (c)
M1 balance = value of the remaining 10 payments, N = 10, PMT = −214351.9961…, FV = 0, I% = 3.9	✓ 1
A1 correct solver entry for PV (or 2400000 × 1.039⁵ minus the future value of 5 payments)	✓ 1
A1 €1747274.01	✓ 1
Part (d)
M1 (1 + 0.037/4)⁴ − 1	✓ 1
A1 3.75 (%)	✓ 1
A1 lender Y is lower (3.75 % < 3.9 %)	✓ 1
Part (e)
M1 solver with N = 60, I% = 3.7, PV = 2400000, FV = 0, P/Y = 4, C/Y = 4 gives quarterly payment €52301.59	✓ 1
M1 interest with Y = 60 × 52301.5852… − 2400000 = €738095.11	✓ 1
A1 saving = 815279.94 − 738095.11 = €77184.83	✓ 1

Exam tip: To compare loans with different compounding periods, convert each to an effective annual rate.

+ Add to test✓ In your test493IBMI3S-1-02r28-04Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard12 marks

A data centre sets up a fund for replacing equipment. It pays €400 into an account at the end of each month. The account pays a nominal annual interest rate of 2.4%, compounded monthly.

(a)

Find the value of the fund after 5 years, in € to 2 decimal places.

(3)

(b)

Find the total interest earned in the 5 years.

(2)

(c)

The data centre wants the fund to be worth exactly €30 000 after 5 years. Find the monthly payment needed, in € to 2 decimal places.

(2)

(d)

With payments of €400 per month, find the number of monthly payments needed for the fund to reach €30 000.

(3)

(e)

Find the effective annual interest rate of the account, as a percentage to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 monthly rate 0.2% and N = 60	✓ 1
M1 use of the finance solver with PMT = −400, PV = 0	✓ 1
A1 € 25 472.35	✓ 1
Part (b)
M1 total paid in 60 × 400 = 24 000	✓ 1
A1 25 472.35 − 24 000 = € 1472.35 (ECF from (a))	✓ 1
Part (c)
M1 finance solver with FV = 30 000, N = 60	✓ 1
A1 € 471.10	✓ 1
Part (d)
M1 finance solver with FV = 30 000, PMT = −400	✓ 1
A1 N = 69.95	✓ 1
R1 70 payments, since a whole number of payments is needed and 69 is not enough	✓ 1
Part (e)
M1 (1 + 0.024/12)¹² − 1	✓ 1
A1 2.43%	✓ 1

Common error: The number of payments must be a whole number: if the solver gives 69.95, round up, because 69 payments leave the fund just short of the target.

+ Add to test✓ In your test494IBMI3S-1-02r28-08Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard13 marks

A cyclist considers two training plans.
Plan A: 30 km in week 1, and each week 5 km more than the previous week.
Plan B: 20 km in week 1, and each week 10% more than the previous week.

(a)

State the type of sequence for each plan, and find the distance in week 10 for each plan. Give answers to 3 significant figures.

(3)

(b)

Find the first week in which the weekly distance of Plan B is greater than that of Plan A.

(3)

(c)

Find the first week by which the total distance of Plan B is greater than the total distance of Plan A.

(3)

(d)

Find the total distance of each plan over the first 20 weeks. Give each answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Plan A is arithmetic, Plan B is geometric	✓ 1
A1 Plan A: 30 + 9 × 5 = 75 km	✓ 1
A1 Plan B: 20 × 1.1⁹ = 47.2 km	✓ 1
Part (b)
M1 compare 20 × 1.1^{n − 1} with 30 + 5(n − 1) using a table or graph	✓ 1
A1 week 20: Plan A 125 km, Plan B 122.3 km (Plan B is not yet greater)	✓ 1
A1 week 21: Plan A 130 km, Plan B 134.5 km, so week 21	✓ 1
Part (c)
M1 set up S_B > S_A using the two sum formulae, with a table or graph	✓ 1
A1 week 28: Plan A 2730 km, Plan B 2684 km (Plan B is not yet greater)	✓ 1
A1 week 29: Plan A 2900 km, Plan B 2973 km, so week 29	✓ 1
Part (d)
M1 S₂₀ for the arithmetic plan: 20/2 (2 × 30 + 19 × 5)	✓ 1
A1 1550 km	✓ 1
M1 S₂₀ for the geometric plan: 20(1.1²⁰ − 1)/0.1	✓ 1
A1 1150 km	✓ 1

Exam tip: To compare an arithmetic and a geometric model use a GDC table of both rules side by side, then check the two neighbouring values to confirm the crossover.

+ Add to test✓ In your test495IBMI3S-1-02r28-12Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard13 marks

After a cooling failure, the temperature T °C of a server rack t minutes after the failure is repaired is modelled by T = 22 + 58e^−0.12t, t ≥ 0.

(a)

Find the temperature of the rack at the moment the failure is repaired.

(1)

(b)

Find the temperature after 10 minutes, to 3 significant figures.

(2)

(c)

Find the time at which the rack reaches 30 °C, to 3 significant figures.

(3)

(d)

State the temperature that the model predicts the rack approaches in the long term.

(1)

(e)

The model can be written as T = 22 + 58b^t. Find b to 3 significant figures and hence the percentage by which the difference above 22 °C falls each minute, to 3 significant figures.

(3)

(f)

Find the time at which the temperature is within 0.5 °C of its long-term value. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 T(0) = 22 + 58 = 80 °C	✓ 1
Part (b)
M1 22 + 58e^−1.2	✓ 1
A1 39.5 °C	✓ 1
Part (c)
M1 set 22 + 58e^−0.12t = 30 so e^−0.12t = 8/58	✓ 1
M1 take natural logarithms: t = −ln(8/58) / 0.12	✓ 1
A1 t = 16.5 minutes	✓ 1
Part (d)
A1 22 °C	✓ 1
Part (e)
M1 b = e^−0.12	✓ 1
A1 b = 0.887	✓ 1
A1 1 − 0.88692… = 11.3% per minute	✓ 1
Part (f)
M1 set 58e^−0.12t = 0.5	✓ 1
M1 t = ln(116) / 0.12	✓ 1
A1 t = 39.6 minutes	✓ 1

Exam tip: In a model of the form A + Be⁻kt, the constant A is the long-term value; solve equations by isolating the exponential term first.

+ Add to test✓ In your test496IBMI3S-1-01r29-12Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

Dilnoza takes out a mortgage of £180 000 to buy a studio for her fashion label. The nominal annual interest rate is 3.6%, compounded monthly. The mortgage is to be repaid by equal payments made at the end of each month for 25 years.

(a)

Calculate the monthly payment, to the nearest penny.

(3)

(b)

Calculate the total interest paid if the mortgage runs for the full 25 years. Use the payment from part (a).

(2)

(c)

Calculate the balance owed immediately after the 120th payment. Use the payment from part (a), and give your answer to the nearest penny.

(3)

(d)

After the 120th payment the interest rate changes to a nominal annual rate of 4.8%, still compounded monthly. The balance from part (c) is repaid by equal monthly payments over the remaining 15 years. Calculate the new monthly payment, to the nearest penny.

(3)

(e)

Calculate how much more Dilnoza pays in total over the 25 years than she would have paid if the rate had stayed at 3.6%. Use the rounded payments from (a) and (d).

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of TVM solver with N = 300, I% = 3.6, PV = 180000, FV = 0, C/Y = 12	✓ 1
M1 correct signs or correct use of the annuity formula	✓ 1
A1 £910.80	✓ 1
Part (b)
M1 300 × 910.8 − 180000	✓ 1
A1 £93,240.00 (ECF from (a))	✓ 1
Part (c)
M1 use of TVM solver with N = 120, PV = 180000, PMT = −910.8	✓ 1
M1 FV found with correct signs, or 180000 × 1.003¹²⁰ − 910.8 × (1.003¹²⁰ − 1) ÷ 0.003	✓ 1
A1 £126,535.93 (ECF from (a))	✓ 1
Part (d)
M1 N = 180, I% = 4.8, PV = 126535.93	✓ 1
M1 correct signs or correct use of the annuity formula	✓ 1
A1 £987.50 (ECF from (c))	✓ 1
Part (e)
M1 total paid = 120 × 910.8 + 180 × 987.5	✓ 1
M1 subtract 300 × 910.8	✓ 1
A1 £13,806.00 (ECF from (a) and (d))	✓ 1

Exam tip: In multi-part finance questions, label the TVM values you enter for each stage; the new loan starts with the balance from the previous stage, not the original amount.

+ Add to test✓ In your test497IBMI3S-1-01r29-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard12 marks

Two boats, A and B, fish in the same waters. The weekly catches of boat A, in tonnes, form an arithmetic sequence with a first-week catch of 5 tonnes and an increase of 0.8 tonnes each week. The weekly catches of boat B form a geometric sequence with a first-week catch of 3 tonnes and an increase of 12% each week.

(a)

Find the catch of boat A in week 10.

(2)

(b)

Find the catch of boat B in week 10. Give your answer to 3 significant figures.

(2)

(c)

Find the total catch of boat A over the first 12 weeks.

(2)

(d)

Find the first week in which the catch of boat B is greater than the catch of boat A.

(3)

(e)

Find the total catch of boat B over the first 17 weeks. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 5 + 9 × 0.8	✓ 1
A1 12.2 tonnes	✓ 1
Part (b)
M1 3 × 1.12⁹	✓ 1
A1 8.32 tonnes	✓ 1
Part (c)
M1 $\frac{12}{2}(2\times5+11\times0.8)$	✓ 1
A1 112.8 tonnes	✓ 1
Part (d)
M1 3 × 1.12ⁿ⁻¹ > 5 + 0.8(n − 1)	✓ 1
M1 attempt to solve using a table of values or graph	✓ 1
A1 week 17	✓ 1
Part (e)
M1 use of $S_n=\frac{u_1(r^n-1)}{r-1}$ with u₁ = 3, r = 1.12	✓ 1
M1 correct substitution, n = 17	✓ 1
A1 147 tonnes (ECF from (d))	✓ 1

Common error: A 12% weekly increase is a common ratio of 1.12; do not add 12% of the first-week catch each week, which would give an arithmetic sequence.

+ Add to test✓ In your test498IBMI3S-1-02r29-12Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard14 marks

Sofia borrows £40 000 to open a fashion studio. The nominal annual interest rate is 7.2%, compounded monthly. She repays the loan by making payments of £800 at the end of each month, with a smaller final payment to clear the loan exactly.

(a)

Find the number of payments of £800 needed before the remaining balance is less than £800, giving your answer as a whole number. Hence state how many payments in total Sofia makes.

(3)

(b)

Calculate the amount of the final payment, to the nearest penny.

(3)

(c)

Calculate the total interest Sofia pays. Use the final payment from part (b).

(2)

(d)

Sofia considers instead paying £1000 each month, again with a smaller final payment. Find the number of payments of £1000 she would make in total.

(3)

(e)

With payments of £1000, the final payment is £876.88 (to the nearest penny). Calculate how much less interest Sofia pays than in part (c).

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM solver with I% = 7.2, PV = 40000, PMT = −800, FV = 0, C/Y = 12 and solve for N	✓ 1
A1 N = 59.62… so 59 full payments of £800	✓ 1
A1 60 payments in total	✓ 1
Part (b)
M1 balance after 59 payments = 40000 × 1.006⁵⁹ − 800 × (1.006⁵⁹ − 1) ÷ 0.006 = £496.77	✓ 1
M1 add one month's interest, balance × 1.006	✓ 1
A1 £499.75	✓ 1
Part (c)
M1 59 × 800 + 499.75 − 40000	✓ 1
A1 £7,699.75 (ECF from (b))	✓ 1
Part (d)
M1 TVM solver with PMT = −1000 and solve for N	✓ 1
A1 N = 45.88…	✓ 1
A1 46 payments in total	✓ 1
Part (e)
M1 total paid = 45 × 1000 + 876.88	✓ 1
M1 interest = total paid − 40000 = £5,876.88, then subtract from the answer to (c)	✓ 1
A1 £1,822.87 (ECF from (c))	✓ 1

Exam tip: When a TVM solver gives a non-integer number of payments, the loan is cleared on the next whole payment, and that last payment is smaller than the others.

+ Add to test✓ In your test499IBMI3S-1-02r29-13Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

A fish-tank treatment adds 200 mg of a medicine to the water every 6 hours, with the first dose at time 0. Between doses, 70% of the medicine in the tank is removed by filtration, so 30% of the amount present just before a dose remains when the next dose is added.

(a)

Show that the amount of medicine just after the 2nd dose is 260 mg.

(2)

(b)

Find the amount of medicine just after the 4th dose. Give your answer to 4 significant figures.

(3)

(c)

Find the smallest number of doses after which the amount just after a dose is greater than 284 mg.

(4)

(d)

Find the amount of medicine in the tank just before the 6th dose. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 200 × 0.3 + 200	✓ 1
A1 = 260 mg (AG)	✓ 1
Part (b)
M1 recognise a geometric sequence, with the amount after the nth dose equal to 200 + 200(0.3) + … + 200(0.3)ⁿ⁻¹	✓ 1
M1 $\frac{200(1-0.3^4)}{1-0.3}$	✓ 1
A1 283.4 mg	✓ 1
Part (c)
M1 $\frac{200(1-0.3^n)}{0.7}>284$	✓ 1
M1 0.3ⁿ < 0.006	✓ 1
M1 attempt to solve, for example n > ln 0.006 ÷ ln 0.3 = 4.2…	✓ 1
A1 5 doses	✓ 1
Part (d)
M1 amount just after the 5th dose = 285.02 mg	✓ 1
M1 multiply by 0.3	✓ 1
A1 85.5 mg	✓ 1

Common error: The amount just after a dose is the new dose plus 30% of what was present, not 30% of the new total.

+ Add to test✓ In your test500IBMI3S-1-01r30-09Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; 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A company that runs data centres borrows €250 000 at a nominal annual interest rate of 5.4%, compounded monthly. The loan is to be repaid by equal payments at the end of each month for 15 years.

(a)

Calculate the monthly payment, to the nearest cent.

(3)

(b)

Using the payment from (a), find the balance owed immediately after the 60th payment, to the nearest cent.

(3)

(c)

Immediately after the 60th payment the company refinances. It borrows the balance from (b) at a nominal annual interest rate of 4.2%, compounded monthly, and repays it with equal monthly payments over a further 10 years. Find the new monthly payment, to the nearest cent.

(2)

(d)

Using the rounded payments from (a) and (c), find how much less the company pays in total over the 15 years by refinancing than by keeping the original loan.

(3)

(e)

The refinancing costs a one-off fee of 1.2% of the balance in (b). Find the net saving after paying the fee, and state whether refinancing is still worthwhile.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of a finance solver or annuity formula with N = 180	✓ 1
A1 correct inputs: I% = 5.4, PV = 250000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €2029.47	✓ 1
Part (b)
M1 use of solver with N = 60, PMT = −2029.47, FV unknown	✓ 1
A1 correct inputs: I% = 5.4, PV = 250000, P/Y = C/Y = 12	✓ 1
A1 €187858.64	✓ 1
Part (c)
M1 N = 120, I% = 4.2, PV = 187858.64, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €1919.88	✓ 1
ECF from (b).	—
Part (d)
M1 refinanced total = 60 × 2029.47 + 120 × 1919.88 = 352153.80	✓ 1
M1 original total = 180 × 2029.47 = 365304.60	✓ 1
A1 €13150.80	✓ 1
Part (e)
M1 fee = 0.012 × 187858.64 = 2254.30	✓ 1
A1 net saving = €10896.50	✓ 1
R1 saving is positive, so refinancing is still worthwhile	✓ 1
ECF from (d).	—

Exam tip: When a later part says to use a rounded payment, keep to that rule throughout; mixing rounded and unrounded values can change the final cents.

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A retired archaeologist places €120 000 in a fund that pays a nominal annual interest rate of 3%, compounded monthly. At the end of each month, after interest has been added, €1000 is withdrawn.

(a)

Find the balance in the fund immediately after the 12th withdrawal. Give your answer to the nearest cent.

(2)

(b)

Find the number of withdrawals of €1000 that can be made, and show that a smaller final withdrawal is needed to empty the fund.

(3)

(c)

Find the amount of the final withdrawal, to the nearest cent.

(3)

(d)

Find the total amount withdrawn, and hence the total interest earned by the fund.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of solver with N = 12, I% = 3, PV = 120000, PMT = −1000, P/Y = C/Y = 12	✓ 1
A1 €111483.53	✓ 1
Part (b)
M1 solve for N with PV = 120000, PMT = −1000, FV = 0	✓ 1
A1 N = 142.85…	✓ 1
R1 so 142 full withdrawals are possible, and a 143rd, smaller, withdrawal empties the fund	✓ 1
Part (c)
M1 balance after 142 withdrawals = 846.2857… (unrounded)	✓ 1
M1 add one month of interest: 846.2857… × 1.0025	✓ 1
A1 €848.40	✓ 1
Part (d)
A1 142 × 1000 + 848.40 = €142848.40	✓ 1
A1 interest = 142848.40 − 120000 = €22848.40	✓ 1
ECF from (c).	—

Exam tip: Before using a solver for the number of payments, check that the result is not a whole number; a decimal answer means the last payment is smaller.

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Coolant leaves a data-centre server at a high temperature and then cools in a room. Its temperature T, in °C, after t minutes is modelled by $T = 22 + 58e^{-0.12t}$.

(a)

Write down the temperature of the coolant when it leaves the server.

(1)

(b)

Find the temperature of the coolant after 10 minutes. Give your answer to 3 significant figures.

(2)

(c)

Find the time for the coolant to cool to 30 °C. Give your answer to 3 significant figures.

(3)

(d)

State the temperature predicted by the model as t becomes very large, and explain why this is sensible.

(2)

(e)

The model shows that the excess of the temperature above 22 °C halves in a constant time. Find this time, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 80 °C	✓ 1
Part (b)
M1 22 + 58e^−1.2	✓ 1
A1 39.5 °C	✓ 1
Part (c)
M1 22 + 58e^−0.12t = 30	✓ 1
M1 $e^{-0.12t} = \frac{8}{58}$, then take natural logarithms	✓ 1
A1 16.5 minutes	✓ 1
Part (d)
A1 22 °C	✓ 1
R1 e^−0.12t tends to 0, and the coolant cannot cool below the room temperature	✓ 1
Part (e)
M1 e^−0.12t = 0.5	✓ 1
A1 $t = \frac{\ln 2}{0.12}$ = 5.78 minutes	✓ 1

Common error: Subtracting 22 from only one side when solving 22 + 58e^(−0.12t) = 30. Isolate the exponential term first, then take logarithms.

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A village committee saves for a new well. It pays €320 at the end of every month into an account with a nominal annual interest rate of 4.8%, compounded monthly. The first payment is made at the end of month 1 and there are no withdrawals.

(a)

Find the value of the account immediately after the 96th payment (8 years). Give your answer to 2 decimal places.

(3)

(b)

The committee needs €40 000. Find the smallest number of monthly payments after which the account holds at least €40 000.

(3)

(c)

Alternatively, the committee could pay a single lump sum into the same account now, with no further payments, so that it holds €40 000 after 8 years. Find the size of this lump sum, to the nearest euro.

(2)

(d)

Use your answer to part (a) to find the total interest earned over the 8 years. Give your answer to 2 decimal places.

(2)

(e)

Find the interest from part (d) as a percentage of the value of the account from part (a). Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 identify $N=96$, $I\%=4.8$, $PMT=-320$, $PV=0$, $P/Y=C/Y=12$	✓ 1
M1 use of TVM solver or $320\times\frac{1.004^{96}-1}{0.004}$	✓ 1
A1 €37361.71	✓ 1
Part (b)
M1 set up $320\times\frac{1.004^{N}-1}{0.004}=40000$ (or TVM with $FV=40000$)	✓ 1
A1 $N=101.57\ldots$	✓ 1
A1 102 payments	✓ 1
Part (c)
M1 $PV\times1.004^{96}=40000$ (or TVM with $N=96$, $FV=40000$, $PMT=0$)	✓ 1
A1 €27266 (accept 27 266.13)	✓ 1
Part (d)
M1 total paid in is $96\times320=30720$	✓ 1
A1 €6641.71 (37361.71 − 30720)	✓ 1
Part (e)
M1 $\frac{\text{interest}}{\text{value}}\times100$	✓ 1
A1 17.8%	✓ 1

Common error: A regular-payment account needs PMT entered with the opposite sign to FV on a TVM solver; the lump sum case has PMT = 0.

+ Add to test✓ In your test504IBMI3S-1-03r30-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard12 marks

A patient takes a 200 mg tablet every 8 hours. Between tablets the body removes 15% of the drug present in the blood each hour. Assume each tablet is absorbed instantly when it is taken.

(a)

Show that the fraction of the drug present immediately after a dose that is still present 8 hours later is 0.272, correct to 3 significant figures.

(2)

(b)

Let $A_n$ be the mass of the drug, in mg, in the blood immediately after the nth tablet is taken. Show that $A_2=200(1+r)$, where $r=0.85^{8}$, and find $A_3$. Use the unrounded value of $r$ and give $A_3$ to the nearest mg.

(3)

(c)

In the long term, the mass immediately after a tablet approaches a limiting value. Find this value, to the nearest mg.

(2)

(d)

Find the smallest number of tablets after which the mass immediately after the dose first exceeds 99% of the limiting value. Use the unrounded limiting value.

(3)

(e)

Find the mass of the drug immediately after the dose found in part (d). Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $0.85^{8}$	✓ 1
A1 0.272 (0.2724...) (AG)	✓ 1
Part (b)
M1 $A_2=200+200r$	✓ 1
M1 $A_3=200(1+r+r^2)$ (or $200\times\frac{1-r^3}{1-r}$)	✓ 1
A1 $A_3=269$ mg	✓ 1
Part (c)
M1 $\frac{200}{1-r}$ (sum to infinity)	✓ 1
A1 275 mg (274.9...)	✓ 1
Part (d)
M1 $200\times\frac{1-r^{n}}{1-r}>0.99\times\frac{200}{1-r}$ so $r^{n}<0.01$	✓ 1
M1 solve using logs, table or graph: $n>\frac{\ln0.01}{\ln r}=3.54\ldots$	✓ 1
A1 $n=4$	✓ 1
Part (e)
M1 $A_4=200\times\frac{1-r^4}{1-r}$	✓ 1
A1 273 mg (273.4...) (ECF from (d))	✓ 1

Exam tip: Write the amount after each dose as a sum, $200+200r+200r^2+\dots$, before using a formula; it shows you whether it is a geometric series and which term count to use.

+ Add to test✓ In your test505IBMI3S-1-03r30-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard11 marks

A water pump costs €18 500 when new. Its book value falls by 12% each year. The cost of a replacement pump rises by 3.5% each year from the same starting price of €18 500.

(a)

Find the book value of the pump after 5 years, to the nearest euro.

(2)

(b)

Find the cost of a replacement pump after 5 years, to the nearest euro.

(2)

(c)

The utility replaces the pump in the first year in which the book value is less than 40% of the replacement cost. Find this year (the number of whole years after purchase).

(4)

(d)

The utility pays equal amounts at the end of each of the 6 years into an account paying 2.5% per year, compounded annually, so that it holds the replacement cost at the end of year 6 (€22 741.22 to 2 decimal places). Find the annual payment, to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $18500\times0.88^{5}$	✓ 1
A1 €9763	✓ 1
Part (b)
M1 $18500\times1.035^{5}$	✓ 1
A1 €21 972	✓ 1
Part (c)
M1 $18500\times0.88^{n}<0.4\times18500\times1.035^{n}$	✓ 1
M1 $\left(\frac{0.88}{1.035}\right)^{n}<0.4$ and logs (or table/graph)	✓ 1
A1 $n>5.65\ldots$ (or ratios 0.444 at $n=5$ and 0.378 at $n=6$)	✓ 1
A1 $n=6$	✓ 1
Part (d)
M1 identify $N=6$, $I\%=2.5$, $PV=0$, $FV=22741.22$, $P/Y=C/Y=1$	✓ 1
M1 use of TVM solver or $PMT=\frac{22741.22\times0.025}{1.025^{6}-1}$	✓ 1
A1 €3560.14	✓ 1

Exam tip: Compare the two growth factors by dividing them; one inequality in a single power then solves with logarithms.

+ Add to test✓ In your test506IBMI3S-1-03r32-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard10 marks

An archaeology trust holds a fund of £250 000 that earns interest at 4.2% per year, compounded annually. At the end of each year, immediately after the interest is added, the trust withdraws £15 000 to pay for excavations.

(a)

Show that the balance immediately after the first withdrawal is £245 500.

(2)

(b)

Find the balance immediately after the 10th withdrawal. Give your answer to the nearest pound.

(2)

(c)

Find the number of years for which the fund can pay the full £15 000 withdrawal, and state the number of complete withdrawals that can be made.

(3)

(d)

The trust now wants the fund to last exactly 25 years, with equal withdrawals at the end of each year. Find the size of the annual withdrawal. Give your answer to the nearest pound.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $250000\times1.042 - 15000$	✓ 1
A1 $260500 - 15000 = 245500$ (answer given, correct working shown)	✓ 1
Part (b)
M1 finance solver with N = 10, I% = 4.2, PV = −250000, PMT = 15000, P/Y = C/Y = 1 (or repeated use of $B_{n} = 1.042B_{n-1} - 15000$)	✓ 1
A1 £195469	✓ 1
Part (c)
M1 finance solver with FV = 0, PV = −250000, PMT = 15000 (or solve $B_n = 0$)	✓ 1
A1 $N = 29.26$	✓ 1
A1 29 complete withdrawals (the 30th would not be fully paid)	✓ 1
Part (d)
M1 finance solver with N = 25, I% = 4.2, PV = −250000, FV = 0	✓ 1
M1 attempt to find PMT, or $W=\dfrac{250000\times0.042}{1-1.042^{-25}}$	✓ 1
A1 £16343	✓ 1

Exam tip: Check the direction of the answer: a longer lifetime for the fund must mean a smaller withdrawal, which is a quick test for a wrong sign or misplaced value.

+ Add to test✓ In your test507IBMI3S-1-03r33-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard12 marks

A marine reserve is restoring a reef by planting coral fragments each month. Under Plan A, 40 fragments are planted in month 1 and the number planted increases by 15 each month, so that the monthly totals form an arithmetic sequence.

(a)

Find the number of fragments planted in month 12.

(2)

(b)

Find the total number of fragments planted in the first 12 months.

(2)

(c)

Find the least number of months needed for the total number of fragments planted to exceed 2000.

(3)

(d)

Plan B plants 40 fragments in month 1, and the number planted increases by 12% each month. Find the total number of fragments planted under Plan B in the first 12 months. Give your answer to 3 significant figures.

(3)

(e)

Find how many more fragments Plan A plants than Plan B in the first 12 months. Use the unrounded value from (d) and give your answer to the nearest whole number.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $u_{12}=40+11\times15$	✓ 1
A1 205 fragments	✓ 1
Part (b)
M1 use of $S_{12}=\frac{12}{2}(40+205)$ (ECF from (a))	✓ 1
A1 1470 fragments	✓ 1
Part (c)
M1 use of $S_n=\frac{n}{2}(80+15(n-1))$ with $S_n>2000$ (or an equation)	✓ 1
M1 solve using a solver, table or quadratic inequality	✓ 1
A1 15 months (total after 14 months is 1925, after 15 months is 2175)	✓ 1
Part (d)
M1 recognise a geometric sequence with $u_1=40$, $r=1.12$	✓ 1
M1 use of $S_{12}=\frac{40(1.12^{12}-1)}{1.12-1}$	✓ 1
A1 965 fragments (unrounded value 965.3)	✓ 1
Part (e)
M1 use of 1470 − 965.3 (the total from (b) minus the unrounded total from (d))	✓ 1
A1 505 fragments	✓ 1

Exam tip: State which formula (arithmetic or geometric) you are using and the values of $u_1$, $d$ or $r$ and $n$ before substituting.

+ Add to test✓ In your test508IBMI3S-1-03r33-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context Paper 2Hard12 marks

The acidity of sea water is measured by its pH, where $\text{pH}=-\log_{10}[\text{H}^+]$ and $[\text{H}^+]$ is the concentration of hydrogen ions in mol dm⁻³.

(a)

Calculate $[\text{H}^+]$ when the pH is 8.20. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a correct to 3 significant figures.

(2)

(b)

A sample of sea water has $[\text{H}^+]=8.91\times10^{-9}$ mol dm⁻³. Find its pH, to 2 decimal places.

(2)

(c)

The pH of a region of ocean falls by 0.4. Find the factor by which $[\text{H}^+]$ is multiplied. Give your answer to 3 significant figures.

(2)

(d)

Show that if $[\text{H}^+]$ is multiplied by a factor $k$, the pH decreases by $\log_{10}k$.

(3)

(e)

A model for the pH of a region of ocean is $\text{pH}=8.20-0.0018t$, where t is the time in years from now. Find the time for $[\text{H}^+]$ to increase by 50%. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of $[\text{H}^+]=10^{-8.2}$	✓ 1
A1 6.31 × 10⁻⁹ mol dm⁻³	✓ 1
Part (b)
M1 use of $-\log_{10}(8.91\times10^{-9})$	✓ 1
A1 8.05	✓ 1
Part (c)
M1 use of $10^{0.4}$ (or equivalent, e.g. $\frac{10^{-(\text{pH}-0.4)}}{10^{-\text{pH}}}$)	✓ 1
A1 2.51	✓ 1
Part (d)
M1 write new pH $=-\log_{10}(k\,h)$ where $h$ is the original concentration	✓ 1
M1 use of $\log_{10}(kh)=\log_{10}k+\log_{10}h$	✓ 1
A1 new pH $=-\log_{10}h-\log_{10}k=$ original pH $-\log_{10}k$ (shown)	✓ 1
Part (e)
M1 use of the result in (d) with $k=1.5$, so the pH must fall by $\log_{10}1.5\ (=0.176)$	✓ 1
M1 set up $0.0018t=\log_{10}1.5$	✓ 1
A1 97.8 years	✓ 1

Exam tip: A fall of 1 in pH means the hydrogen ion concentration is multiplied by 10, so keep track of the direction of every change.

+ Add to test✓ In your test509IBMI3S-1-04r34-12Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; 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A fashion label borrows €32 000 for new equipment. Loan 1 has a nominal annual interest rate of 5.4%, compounded monthly, repaid by equal payments at the end of each month for 5 years. Give money answers to the nearest cent unless stated otherwise.

(a)

Calculate the monthly payment on Loan 1.

(2)

(b)

Find the total interest paid over the 5 years, using your answer to part (a).

(2)

(c)

Find the amount still owed after the 24th payment, to the nearest euro.

(2)

(d)

Immediately after the 24th payment, the label pays an extra €10 000 and continues to pay the same monthly payment. Find the number of further monthly payments needed to clear the loan, giving your answer as a whole number (the final payment may be smaller).

(3)

(e)

Loan 2 is for the same amount over the same 5 years with equal monthly payments, but its nominal annual rate is 5.1% compounded quarterly. Find the monthly payment on Loan 2, and state which loan has the smaller monthly payment.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finance app with N = 60, PV = 32000, FV = 0, I% = 5.4, P/Y = C/Y = 12	✓ 1
A1 €609.76	✓ 1
Part (b)
M1 60 × 609.76 − 32 000	✓ 1
A1 €4585.60 (ECF from (a))	✓ 1
Part (c)
M1 finance app with N = 24 and the same PV, I% and PMT	✓ 1
A1 €20 224	✓ 1
Part (d)
M1 new balance 20 223.71… − 10 000 = 10 223.71…	✓ 1
M1 solving for N using the finance app with PMT = −609.76	✓ 1
A1 N = 17.47…, so 18 payments	✓ 1
Part (e)
M1 finance app with I% = 5.1, P/Y = 12, C/Y = 4, N = 60	✓ 1
A1 €605.03	✓ 1
R1 Loan 2 has the smaller monthly payment (compared with €609.76 for Loan 1)	✓ 1

Common error: When payments are monthly but interest is compounded quarterly, set P/Y = 12 and C/Y = 4 separately; using 12 for both gives the wrong answer.

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A fashion label compares two revenue plans for a new collection over 24 months. Plan A: the revenue in month 1 is €2000 and it rises by €350 each month (arithmetic sequence). Plan B: the revenue in month 1 is €1800 and it rises by 9.5% each month (geometric sequence). Give money answers to the nearest cent.

(a)

Find the revenue in month 12 under Plan A.

(2)

(b)

Find the total revenue over the 24 months under Plan A.

(2)

(c)

Find the revenue in month 12 under Plan B.

(2)

(d)

Find the first month in which the monthly revenue under Plan B is greater than under Plan A.

(3)

(e)

Find the total revenue over the 24 months under Plan B.

(2)

(f)

Find how much more total revenue Plan B gives than Plan A over the 24 months.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2000 + 11 × 350	✓ 1
A1 €5850.00	✓ 1
Part (b)
M1 S₂₄ = 24/2 × (2 × 2000 + 23 × 350)	✓ 1
A1 €144 600.00	✓ 1
Part (c)
M1 1800 × 1.095¹¹	✓ 1
A1 €4884.59	✓ 1
Part (d)
M1 setting up 1800 × 1.095^{n − 1} > 2000 + 350(n − 1) (or a table of both sequences)	✓ 1
M1 comparing consecutive months, e.g. month 16: Plan B €7022.38 is less than Plan A €7250; month 17: Plan B €7689.51 is greater than Plan A €7600	✓ 1
A1 month 17	✓ 1
Part (e)
M1 S₂₄ = 1800(1.095²⁴ − 1)/(1.095 − 1)	✓ 1
A1 €148 349.48	✓ 1
Part (f)
A1 €3749.48 (ECF from (b) and (e))	✓ 1

Exam tip: A geometric plan can start below an arithmetic plan and still win overall, because its growth compounds; compare both the month-by-month and the total values.

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A renewable-energy co-operative saves for a new turbine. It pays €320 into an account at the end of each month. The account pays a nominal annual interest rate of 3.6%, compounded monthly, and the first payment is made at the end of month 1.

(a)

Calculate the effective annual interest rate. Give your answer as a percentage to 3 significant figures.

(2)

(b)

Find the value of the account immediately after the 60th payment. Give your answer to the nearest euro.

(3)

(c)

Find the number of payments needed for the value of the account to first reach €25 000.

(3)

(d)

Instead of making monthly payments, the co-operative could deposit a single amount today so that it grows to the value found in (b) after 60 months. Find this amount, to the nearest euro. Use the unrounded value from (b).

(2)

(e)

Find the total interest earned by the monthly payments over 60 months, to the nearest euro. Use the unrounded value from (b).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (1 + 0.036/12)¹² − 1	✓ 1
A1 3.66%	✓ 1
Part (b)
M1 N = 60, I% = 3.6, PMT = −320, PV = 0, P/Y = 12, C/Y = 12	✓ 1
M1 solving for FV (payments at end of period)	✓ 1
A1 €21002	✓ 1
Part (c)
M1 sets up N unknown with FV = 25 000 (or 320((1.003)^N − 1)/0.003 = 25 000)	✓ 1
A1 N = 70.29 (accept 70.3, or a check of N = 70 and N = 71)	✓ 1
A1 71 payments	✓ 1
Part (d)
M1 N = 60, I% = 3.6, FV = value from (b), PMT = 0, C/Y = 12 (or FV/1.003⁶⁰)	✓ 1
A1 €17547	✓ 1
Part (e)
M1 FV − 60 × 320 (or FV − 19 200)	✓ 1
A1 €1802	✓ 1

Exam tip: With a TVM solver, payments you make are entered as negative numbers and the amount you receive is positive; check the sign convention before reading off an answer.

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The number of people infected in an outbreak is modelled by I = a × b^t, where t is the number of days after the first report and a and b are positive constants. The model gives I = 54 when t = 3 and I = 400 when t = 9.

(a)

Show that b⁶ = 400/54, and hence find b. Give your answer to 4 significant figures.

(3)

(b)

Find the value of a. Give your answer to 1 decimal place, and use b = 1.3962 in this part.

(2)

(c)

Using a = 19.8 and b = 1.3962, find the number of people the model predicts are infected when t = 14. Give your answer to the nearest whole number.

(2)

(d)

Using the same values of a and b, find the value of t at which the model predicts that 5000 people are infected. Give your answer to 3 significant figures.

(3)

(e)

Explain why this model becomes unrealistic for large values of t, and suggest one change that would improve it.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 54 = ab³ and 400 = ab⁹	✓ 1
M1 divides to obtain b⁶ = 400/54	✓ 1
A1 b = 1.3962	✓ 1
Part (b)
M1 a = 54 / 1.3962³	✓ 1
A1 a = 19.8	✓ 1
Part (c)
M1 19.8 × 1.3962¹⁴	✓ 1
A1 2118	✓ 1
Part (d)
M1 19.8 × 1.3962^t = 5000	✓ 1
M1 takes logarithms and solves	✓ 1
A1 16.6 days	✓ 1
Part (e)
R1 the model grows without limit, but the number infected cannot exceed the size of the population (accept: recovery, vaccination and isolation are ignored)	✓ 1
R1 suggests a sensible change, for example a model with an upper limit or one that allows the growth rate to fall over time	✓ 1

Exam tip: When the same constant appears in two equations, dividing one equation by the other removes it and often leaves a single power to solve.

+ Add to test✓ In your test513IBMI3S-1-01r37-09Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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Calculator

Two players earn coins in a game over consecutive days. Player A's daily coins form an arithmetic sequence; A earns 70 coins on day 3 and 145 coins on day 8. Player B earns 10 coins on day 1, and each day after that B earns 20% more than on the previous day. Treat all coin amounts as exact values (no rounding to whole coins).

(a)

Find the number of coins A earns on day 1, and the common difference of A's sequence.

(3)

(b)

Find the number of coins B earns on day 15, to the nearest whole number.

(2)

(c)

Find the total number of coins A has earned by the end of day 15.

(2)

(d)

Find the first day by the end of which B's total coins earned exceeds A's total coins earned.

(3)

(e)

Find A's total coins at the end of that day.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 u₁ + 2d = 70 and u₁ + 7d = 145	✓ 1
A1 d = 15	✓ 1
A1 u₁ = 40	✓ 1
Part (b)
M1 10 × 1.2¹⁴	✓ 1
A1 128 coins	✓ 1
Part (c)
M1 S₁₅ = 15/2 (2 × 40 + 14 × 15)	✓ 1
A1 2175 coins	✓ 1
Part (d)
M1 set up 10(1.2ⁿ − 1)/0.2 > n/2 (80 + 15(n − 1))	✓ 1
M1 use of table or graph to compare the two totals (at n = 26 B's total, 5673.8…, is below A's total, 5915)	✓ 1
A1 day 27	✓ 1
Part (e)
M1 S₂₇ = 27/2 (2 × 40 + 26 × 15)	✓ 1
A1 6345 coins (ECF from (d))	✓ 1

Common error: For sequence totals, compare Sn for both players; do not compare single-day amounts.

+ Add to test✓ In your test514IBMI3S-1-01r37-13Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; 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Calculator

The annual water demand of a town is 2.4 × 10⁶ m³ in year 1 and grows by 3% each year, so the yearly demands form a geometric sequence. The water supply capacity increases by a fixed amount each year, so the yearly capacities form an arithmetic sequence. The capacity is 3.27 × 10⁶ m³ in year 4 and 3.72 × 10⁶ m³ in year 9.

(a)

Find the demand in year 10. Give your answer to three significant figures.

(2)

(b)

Find the common difference and the capacity in year 1.

(3)

(c)

Find the total demand over the first 10 years. Give your answer to three significant figures.

(2)

(d)

Find the first year in which the demand is greater than the capacity.

(3)

(e)

Find by how much the demand exceeds the capacity in that year. Give your answer to three significant figures.

(2)

(f)

Give one reason why a geometric model for demand may stop being valid over many years.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2.4 × 10⁶ × 1.03⁹	✓ 1
A1 3.13 × 10⁶ m³	✓ 1
Part (b)
M1 5d = 3.72 × 10⁶ − 3.27 × 10⁶	✓ 1
A1 d = 9 × 10⁴ m³	✓ 1
A1 capacity in year 1 = 3.0 × 10⁶ m³	✓ 1
Part (c)
M1 2.4 × 10⁶(1.03¹⁰ − 1)/(1.03 − 1)	✓ 1
A1 2.75 × 10⁷ m³	✓ 1
Part (d)
M1 2.4 × 10⁶ × 1.03ⁿ⁻¹ > 3.0 × 10⁶ + 9 × 10⁴(n − 1)	✓ 1
M1 use of table or graph to compare the sequences (at n = 29 demand is below capacity)	✓ 1
A1 year 30	✓ 1
Part (e)
M1 2.4 × 10⁶ × 1.03²⁹ − (3.0 × 10⁶ + 9 × 10⁴ × 29)	✓ 1
A1 4.58 × 10⁴ m³ (ECF from (d))	✓ 1
Part (f)
R1 e.g. population growth or demand per person cannot grow at a fixed percentage indefinitely; any valid reason	✓ 1

Exam tip: Write both sequences as formulas in n before comparing them in a table.

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Calculator

Two drugs, A and B, are given to a patient at the same time. The concentrations in the blood, in mg per litre, t hours later are modelled by A(t) = 60e^−0.25t and B(t) = 24e^−0.08t.

(a)

Show that the two concentrations are equal when e^−0.17t = 0.4.

(2)

(b)

Hence find the time at which the concentrations are equal. Give your answer in hours to three significant figures.

(2)

(c)

Find the common concentration at this time. Use the unrounded value of t and give your answer to three significant figures.

(2)

(d)

Write A(t) in the form 60 × b^t, giving b to three significant figures.

(1)

(e)

Find the time at which the concentration of A is 10% of the concentration of B. Give your answer in hours to three significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 60e^−0.25t = 24e^−0.08t	✓ 1
A1 divide by 60e^−0.08t and use e^−0.25t ÷ e^−0.08t = e^−0.17t, giving e^−0.17t = 24/60 = 0.4	✓ 1
Part (b)
M1 t = ln 0.4 ÷ (−0.17)	✓ 1
A1 5.39 hours	✓ 1
Part (c)
M1 substitute t = 5.3899… into A(t) or B(t)	✓ 1
A1 15.6 mg per litre	✓ 1
Part (d)
A1 b = e^−0.25 = 0.779	✓ 1
Part (e)
M1 60e^−0.25t = 0.1 × 24e^−0.08t	✓ 1
M1 e^−0.17t = 0.04, so t = ln 0.04 ÷ (−0.17)	✓ 1
A1 18.9 hours	✓ 1

Exam tip: Combine terms with the same base: ea ÷ eb = ea⁻b.

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Calculator

A company builds a wind farm in stages. It installs 8 turbines in year 1. In each later year it installs 3 more turbines than in the previous year, so the number installed each year forms an arithmetic sequence.
Each turbine generates 6.2 GWh of electricity in the first year after it is installed. Its yearly output then decreases by 1.5% each year, forming a geometric sequence.

(a)

Find the number of turbines installed in year 10.

(2)

(b)

Find the total number of turbines installed in the first 10 years.

(2)

(c)

The company's target is a wind farm of at least 500 turbines. Find the smallest number of years in which the total installed reaches this target.

(3)

(d)

Find the output of one turbine in its 12th year of operation. Give your answer to three significant figures.

(2)

(e)

Find the total energy generated by one turbine in its first 20 years of operation. Give your answer to three significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $8+9\times3$	✓ 1
A1 35	✓ 1
Part (b)
M1 $S_{10}=\frac{10}{2}(8+35)$ or $\frac{10}{2}(2\times8+9\times3)$	✓ 1
A1 215 (ECF from (a))	✓ 1
Part (c)
M1 $\frac{n}{2}(16+3(n-1))\ge500$	✓ 1
M1 solving by calculator, table or graph (the total is 488 after 16 years)	✓ 1
A1 17 years (total 544)	✓ 1
Part (d)
M1 $6.2\times0.985^{11}$	✓ 1
A1 $5.25$ GWh	✓ 1
Part (e)
M1 $S_{20}=\frac{6.2(1-0.985^{20})}{1-0.985}$	✓ 1
M1 correct substitution of $r=0.985$, $n=20$	✓ 1
A1 $108$ GWh	✓ 1

Exam tip: Decide first whether each situation is arithmetic (constant addition) or geometric (constant percentage change); the same question can use both formulas in different parts.

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Calculator

A leisure centre takes out a loan of €60 000 to refurbish its swimming pool. Interest is charged at a nominal annual rate of 6%, compounded monthly (that is, 0.5% per month). The centre makes a payment of €1200 at the end of each month, starting one month after the loan is taken out. Let $B_n$ be the amount owed, in euros, immediately after the $n$th payment.

(a)

Find the amount owed immediately after the first payment.

(2)

(b)

Show that $B_n=240\,000-180\,000\times1.005^{n}$.

(3)

(c)

Hence find the number of monthly payments needed to repay the loan completely, given that the final payment may be smaller than €1200.

(3)

(d)

Find the amount of the final payment. Give your answer to the nearest cent.

(3)

(e)

Find the total interest paid over the whole loan. Use the final payment from part (d) and give your answer to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 interest for month 1 $=60\,000\times0.005=300$	✓ 1
A1 €59 100	✓ 1
Part (b)
M1 $B_n=60\,000\times1.005^n-1200\left(1+1.005+\dots+1.005^{n-1}\right)$	✓ 1
M1 sum of the geometric series $=\frac{1.005^n-1}{0.005}$ so $B_n=60\,000\times1.005^n-240\,000(1.005^n-1)$	✓ 1
A1 simplification to the given result with no errors	✓ 1
Part (c)
M1 set $B_n\le0$, giving $1.005^n\ge\frac{4}{3}$	✓ 1
M1 use of logarithms: $n\ge\frac{\ln(4/3)}{\ln1.005}=57.68\ldots$	✓ 1
A1 58 payments (57 full payments and a smaller final payment)	✓ 1
Part (d)
M1 $B_{57}=240\,000-180\,000\times1.005^{57}=812.75\ldots$	✓ 1
M1 final payment $=B_{57}\times1.005$ (one more month of interest)	✓ 1
A1 €816.81	✓ 1
Part (e)
M1 total paid $=57\times1200+816.81$	✓ 1
A1 total interest $=€9216.81$	✓ 1

Common error: In the last month the payment must also cover that month's interest: the final payment is the remaining balance multiplied by 1.005, not the balance itself.

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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 2Hard12 marks

A company borrows €480 000 to build a wind farm. The loan is repaid by equal payments made at the end of every quarter for 10 years. Interest is charged at a nominal annual rate of 5.2%, compounded quarterly. Use unrounded values in your working and give money answers to the nearest cent unless told otherwise.

(a)

Find the quarterly payment.

(3)

(b)

Find the amount still owed immediately after the 16th payment (4 years).

(3)

(c)

Immediately after the 16th payment the company repays an extra €50 000. The remaining 24 payments are recalculated so that the loan is still cleared after 10 years. Find the new quarterly payment.

(3)

(d)

Find how much less the company pays in total over the 10 years, compared with making no extra repayment. Give your answer to the nearest euro.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM with N = 40, I% = 5.2, PV = 480 000, FV = 0	✓ 1
M1 P/Y = C/Y = 4	✓ 1
A1 €15 465.30	✓ 1
Part (b)
M1 balance by TVM with N = 16 (or the present value of the remaining 24 payments)	✓ 1
M1 correct entries, payment as in (a)	✓ 1
A1 €317 092.55	✓ 1
Part (c)
M1 new balance = (answer to (b)) − 50 000	✓ 1
M1 TVM with N = 24, PV = new balance, FV = 0	✓ 1
A1 €13 026.69 (ECF from (b))	✓ 1
Part (d)
M1 original total = 40 × payment in (a); new total = 16 × payment + 50 000 + 24 × new payment	✓ 1
M1 subtract (or 24 × (old − new payment) − 50 000)	✓ 1
A1 €8527 (saving of interest)	✓ 1

Common error: The extra €50 000 repayment is part of the total paid, so subtract it back out; the saving is only the interest avoided, not the reduction in payments alone.

+ Add to test✓ In your test519IBMI3S-1-02r39-13Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context Paper 2Hard12 marks

A library's digital loans are modelled by L = a × b^t, where L is the number of digital loans in a year and t is the number of years since 2020 (so t = 0 is 2020). There were 3200 loans in 2020 and 5400 loans in 2023. Use the unrounded value of b in all later parts.

(a)

Write down the value of a.

(1)

(b)

Find the value of b, to 3 significant figures.

(3)

(c)

Find the annual percentage increase in loans according to the model, to 3 significant figures.

(1)

(d)

Find the predicted number of loans in 2027, to the nearest whole number.

(2)

(e)

The relationship can be written as ln L = p + qt. Find p and q, each to 3 significant figures.

(2)

(f)

The library's server can handle 20 000 loans per year. Find the value of t at which the model predicts 20 000 loans, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = 3200	✓ 1
Part (b)
M1 3200b³ = 5400	✓ 1
M1 b = (5400/3200)^1/3	✓ 1
A1 b = 1.19	✓ 1
Part (c)
A1 19.1%	✓ 1
Part (d)
M1 3200b⁷	✓ 1
A1 10 849	✓ 1
Part (e)
M1 ln L = ln 3200 + t ln b	✓ 1
A1 p = 8.07 and q = 0.174	✓ 1
Part (f)
M1 3200b^t = 20 000, so b^t = 6.25	✓ 1
M1 take logs: t = ln 6.25 / ln b	✓ 1
A1 t = 10.5	✓ 1

Exam tip: Taking logs of both sides turns a×bt into a straight line in t: the gradient is ln b and the intercept is ln a.

+ Add to test✓ In your test520IBMI3S-1-03r40-06Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard13 marks

In the equal-tempered musical scale, the frequencies of notes one semitone apart form a geometric sequence with common ratio r. Let f_k be the frequency, in Hz, of the note k semitones above the note A4, so that f₀ = 440. The note 12 semitones above A4 has exactly twice the frequency of A4.

(a)

Show that r = 2^1/12.

(2)

(b)

The note C5 is 3 semitones above A4. Find its frequency to 3 significant figures.

(2)

(c)

A piccolo can play notes up to a frequency of 3000 Hz. Find the greatest value of k for which f_k is less than 3000 Hz.

(3)

(d)

The 12 notes from f₀ to f₁₁ are played together as a chord cluster. Find the sum of their frequencies, to the nearest Hz.

(3)

(e)

An orchestra retunes so that A4 is 442 Hz instead of 440 Hz. All other notes keep the same ratios. Find the percentage increase in the frequency of the note C5, and state whether the percentage increase is the same for every note. Give a reason.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 f₁₂ = 440r¹² = 880	✓ 1
A1 r¹² = 2, so r = 2^1/12 (AG)	✓ 1
Part (b)
M1 440 × 2^3/12	✓ 1
A1 523 Hz	✓ 1
Part (c)
M1 440 × 2^k/12 = 3000 (or an inequality)	✓ 1
A1 k = 33.2 (from k = 12 log₂(3000/440))	✓ 1
A1 greatest whole number k is 33 (note 34 would exceed 3000 Hz)	✓ 1
Part (d)
M1 S₁₂ = 440(r¹² − 1)/(r − 1)	✓ 1
A1 440(2 − 1)/(2^1/12 − 1) = 440/0.059463	✓ 1
A1 7400 Hz	✓ 1
Part (e)
M1 (442 − 440) ÷ 440 × 100 (or 442 × 2^3/12 compared with 440 × 2^3/12)	✓ 1
A1 0.455%	✓ 1
R1 yes: every frequency is a fixed multiple (r^k) of the A4 frequency, so every frequency is scaled by the same factor 442/440	✓ 1

Common error: A geometric sequence has a constant ratio, so the frequency difference between neighbouring notes is not constant; do not use an arithmetic formula.

+ Add to test✓ In your test521IBMI3S-1-03r40-11Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard14 marks

A pesticide, A, is applied to a field. The concentration of A in the soil, C mg kg⁻¹, t days after application is modelled by C = C₀e^−kt, where C₀ and k are positive constants. Immediately after application C = 24 and the half-life of A in the soil is 18 days.

(a)

Show that k = 0.0385, correct to 3 significant figures.

(2)

(b)

Find the concentration of A after 50 days. Use the unrounded value of k and give your answer to 3 significant figures.

(2)

(c)

The soil is safe for planting when C is below 2.5 mg kg⁻¹. Find the time after application at which the soil first becomes safe, in days to 3 significant figures. Use the unrounded value of k.

(3)

(d)

The farmer plots ln C against t for pesticide A. The points lie on a straight line. State, to 3 significant figures, the gradient and the vertical intercept of this line.

(3)

(e)

A second pesticide, B, has concentration D = 9.6e^−0.021t mg kg⁻¹ in the same soil. Find the value of t at which the two concentrations are equal. Use the unrounded value of k and give your answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 12 = 24e^−18k (or e^−18k = 1/2)	✓ 1
A1 k = ln 2 ÷ 18 = 0.0385 (AG)	✓ 1
Part (b)
M1 24e^−50k (= 24 × 2^−50/18)	✓ 1
A1 3.50 mg kg⁻¹	✓ 1
Part (c)
M1 24e^−kt = 2.5	✓ 1
M1 t = ln(24/2.5) ÷ k	✓ 1
A1 58.7 days	✓ 1
Part (d)
M1 ln C = ln 24 − kt	✓ 1
A1 gradient −0.0385 (accept −ln 2 ÷ 18)	✓ 1
A1 intercept 3.18	✓ 1
Part (e)
M1 24e^−kt = 9.6e^−0.021t	✓ 1
M1 e^{(0.021 − k)t} = 0.4 (or ln 2.5 = (k − 0.021)t)	✓ 1
A1 t = ln 2.5 ÷ (k − 0.021)	✓ 1
A1 52.3 days	✓ 1

Common error: Keep the unrounded k in the calculator memory for later parts; replacing it with 0.0385 changes the last answer in the third significant figure.

+ Add to test✓ In your test522IBMI3S-1-03r40-12Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

A violinist buys a violin for €6 400. She pays a deposit of €400 and borrows the rest. The loan is repaid by 36 equal monthly payments of €187.50, the first made one month after the loan is taken out. Interest is compounded monthly.

(a)

State the amount borrowed.

(1)

(b)

Find the nominal annual interest rate charged on the loan. Give your answer as a percentage correct to 3 significant figures.

(2)

(c)

Calculate the effective annual interest rate on the loan as a percentage correct to 3 significant figures. Use the unrounded nominal rate from (b).

(2)

(d)

Find the total interest paid on this loan.

(2)

(e)

A bank offers to lend the same €6000 over the same 36 months at a nominal annual rate of 3.9%, compounded monthly, with equal monthly payments. Find the monthly payment, to the nearest cent.

(2)

(f)

Calculate how much less interest the violinist would pay with the bank loan than with the original loan. Use the unrounded monthly payment from (e) and give your answer to the nearest cent.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €6000	✓ 1
Part (b)
M1 TVM with N = 36, PV = 6000, PMT = −187.5, FV = 0, P/Y = C/Y = 12	✓ 1
A1 I% = 7.81	✓ 1
Part (c)
M1 (1 + 7.81263/1200)¹² − 1	✓ 1
A1 8.10%	✓ 1
Part (d)
M1 36 × 187.5 − 6000 (= 6750 − 6000)	✓ 1
A1 €750	✓ 1
Part (e)
M1 TVM with N = 36, I% = 3.9, PV = 6000, FV = 0, P/Y = C/Y = 12	✓ 1
A1 €176.88	✓ 1
Part (f)
M1 36 × 176.8771 − 6000	✓ 1
A1 interest on bank loan €367.58	✓ 1
A1 difference €382.42	✓ 1

Exam tip: 'Nominal rate' and 'effective rate' are different questions: use the TVM solver for the nominal rate, then convert with (1 + r/n)ⁿ − 1.

+ Add to test✓ In your test523IBMI3S-1-01r41-10Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard12 marks

During a stint, the time taken for each lap of a Formula 1 car increases as the tyres wear. The first lap takes 91.2 s and each lap takes 0.08 s longer than the one before, so the lap times form an arithmetic sequence.

(a)

Find the time for lap 30.

(2)

(b)

Find the total time, in seconds, for the first 30 laps.

(2)

(c)

Find the smallest number of laps for which the total time exceeds 3000 s.

(3)

(d)

A team compares two strategies for a race of 56 laps. Strategy A: one stint of 56 laps. Strategy B: two stints of 28 laps, each starting with new tyres and a first lap of 91.2 s as above, plus one pit stop costing 22.5 s. Find which strategy gives the shorter total race time, and by how many seconds.

(5)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $91.2 + 29\times0.08$	✓ 1
A1 93.52 s	✓ 1
Part (b)
M1 $S_{30}=\frac{30}{2}(91.2+93.52)$ (ECF from (a))	✓ 1
A1 2770.8 s	✓ 1
Part (c)
M1 $\frac{n}{2}\left(2\times91.2+0.08(n-1)\right)>3000$ or an equation with equality	✓ 1
M1 use of a GDC to solve, or tabulate values of $S_n$, giving $n\approx32.4$ or $S_{32}=2958.08$ and $S_{33}=3051.84$	✓ 1
A1 $n=33$	✓ 1
Part (d)
M1 $S_{56}=\frac{56}{2}(2\times91.2+55\times0.08)$	✓ 1
A1 $S_{56}=5230.4$ s	✓ 1
M1 $S_{28}=\frac{28}{2}(2\times91.2+27\times0.08)=2583.84$ s	✓ 1
A1 strategy B total $2\times2583.84+22.5=5190.18$ s	✓ 1
A1 strategy B is quicker by 40.2 s	✓ 1

Exam tip: In sequence problems, write down $u_1$, $d$ (or $r$) and $n$ before choosing a formula.

+ Add to test✓ In your test524IBMI3S-1-01r41-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; 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exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; 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exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; 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Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

The number of bees, P, in a colony is modelled by $P = a\times b^{t}$, where t is the time in months after the colony was first counted, $a>0$ and $b>1$. After 1 month there are 8000 bees and after 4 months there are 15 625 bees.

(a)

Find the value of b.

(3)

(b)

Find the value of a.

(2)

(c)

Find the time at which the colony first reaches 50 000 bees. Give your answer in months to 3 significant figures.

(3)

(d)

Taking logarithms to base 10, the model can be written as $\log_{10}P = mt + c$. Find the exact form and the value to 3 significant figures of m and of c.

(3)

(e)

Find the time, in months, for the number of bees to double. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $ab=8000$ and $ab^4=15625$	✓ 1
M1 $b^3=\frac{15625}{8000}=1.953125$	✓ 1
A1 $b=1.25$	✓ 1
Part (b)
M1 $a=\frac{8000}{1.25}$	✓ 1
A1 $a=6400$	✓ 1
Part (c)
M1 $6400\times1.25^t=50000$	✓ 1
M1 use of logarithms or a GDC solver	✓ 1
A1 9.21 months	✓ 1
Part (d)
M1 $\log_{10}P=\log_{10}a+t\log_{10}b$	✓ 1
A1 $m=\log_{10}1.25=0.0969$	✓ 1
A1 $c=\log_{10}6400=3.81$	✓ 1
Part (e)
M1 $b^{T}=2$ (the doubling time does not depend on the starting value)	✓ 1
M1 $T=\frac{\ln2}{\ln1.25}$	✓ 1
A1 3.11 months	✓ 1

Exam tip: If an exponential model $ab^t$ is given two data points, divide the two equations to eliminate a.

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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard14 marks

A hospital borrows €180 000 for new equipment at a nominal annual interest rate of 6%, compounded monthly. It plans to repay the loan with payments of €2500 at the end of each month until the loan is cleared, with the final payment being smaller.

(a)

Find the number of months of payments needed to repay the loan. Give your answer to 3 significant figures.

(3)

(b)

Hence there are 89 full payments of €2500 followed by a smaller 90th payment. Find the balance owed immediately after the 89th payment, to the nearest cent.

(3)

(c)

Find the amount of the 90th payment, to the nearest cent.

(2)

(d)

Find the total interest paid on the loan.

(2)

(e)

The hospital considers instead repaying the loan with equal monthly payments over exactly 7 years. Find the monthly payment, to the nearest cent, and show that the total interest paid would be less than in part (d).

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 TVM solver with $I\%=6$, $PV=180000$, $PMT=-2500$, $FV=0$, $P/Y=C/Y=12$	✓ 1
M1 attempt to solve for N	✓ 1
A1 $N=89.5$	✓ 1
Part (b)
M1 TVM solver with $N=89$ and the same entries	✓ 1
M1 attempt to solve for FV	✓ 1
A1 €1196.52	✓ 1
Part (c)
M1 add one month's interest: $1196.52\times1.005$	✓ 1
A1 €1202.50	✓ 1
Part (d)
M1 $89\times2500+1202.50-180000$	✓ 1
A1 €43 702.50	✓ 1
Part (e)
M1 TVM solver with $N=84$, $I\%=6$, $PV=180000$, $FV=0$, $P/Y=C/Y=12$	✓ 1
M1 attempt to solve for PMT	✓ 1
A1 €2629.54	✓ 1
R1 total interest $84\times2629.54-180000=€40\,881.36$, which is less than €43 702.50	✓ 1

Common error: The last payment of a loan is usually smaller than the others: do not assume it equals the regular payment.

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Calculator

Two players, Alex and Bea, collect points in a game each day. Alex collects 30 points on day 1, and then 12 more points each day than on the previous day. Bea collects 20 points on day 1, and then 25% more points each day than on the previous day.

(a)

Find the number of points Alex collects on day 20.

(2)

(b)

Find the total number of points Alex has collected after 20 days.

(2)

(c)

Find the number of points Bea collects on day 20. Give your answer correct to 3 significant figures.

(2)

(d)

Find the first day on which Bea collects more than 1000 points in that day.

(3)

(e)

Find the smallest number of days n for which Bea's total number of points collected first exceeds Alex's total.

(3)

(f)

Find Bea's total number of points after the number of days found in part (e), to the nearest whole point.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 30 + 19 × 12	✓ 1
A1 258 points	✓ 1
Part (b)
M1 20/2 (30 + 258) (or 10(60 + 19 × 12))	✓ 1
A1 2880 points	✓ 1
Part (c)
M1 20 × 1.25¹⁹	✓ 1
A1 1390 points (1387.7…)	✓ 1
Part (d)
M1 20 × 1.25^{(n − 1)} > 1000	✓ 1
A1 n − 1 > 17.5… (or 1.25¹⁷ and 1.25¹⁸ compared with 50)	✓ 1
A1 day 19	✓ 1
Part (e)
A1 Alex's total after n days is 6n² + 24n; Bea's is 80(1.25ⁿ − 1)	✓ 1
M1 compares totals using a table, graph or inequality (e.g. 1152 against 1084 for n = 12)	✓ 1
A1 n = 13	✓ 1
Part (f)
A1 80(1.25¹³ − 1) = 1375 points (ECF from (e))	✓ 1

Exam tip: When comparing two totals, show the values on either side of the crossover (here 12 and 13 days) to justify that it is the first.

+ Add to test✓ In your test527IBMI3S-1-02r43-08Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard14 marks

A fisheries agency models the stock N (in tonnes) of a cod population t years after a fishing ban begins as N = 540e^−0.07t in an over-fished region. In a protected region the stock of the same species is modelled by M = 180e^0.11t.

(a)

Calculate the stock in the over-fished region after 6 years. Give your answer to 3 significant figures.

(2)

(b)

Find the time taken for the stock in the over-fished region to halve. Give your answer to 3 significant figures.

(3)

(c)

The stocks in the two regions are equal at time T. Show that e^0.18T = 3.

(3)

(d)

Hence find T. Give your answer to 3 significant figures.

(2)

(e)

Write M in the form 180a^t, and find the value of a to 3 significant figures.

(2)

(f)

Find the common stock, in tonnes, at time T. Use the unrounded value of T and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 540e^{−0.07 × 6}	✓ 1
A1 355 tonnes (354.8… before rounding)	✓ 1
Part (b)
M1 540e^−0.07t = 270 (or e^−0.07t = 0.5)	✓ 1
M1 taking natural logarithms: −0.07t = ln 0.5	✓ 1
A1 t = 9.90 years (9.9021… before rounding)	✓ 1
Part (c)
M1 540e^−0.07T = 180e^0.11T	✓ 1
M1 dividing by 180e^−0.07T and combining powers: 3 = e^{0.11T + 0.07T}	✓ 1
A1 e^0.18T = 3 (AG)	✓ 1
Part (d)
M1 0.18T = ln 3	✓ 1
A1 T = 6.10 years (6.1034… before rounding)	✓ 1
Part (e)
M1 e^0.11t = (e^0.11)^t	✓ 1
A1 a = 1.12 (1.1163… before rounding)	✓ 1
Part (f)
M1 substituting T = (ln 3)/0.18 into either model	✓ 1
A1 352 tonnes (352.25… before rounding)	✓ 1

Exam tip: When two exponential models are equated, divide to collect the exponentials into one power before taking logarithms; this keeps the algebra short and avoids sign errors.

+ Add to test✓ In your test528IBMI3S-1-02r43-09Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

A fashion label sells 240 jackets in its first month, and sales increase by 18 jackets each month (an arithmetic sequence). A second label sells 150 jumpers in its first month, and sales increase by 8% each month (a geometric sequence).

(a)

Find the number of jackets sold in month 12.

(2)

(b)

Find the total number of jackets sold in the first 12 months.

(2)

(c)

Find the number of jumpers sold in month 12. Give your answer to the nearest whole number.

(2)

(d)

Find the first month in which the number of jumpers sold exceeds the number of jackets sold.

(3)

(e)

Find the total number of jumpers sold in the first 18 months. Give your answer to the nearest whole number.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 240 + 11 × 18	✓ 1
A1 438	✓ 1
Part (b)
M1 S₁₂ = 12/2 × (2 × 240 + 11 × 18)	✓ 1
A1 4068	✓ 1
Part (c)
M1 150 × 1.08¹¹	✓ 1
A1 350 (349.7… before rounding)	✓ 1
Part (d)
M1 comparing 150 × 1.08ⁿ⁻¹ with 240 + 18(n − 1) (table, graph or solver)	✓ 1
A1 month 17: jumpers 513.9…, jackets 528 (jumpers fewer)	✓ 1
A1 month 18: jumpers 555.0…, jackets 546, so month 18	✓ 1
Part (e)
M1 S₁₈ = 150(1.08¹⁸ − 1)/(1.08 − 1)	✓ 1
A1 substitution correct	✓ 1
A1 5618 (5617.5… before rounding)	✓ 1

Common error: In a geometric sequence the nth term is u₁rⁿ⁻¹, not u₁rⁿ; an 8% increase means r = 1.08, not 0.08.

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A batch of food contains 1200 bacteria at time t = 0, where t is in hours. The number of bacteria is modelled by N = 1200e^kt, where k is a constant. At t = 4 the number of bacteria is 8600.

(a)

Find the value of k. Give your answer to 3 significant figures.

(3)

(b)

Find the time taken for the number of bacteria to double. Give your answer to 3 significant figures.

(2)

(c)

Find the time at which the number of bacteria reaches 1.0 × 10⁶, if the model continues to hold. Give your answer to 3 significant figures.

(3)

(d)

At t = 5 the batch is refrigerated. Find the number of bacteria at t = 5. Use the unrounded value of k and give your answer to 3 significant figures.

(2)

(e)

After refrigeration the number of bacteria is modelled by N = N₅e^0.08(t−5), where N₅ is the unrounded value from (d). Find the time t at which the number of bacteria reaches 1.0 × 10⁶. Give your answer to 3 significant figures.

(3)

(f)

Find how many hours later the number reaches 1.0 × 10⁶ because of the refrigeration, compared with the model in (c). Use the unrounded values from (c) and (e) and give your answer to 3 significant figures.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 8600 = 1200e^4k	✓ 1
M1 k = ln(8600/1200)/4	✓ 1
A1 k = 0.492	✓ 1
Part (b)
M1 e^kt = 2, so t = ln 2/k	✓ 1
A1 1.41 hours	✓ 1
Part (c)
M1 1 000 000 = 1200e^kt	✓ 1
M1 t = ln(1 000 000/1200)/k	✓ 1
A1 13.7 hours	✓ 1
Part (d)
M1 1200e^5k	✓ 1
A1 14 100 (14 071…)	✓ 1
Part (e)
M1 1 000 000 = 14 071…e^0.08(t−5)	✓ 1
M1 0.08(t − 5) = ln(1 000 000/14 071…)	✓ 1
A1 t = 58.3 hours	✓ 1
Part (f)
A1 58.295… − 13.659… = 44.6 hours	✓ 1

Exam tip: Keep unrounded values in your calculator memory when a later part builds on an earlier answer, and round only at the end.

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A glacier on a mountain is approximately rectangular, measuring 2.84 km by 1.37 km. Its average ice thickness is 45 m.

(a)

Calculate the area of the glacier in km². Give your answer to 3 significant figures.

(2)

(b)

A climber estimates the lengths to 1 significant figure and calculates the area using these values. Find the percentage error in her area. Use the unrounded area from (a), taken as the exact value, and give your answer to 3 significant figures.

(3)

(c)

Find the volume of ice in the glacier in m³. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k is an integer, with a to 3 significant figures. Use the unrounded area from (a).

(2)

(d)

The density of the ice is 917 kg m⁻³. Find the mass of the ice in the form a × 10^k kg, with a to 3 significant figures. Use the unrounded volume from (c).

(2)

(e)

The glacier is melting at a constant rate of 2.1 × 10⁵ m³ of ice per day. Find how many years it takes to melt completely, assuming a year has 365 days. Use the unrounded volume from (c) and give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2.84 × 1.37	✓ 1
A1 3.89 km²	✓ 1
Part (b)
A1 estimated lengths 3 km and 1 km, so estimated area 3 km²	✓ 1
M1 \|3 − 3.8908\|/3.8908 × 100	✓ 1
A1 22.9%	✓ 1
Part (c)
M1 area = 3.8908 × 10⁶ m² (1 km² = 10⁶ m²)	✓ 1
A1 volume = 3.8908 × 10⁶ × 45 = 1.75 × 10⁸ m³	✓ 1
Part (d)
M1 1.7509… × 10⁸ × 917	✓ 1
A1 1.61 × 10¹¹ kg	✓ 1
Part (e)
M1 days = 1.7509… × 10⁸/(2.1 × 10⁵) = 833.7…	✓ 1
M1 years = 833.7…/365	✓ 1
A1 2.28 years	✓ 1

Common error: Convert km² to m² by multiplying by 10⁶, not 10³, because both lengths change by a factor of 1000.

+ Add to test✓ In your test531IBMI3S-1-02r45-07Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard13 marks

Beekeeper A collects 18 kg of honey in week 1. Her weekly collection increases by 12% each week, forming a geometric sequence.
Beekeeper B collects 20 kg in week 1 and her weekly collection increases by a constant amount of d kg each week, forming an arithmetic sequence.

(a)

Find the mass collected by beekeeper A in week 8. Give your answer correct to 3 significant figures.

(2)

(b)

Find the total mass collected by beekeeper A in the first 8 weeks. Give your answer correct to 3 significant figures.

(2)

(c)

Find the least number of weeks for which beekeeper A's total collection exceeds 1000 kg.

(3)

(d)

In week 8, beekeeper B collects the same mass as beekeeper A. Use the unrounded value of A's week 8 mass to find d, correct to 3 significant figures.

(3)

(e)

Find the difference between the total masses collected by the two beekeepers in the first 8 weeks. State who collects more. Use unrounded values and give your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18 × 1.12⁷	✓ 1
A1 39.8 kg	✓ 1
Part (b)
M1 S₈ = 18(1.12⁸ − 1)/(1.12 − 1)	✓ 1
A1 221 kg	✓ 1
Part (c)
M1 18(1.12ⁿ − 1)/0.12 > 1000	✓ 1
M1 attempt to solve using logarithms or a table (n > 17.97…)	✓ 1
A1 18 weeks	✓ 1
Part (d)
M1 20 + 7d = 18 × 1.12⁷	✓ 1
A1 7d = 19.79… (ECF from (a))	✓ 1
A1 d = 2.83	✓ 1
Part (e)
M1 beekeeper B total = (8/2)(20 + 39.79…) = 239.2	✓ 1
A1 B collects more	✓ 1
A1 difference = 239.2 − 221.4 = 17.8 kg	✓ 1

Common error: Geometric growth of 12% means multiplying by 1.12, not by 0.12; and the formula for the sum needs the ratio minus one in the denominator.

+ Add to test✓ In your test532IBMI3S-1-02r45-14Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities Paper 2Hard13 marks

A patient takes a 200 mg dose of a drug at the start of every 8-hour period. At the end of each 8-hour period, 30% of the drug present at the start of that period remains in the body.

(a)

Find the amount of the drug in the body just before the second dose is taken.

(1)

(b)

Find the amount of the drug in the body just after the second dose is taken.

(1)

(c)

Find the amount of the drug in the body just after the third dose is taken.

(2)

(d)

Show that the amount in the body just after the nth dose is 200(1 − 0.3ⁿ)/0.7 mg.

(3)

(e)

Find the limit of the amount just after a dose as n becomes very large. Give your answer correct to 3 significant figures.

(2)

(f)

The level of the drug becomes unsafe above 300 mg. State, with a reason, whether this regimen will ever exceed 300 mg.

(2)

(g)

Find the amount just after the 4th dose as a percentage of the limit from part (e). Use unrounded values and give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.3 × 200 = 60 mg	✓ 1
Part (b)
A1 260 mg	✓ 1
Part (c)
M1 0.3 × 260 + 200	✓ 1
A1 278 mg	✓ 1
Part (d)
M1 amount = 200 + 200(0.3) + 200(0.3)² + … + 200(0.3)ⁿ⁻¹, a geometric series with first term 200 and ratio 0.3	✓ 1
M1 use of S_n = 200(1 − 0.3ⁿ)/(1 − 0.3)	✓ 1
A1 200(1 − 0.3ⁿ)/0.7 (AG)	✓ 1
Part (e)
M1 0.3ⁿ → 0, so the limit is 200/0.7	✓ 1
A1 286 mg	✓ 1
Part (f)
R1 the amount just after a dose increases with n but never exceeds its limit of 285.7… mg	✓ 1
A1 no, it never exceeds 300 mg	✓ 1
Part (g)
M1 (1 − 0.3⁴) × 100 (the ratio of 200(1 − 0.3⁴)/0.7 to 200/0.7)	✓ 1
A1 99.2%	✓ 1

Exam tip: For an infinite geometric series with |r| < 1 the sum is the first term divided by (1 − r); say why the powers of r vanish.

+ Add to test✓ In your test533IBMI3S-1-01r47-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Exponents and logarithms; Approximation, percentage error and standard form; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; 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arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard10 marks

The sound level, L decibels (dB), of a sound with intensity I watts per square metre is given by L = 10 log₁₀(I ÷ 10⁻¹²).
A Formula 1 engine, at a certain point in the pits, produces a sound level of 94 dB.

(a)

Find the intensity of the sound from one engine. Give your answer in the form a × 10^k with a correct to 3 significant figures.

(2)

(b)

n identical engines each produce intensity I at that point and their intensities add. Show that the sound level is then 94 + 10 log₁₀ n dB.

(3)

(c)

Find the smallest number of engines needed for the sound level to be at least 112 dB.

(3)

(d)

Find the sound level produced by this number of engines. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 sets 94 = 10 log₁₀(I/10⁻¹²) and writes I = 10⁻¹² × 10^9.4	✓ 1
A1 2.51 × 10⁻³ W m⁻²	✓ 1
Part (b)
M1 total intensity = nI, so L = 10 log₁₀(nI/10⁻¹²)	✓ 1
M1 uses log(ab) = log a + log b to separate 10 log₁₀ n	✓ 1
R1 identifies the other term as 10 log₁₀(I/10⁻¹²) = 94 (AG)	✓ 1
Part (c)
M1 sets 94 + 10 log₁₀ n ≥ 112	✓ 1
A1 log₁₀ n ≥ 1.8 so n ≥ 63.1	✓ 1
A1 64 engines	✓ 1
Part (d)
M1 94 + 10 log₁₀ 64	✓ 1
A1 112 dB	✓ 1

Exam tip: Doubling the intensity adds about 3 dB, whatever the starting level; this gives a quick check on the answer to a log-scale question.

+ Add to test✓ In your test534IBMI3S-1-01r48-11Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; 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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

Calculator

Two riders, Alma and Boris, train over a number of consecutive days. Alma rides 40 km on day 1 and increases her distance by 6 km each day. Boris rides 25 km on day 1 and increases his distance by 20% each day.

(a)

Find the distance Alma rides on day 10.

(2)

(b)

Find the total distance Alma rides in the first 10 days.

(2)

(c)

Show that Boris rides a total of approximately 649 km in the first 10 days.

(3)

(d)

Find the smallest number of days for which Boris's total distance exceeds Alma's total distance.

(3)

(e)

Find by how many kilometres Boris's total distance exceeds Alma's total distance after that number of days. Give your answer correct to 1 decimal place.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $u_{10}=40+9\times6$	✓ 1
A1 94 km	✓ 1
Part (b)
M1 $\frac{10}{2}(40+94)$ or $\frac{10}{2}(2\times40+9\times6)$	✓ 1
A1 670 km	✓ 1
Part (c)
M1 geometric sequence with $u_1=25$, $r=1.2$	✓ 1
M1 $S_{10}=\frac{25(1.2^{10}-1)}{1.2-1}$	✓ 1
A1 648.97, which rounds to 649 km	✓ 1
Part (d)
M1 compare $\frac{25(1.2^n-1)}{0.2}$ with $\frac{n}{2}(80+6(n-1))$ using a table or graph	✓ 1
M1 evidence of values either side of the crossover, e.g. day 10: Boris 649.0 km, Alma 670 km	✓ 1
A1 11 days	✓ 1
Part (e)
M1 $\frac{25(1.2^{11}-1)}{0.2}-\frac{11}{2}(80+6\times10)$ (Boris 803.8, Alma 770)	✓ 1
A1 33.8 km (ECF from (d))	✓ 1

Exam tip: For comparisons between sequences, tabulate both running totals on your calculator and quote the values either side of the crossover.

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financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; Arithmetic and geometric sequences in context; Financial mathematics, loans and annuities; Arithmetic and geometric sequences in context; Arithmetic and geometric sequences in context; Exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Financial mathematics, loans and annuities; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

Calculator

The data traffic through a network hub is modelled by $T=120e^{kt}$, where $T$ is the traffic in gigabits per second and $t$ is the time in hours after 08:00. At 12:00 the traffic is 300 gigabits per second.

(a)

Find the value of $k$. Give your answer to 3 significant figures.

(3)

(b)

Use the unrounded value of $k$ to find the traffic at 18:00. Give your answer to the nearest gigabit per second.

(2)

(c)

Find the time, in hours to 3 significant figures, for the traffic to double.

(3)

(d)

The hub can carry at most 500 gigabits per second. Find the time after 08:00, in hours to 3 significant figures, at which the traffic first reaches this limit. Use the unrounded value of $k$.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $t=4$, $T=300$: $300=120e^{4k}$	✓ 1
M1 $k=\frac{1}{4}\ln\left(\frac{300}{120}\right)$	✓ 1
A1 $k=0.229$	✓ 1
Part (b)
M1 $T=120e^{10k}$	✓ 1
A1 1186 gigabits per second	✓ 1
Part (c)
M1 $2=e^{kt}$	✓ 1
M1 $t=\frac{\ln2}{k}$	✓ 1
A1 3.03 hours	✓ 1
Part (d)
M1 $120e^{kt}=500$	✓ 1
M1 $e^{kt}=\frac{500}{120}$	✓ 1
M1 $t=\frac{\ln(500/120)}{k}$	✓ 1
A1 6.23 hours	✓ 1

Common error: Do not use the rounded value of $k$ in later parts when told to use the unrounded value; store it in your calculator.

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arithmetic and geometric sequences in context; exponents and logarithms; Approximation, percentage error and standard form; financial mathematics, loans and annuities; arithmetic and geometric sequences in context; exponents and logarithms Paper 2Hard12 marks

Calculator

An archaeology trust wants to fund a research programme by paying out €4000 at the end of each quarter for 5 years. The money is held in an account that pays a nominal annual interest rate of 4%, compounded quarterly, and the account balance is zero after the final payment.

(a)

Find the amount the trust must deposit at the start. Give your answer to the nearest cent.

(3)

(b)

Find the total interest the account earns over the 5 years. Use your answer to part (a).

(2)

(c)

Find the effective annual interest rate. Give your answer as a percentage to 3 significant figures.

(3)

(d)

Find the balance in the account immediately after the 8th payment. Give your answer to the nearest euro.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 N=20, I%=4, FV=0, PMT=−4000, P/Y=C/Y=4	✓ 1
M1 PV found from TVM solver or $4000\times\frac{1-1.01^{-20}}{0.01}$	✓ 1
A1 €72,182.21	✓ 1
Part (b)
M1 $20\times4000-72,182.21$	✓ 1
A1 €7,817.79 (ECF from (a))	✓ 1
Part (c)
M1 quarterly growth factor $1+\frac{4}{400}=1.01$	✓ 1
M1 $1.01^4-1$	✓ 1
A1 4.06%	✓ 1
Part (d)
M1 recognise remaining payments: 12 payments of €4000 remain	✓ 1
M1 N=12, I%=4, PMT=−4000, FV=0, P/Y=C/Y=4	✓ 1
M1 $4000\times\frac{1-1.01^{-12}}{0.01}$	✓ 1
A1 €45,020	✓ 1

Exam tip: The balance after $k$ payments equals the present value of the payments that remain.

Marking point	Mark	Notes
Part (a)
M1 for \(\left\|\frac{85-82.4}{82.4}\right\|\times100\)	✓ 1
A1 3.16 % (accept 3.155…)	✓ 1
Part (b)
M1 for 82.4 × 0.95 (or 82.4 − 0.05 × 82.4)	✓ 1
A1 78.28 m (accept 78.3 m)	✓ 1

Marking point	Mark	Notes
Part (a)
M1 for \(\frac{2.85\times10^{12}}{6.72\times10^{7}}\)	✓ 1
A1 42 400 USD	✓ 1
Part (b)
A1 4.24×10⁴ USD	✓ 1
Part (c)
M1 for \(\frac{6.72\times10^{7}}{2.43\times10^{5}}\)	✓ 1
A1 2.77×10² people per km²	✓ 1

Marking point	Mark	Notes
Part (a)
Sets up \(4500\left(1+\frac{0.032}{12}\right)^n > 6000\) or equivalent TVM entry (PV = −4500, FV = 6000, I% = 3.2, C/Y = 12) M1	✓ 1
\(n = 108.02...\) A1	✓ 1
109 months A1	✓ 1
Part (b)
\(4500\left(1+\frac{0.032}{12}\right)^{72}\) or TVM entry with N = 72 M1	✓ 1
Value = 5451.12 euros A1	✓ 1
Interest = 5451.12 − 4500 = 951.12 euros A1	✓ 1

Marking point	Mark	Notes
Part (a)
Uses \(u_{25}=18+(25-1)\times3\) M1 ; 90 seats A1	✓ 2
Part (b)
Forms \(18+3(n-1)>60\) or lists terms. Row 15 has exactly 60 seats M1 ; row 16, with 63 seats A1	✓ 2
Part (c)
Uses \(S_{25}=\frac{25}{2}(18+90)\) or an equivalent formula M1 ; 1350 seats A1	✓ 2

Marking point	Mark	Notes
Part (a)
Uses \(2400\times1.15^{7}\) M1 ; 6380 downloads (6384.05…) A1	✓ 2
Part (b)
Forms \(2400\times1.15^{n-1}>10000\) or uses a table on a GDC M1 ; gives \(n-1>10.2\), or week 11 ≈ 9709 and week 12 ≈ 11 166 A1 ; week 12 A1	✓ 3
Part (c)
Uses \(S_{12}=\frac{2400(1.15^{12}-1)}{1.15-1}\) M1 ; 69 600 downloads (69 604.0…) A1	✓ 2

Marking point	Mark	Notes
Part (a)
Solve \(4500\left(1+\frac{0.032}{12}\right)^{n}=6000\), or TVM with PV = −4500, FV = 6000, I% = 3.2, P/Y = C/Y = 12	✓ 1
n = 108.02…	✓ 1
109 months, since after 108 months the value is still just below EUR 6000	✓ 1
Part (b)
\(4500\left(1+\frac{0.032}{12}\right)^{72}\)	✓ 1
= EUR 5451.12	✓ 1
Interest = 5451.12 − 4500 = EUR 951.12	✓ 1

Marking point	Mark	Notes
Part (a)
\(u_{24} = 18 + (24-1)(3)\) M1	✓ 1
\(= 87\) seats A1	✓ 1
Part (b)
\(S_{24} = \frac{24}{2}(18 + 87)\) or \(\frac{24}{2}(2\times18 + 23\times3)\) M1	✓ 1
\(= 1260\) seats A1	✓ 1
Part (c)
\(18 + 3(n-1) > 60\), or noting that row 15 has exactly 60 seats M1	✓ 1
Row 16, which has 63 seats A1	✓ 1

Marking point	Mark	Notes
Part (a)
M1: substitutes correctly, \(L = 10\log_{10}\left(\frac{3.2\times10^{-5}}{1.0\times10^{-12}}\right) = 10\log_{10}(3.2\times10^{7})\)	✓ 1
A1: \(L = 75.1\) dB	✓ 1
Part (b)
M1: substitutes correctly, \(88 = 10\log_{10}\left(\frac{I}{10^{-12}}\right)\)	✓ 1
M1: rearranges correctly, \(\frac{I}{10^{-12}} = 10^{8.8}\), or solves with GDC	✓ 1
A1: \(I = 6.31\times10^{-4}\) W m⁻²	✓ 1

Marking point	Mark	Notes
Part (a)
\(u_{25} = 18 + (25-1)\times 3\) M1	✓ 1
= 90 seats A1	✓ 1
Part (b)
Sets up \(18 + 3(n-1) > 60\), or lists terms using a GDC table M1	✓ 1
\(n > 15\), so row 16 (which has 63 seats) A1	✓ 1
Part (c)
\(S_{25} = \frac{25}{2}(18 + 90)\) or \(\frac{25}{2}(2\times 18 + 24\times 3)\) M1	✓ 1
= 1350 seats A1	✓ 1

Marking point	Mark	Notes
Part (a)
\(r = 1.15\) A1	✓ 1
Part (b)
\(u_{10} = 40 \times 1.15^{9}\) M1	✓ 1
= 140.7… ≈ 141 shares A1	✓ 1
Part (c)
\(S_{10} = \frac{40(1.15^{10} - 1)}{1.15 - 1}\) M1	✓ 1
= 812.1… ≈ 812 shares A1	✓ 1

Marking point	Mark	Notes
Part (a)
M1: substitutes \(t=4\), \(28\,500\times0.84^{4}\)	✓ 1
A1: $14 189.33	✓ 1
Part (b)
M1: sets up the equation \(28\,500\times0.84^{t} = 8000\)	✓ 1
M1: uses logarithms or technology, e.g. \(t = \dfrac{\log(8000/28\,500)}{\log 0.84}\), or finds the intersection on a GDC	✓ 1
A1: \(t = 7.29\) years	✓ 1

Marking point	Mark	Notes
Part (a)
Correct substitution: \(4500\left(1+\frac{0.036}{12}\right)^{72}\), or TVM with N = 72, I% = 3.6, PV = −4500, PMT = 0, P/Y = C/Y = 12	✓ 1
$5583.16	✓ 1
Part (b)
Sets up \(4500(1.003)^n > 6000\) or equivalent, e.g. TVM solving for N with FV = 6000	✓ 1
\(n = 96.04...\)	✓ 1
97 months, because the value has not reached $6000 after 96 months	✓ 1
Part (c)
\(4500(1.003)^{97}\) = $6017.32	✓ 1

Marking point	Mark	Notes
Part (a)
A1: \(r=1.025\)	✓ 1
Part (b)
M1: sets up \(42000(1.025)^{n-1}>60000\) and solves it, e.g. with a GDC or logarithms, giving \(n-1>14.4\)	✓ 1
A1: \(n=16\), so the start of 2035	✓ 1
Part (c)
M1: \(42000\times1.025^{10}\), i.e. the correct power for \(u_{11}\)	✓ 1
A1: 53764 (accept 53800 or 53763)	✓ 1

Marking point	Mark	Notes
Part (a)
M1: use of \(u_n=u_1+(n-1)d\) with \(u_1=18\), \(d=3\), \(n=20\), giving \(18+19\times3\)	✓ 1
A1: \(u_{20}=75\) seats	✓ 1
Part (b)
M1: sets up \(18+3(n-1)>100\), or an equivalent equation or table of values	✓ 1
A1: \(n>28.3\), so row 29 (which has 102 seats)	✓ 1
Part (c)
M1: use of \(S_n=\frac{n}{2}(2u_1+(n-1)d)\) or \(\frac{n}{2}(u_1+u_n)\) with \(n=20\)	✓ 1
A1: 930 seats	✓ 1

Marking point	Mark	Notes
Part (a)
\(28000 \times 0.85^4\) M1	✓ 1
14 616.18 dollars (accept 14 600 dollars to 3 s.f.) A1	✓ 1
Part (b)
\(28000 \times 0.85^n < 5000\) or equivalent, e.g. a table of values or a GDC solver M1	✓ 1
\(n = 10.6...\), OR values after 10 and 11 years found as 5512 and 4686 dollars A1	✓ 1
11 years A1	✓ 1
Part (c)
\((1-r)^4 = 0.5\) M1	✓ 1
\(r = 1 - 0.5^{0.25} = 0.159\), so 15.9% A1	✓ 1

Marking point	Mark	Notes
Part (a)
\(u_8 = 1200 \times 1.15^{7}\) M1	✓ 1
\(= 3190\) views (3192.02…; accept 3192) A1	✓ 1
Part (b)
Recognises a geometric series with \(u_1 = 1200\) and \(r = 1.15\) M1	✓ 1
\(S_{14} = \frac{1200(1.15^{14}-1)}{1.15-1}\) A1	✓ 1
\(= 48600\) views (48605.6…; accept 48606) A1	✓ 1

Marking point	Mark	Notes
Part (a)
M1 attempt to use \(\frac{\|v_A - v_E\|}{\|v_E\|}\times 100\) with 2.4 and 2.5	✓ 1
A1 4 %	✓ 1
Part (b)
M1 \(12\,000\times2.5 = 30\,000\) mg, then convert to grams (divide by 1000)	✓ 1
A1 \(3\times10^{1}\) g	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(u_{10}=3.0+9(0.4)\)	✓ 1
A1 6.6 km	✓ 1
Part (b)
M1 \(S_{15}=\frac{15}{2}\left(2(3.0)+14(0.4)\right)\)	✓ 1
A1 87 km	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(\frac{6.4\times10^{9}}{17.2}\)	✓ 1
A1 3.72 × 10⁸ s	✓ 1
Part (b)
M1 \(\frac{3.72\times10^{8}}{3.156\times10^{7}}\) (or the unrounded value from (a))	✓ 1
A1 11.8 years	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(165+7.5(n-1)>215\)	✓ 1
A1 \(n>7.67\) (or \(n-1>6.67\))	✓ 1
A1 \(n=8\) (temperature 217.5 °C at the end of minute 8)	✓ 1
Part (b)
M1 \(165+9\times7.5\)	✓ 1
A1 232.5 °C	✓ 1

Marking point	Mark	Notes
Part (a)
A1 common ratio 0.99 (so \(P_n=480\times0.99^n\))	✓ 1
A1 434 W	✓ 1
Part (b)
M1 \(480\times0.99^{t}=400\)	✓ 1
M1 take logarithms: \(t=\frac{\ln(400/480)}{\ln0.99}\)	✓ 1
A1 18.1 years	✓ 1

Marking point	Mark	Notes
Part (a)
M1 use of \(u_n = u_1 + (n-1)d\) with \(n=12\), \(d=2.5\)	✓ 1
A1 \(u_{12} = 45.5\) kg	✓ 1
Part (b)
M1 use of \(S_n=\frac{n}{2}(2u_1+(n-1)d)\) with an inequality or by trying values of \(n\)	✓ 1
A1 \(S_{16}=588\) (below 600) or \(S_{17}=646\)	✓ 1
A1 week 17 (since \(S_{16}<600	✓ 1
Part (c)
M1 \(\frac{45.5-18}{18}\times100\)	✓ 1
A1 \(153\%\) (3 s.f.)	✓ 1

Marking point	Mark	Notes
Part (a)
M1 substitute \(t=6\) into \(P=1.85\times0.992^{t}\)	✓ 1
A1 \(P=1.76\) bar	✓ 1
Part (b)
M1 \(1.85\times0.992^{t}=1.60\)	✓ 1
M1 take logarithms: \(t=\frac{\ln(1.60/1.85)}{\ln 0.992}\)	✓ 1
A1 \(t=18.1\) hours	✓ 1

Marking point	Mark	Notes
Part (a)
M1 convert: 5.1 litres = \(5.1\times10^{6}\) microlitres	✓ 1
M1 \(5.2\times10^{6}\times5.1\times10^{6}\)	✓ 1
A1 \(2.65\times10^{13}\)	✓ 1
Part (b)
M1 \(\frac{\|2.5\times10^{13}-2.652\times10^{13}\|}{2.652\times10^{13}}\times100\)	✓ 1
A1 \(5.73\%\)	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(u_{20}=45+19\times3\)	✓ 1
A1 102 patients	✓ 1
Part (b)
M1 \(S_{20}=\frac{20}{2}(45+102)\)	✓ 1
A1 1470 patients	✓ 1

Marking point	Mark	Notes
Part (a)
M1 finance solver with \(N=24\) (or \(N=6\) with \(C/Y=4\)), \(I=3.2\), \(PV=-8000\), \(PMT=0\), \(P/Y=C/Y=4\)	✓ 1
A1 £9685.96	✓ 1
Part (b)
M1 simple interest \(=8000\times0.032\times6=1536\) and compound interest \(=9685.96-8000\)	✓ 1
A1 £149.96	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(200\times3^{4}\)	✓ 1
A1 16 200	✓ 1
Part (b)
M1 \(200\times3^{t}=150000\), so \(3^{t}=750\)	✓ 1
M1 \(t=\frac{\ln750}{\ln3}\)	✓ 1
A1 6.03 days	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(u_{12} = 4 + 11 \times 0.75\)	✓ 1
A1 12.25 km	✓ 1
Part (b)
M1 4 + 0.75(n − 1) > 15	✓ 1
A1 n > 14.67 (14.666…)	✓ 1
A1 week 16	✓ 1
Part (c)
M1 \(S_{16} = \dfrac{16}{2}(2 \times 4 + 15 \times 0.75)\)	✓ 1
A1 154 km	✓ 1

Marking point	Mark	Notes
Part (a)
M1 use of \(u_n=u_1+(n-1)d\) with \(u_1=12\), \(d=5\), \(n=8\)	✓ 1
A1 47 units	✓ 1
Part (b)
M1 use of \(S_n=\frac{n}{2}(2u_1+(n-1)d)\) or \(\frac{8}{2}(12+47)\)	✓ 1
A1 236 units	✓ 1

Marking point	Mark	Notes
Part (a)
M1 substitute \(t=3\): \(240\times0.82^3\)	✓ 1
A1 132 mg (132.3...)	✓ 1
Part (b)
M1 set up \(240\times0.82^t=20\) or \(0.82^t=\frac{1}{12}\)	✓ 1
M1 take logarithms: \(t=\frac{\ln(1/12)}{\ln 0.82}\) (or use of a graph or solver)	✓ 1
A1 12.5 hours	✓ 1