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IB Maths Applications & Interpretation · SL · Topic 2

Functions

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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-031-00-01Linear models and piecewise models Paper 1Easy5 marks

A taxi company charges a fixed fee plus a charge for each kilometre travelled. The cost, C dollars, of a journey of d kilometres is modelled by
$C = 3.50 + 1.85d$.

(a)

Find the cost of a 12 km journey.

(2)

(b)

Interpret the value 1.85 in the context of the model.

(1)

(c)

Priya has 40.00 dollars. Find the greatest distance she can travel.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substituting d = 12: 3.50 + 1.85 × 12	✓ 1
A1 25.70 dollars	✓ 1
Part (b)
R1 each additional kilometre travelled adds 1.85 dollars to the cost (cost per kilometre)	✓ 1
Part (c)
M1 forming 3.50 + 1.85d = 40 (or 3.50 + 1.85d ≤ 40)	✓ 1
A1 d = 19.7 km (accept 19.7297…; do not accept 19.8, because that would cost more than 40 dollars)	✓ 1

Exam tip: When asked to interpret a gradient, give its meaning with units and context, for example 'dollars per kilometre', not just 'the slope'.

+ Add to test✓ In your test2IBMI-031-00-03Linear models and piecewise models Paper 1Easy5 marks

A mobile phone contract charges a monthly cost, C dollars, for x gigabytes (GB) of data used, given by
$C(x)=\begin{cases}15, & 0\le x\le 10\\ 15+4.5(x-10), & x>10\end{cases}$

(a)

Write down the cost for a month in which 7 GB of data is used.

(1)

(b)

Find the cost for a month in which 16.4 GB of data is used.

(2)

(c)

In one month the bill was $60.45. Find the amount of data used.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 $15.00	✓ 1
Part (b)
M1 15 + 4.5(16.4 − 10)	✓ 1
A1 $43.80	✓ 1
Part (c)
M1 15 + 4.5(x − 10) = 60.45, using the x > 10 rule because the bill exceeds $15	✓ 1
A1 x = 20.1 GB	✓ 1

Exam tip: Check which piece of a piecewise function applies before substituting: the domain condition tells you which rule to use.

+ Add to test✓ In your test3IBMI-031-00-04Linear models and piecewise models Paper 1Easy6 marks

A water tank initially contains 4800 litres. A valve is opened and water drains out at a constant rate of 65 litres per minute.
The volume of water remaining in the tank after t minutes is V litres.

(a)

Write down an equation for V in terms of t.

(1)

(b)

Find the volume remaining after 40 minutes.

(2)

(c)

Find the time taken for the tank to empty.

(2)

(d)

Hence state a suitable domain for the model.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 V = 4800 − 65t	✓ 1
Part (b)
M1 Substituting t = 40: 4800 − 65 × 40	✓ 1
A1 2200 litres	✓ 1
Part (c)
M1 Setting 4800 − 65t = 0	✓ 1
A1 t = 73.8 minutes (73.846…, exact value 960/13)	✓ 1
Part (d)
A1 0 ≤ t ≤ 73.8. Accept equivalent endpoints such as 73.846…, 960/13 or 4800/65.	✓ 1

Exam tip: A model's domain should stop where the context stops making sense. A negative volume is impossible, so the domain ends at the moment the tank is empty.

+ Add to test✓ In your test4IBMI-031-07-01Exponential and sinusoidal models Paper 1Easy6 marks

A cup of coffee is left to cool in a room. Its temperature, T °C, t minutes after it is poured, is modelled by
$T(t) = 20 + 70(0.92)^t$, $t \ge 0$.

(a)

Write down the temperature of the coffee when it is poured.

(1)

(b)

Find the temperature of the coffee 10 minutes after it is poured.

(2)

(c)

Find the time taken for the coffee to cool to 40 °C.

(2)

(d)

The graph of T has a horizontal asymptote at T = 20. Interpret this value in context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $T(0) = 20 + 70 = 90$ °C	✓ 1
Part (b)
M1: substituting $t = 10$: $20 + 70(0.92)^{10}$	✓ 1
A1: 50.4 °C	✓ 1
Part (c)
M1: setting $20 + 70(0.92)^t = 40$ or using a graph / solver	✓ 1
A1: $t = 15.0$ minutes	✓ 1
Part (d)
R1: 20 °C is the room temperature; the coffee approaches this temperature over time but never cools below it	✓ 1

Exam tip: For 'find the time' questions, a GDC graph intersection or solver is fully acceptable, but write down the equation you are solving to earn the method mark.

+ Add to test✓ In your test5IBMI-031-07-02Exponential and sinusoidal models Paper 1Easy6 marks

The height, h metres, of a passenger capsule on an observation wheel above the ground is modelled by
$h(t) = 26 - 24\cos(12t)$,
where t is the time in minutes after the passenger boards and the angle is measured in degrees.

(a)

Write down the maximum and minimum heights of the capsule.

(2)

(b)

Find the time taken for the wheel to complete one full revolution.

(2)

(c)

Find the height of the capsule 8 minutes after the passenger boards.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: maximum height $26 + 24 = 50$ m	✓ 1
A1: minimum height $26 - 24 = 2$ m	✓ 1
Part (b)
M1: period $= \frac{360}{12}$	✓ 1
A1: 30 minutes	✓ 1
Part (c)
M1: $h(8) = 26 - 24\cos(96°)$	✓ 1
A1: 28.5 m	✓ 1

Common error: Check your GDC is in degree mode when the model uses degrees; in radian mode h(8) gives a completely different (and wrong) height.

+ Add to test✓ In your test6IBMI-031-07-03Exponential and sinusoidal models Paper 1Easy5 marks

Mei invests 5000 US dollars (USD) in a savings account. The value of the investment, V USD, after t years is modelled by
$V = 5000(1.035)^t$.

(a)

Find the number of complete years after which the value of the investment first exceeds 8000 USD.

(3)

(b)

Find the value of the investment after 6 years. Give your answer correct to two decimal places.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $5000(1.035)^t = 8000$ or $(1.035)^t = 1.6$	✓ 1
A1: $t = 13.66\ldots$	✓ 1
A1: 14 complete years	✓ 1
Part (b)
M1: $5000(1.035)^6$	✓ 1
A1: 6146.28 USD	✓ 1

Exam tip: When the question asks for complete years, round the time up to the next whole number, since the value only exceeds the target after that point.

+ Add to test✓ In your test7IBMI-031-07-04Exponential and sinusoidal models Paper 1Easy6 marks

The depth of water in a harbour varies between a minimum of 2.4 m and a maximum of 8.8 m. The time between consecutive high tides is 12.4 hours. The depth, d metres, t hours after midnight is modelled by
$d(t) = a\sin(bt) + c$,
where a, b, c > 0 and the angle is in degrees.

(a)

Find the value of a.

(1)

(b)

Find the value of c.

(1)

(c)

Find the value of b.

(2)

(d)

Find the depth of water at 05:00.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $a = \frac{8.8 - 2.4}{2} = 3.2$	✓ 1
Part (b)
A1: $c = \frac{8.8 + 2.4}{2} = 5.6$	✓ 1
Part (c)
M1: $b = \frac{360}{12.4}$	✓ 1
A1: $b = 29.0$ (29.032…)	✓ 1
Part (d)
M1: $d(5) = 3.2\sin(29.032\ldots \times 5) + 5.6$	✓ 1
A1: 7.43 m (7.428…)	✓ 1

Common error: Using the rounded value b = 29.0 in part (d) gives 7.44 m instead of 7.43 m. Keep the full GDC value of b in memory when you substitute.

+ Add to test✓ In your test8IBMI-031-03-r1-01Choosing and evaluating models Paper 1Easy5 marks

A taxi company charges a fixed amount plus a constant rate per kilometre travelled. A 4 km journey costs $9.80 and a 10 km journey costs $17.60. The cost, $C$ dollars, of a journey of $k$ km is modelled by $C = mk + c$.

(a)

Find the value of m and the value of c.

(3)

(b)

Interpret the value of c in the context of the question.

(1)

(c)

Use the model to find the cost of a 23 km journey.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to find gradient, e.g. (17.60 − 9.80)/(10 − 4)	✓ 1
A1 m = 1.30	✓ 1
A1 c = 4.60	✓ 1
Part (b)
R1 c = 4.60 is the fixed charge in dollars (the cost before any distance is travelled)	✓ 1
Part (c)
A1 C = 1.30 × 23 + 4.60 = $34.50	✓ 1

Exam tip: When interpreting a parameter, give its value, its units and its meaning in the context. Saying only 'the y-intercept' does not earn the mark.

+ Add to test✓ In your test9IBMI-031-02-r1-01Exponential and sinusoidal models Paper 1Easy5 marks

The mass of caffeine, C mg, remaining in a person's bloodstream t hours after drinking a large coffee is modelled by
$C(t) = 180(0.87)^t$, $t \ge 0$.

(a)

Write down the mass of caffeine in the bloodstream immediately after the coffee is drunk.

(1)

(b)

Find the mass of caffeine remaining after 5 hours.

(2)

(c)

The person can sleep comfortably once the mass of caffeine falls below 20 mg. Find the time, to the nearest tenth of an hour, after which the person can sleep comfortably.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
180 mg	✓ 1
Part (b)
Substitutes t = 5: 180(0.87)⁵ (M1)	✓ 1
= 89.7 mg (A1)	✓ 1
Part (c)
Sets up 180(0.87)^t = 20, e.g. using a graph intersection, a solver or logarithms (M1)	✓ 1
t = 15.8 hours (A1)	✓ 1

Exam tip: The multiplier 0.87 means the mass falls by 13% each hour. Equations such as $180(0.87)^t = 20$ can be solved with the GDC's solver or by intersecting graphs; you do not need logarithms, but you must show the equation you solved.

+ Add to test✓ In your test10IBMI-031-02-r1-02Exponential and sinusoidal models Paper 1Easy6 marks

The depth of water, D metres, at the entrance to a small fishing harbour is modelled by
$D(t) = 1.6\sin\left(\frac{\pi t}{6.2}\right) + 5.0$,
where t is the number of hours after midnight, $0 \le t \le 24$.

(a)

Write down the maximum depth and the minimum depth of water predicted by the model.

(2)

(b)

Find the period of the model, and interpret this value in context.

(2)

(c)

Find the depth of water at 03:00.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Maximum depth = 6.6 m (A1)	✓ 1
Minimum depth = 3.4 m (A1)	✓ 1
Part (b)
Period = $\frac{2\pi}{\pi/6.2}$ = 12.4 hours (A1)	✓ 1
The time between two successive high tides (or low tides) is 12.4 hours (R1)	✓ 1
Part (c)
Substitutes t = 3: 1.6 sin(3π/6.2) + 5.0 (M1)	✓ 1
= 6.60 m (A1)	✓ 1

Common error: This model uses radians. Evaluating $\sin\left(\frac{3\pi}{6.2}\right)$ with the calculator in degree mode gives about 5.04 m instead of 6.60 m – check the mode before substituting.

+ Add to test✓ In your test11IBMI-031-01-r2-01Quadratic and cubic models Paper 1Easy5 marks

A netball is thrown upwards from a player's hand. Its height above the ground, h metres, t seconds after it is released is modelled by
$h(t) = -4.9t^2 + 14.7t + 1.8$.

(a)

Write down the height of the ball at the moment it is released.

(1)

(b)

Find the maximum height reached by the ball.

(2)

(c)

The ball is not caught and lands on the ground. Find the time taken for the ball to reach the ground.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
1.8 m	✓ 1
Part (b)
Uses vertex, e.g. $t = \frac{14.7}{2 \times 4.9} = 1.5$, or the GDC maximum (M1)	✓ 1
Maximum height = 12.825, so 12.8 m (A1)	✓ 1
Part (c)
Sets $-4.9t^2 + 14.7t + 1.8 = 0$ and solves, e.g. with the GDC (M1)	✓ 1
Rejects the negative root; t = 3.12 s (A1)	✓ 1

Exam tip: A quadratic model usually gives two roots. In context only one makes sense here, so state which root you reject and why: time cannot be negative.

+ Add to test✓ In your test12IBMI-031-01-r2-02Quadratic and cubic models Paper 1Easy6 marks

An open-topped tray is made from a rectangular sheet of card measuring 30 cm by 20 cm. A square of side x cm is cut from each corner. The sides are then folded up to form the tray.

(a)

Show that the volume of the tray, V cm³, is given by $V = 4x^3 - 100x^2 + 600x$.

(2)

(b)

State the set of possible values of x in this context.

(1)

(c)

Find the value of x that gives the maximum volume, and state this maximum volume.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Base dimensions $(30-2x)$ and $(20-2x)$, height $x$, so $V = x(30-2x)(20-2x)$ (M1)	✓ 1
Expands correctly: $x(600 - 100x + 4x^2) = 4x^3 - 100x^2 + 600x$ (A1) (AG)	✓ 1
Part (b)
0 < x < 10, because the 20 cm side requires 20 − 2x > 0 (A1)	✓ 1
Part (c)
Uses the GDC maximum on the graph of V, or solves V′(x) = 12x² − 200x + 600 = 0 (M1)	✓ 1
x = 3.92 cm (A1)	✓ 1
V = 1060 cm³ (1056.3…) (A1)	✓ 1

Common error: The cubic has a second turning point at x ≈ 12.7. It lies outside the domain 0 < x < 10, so it does not give a real tray. Always check turning points against the domain.

+ Add to test✓ In your test13IBMI-031-03-r2-01Choosing and evaluating models Paper 1Easy5 marks

A taxi company charges a fixed booking fee plus a constant rate per kilometre. A journey of 4 km costs $9.80 and a journey of 10 km costs $18.20. The cost, $C$ dollars, of a journey of $d$ km is modelled by $C = md + c$.

(a)

Find the value of $m$ and the value of $c$.

(3)

(b)

Interpret the value of $c$ in the context of the question.

(1)

(c)

Use the model to find the cost of a 25 km journey.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt at gradient: $m = \frac{18.20-9.80}{10-4}$	✓ 1
A1 $m = 1.40$	✓ 1
A1 $c = 9.80 - 4(1.40) = 4.20$	✓ 1
Part (b)
R1 $c$ is the fixed booking fee of $4.20, charged before any distance is travelled	✓ 1
Part (c)
A1 $C = 1.40(25) + 4.20 = $39.20	✓ 1

Exam tip: Interpretation marks need the context: say what the number means for the taxi fare, not just that it is the y-intercept.

+ Add to test✓ In your test14IBMI-031-05-r1-02Linear models and piecewise models Paper 1Easy6 marks

A decorative candle burns at a constant rate. Its height, h cm, t hours after it was lit, is modelled by $h=mt+c$. After 2 hours the height is 18.4 cm, and after 5 hours the height is 13.9 cm.

(a)

Find the value of m.

(2)

(b)

Find the value of c.

(1)

(c)

Interpret the value of m in context.

(1)

(d)

Find how long after it was lit the candle burns down completely.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$m=\frac{13.9-18.4}{5-2}$ M1 ; $m=-1.5$ A1	✓ 2
Part (b)
$18.4=-1.5(2)+c \Rightarrow c=21.4$ A1	✓ 1
Part (c)
The height of the candle decreases by 1.5 cm every hour R1	✓ 1
Part (d)
Sets $-1.5t+21.4=0$ M1 ; $t=14.3$ hours (14.266…) A1	✓ 2

Common error: A negative gradient must be interpreted as a decrease. Saying 'the height changes by −1.5 cm' without stating units per hour does not fully interpret the value in context.

+ Add to test✓ In your test15IBMI-031-06-r2-01Quadratic and cubic models Paper 1Easy5 marks

At a school science fair, a small catapult on a raised platform launches a ball upwards. The height of the ball above the ground, h metres, t seconds after launch is modelled by
$h(t) = -4.9t^2 + 14.7t + 1.8$, for $t \geq 0$.

(a)

Write down the height of the platform from which the ball is launched.

(1)

(b)

Find the maximum height reached by the ball.

(2)

(c)

Find the time taken for the ball to reach the ground.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
h(0) = 1.8 m A1	✓ 1
Part (b)
Attempt to find the vertex, e.g. $t = \frac{-14.7}{2(-4.9)} = 1.5$ or use of GDC maximum M1	✓ 1
Maximum height = 12.8 m (12.825) A1	✓ 1
Part (c)
Setting h(t) = 0, or use of GDC to find the positive zero M1	✓ 1
t = 3.12 s (3.1178…), rejecting the negative root A1	✓ 1

Exam tip: When a quadratic model gives two roots, only one usually makes sense in context. State clearly which root you reject and why, e.g. time cannot be negative.

+ Add to test✓ In your test16IBMI-031-06-r2-02Quadratic and cubic models Paper 1Easy6 marks

A footbridge arch spans a river. The arch is modelled by $y = ax^2 + bx + c$, where x metres is the horizontal distance from the left base of the arch and y metres is the height of the arch above the base. The arch meets the ground at (0, 0) and (40, 0), and passes through the point (10, 9).

(a)

Write down the value of c.

(1)

(b)

Find the values of a and b.

(3)

(c)

Find the maximum height of the arch.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
c = 0 A1	✓ 1
Part (b)
Substituting both points to form equations, e.g. 1600a + 40b = 0 and 100a + 10b = 9 M1	✓ 1
a = −0.03 A1	✓ 1
b = 1.2 A1	✓ 1
Part (c)
Use of axis of symmetry x = 20, or GDC maximum M1	✓ 1
Maximum height = 12 m (12.0 m) A1	✓ 1

Common error: A common mistake is to substitute the coordinates the wrong way round, e.g. putting 9 in place of x. Write each point as (x, y) before you substitute.

+ Add to test✓ In your test17IBMI-031-05-r2-01Linear models and piecewise models Paper 1Easy5 marks

A taxi company in a city charges fares using the linear model $C = 3.50 + 1.85d$, where $C$ is the fare in dollars and $d$ is the distance travelled in kilometres.

(a)

Find the fare for a journey of 12.4 km.

(2)

(b)

Interpret the value 1.85 in the context of the model.

(1)

(c)

A passenger pays $41.30 for a journey. Find the distance travelled.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes $d = 12.4$: $C = 3.50 + 1.85 \times 12.4$	✓ 1
A1: $C = $26.44$	✓ 1
Part (b)
A1: the fare increases by $1.85 for each additional kilometre travelled (cost per km)	✓ 1
Part (c)
M1: sets up $3.50 + 1.85d = 41.30$	✓ 1
A1: $d = 20.4$ km (20.432…)	✓ 1

Exam tip: When interpreting a gradient, always state the units of both variables: 'dollars per kilometre', not just 'the rate'.

+ Add to test✓ In your test18IBMI-031-01-r1-02Quadratic and cubic models Paper 1Easy7 marks

A small bakery sells a speciality loaf at a price of x dollars. Its daily profit from these loaves, P dollars, is modelled by $P(x) = ax^2 + bx + c$.
When the price is $2 the daily profit is $60, when the price is $4 the daily profit is $140, and when the price is $6 the daily profit is $156.

(a)

Find the values of a, b and c.

(3)

(b)

Find the price that maximises the daily profit and state this maximum profit.

(2)

(c)

Find the range of prices for which the bakery makes a profit on these loaves.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: forms three equations, e.g. 4a + 2b + c = 60, 16a + 4b + c = 140, 36a + 6b + c = 156	✓ 1
A1: a = −8, b = 88	✓ 1
A1: c = −84	✓ 1
Part (b)
A1: price = $5.50	✓ 1
A1: maximum daily profit = $158.00	✓ 1
Part (c)
A1: solves −8x² + 88x − 84 = 0 to obtain x = 1.06 and x = 9.94	✓ 1
A1: profit made for $1.06 < x < $9.94	✓ 1

Exam tip: Money answers should be given to 2 decimal places, so write $5.50 and $158.00 rather than 5.5 and 158.

+ Add to test✓ In your test19IBMI-031-02-r2-01Exponential and sinusoidal models Paper 1Easy5 marks

A used car is bought for $24 500. Its value, $V$ dollars, $t$ years after purchase is modelled by $V(t) = 24500 \times 0.86^{t}$.

(a)

Find the value of the car 3 years after purchase. Give your answer to 2 decimal places.

(2)

(b)

Find the value of $t$ at which the value of the car first falls below $8000, and hence state in which year after purchase this happens.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitutes t = 3: 24500 × 0.86³ (M1)	✓ 1
$15 583.37 (A1)	✓ 1
Part (b)
Sets up 24500 × 0.86^t = 8000 (or uses a graph/solver with V = 8000) (M1)	✓ 1
t = 7.42 years (A1)	✓ 1
The value first falls below $8000 during the 8th year after purchase (A1)	✓ 1

Exam tip: When a question asks 'in which year', a time of 7.42 means the event happens during the 8th year, not the 7th: year 1 runs from t = 0 to t = 1.

+ Add to test✓ In your test20IBMI-031-02-r2-02Exponential and sinusoidal models Paper 1Easy7 marks

The height, $h$ metres, of a passenger capsule on an observation wheel above the ground is modelled by
$h(t) = 21 - 18\cos(12t)$, $t \ge 0$,
where $t$ is the time in minutes after the passenger boards and the angle is measured in degrees.

(a)

Write down the maximum and minimum heights of the capsule.

(2)

(b)

Find the time taken for the wheel to complete one full rotation.

(1)

(c)

Find the height of the capsule 7 minutes after boarding.

(2)

(d)

Find the first time after boarding at which the capsule is 30 m above the ground.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Maximum height 39 m (A1)	✓ 1
Minimum height 3 m (A1)	✓ 1
Part (b)
360 ÷ 12 = 30 minutes (A1)	✓ 1
Part (c)
Substitutes t = 7: 21 − 18cos(84°) (M1)	✓ 1
h = 19.1 m (A1)	✓ 1
Part (d)
Solves 21 − 18cos(12t) = 30, e.g. cos(12t) = −0.5 or uses graph intersection (M1)	✓ 1
t = 10 minutes (A1)	✓ 1

Common error: Check your calculator is in degree mode. In radian mode cos(84) gives a completely different height, and the answer will not match the physical situation.

+ Add to test✓ In your test21IBMI-031-04-r2-02Transformations of graphs and logarithmic scales Paper 1Easy5 marks

The sound level $L$, in decibels (dB), of a sound with intensity $I$ W m⁻² is given by
$L=10\log_{10}\left(\frac{I}{I_0}\right)$, where $I_0=1.0\times10^{-12}$ W m⁻².
The sound from a busy café has an intensity of $3.2\times10^{-5}$ W m⁻².

(a)

Find the sound level of the café.

(2)

(b)

A leaf blower has a sound level of 92 dB. Find its intensity.

(2)

(c)

Find how many times greater the intensity of the leaf blower is than the intensity of the café.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $10\log_{10}\left(\frac{3.2\times10^{-5}}{1.0\times10^{-12}}\right)$	✓ 1
A1: 75.1 dB (75.05...)	✓ 1
Part (b)
M1: $\frac{I}{I_0}=10^{9.2}$	✓ 1
A1: $I=1.58\times10^{-3}$ W m⁻² (1.5849...×10⁻³)	✓ 1
Part (c)
A1: $\frac{1.585\times10^{-3}}{3.2\times10^{-5}}=49.5$ times. Also accept $10^{(92-75.05)/10}\approx49.5$. Accept 49.0 to 49.6 where it follows from rounded earlier answers.	✓ 1

Common error: A difference of about 17 dB does not mean 17 times the intensity. Every 10 dB multiplies the intensity by 10, so compare the intensities, not the decibel values. Keep full calculator values between parts to avoid rounding drift.

+ Add to test✓ In your test22IBMI-031-06-r1-01Quadratic and cubic models Paper 1Easy6 marks

A netball is thrown upwards from a player's hand. Its height above the ground, h metres, t seconds after release is modelled by
$h(t) = -4.9t^2 + 14.7t + 1.8$, for $t \ge 0$, until the ball reaches the ground.

(a)

Find the maximum height reached by the ball.

(3)

(b)

Find the time at which the ball hits the ground, assuming nothing stops it.

(2)

(c)

Interpret the value 1.8 in the context of the model.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Uses a vertex method, e.g. $t = -\frac{14.7}{2(-4.9)}$, or the GDC maximum	✓ 1
$t = 1.5$ s	✓ 1
Maximum height = 12.8 m (12.825)	✓ 1
Part (b)
Sets $h(t) = 0$ and solves, e.g. by GDC or the quadratic formula	✓ 1
$t = 3.12$ s, rejecting the negative root	✓ 1
Part (c)
The ball is 1.8 m above the ground at the moment it is released (at $t = 0$)	✓ 1

Exam tip: When solving $h(t)=0$, state clearly that the negative root is rejected because time cannot be negative in this model.

+ Add to test✓ In your test23IBMI-031-05-r1-01Linear models and piecewise models Paper 1Easy5 marks

A taxi company in a city charges a fixed fee of USD 3.50 for every journey plus USD 1.85 per kilometre travelled. The cost, in USD, of a journey of d kilometres is $C(d)$.

(a)

Write down an expression for $C(d)$ in terms of d.

(2)

(b)

Find the cost of a journey of 12.4 km.

(1)

(c)

Amira has USD 40.00 to spend on a single taxi journey. Find the greatest distance she can travel.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$C(d)=1.85d+3.50$: coefficient of d is 1.85 A1 ; constant term is 3.50 A1	✓ 2
Part (b)
$C(12.4)=3.50+1.85\times12.4=26.44$, so the cost is USD 26.44 A1	✓ 1
Part (c)
Sets $1.85d+3.50=40$ (or $\le 40$) M1 ; $d=19.7$ km (19.729…), also accept 19.72 km, but not a value rounded up to 19.73 km A1	✓ 2

Exam tip: In a linear cost model, the gradient is the rate (cost per km) and the vertical intercept is the fixed charge. Identify each before writing the equation. When a budget is a maximum, round the distance down so the cost stays within budget.

+ Add to test✓ In your test24IBMI-031-08-r2-02Choosing and evaluating models Paper 1Easy5 marks

A food truck sells fruit smoothies. The owner models the daily profit, P dollars, when each smoothie is sold at a price of x dollars by
$P(x)=-40x^2+560x-1000$.
Give all monetary answers correct to the nearest cent.

(a)

Find the price that maximises the daily profit and the corresponding maximum profit.

(2)

(b)

Find the two prices at which the food truck makes zero profit.

(2)

(c)

Suggest one reason why this model may not be realistic for prices well above 12 dollars.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: price = 7.00 dollars (from $x=\frac{-560}{2(-40)}$ or using GDC)	✓ 1
A1: maximum profit = 960.00 dollars	✓ 1
Part (b)
M1: attempt to solve $-40x^2+560x-1000=0$ (e.g. GDC or quadratic formula)	✓ 1
A1: $x=2.10$ dollars and $x=11.90$ dollars	✓ 1
Part (c)
R1: e.g. the model predicts an ever-increasing loss, whereas in reality almost no smoothies would be sold, so the loss would be limited to roughly the fixed costs (or the owner would stop selling); OR such prices are likely outside the range of data used to build the model	✓ 1

Exam tip: When a question asks for answers to the nearest cent, give exactly 2 decimal places, e.g. 2.10 rather than 2.1, and do not round until the final step.

+ Add to test✓ In your test25IBMI-031-03-r2-02Choosing and evaluating models Paper 1Easy6 marks

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

(a)

Write down the initial mass of caffeine in the bloodstream.

(1)

(b)

Write down the percentage of caffeine removed from the bloodstream each hour.

(1)

(c)

Find the mass of caffeine remaining after 5 hours.

(2)

(d)

The person wants to sleep once the mass of caffeine has fallen to 25 mg. Find the time after drinking the coffee at which the mass of caffeine first equals 25 mg.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 200 mg	✓ 1
Part (b)
A1 13%	✓ 1
Part (c)
M1 substituting $t = 5$: $200(0.87)^5$	✓ 1
A1 99.7 mg	✓ 1
Part (d)
M1 setting $200(0.87)^t = 25$ or equivalent, e.g. graph intersection, a table of values or logarithms	✓ 1
A1 $t = 14.9$ hours (accept 14.93; 15 alone does not earn this mark)	✓ 1

Common error: A base of 0.87 means 13% is lost each hour, not 87%. The base is the fraction that remains.

+ Add to test✓ In your test26IBMI-031-08-r2-01Choosing and evaluating models Paper 1Easy6 marks

A candle is lit and its height is recorded. After 10 minutes the candle is 23.5 cm tall and after 40 minutes it is 17.5 cm tall. The height, $h$ cm, of the candle $t$ minutes after it was lit is modelled by a linear function $h(t)=mt+c$.

(a)

Find the value of $m$ and the value of $c$.

(3)

(b)

Interpret the value of $m$ in the context of the question.

(1)

(c)

Use the model to estimate how long after being lit the candle will have burnt down completely.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: attempt at gradient using both points, $m=\frac{17.5-23.5}{40-10}$	✓ 1
A1: $m=-0.2$	✓ 1
A1: $c=25.5$ (from $23.5=-0.2(10)+c$)	✓ 1
Part (b)
R1: the candle's height decreases by 0.2 cm every minute (it burns down at 0.2 cm per minute)	✓ 1
Part (c)
M1: setting $25.5-0.2t=0$	✓ 1
A1: $t=127.5$ minutes (accept 128 minutes)	✓ 1

Exam tip: When interpreting a gradient, always give its units and say what changes per unit of the input — a bare number like 'the gradient is −0.2' earns no interpretation mark.

+ Add to test✓ In your test27IBMI-031-03-r1-02Choosing and evaluating models Paper 1Easy6 marks

The value, $V$ dollars, of a car $t$ years after it is bought new is modelled by
$V(t) = 24000(0.85)^t$, $t \ge 0$.

(a)

State the annual percentage decrease in the value of the car.

(1)

(b)

Find the value of the car after 3 years.

(2)

(c)

Find the time taken for the value of the car to fall to 8000 dollars.

(2)

(d)

With reference to the model, explain why it may not be suitable for predicting the value of the car many years after purchase.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 15%	✓ 1
Part (b)
M1 substituting t = 3: 24000(0.85)³	✓ 1
A1 14 739 dollars	✓ 1
Part (c)
M1 setting up 24000(0.85)^t = 8000, solved by GDC or logarithms	✓ 1
A1 t = 6.76 years	✓ 1
Part (d)
R1 as t increases, V approaches 0 (horizontal asymptote V = 0), giving ever-smaller values, whereas a real car keeps a minimum value, e.g. its scrap value; or the model ignores factors such as condition, mileage or demand (e.g. classic cars may rise in value)	✓ 1

Common error: Reading 0.85 as an 85% decrease. The multiplier 0.85 means 85% of the value is kept each year, so the decrease is 15%.

+ Add to test✓ In your test28IBMI-031-04-r1-01Transformations of graphs and logarithmic scales Paper 1Easy6 marks

The sound level L, in decibels (dB), of a sound of intensity I W m⁻² is given by $L = 10\log_{10}\left(\frac{I}{10^{-12}}\right)$.
A leaf blower produces sound of intensity 3.2×10⁻⁵ W m⁻² at the position of its operator.

(a)

Find the sound level experienced by the operator.

(2)

(b)

A motorbike engine produces a sound level of 92 dB. Find the intensity of this sound.

(2)

(c)

The sound level at a concert increases by 15 dB. Find the factor by which the sound intensity increases.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes correctly, $L = 10\log_{10}(3.2\times10^{-5}/10^{-12}) = 10\log_{10}(3.2\times10^{7})$	✓ 1
A1: L = 75.1 dB (accept 75.05)	✓ 1
Part (b)
M1: rearranges to $I = 10^{-12}\times10^{9.2}$	✓ 1
A1: I = 1.58×10⁻³ W m⁻² (accept 1.585×10⁻³)	✓ 1
Part (c)
M1: recognises that an increase of 15 dB multiplies the intensity by $10^{15/10}$	✓ 1
A1: factor = 31.6	✓ 1

Common error: An increase of 15 dB does not mean the intensity increases 15 times or 1.5 times. Each 10 dB step multiplies the intensity by 10, so 15 dB multiplies it by 10¹.⁵.

+ Add to test✓ In your test29IBMI-031-08-r1-02Choosing and evaluating models Paper 1Easy6 marks

The value $V$, in dollars, of a new electric car $t$ years after purchase is modelled by
$V(t) = 32000 \times 0.85^{t}$, $t \geq 0$.

(a)

Find the value of the car after 4 years. Give your answer in dollars to two decimal places.

(2)

(b)

Interpret the number 0.85 in this context.

(1)

(c)

Find the time, in years, at which the model predicts the value of the car falls to 5000 dollars. Give your answer to three significant figures.

(2)

(d)

State one reason why this model may not be appropriate for predicting the car's value after 30 years.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitutes $t = 4$: $32000 \times 0.85^4$ M1	✓ 1
16704.20 dollars A1	✓ 1
Part (b)
Each year the car keeps 85% of its value from the previous year, so it loses 15% of its value per year. R1	✓ 1
Part (c)
Sets up $32000 \times 0.85^{t} = 5000$ and solves by GDC or logarithms M1	✓ 1
$t = 11.4$ years A1	✓ 1
Part (d)
Any valid reason, e.g. a very old car may be scrapped, so its value is only its scrap value; or a rare model may gain value as a classic; or the rate of depreciation is unlikely to stay at 15% per year for 30 years. R1	✓ 1

Common error: Reading 0.85 as 'loses 85% per year' confuses the fraction of value kept with the percentage lost.

+ Add to test✓ In your test30IBMI-031-06-r1-02Quadratic and cubic models Paper 1Easy6 marks

An open-topped tray is made from a rectangular sheet of card measuring 30 cm by 20 cm. A square of side x cm is cut from each corner and the sides are folded up.

(a)

Show that the volume of the tray, V cm³, is given by $V = 4x^3 - 100x^2 + 600x$.

(2)

(b)

State the set of possible values of x for which the model is valid.

(1)

(c)

Find the value of x that maximises the volume, and the maximum volume.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Base dimensions $(30-2x)$ and $(20-2x)$ with height $x$, so $V = x(30-2x)(20-2x)$	✓ 1
Expands correctly: $x(600 - 100x + 4x^2) = 4x^3 - 100x^2 + 600x$	✓ 1
Part (b)
$0 < x < 10$	✓ 1
Part (c)
Valid method: either a GDC to locate the local maximum of V in \(0	✓ 1
$x = 3.92$ cm	✓ 1
$V_{max} = 1060$ cm³ (3 s.f.; 1056.3… also accepted)	✓ 1

Common error: Do not give the other turning point, x ≈ 12.7, as an answer. It lies outside 0 < x < 10 because the 20 cm side would then have negative length. Always check a turning point against the domain.

+ Add to test✓ In your test31IBMI-031-01-r1-01Quadratic and cubic models Paper 1Easy5 marks

A footballer kicks a ball from a raised platform. The height of the ball above the ground, h metres, t seconds after it is kicked is modelled by
$h(t) = -4.9t^2 + 12t + 0.8$, for $t \ge 0$.

(a)

Find the maximum height reached by the ball.

(2)

(b)

Find the time taken for the ball to reach the ground. Justify your choice of solution.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: uses the GDC maximum, or finds the vertex $t = \frac{-12}{2(-4.9)} = 1.22$ s and substitutes it into h	✓ 1
A1: maximum height = 8.15 m (8.14694…)	✓ 1
Part (b)
M1: sets $-4.9t^2 + 12t + 0.8 = 0$, using the graph meeting the t-axis or the quadratic formula	✓ 1
A1: t = 2.51 s (2.51393…)	✓ 1
R1: rejects t = −0.0649 (−0.06494…) because time cannot be negative in this context	✓ 1

Exam tip: When a quadratic gives two roots, state which one you are rejecting and explain why in terms of the context. The reasoning mark is only earned if the reason refers to the situation, which here is that time cannot be negative.

+ Add to test✓ In your test32IBMI3S-2-03-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy5 marks

An electric scooter hire company charges a fixed unlock fee of $1.50 plus $0.28 for every minute of a ride. The cost of a ride lasting t minutes is $C.

(a)

Write down a linear model for C in terms of t.

(2)

(b)

Find the cost of a 25-minute ride.

(1)

(c)

A ride costs exactly $12.00. Find its duration in minutes.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 gradient 0.28 identified as the rate per minute	✓ 1
A1 C = 1.50 + 0.28t	✓ 1
Part (b)
A1 $8.50	✓ 1
Part (c)
M1 1.50 + 0.28t = 12	✓ 1
A1 t = 37.5 minutes	✓ 1

Common error: Leaving out the fixed fee when substituting, so that the cost is treated as directly proportional to time.

+ Add to test✓ In your test33IBMI3S-2-03-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy5 marks

A courier charges for parcels using the model
$P(w) = 4.50$ for $0 < w \le 2$
$P(w) = 4.50 + 1.20(w-2)$ for $2 < w \le 12$
where $P$ is the price in euros and w is the mass of the parcel in kg.

(a)

Write down the price of a parcel of mass 1.5 kg.

(1)

(b)

Find the price of a parcel of mass 6.5 kg.

(2)

(c)

A parcel costs €15.30. Find its mass.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €4.50	✓ 1
Part (b)
M1 4.50 + 1.20(6.5 − 2)	✓ 1
A1 €9.90	✓ 1
Part (c)
M1 4.50 + 1.20(w − 2) = 15.30	✓ 1
A1 w = 11 kg (inside the interval 2 < w ≤ 12, so the second rule is valid)	✓ 1

Exam tip: Choose the piece of the function by checking which interval the given input or output belongs to, then confirm the answer lies in that interval.

+ Add to test✓ In your test34IBMI3S-2-03-08Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy7 marks

A cyclist accelerates from rest, rides at a steady speed and then brakes to a stop. Her speed v m s⁻¹ at time t seconds is modelled by
$v(t) = 1.5t$ for $0 \le t \le 8$
$v(t) = 12$ for $8 < t \le 20$
$v(t) = 12 - 0.8(t-20)$ for $20 < t \le T$.

(a)

Find her speed when t = 5.

(1)

(b)

Show that the model gives the same speed at t = 8 from the first two rules.

(1)

(c)

Given that she is at rest at t = T, find T.

(2)

(d)

Find the time during the braking phase at which her speed is 9 m s⁻¹.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 7.5 m s⁻¹	✓ 1
Part (b)
R1 1.5 × 8 = 12, which equals the constant 12 (so there is no jump in speed)	✓ 1
Part (c)
M1 12 − 0.8(T − 20) = 0	✓ 1
A1 T = 35	✓ 1
Part (d)
M1 12 − 0.8(t − 20) = 9	✓ 1
A1 t = 23.75 s	✓ 1
A1 this lies in 20 < t ≤ 35, so it is valid	✓ 1

Exam tip: For each piecewise rule, solve the equation and then confirm the answer falls in that rule's interval.

+ Add to test✓ In your test35IBMI3S-2-03b-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy5 marks

Researchers tracking juvenile green turtles in a marine nursery model the shell length, L cm, of one turtle by L = 6.8 + 2.35t, where t is the time in months since hatching, for 0 ≤ t ≤ 24.

(a)

Calculate the shell length of the turtle after 8 months.

(2)

(b)

Interpret the value 2.35 in this context.

(1)

(c)

Find the age at which the shell length first reaches 30 cm.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 8 into the model	✓ 1
A1 L = 25.6 cm	✓ 1
Part (b)
R1 the shell length increases by 2.35 cm each month (accept per month)	✓ 1
Part (c)
M1 sets 6.8 + 2.35t = 30 (or equivalent)	✓ 1
A1 t = 9.87 months	✓ 1

Common error: Quoting the gradient without its rate unit. A gradient is a change in the output per one unit of the input, so say "per month".

+ Add to test✓ In your test36IBMI3S-2-03b-03Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy5 marks

The parabolic entrance arch of a museum has height h metres at a horizontal distance x metres from its left-hand base, where h = −0.25x² + 2x for 0 ≤ x ≤ 8.

(a)

Find the maximum height of the arch.

(2)

(b)

A rectangular crate is moved through the arch. The crate is 3 m tall. Find the greatest width of crate that fits through the arch, assuming it is placed symmetrically.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finds the x-coordinate of the vertex, x = 4 (or uses technology)	✓ 1
A1 maximum height 4 m	✓ 1
Part (b)
M1 sets −0.25x² + 2x = 3	✓ 1
A1 x = 2 and x = 6	✓ 1
A1 width = 6 − 2 = 4 m	✓ 1

Common error: Giving the two x-values (2 and 6) as the width. The width is the distance between them.

+ Add to test✓ In your test37IBMI3S-2-03b-08Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy6 marks

A smartwatch battery drains steadily when in use. The charge level is 82% after 2 hours and 58% after 6 hours. The charge level C% is modelled as a linear function of the time t hours.

(a)

Find the gradient of the model and state what it means.

(2)

(b)

Find the equation of the model in the form C = mt + c.

(2)

(c)

Find the time at which the charge level is predicted to reach 10%.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (58 − 82) ÷ (6 − 2)	✓ 1
A1 −6, meaning the charge falls by 6 percentage points each hour	✓ 1
Part (b)
M1 substitutes a point, for example 82 = −6 × 2 + c	✓ 1
A1 C = −6t + 94	✓ 1
Part (c)
M1 sets −6t + 94 = 10	✓ 1
A1 t = 14 hours	✓ 1

Common error: Writing a positive gradient for a falling quantity. A decrease gives a negative gradient.

+ Add to test✓ In your test38IBMI3S-2-03b-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy5 marks

A buoy bobs on the sea. Its height above mean sea level is modelled by h = 0.85 sin(0.75t) metres, where t is the time in seconds and the angle is in radians.

(a)

State the greatest height of the buoy above mean sea level.

(1)

(b)

Find the time taken for one complete bob.

(2)

(c)

Find the height of the buoy when t = 2.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.85 m	✓ 1
Part (b)
M1 period = 2π ÷ 0.75	✓ 1
A1 8.38 s	✓ 1
Part (c)
M1 substitutes t = 2, 0.85 sin(1.5)	✓ 1
A1 0.848 m	✓ 1

Common error: Leaving the calculator in degree mode. With a sine of 1.5 radians the answer is very different from sin 1.5°.

+ Add to test✓ In your test39IBMI3S-2-01r9-01Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

A space probe is moving away from Earth. Its distance from Earth, D million km, is modelled by D = 48 + 3.6t, where t is the number of days since the probe passed a tracking marker.

(a)

Interpret the value 3.6 in this model.

(1)

(b)

Calculate the distance of the probe from Earth when t = 25.

(2)

(c)

Find the value of t at which the probe is 300 million km from Earth.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the probe moves away from Earth at 3.6 million km per day (rate of change of distance)	✓ 1
Part (b)
M1 substitute t = 25 into the model	✓ 1
A1 D = 138 million km	✓ 1
Part (c)
M1 set 48 + 3.6t = 300	✓ 1
A1 t = 70 days	✓ 1

Exam tip: State the unit in any interpretation: a gradient is always a change in one quantity per unit of the other.

+ Add to test✓ In your test40IBMI3S-2-01r9-02Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

A coffee roaster compares two suppliers of green coffee beans. Supplier A charges a fixed delivery fee of £60 plus £3.40 per kg. Supplier B has no delivery fee and charges £5.20 per kg. Let n be the mass of beans bought, in kg.

(a)

Find the cost, in £, of buying 25 kg of beans from Supplier A.

(2)

(b)

Find the value of n for which the two suppliers charge the same amount. Give your answer to 3 significant figures.

(2)

(c)

For an order of 50 kg, state which supplier is cheaper and by how much.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 60 + 3.4 × 25	✓ 1
A1 £145	✓ 1
Part (b)
M1 60 + 3.4n = 5.2n	✓ 1
A1 n = 33.3 kg	✓ 1
Part (c)
M1 compare £230 (A) and £260 (B)	✓ 1
A1 Supplier A is cheaper by £30	✓ 1

Common error: Do not compare the per-kg prices alone; the fixed fee changes which supplier is cheaper for small orders.

+ Add to test✓ In your test41IBMI3S-2-01r9-03Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

During roasting, green coffee beans lose water. For a particular roast the mass of roasted beans, R kg, is modelled by R = 0.85G, where G kg is the mass of green beans loaded into the roaster.

(a)

Interpret the value 0.85 in this context.

(1)

(b)

Calculate the mass of roasted beans produced from 18.6 kg of green beans. Give your answer to 3 significant figures.

(2)

(c)

A café needs 22.1 kg of roasted beans. Find the mass of green beans that must be loaded.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the roasted beans have 85% of the mass of the green beans (15% of mass is lost)	✓ 1
Part (b)
M1 0.85 × 18.6	✓ 1
A1 15.8 kg	✓ 1
Part (c)
M1 22.1 = 0.85G or 22.1 ÷ 0.85	✓ 1
A1 G = 26 kg	✓ 1

Exam tip: To find the input from the output, rearrange the model first (divide), rather than multiplying by 0.85 again.

+ Add to test✓ In your test42IBMI3S-2-01r9-04Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

The cost, C euros (€), of a bridge inspection that lasts h hours is modelled by
C = 120 + 45h for 0 ≤ h ≤ 8
C = 60h for h > 8.

(a)

Find the cost of an inspection lasting 5 hours.

(1)

(b)

Find the cost of an inspection lasting 11.5 hours.

(2)

(c)

A client has a budget of €1020. Find the longest inspection, in hours, that the client can afford.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €345	✓ 1
Part (b)
M1 use the second rule, 60 × 11.5	✓ 1
A1 €690	✓ 1
Part (c)
M1 recognise that €1020 > €480 so use 60h = 1020	✓ 1
A1 h = 17 hours	✓ 1

Common error: Choose the rule by checking which interval the input lies in, before substituting.

+ Add to test✓ In your test43IBMI3S-2-01r10-01Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy5 marks

The distance, d metres, of the front of a lava flow from the vent is modelled by d = 40 + 2.5t, where t is the time in minutes after the flow was first measured, for 0 ≤ t ≤ 120.

(a)

Interpret the value 2.5 in this model.

(1)

(b)

Calculate the distance of the lava front from the vent when t = 30.

(2)

(c)

Find the time at which the lava front is 300 m from the vent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
The lava front advances at 2.5 metres per minute (rate of change of distance / speed)	✓ 1
Part (b)
Substitution 40 + 2.5 × 30	✓ 1
d = 115 m	✓ 1
Part (c)
Set 40 + 2.5t = 300 (or equivalent rearrangement)	✓ 1
260 ÷ 2.5 = 104 minutes	✓ 1

Common error: When a model is written as y = a + bx, the constant a is the starting value and b is the rate of change; do not swap their meanings.

+ Add to test✓ In your test44IBMI3S-2-01r10-02Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy6 marks

A theatre sells tickets for a one-night show. The revenue, R euros, when each ticket is priced at p euros is modelled by R = −40p² + 1200p, for 0 ≤ p ≤ 30.

(a)

Calculate the revenue when the ticket price is €10.

(2)

(b)

Find the ticket price that maximises the revenue.

(2)

(c)

Write down the maximum revenue.

(1)

(d)

Find the number of tickets sold at the price that maximises the revenue.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitution of p = 10	✓ 1
R = €8000	✓ 1
Part (b)
Use of axis of symmetry p = −b/(2a) or GDC maximum	✓ 1
p = €15	✓ 1
Part (c)
R = €9000	✓ 1
Part (d)
Number of tickets = revenue ÷ price = 9000 ÷ 15 = 600 tickets	✓ 1

Exam tip: Check the answer to a maximum problem by substituting the price back into the model rather than trusting the vertex formula alone.

+ Add to test✓ In your test45IBMI3S-2-01r10-03Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy6 marks

The value, V million pounds, of a cargo ship t years after it was bought is modelled by V = 48 × 0.92^t.

(a)

Calculate the value of the ship after 5 years. Give your answer to 3 significant figures.

(2)

(b)

State what the number 0.92 tells you about how the value changes each year.

(1)

(c)

Find the value of t at which the ship is worth £25 million. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitution t = 5 into 48 × 0.92⁵	✓ 1
V = £31.6 million	✓ 1
Part (b)
The value is multiplied by 0.92 each year, so it falls by 8% per year	✓ 1
Part (c)
Set 48 × 0.92^t = 25 (or 0.92^t = 25/48)	✓ 1
Use of logarithms or GDC solver	✓ 1
t = 7.82 years	✓ 1

Common error: A multiplier of 0.92 means an 8% decrease, not a decrease of 0.92 or 92%.

+ Add to test✓ In your test46IBMI3S-2-01r10-04Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy6 marks

The depth of water, h metres, at the entrance to a harbour is modelled by h = 4 + 1.5 sin(30t)°, where t is the time in hours after midnight. Use your calculator in degree mode.

(a)

Write down the maximum depth and the minimum depth predicted by the model.

(2)

(b)

Find the time between two successive high tides.

(2)

(c)

Calculate the depth at t = 2.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Maximum depth = 4 + 1.5 = 5.5 m	✓ 1
Minimum depth = 4 − 1.5 = 2.5 m	✓ 1
Part (b)
Period = 360 ÷ 30	✓ 1
12 hours	✓ 1
Part (c)
Substitution 4 + 1.5 sin 60°	✓ 1
h = 5.30 m	✓ 1

Exam tip: Switch the calculator to the angle mode the model uses before evaluating any sine value; the wrong mode is the most common source of lost accuracy.

+ Add to test✓ In your test47IBMI3S-2-02r11-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy4 marks

A cycle-hire shop charges a fixed fee of €8 plus €3.50 for each hour the bike is hired. The cost, €C, of hiring a bike for h hours is modelled as a linear function of h.

(a)

Write down an equation for C in terms of h.

(2)

(b)

Find the cost of hiring a bike for 6 hours.

(1)

(c)

A customer pays €50. Find the number of hours for which the bike was hired.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
gradient 3.50 (rate per hour)	✓ 1
C = 3.5h + 8	✓ 1
Part (b)
C = 3.5 × 6 + 8 = €29	✓ 1
Part (c)
50 = 3.5h + 8, so h = 42 ÷ 3.5 = 12 hours	✓ 1

Exam tip: The fixed fee is the intercept (the value when h = 0) and the hourly rate is the gradient.

+ Add to test✓ In your test48IBMI3S-2-02r11-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy5 marks

A data centre is billed for electricity using a piecewise tariff. For a monthly usage of x kWh, the cost in euros is
C(x) = 0.12x for 0 ≤ x ≤ 5000
C(x) = 600 + 0.09(x − 5000) for x > 5000.

(a)

Find the cost of using 4200 kWh in a month.

(1)

(b)

Find the cost of using 7800 kWh in a month.

(2)

(c)

One month's bill is €1000. Find the usage, to the nearest kWh.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
C = 0.12 × 4200 = €504	✓ 1
Part (b)
C = 600 + 0.09 × (7800 − 5000)	✓ 1
= €852	✓ 1
Part (c)
600 + 0.09(x − 5000) = 1000 (second rule, since 1000 > 600)	✓ 1
x = 9444 kWh	✓ 1

Common error: Choose the correct rule by checking which interval the input lies in; a bill above €600 means the usage is in the second interval.

+ Add to test✓ In your test49IBMI3S-2-02r11-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy5 marks

The amount of data stored by a company is modelled by S = 40 × 1.18^t terabytes (TB), where t is the number of years after 1 January 2022.

(a)

Write down the amount of data stored on 1 January 2022.

(1)

(b)

State the annual percentage increase in the amount of data stored according to the model.

(1)

(c)

Find the amount stored 4.5 years after 1 January 2022, correct to 3 significant figures.

(1)

(d)

Find the value of t when the amount stored first exceeds 200 TB, correct to 1 decimal place.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
40 TB	✓ 1
Part (b)
18%	✓ 1
Part (c)
S = 40 × 1.18^4.5 = 84.2 TB	✓ 1
Part (d)
40 × 1.18^t = 200	✓ 1
t = 9.7	✓ 1

Exam tip: In a × bt, a is the starting value and b = 1 + (percentage rate ÷ 100).

+ Add to test✓ In your test50IBMI3S-2-02r11-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy5 marks

The height of a cycle road, h metres above a reference level, at distance x km from its start is modelled for road A by h_A(x) = −0.4x² + 6x, for 0 ≤ x ≤ 15. The profile of road B is the graph of road A translated 3 km to the right and 40 m up.

(a)

Write down an expression for h_B(x) in terms of x.

(1)

(b)

The highest point of road A is at (7.5, 22.5). Write down the coordinates of the highest point of road B.

(2)

(c)

Road C is the reflection of road A in the horizontal axis. Write down its equation and find its lowest height.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
h_B(x) = −0.4(x − 3)² + 6(x − 3) + 40	✓ 1
Part (b)
x = 7.5 + 3 = 10.5	✓ 1
h = 22.5 + 40 = 62.5, so (10.5, 62.5)	✓ 1
Part (c)
h_C(x) = 0.4x² − 6x	✓ 1
lowest height = −22.5 m	✓ 1

Common error: A translation to the right by 3 replaces x with x − 3, not x + 3.

+ Add to test✓ In your test51IBMI3S-2-01r12-01Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

A rowing ergometer records the mean power P watts of an athlete at stroke rate r strokes per minute. For 20 ≤ r ≤ 40 the sports scientist uses the linear model P = 9.2r − 118.

(a)

Interpret the value 9.2 in this model.

(1)

(b)

Calculate the predicted power at a stroke rate of 28 strokes per minute.

(2)

(c)

Find the stroke rate at which the model predicts a power of 200 W. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 each extra stroke per minute increases the mean power by 9.2 W (accept: rate of change of power with stroke rate)	✓ 1
Part (b)
M1 substitutes r = 28 into the model	✓ 1
A1 P = 139.6 W (accept 140 W)	✓ 1
Part (c)
M1 200 = 9.2r − 118	✓ 1
A1 r = 34.6 strokes per minute	✓ 1

Exam tip: When a model has a stated domain, check that your answer lies inside it before finishing.

+ Add to test✓ In your test52IBMI3S-2-01r12-02Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy7 marks

A rail operator charges a fare F euros (€) for a journey of d km using the piecewise model:
F(d) = 2.40 + 0.18d for 0 < d ≤ 40
F(d) = 9.60 + 0.11(d − 40) for d > 40

(a)

Calculate the fare for a journey of 25 km.

(2)

(b)

Calculate the fare for a journey of 120 km.

(2)

(c)

A passenger pays €15.00. Find the distance travelled, correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses the first rule with d = 25	✓ 1
A1 €6.90	✓ 1
Part (b)
M1 uses the second rule with d = 120	✓ 1
A1 €18.40	✓ 1
Part (c)
R1 recognises that €15.00 is greater than €9.60, so the second rule applies	✓ 1
M1 9.60 + 0.11(d − 40) = 15	✓ 1
A1 d = 89.1 km	✓ 1

Common error: Choosing the wrong piece of a piecewise function. Compare the given output with the value at the break point first.

+ Add to test✓ In your test53IBMI3S-2-01r12-03Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

The cross-section of a parabolic telescope mirror is modelled by y = 0.05x², where x metres is the horizontal distance from the central axis and y metres is the height of the mirror surface above its centre point. The mirror has diameter 1.2 m, so −0.6 ≤ x ≤ 0.6.

(a)

Calculate the height of the edge of the mirror above its centre. Give your answer in centimetres.

(2)

(b)

Write down the equation of the axis of symmetry of the graph.

(1)

(c)

Find the positive value of x at which the height above the centre is 0.010 m. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes x = 0.6	✓ 1
A1 height = 0.018 m = 1.8 cm	✓ 1
Part (b)
A1 x = 0	✓ 1
Part (c)
M1 0.010 = 0.05x²	✓ 1
A1 x = 0.447 m	✓ 1

Exam tip: Keep units consistent: here the model uses metres, so convert to centimetres only at the end.

+ Add to test✓ In your test54IBMI3S-2-01r12-09Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

Astronomers model the recession speed v km s⁻¹ of galaxies at distance d megaparsecs (Mpc) by a linear function v = md + c. One galaxy is at d = 40 with v = 3100, and another is at d = 120 with v = 8700.

(a)

Find the value of m.

(2)

(b)

Find the equation of the model.

(2)

(c)

Use the model to estimate the distance of a galaxy with recession speed 5000 km s⁻¹. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (8700 − 3100)/(120 − 40)	✓ 1
A1 m = 70	✓ 1
Part (b)
M1 3100 = 70 × 40 + c, or equivalent	✓ 1
A1 v = 70d + 300	✓ 1
Part (c)
M1 5000 = 70d + 300	✓ 1
A1 d = 67.1 Mpc	✓ 1

Exam tip: Substitute a data point into your equation to confirm it reproduces the other point before using the model.

+ Add to test✓ In your test55IBMI3S-2-02r17-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy4 marks

A library charges an annual membership fee plus a charge for each DVD borrowed. The total annual cost, £C, for borrowing n DVDs is modelled by C = 18 + 0.40n.

(a)

State what the value 0.40 represents in this model.

(1)

(b)

Calculate the annual cost for a member who borrows 35 DVDs.

(1)

(c)

A member has a budget of £46 for the year. Find the number of DVDs this member can borrow.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the charge (£0.40) for each DVD borrowed	✓ 1
Part (b)
A1 C = 18 + 0.40 × 35 = £32	✓ 1
Part (c)
M1 46 = 18 + 0.40n	✓ 1
A1 n = 70	✓ 1

Exam tip: In a linear model, the gradient is the change in the output for each one-unit increase in the input, and the intercept is the value when the input is zero.

+ Add to test✓ In your test56IBMI3S-2-02r17-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy6 marks

The power output, P MW, of a wind turbine for wind speed v m s⁻¹ is modelled by
P = 0 for 0 ≤ v < 3
P = 0.3(v − 3) for 3 ≤ v < 11
P = 2.4 for 11 ≤ v ≤ 25

(a)

Calculate the power output when the wind speed is 8.0 m s⁻¹.

(2)

(b)

Find the wind speed for which the power output is 1.95 MW.

(2)

(c)

A wind farm has 20 identical turbines, each following this model. All turbines experience a wind speed of 15 m s⁻¹. Find the total power output of the farm.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 P = 0.3(8 − 3) (correct piece chosen)	✓ 1
A1 1.5 MW	✓ 1
Part (b)
M1 0.3(v − 3) = 1.95 (a value less than 2.4 so the middle piece applies)	✓ 1
A1 v = 9.5 m s⁻¹	✓ 1
Part (c)
M1 15 lies in the interval 11 ≤ v ≤ 25, so each turbine produces 2.4 MW	✓ 1
A1 20 × 2.4 = 48 MW	✓ 1

Common error: Before substituting into a piecewise function, check which interval the input belongs to.

+ Add to test✓ In your test57IBMI3S-2-02r17-03Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy5 marks

A diver jumps from a board that is 3.0 m above the water. The height, h metres, of the diver's centre of mass above the water t seconds after leaving the board is modelled by h = −4.9t² + 4.2t + 3.0.

(a)

State the height of the diver's centre of mass at the instant of leaving the board.

(1)

(b)

Find the maximum height reached by the centre of mass.

(2)

(c)

Find the time at which the centre of mass reaches the water surface (h = 0). Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 3.0 m	✓ 1
Part (b)
M1 use of the vertex: t = 4.2 ÷ (2 × 4.9) = 0.429… (or use of technology)	✓ 1
A1 3.9 m	✓ 1
Part (c)
M1 −4.9t² + 4.2t + 3.0 = 0 (solved with technology or the quadratic formula)	✓ 1
A1 t = 1.32 s (the negative root is rejected)	✓ 1

Common error: A quadratic equation can have two roots; reject the one that falls outside the context, such as a negative time.

+ Add to test✓ In your test58IBMI3S-2-02r17-04Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy6 marks

The number of swimmers, S, in a public pool t hours after it opens at 09:00 is modelled by S(t) = −3t² + 36t, for 0 ≤ t ≤ 12.

(a)

Find the maximum number of swimmers in the pool and the time, in hours after opening, at which it occurs.

(2)

(b)

On a busy day the pool opens 2 hours later and there are always 10 more swimmers than the original model predicts. The new model is S₂(t) = S(t − 2) + 10. Describe fully the transformation that maps the graph of S onto the graph of S₂.

(2)

(c)

Write down the maximum value of S₂ and the value of t at which it occurs.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 t = 6 (axis of symmetry t = −36 ÷ (2 × (−3)))	✓ 1
A1 maximum S = 108 swimmers	✓ 1
Part (b)
A1 a translation of 2 units in the positive t direction (to the right)	✓ 1
A1 together with a translation of 10 units in the positive S direction (up), that is, a translation by the vector (2, 10)	✓ 1
Part (c)
A1 t = 8	✓ 1
A1 maximum = 118 swimmers	✓ 1

Exam tip: For y = f(x − a) + b, the graph moves a units right and b units up; every point, including the maximum, moves by the same amounts.

+ Add to test✓ In your test59IBMI3S-2-01r20-01Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy6 marks

A company installs rooftop solar panels. The total cost, C euros, of installing n panels is modelled by a linear function. An installation of 6 panels costs €4150 and an installation of 10 panels costs €6150.

(a)

Find the equation of the model in the form C = a + bn.

(3)

(b)

Interpret the values of a and b in this context.

(1)

(c)

A customer has a budget of €9650. Find the greatest number of panels that can be installed within this budget.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to find gradient, e.g. (6150 − 4150)/(10 − 6)	✓ 1
A1 b = 500	✓ 1
A1 C = 1150 + 500n (from substituting a point)	✓ 1
Part (b)
A1 a = €1150 is the fixed cost (cost with no panels); b = €500 is the extra cost of each additional panel	✓ 1
Part (c)
M1 1150 + 500n ≤ 9650 (or equation equal to 9650)	✓ 1
A1 n = 17 panels	✓ 1

Exam tip: When asked to interpret a linear model, always refer to the units and the context, not just the number.

+ Add to test✓ In your test60IBMI3S-2-01r20-02Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy6 marks

In the early stage of an outbreak, the number of reported cases of a virus in a town, N, is modelled by N = 60 × 1.25^d, where d is the number of days after the first report was recorded.

(a)

Find the number of cases predicted after 7 days. Give your answer to the nearest whole number.

(2)

(b)

Write down the percentage increase in the number of cases each day predicted by the model.

(1)

(c)

Find the first whole number of days after which the model predicts more than 1000 cases.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitution of d = 7, i.e. 60 × 1.25⁷	✓ 1
A1 286 cases	✓ 1
Part (b)
A1 25%	✓ 1
Part (c)
M1 60 × 1.25^d = 1000 (or inequality, or table/graph attempt)	✓ 1
A1 d = 12.6 (12.61...) or evidence that N(12) = 873 and N(13) = 1091	✓ 1
A1 13 days	✓ 1

Exam tip: When a question asks for the first whole number of days, check the values either side of your solution.

+ Add to test✓ In your test61IBMI3S-2-01r20-03Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy7 marks

An electricity supplier charges a standing charge of €8 per month plus €0.21 per kWh for the first 250 kWh used in a month, and €0.16 per kWh for each kWh used above 250 kWh.

(a)

Calculate the bill for a month in which 180 kWh are used.

(2)

(b)

Calculate the bill for a month in which 400 kWh are used.

(2)

(c)

In one month a household's bill is exactly €100. Find the number of kWh used, giving your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 8 + 0.21 × 180	✓ 1
A1 €45.80	✓ 1
Part (b)
M1 8 + 0.21 × 250 + 0.16 × (400 − 250)	✓ 1
A1 €84.50	✓ 1
Part (c)
M1 8 + 0.21 × 250 + 0.16(x − 250) = 100	✓ 1
A1 x = 496.875	✓ 1
A1 497 kWh	✓ 1

Exam tip: For piecewise costs, first decide which section of the model applies by comparing with the cost at the boundary.

+ Add to test✓ In your test62IBMI3S-2-01r20-04Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy6 marks

A market stall sells reusable water bottles. The daily profit, P euros, when the selling price of a bottle is €p, is modelled by P = −2p² + 60p − 300, for 0 ≤ p ≤ 30.

(a)

Find the selling price that gives the maximum daily profit, and state this maximum profit.

(3)

(b)

Find the range of selling prices for which the stall makes a profit. Give the end points to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of axis of symmetry p = −60/(2 × −2), or GDC maximum	✓ 1
A1 p = €15	✓ 1
A1 maximum profit €150	✓ 1
Part (b)
M1 solve −2p² + 60p − 300 = 0 (or sketch with GDC)	✓ 1
A1 p = 6.34 and p = 23.7	✓ 1
A1 6.34 < p < 23.7	✓ 1

Exam tip: A profit is made only where the quadratic is above zero, so the answer to a range question is an inequality, not two separate values.

+ Add to test✓ In your test63IBMI3S-2-01r25-01Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy6 marks

A mountaineering team records the air temperature, T °C, at altitude h metres. At base camp, at an altitude of 4200 m, the temperature is 5.0 °C. Above base camp the temperature falls at a constant rate of 6.4 °C for every 1000 m climbed.

(a)

Write down the gradient of the linear model for T in terms of h, and hence find an expression for T in terms of h.

(2)

(b)

Find the temperature at the high camp, which is at an altitude of 5600 m.

(2)

(c)

The team will turn back if the temperature falls to −9.4 °C. Find the altitude at which this happens.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 gradient = −0.0064	✓ 1
A1 T = 31.88 − 0.0064h (or T = 5.0 − 0.0064(h − 4200))	✓ 1
Part (b)
M1 substitution of h = 5600 into the model	✓ 1
A1 T = −3.96 °C	✓ 1
Part (c)
M1 −9.4 = 31.88 − 0.0064h (or 14.4 ÷ 0.0064)	✓ 1
A1 h = 6450 m	✓ 1

Exam tip: State the gradient with its sign: a temperature that falls as altitude rises has a negative gradient.

+ Add to test✓ In your test64IBMI3S-2-01r25-02Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy5 marks

The vitamin C content, V mg per 100 mL, of a fruit juice stored in a clear bottle is modelled by V = 90 × 0.95^t, where t is the number of days after the bottle was opened.

(a)

Find the vitamin C content after 12 days. Give your answer correct to 3 significant figures.

(2)

(b)

The juice is no longer labelled as a source of vitamin C once the content falls below 30 mg per 100 mL. Find the number of complete days after opening that the juice can be labelled as a source of vitamin C.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitution of t = 12: 90 × 0.95¹²	✓ 1
A1 V = 48.6 mg per 100 mL	✓ 1
Part (b)
M1 90 × 0.95^t = 30 (or an equivalent inequality)	✓ 1
A1 t = 21.4 (to 3 s.f.)	✓ 1
A1 21 complete days	✓ 1

Common error: In part (b) rounding 21.4 up to 22 gives a day on which the content is already below 30 mg per 100 mL; the question asks for complete days of valid labelling.

+ Add to test✓ In your test65IBMI3S-2-01r25-09Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy5 marks

The temperature, T °C, inside a baking oven t minutes after the thermostat is set is modelled by T = 180 + 12 sin(36t)°, where the angle is measured in degrees.

(a)

Write down the maximum and minimum temperatures predicted by the model.

(2)

(b)

Find the time taken for the temperature to complete one full cycle.

(1)

(c)

Find the temperature at t = 7. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 192 °C	✓ 1
A1 minimum 168 °C	✓ 1
Part (b)
A1 360 ÷ 36 = 10 minutes	✓ 1
Part (c)
M1 180 + 12 sin(252°)	✓ 1
A1 T = 169 °C	✓ 1

Common error: Set the calculator to degree mode when the angle is given in degrees; radian mode would give a quite different value.

+ Add to test✓ In your test66IBMI3S-2-01r25-13Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Easy5 marks

The power, P milliwatts (mW), of a radio signal received at a distance d km from a transmitter is modelled by P = A × b^d. The power is 12.8 mW at 2 km and 3.2 mW at 6 km.

(a)

Find the value of b and the value of A. Give b correct to 3 significant figures and A as an exact value.

(3)

(b)

The receiver cannot decode the signal when the power is below 0.1 mW. Find the greatest distance at which the signal can be decoded. Use unrounded values from part (a).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 b⁴ = 3.2 ÷ 12.8 = 0.25 (dividing the two equations)	✓ 1
A1 b = 0.707 (= 2^−1/2)	✓ 1
A1 A = 12.8 ÷ b² = 25.6	✓ 1
Part (b)
M1 25.6 × (2^−1/2)^d = 0.1 (or 0.707…^d = 1/256)	✓ 1
A1 d = 16 km	✓ 1

Common error: Use the unrounded value of b in part (b); the rounded 0.707 gives a distance that differs slightly from 16.

+ Add to test✓ In your test67IBMI3S-2-01r28-01Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy5 marks

An archaeologist estimates the height H cm of an adult from the length F cm of a complete femur found at a burial site, using the linear model H = 2.4F + 61.

(a)

Estimate the height of a person whose femur has length 45 cm.

(2)

(b)

State what the value 2.4 represents in this context.

(1)

(c)

A different skeleton is estimated to have been 170 cm tall. Find the femur length, in cm, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute F = 45 into the model	✓ 1
A1 H = 169 cm	✓ 1
Part (b)
A1 each extra 1 cm of femur length corresponds to an increase of 2.4 cm in estimated height	✓ 1
Part (c)
M1 set 2.4F + 61 = 170 and rearrange	✓ 1
A1 F = 109/2.4 = 45.4 cm	✓ 1

Exam tip: In a context question, a gradient needs a unit-based sentence: say how much one quantity changes for a 1-unit change in the other.

+ Add to test✓ In your test68IBMI3S-2-01r28-06Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy6 marks

A server purchased by a data centre for €24 000 loses value according to the model V = 24 000 × 0.85^t, where V is its value in euros after t years.

(a)

Find the value of the server after 3 years, to the nearest euro.

(2)

(b)

State the percentage by which the value falls each year.

(1)

(c)

Find the time, in years to 3 significant figures, for the value to fall to half of its original value.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 3	✓ 1
A1 € 14 739	✓ 1
Part (b)
A1 15%	✓ 1
Part (c)
M1 set 0.85^t = 0.5 (or 24 000 × 0.85^t = 12 000)	✓ 1
M1 solve using logarithms or GDC	✓ 1
A1 t = 4.27 years	✓ 1

Exam tip: In an exponential model a × bt, a multiplier of 0.85 means a 15% fall per period; subtract the multiplier from 1.

+ Add to test✓ In your test69IBMI3S-2-01r28-11Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy6 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)

A server hall has sound intensity 4.0 × 10⁻⁵ W m⁻². Find its sound intensity level to 3 significant figures.

(2)

(b)

A maintenance corridor has sound intensity level 82 dB. Find its sound intensity, to 3 significant figures.

(2)

(c)

Show that doubling the sound intensity increases the sound intensity level by approximately 3 dB.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 10 log₁₀(4.0 × 10⁻⁵ / 1.0 × 10⁻¹²)	✓ 1
A1 76.0 dB	✓ 1
Part (b)
M1 82 = 10 log₁₀(I / I₀), so I = I₀ × 10^8.2	✓ 1
A1 I = 1.58 × 10⁻⁴ W m⁻²	✓ 1
Part (c)
M1 change in level = 10 log₁₀(2I / I₀) − 10 log₁₀(I / I₀) = 10 log₁₀ 2	✓ 1
A1 10 log₁₀ 2 = 3.01… ≈ 3 dB	✓ 1

Exam tip: On a logarithmic scale, multiplying the quantity by a fixed factor adds a fixed amount to the scale reading.

+ Add to test✓ In your test70IBMI3S-2-03r31-01Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Easy4 marks

A warehouse robot has a battery charge, P %, that is modelled by P = 92 − 6.5t, where t is the time in hours since the robot started its shift.

(a)

Interpret the values 92 and −6.5 in this model.

(2)

(b)

The robot must return to its dock when its charge reaches 15 %. Find the value of t at which this happens. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 92 is the initial charge (at the start of the shift) as a percentage	✓ 1
A1 −6.5: the charge falls by 6.5 percentage points each hour	✓ 1
Part (b)
M1 setting 92 − 6.5t = 15	✓ 1
A1 t = 11.8 (hours)	✓ 1

Exam tip: When asked to interpret a linear model, mention both the quantity and its units, and say whether it rises or falls.

+ Add to test✓ In your test71IBMI3S-2-03r31-02Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Easy5 marks

A fashion shop sells silk scarves. When the selling price of a scarf is €x, the weekly profit, €P, is modelled by P(x) = −2x² + 56x − 150, for 0 < x < 40.

(a)

Find the selling price that gives the greatest weekly profit.

(2)

(b)

Write down the greatest weekly profit.

(1)

(c)

Find the lowest selling price at which the shop makes no loss.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of axis of symmetry x = −56/(2 × −2) (or graphing the function)	✓ 1
A1 €14	✓ 1
Part (b)
A1 €242 (ECF from (a))	✓ 1
Part (c)
M1 setting P(x) = 0 (or equivalent, e.g. x² − 28x + 75 = 0)	✓ 1
A1 €3	✓ 1

Common error: The greatest profit is the value of P, not the value of x at which it occurs; label each answer with the correct quantity.

+ Add to test✓ In your test72IBMI3S-2-03r31-08Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Easy5 marks

A welding robot moves along the curve y = f(x), where f(x) = x² − 4x, and x and y are in metres. The vertex of the graph of f is at (2, −4). After reprogramming, the robot follows the curve y = g(x), where g(x) = f(x − 2) + 3.

(a)

Describe the single transformation that maps the graph of f onto the graph of g.

(2)

(b)

Write down the coordinates of the vertex of the graph of g.

(1)

(c)

Find the y-intercept of the graph of g.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a translation	✓ 1
A1 by the vector (2, 3), i.e. 2 right and 3 up	✓ 1
Part (b)
A1 (4, −1)	✓ 1
Part (c)
M1 g(0) = f(−2) + 3 or (0 − 2)² − 4(0 − 2) + 3	✓ 1
A1 15 m	✓ 1

Common error: In f(x − 2) the horizontal shift is to the right by 2, the opposite of what the sign inside the brackets suggests.

+ Add to test✓ In your test73IBMI3S-2-03r31-09Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Easy4 marks

A fashion brand models the weekly demand and supply for a hoodie, where q is the number of hoodies and €p is the price of one hoodie.
Demand: p = 95 − 0.4q
Supply: p = 20 + 0.35q

(a)

Interpret the value 0.35 in the supply model.

(1)

(b)

Find the number of hoodies at which demand equals supply.

(2)

(c)

Find the price at which demand equals supply.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the supplier needs a price increase of €0.35 for each additional hoodie supplied	✓ 1
Part (b)
M1 setting 95 − 0.4q = 20 + 0.35q	✓ 1
A1 q = 100	✓ 1
Part (c)
A1 €55 (ECF from (b))	✓ 1

Exam tip: At the point where two linear models meet, check your answer by substituting into both equations; they must give the same value.

+ Add to test✓ In your test74IBMI3S-2-02r32-01Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy5 marks

A deep-space probe is powered by a radioisotope generator. Its electrical output, P watts, t years after launch is modelled by
P(t) = 480 × 0.985^t.

(a)

Write down the output of the generator at launch.

(1)

(b)

Find the output 12 years after launch. Give your answer to 3 significant figures.

(2)

(c)

The probe's instruments need at least 300 W to operate. Find the number of years after launch at which the output falls to 300 W. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 480 W	✓ 1
Part (b)
M1 substitute t = 12: 480 × 0.985¹²	✓ 1
A1 400 W (400.38…)	✓ 1
Part (c)
M1 set 480 × 0.985^t = 300 and solve (logarithms or technology)	✓ 1
A1 31.1 years	✓ 1

Exam tip: In a model of the form a × bt, a is the starting value and b is the multiplier per time unit; 0.985 means a 1.5% decrease each year.

+ Add to test✓ In your test75IBMI3S-2-02r32-04Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy5 marks

Pedestrians crossing a footbridge cause the middle of the deck to oscillate vertically. The displacement from its rest position, d millimetres, at time t seconds is modelled by
d(t) = 18 sin(bt), where b > 0 and the angle is in radians.
The deck takes 0.8 seconds to complete one oscillation.

(a)

Write down the amplitude of the oscillation.

(1)

(b)

Find the value of b, giving your answer to 3 significant figures.

(2)

(c)

Find the displacement of the deck when t = 0.3. Use the unrounded value of b. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 18 mm	✓ 1
Part (b)
M1 period = 2π/b = 0.8	✓ 1
A1 b = 7.85	✓ 1
Part (c)
M1 substitute t = 0.3 into 18 sin(bt) (calculator in radian mode)	✓ 1
A1 12.7 mm	✓ 1

Exam tip: For y = a sin(bt) the period is 2π/b; make sure the calculator is in radian mode when b is found this way.

+ Add to test✓ In your test76IBMI3S-2-02r32-08Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy5 marks

A small business roasts coffee for other cafés. The cost of roasting m kg of coffee is modelled by
C = 12 + 3.4m,
where C is in euros (€).

(a)

State what the value 3.4 represents in this model.

(1)

(b)

Find the cost of roasting 18 kg of coffee.

(2)

(c)

A café has a budget of €100. Find the mass of coffee that can be roasted for exactly €100. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the cost for each extra kilogram of coffee roasted is €3.40	✓ 1
Part (b)
M1 substitute m = 18: 12 + 3.4 × 18	✓ 1
A1 €73.20	✓ 1
Part (c)
M1 set 12 + 3.4m = 100	✓ 1
A1 25.9 kg	✓ 1

Exam tip: In a linear model y = c + mx the gradient is the change in cost per unit and the constant is the cost when the quantity is zero.

+ Add to test✓ In your test77IBMI3S-2-02r32-11Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy7 marks

The curved arch of a footbridge is modelled by
y = −0.004x(x − 120), for 0 ≤ x ≤ 120,
where x is the horizontal distance in metres from one end of the arch and y is the height of the arch above the water in metres.

(a)

Write down the width of the arch at water level.

(2)

(b)

Find the maximum height of the arch above the water.

(2)

(c)

A boat has a height of 4.5 m above the water. Find the greatest width of the boat that can pass under the arch, assuming it is at the middle of the arch. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 set y = 0 (or identify the roots x = 0 and x = 120)	✓ 1
A1 120 m	✓ 1
Part (b)
M1 use symmetry: maximum at x = 60 and substitute	✓ 1
A1 14.4 m	✓ 1
Part (c)
M1 set −0.004x(x − 120) = 4.5	✓ 1
A1 x = 10.25… and x = 109.75…	✓ 1
A1 width = 109.75… − 10.25… = 99.5 m	✓ 1

Exam tip: For a quadratic written in factorised form, the roots are read straight from the brackets and the maximum lies halfway between them.

+ Add to test✓ In your test78IBMI3S-2-04r32-01Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

The volume V of water, in megalitres (ML), in a small reservoir t days after the start of a dry spell is modelled by $V = 5400 - 45t$.

(a)

Write down the meaning of the value −45 in this context.

(1)

(b)

Find the volume of water in the reservoir after 30 days.

(2)

(c)

Find the number of days after which the reservoir is empty according to the model.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the volume of water falls by 45 ML each day	✓ 1
Part (b)
M1 $5400 - 45\times30$	✓ 1
A1 4050 ML	✓ 1
Part (c)
M1 $5400 - 45t = 0$	✓ 1
A1 120 days	✓ 1

Exam tip: Interpret a gradient with its units: change in the vertical quantity per one unit of the horizontal quantity.

+ Add to test✓ In your test79IBMI3S-2-04r32-02Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

A gaming server charges a monthly fee, C euros, that depends on the number of hours h played in the month:
$C = 6 + 0.8h$ for $0 \le h \le 20$
$C = 12 + 0.5h$ for $h > 20$

(a)

Find the fee for a month in which a player uses the server for 14 hours.

(2)

(b)

Find the fee for a month in which a player uses the server for 50 hours.

(2)

(c)

In one month a player is charged €30. Find the number of hours played in that month.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $6 + 0.8\times14$	✓ 1
A1 €17.20	✓ 1
Part (b)
M1 $12 + 0.5\times50$	✓ 1
A1 €37.00	✓ 1
Part (c)
M1 recognise that €30 is above the fee at 20 hours (€22), so use $12 + 0.5h = 30$	✓ 1
A1 36 hours	✓ 1

Common error: Choose the correct branch of a piecewise function before substituting: check which interval the input lies in first.

+ Add to test✓ In your test80IBMI3S-2-04r32-03Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

A decorative fountain sends a jet of water into the air. The height h metres of the jet above the water surface at a horizontal distance x metres from the nozzle is modelled by $h = -0.5x^{2} + 2x + 0.75$, for $h \ge 0$.

(a)

Write down the height of the nozzle above the water surface.

(1)

(b)

Find the maximum height reached by the jet.

(2)

(c)

Find the horizontal distance from the nozzle at which the jet hits the water surface. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.75 m	✓ 1
Part (b)
M1 use of $x = -\dfrac{b}{2a} = 2$ or GDC maximum	✓ 1
A1 2.75 m	✓ 1
Part (c)
M1 $-0.5x^{2}+2x+0.75 = 0$ (or GDC zero)	✓ 1
A1 $x = 4.3452$, so 4.35 m	✓ 1

Common error: The second root of the quadratic is negative here (x ≈ −0.345) and must be rejected as it lies behind the nozzle.

+ Add to test✓ In your test81IBMI3S-2-04r32-04Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

After an injection, 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 since the injection.

(a)

Find the concentration of the drug after 3 hours. Give your answer to 3 significant figures.

(2)

(b)

Find the time taken for the concentration 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 $80\times0.85^{3}$	✓ 1
A1 49.1 mg per litre	✓ 1
Part (b)
M1 $80\times0.85^{t} = 40$ or $0.85^{t} = 0.5$	✓ 1
M1 take logarithms or use a GDC: $t = \dfrac{\ln0.5}{\ln0.85}$	✓ 1
A1 4.27 hours	✓ 1

Exam tip: The time for a quantity to halve does not depend on the starting amount, so you can solve $0.85^{t} = 0.5$ directly.

+ Add to test✓ In your test82IBMI3S-2-03r34-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Easy5 marks

A warehouse robot drives along a straight rail away from its charging point. Its distance from the charging point is modelled by d = 2.4t + 8.5, where d is in metres and t is the time in seconds after the robot starts moving, t ≥ 0.

(a)

Write down the meaning of the value 8.5 and the meaning of the value 2.4 in this context.

(2)

(b)

Find the time taken for the robot to be 50 m from the charging point. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 8.5 m is the distance from the charging point at the start (when t = 0)	✓ 1
A1 2.4 m s⁻¹ is the (constant) speed of the robot / rate at which the distance increases	✓ 1
Part (b)
M1 setting 2.4t + 8.5 = 50	✓ 1
M1 correct rearrangement, t = (50 − 8.5) ÷ 2.4	✓ 1
A1 t = 17.3 s	✓ 1

Common error: Substituting 50 for t instead of for d; always check which variable the given value belongs to.

+ Add to test✓ In your test83IBMI3S-2-03r34-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Easy6 marks

An online fashion shop calculates the delivery cost C (in euros) of a parcel of mass w kg using
C(w) = 3.80 for 0 < w ≤ 1
C(w) = 3.80 + 1.45(w − 1) for 1 < w ≤ 5
C(w) = 9.60 + 0.90(w − 5) for w > 5

(a)

Find the cost of delivering a parcel of mass 3.2 kg.

(2)

(b)

Find the cost of delivering a parcel of mass 12 kg.

(2)

(c)

A parcel costs €13.20 to deliver. Find its mass.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 using the second piece, 3.80 + 1.45 × 2.2	✓ 1
A1 €6.99	✓ 1
Part (b)
M1 using the third piece, 9.60 + 0.90 × 7	✓ 1
A1 €15.90	✓ 1
Part (c)
M1 setting 9.60 + 0.90(w − 5) = 13.20 (third piece, since 13.20 > 9.60)	✓ 1
A1 w = 9 kg	✓ 1

Exam tip: Before using a piece of a piecewise function, check which interval your input (or output) lies in; the cost at w = 5 tells you which piece applies for a given cost.

+ Add to test✓ In your test84IBMI3S-2-03r34-03Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Easy6 marks

The number of fish, P, in a protected lagoon is modelled by P = 2400 × 0.92^t, where t is the number of years since monitoring began.

(a)

State the initial population and the percentage by which the population decreases each year.

(2)

(b)

Find the population after 6 years, giving your answer to the nearest whole number.

(2)

(c)

Find how many years it takes for the population to fall to 900. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 2400 fish	✓ 1
A1 8 % per year	✓ 1
Part (b)
M1 2400 × 0.92⁶	✓ 1
A1 1455	✓ 1
Part (c)
M1 0.92^t = 900/2400 (or equivalent graphical / logarithmic set-up)	✓ 1
A1 t = 11.8 years	✓ 1

Common error: A decrease of 8% each year means multiplying by 0.92, not by 0.08.

+ Add to test✓ In your test85IBMI3S-2-03r34-13Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Easy4 marks

The area, f(t), in cm², covered by algae on a net after t days is modelled by f(t) = 20 × 1.5^t. A second net is modelled by g(t) = f(t + 2).

(a)

Show that g(t) = 45 × 1.5^t.

(2)

(b)

Find the time for the area on the second net to reach 500 cm². Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 g(t) = 20 × 1.5^{t + 2} = 20 × 1.5² × 1.5^t	✓ 1
A1 20 × 2.25 = 45, so g(t) = 45 × 1.5^t	✓ 1
Part (b)
M1 setting 45 × 1.5^t = 500	✓ 1
A1 t = 5.94 days	✓ 1

Exam tip: A horizontal translation of an exponential graph is equivalent to a vertical stretch, because 1.5t ⁺ ² = 1.5² × 1.5t.

+ Add to test✓ In your test86IBMI3S-2-01r35-01Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 1Easy6 marks

Calculator

A beekeeper weighs the honey stored in one hive. The mass of honey, H kg, is modelled by H = 4.5 + 0.7t, where t is the number of days since the first weighing.

(a)

Interpret the value 0.7 in this context.

(1)

(b)

Find the mass of honey after 12 days.

(2)

(c)

Find how many days it takes for the mass to reach 20 kg. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the mass of honey increases by 0.7 kg each day	✓ 1
Part (b)
M1 substitutes t = 12: 4.5 + 0.7 × 12	✓ 1
A1 H = 12.9 kg	✓ 1
Part (c)
M1 sets 4.5 + 0.7t = 20	✓ 1
A1 t = 15.5 ÷ 0.7 = 22.142…	✓ 1
A1 t = 22.1 days	✓ 1

Exam tip: When a part says 'to 3 significant figures', write the unrounded value first and then the rounded one.

+ Add to test✓ In your test87IBMI3S-2-01r35-02Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 1Easy7 marks

Calculator

The number of patients, P, waiting in a hospital emergency department t hours after 08:00 is modelled by P = −0.5t² + 6t + 4, for 0 ≤ t ≤ 12.

(a)

Write down the number of patients waiting at 08:00.

(1)

(b)

Find the value of t at which the number of patients waiting is greatest.

(2)

(c)

Find the greatest number of patients waiting.

(1)

(d)

Find the length of time, in hours, for which at least 20 patients are waiting.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4	✓ 1
Part (b)
M1 uses t = −6 ÷ (2 × −0.5), or the maximum on the GDC	✓ 1
A1 t = 6	✓ 1
Part (c)
A1 P(6) = −18 + 36 + 4 = 22 (ECF from (b))	✓ 1
Part (d)
M1 solves −0.5t² + 6t + 4 = 20 (or equivalent graph intersection)	✓ 1
A1 t = 4 and t = 8	✓ 1
A1 8 − 4 = 4 hours	✓ 1

Common error: The two solutions of the quadratic are clock times; the question asks for the length of the interval between them.

+ Add to test✓ In your test88IBMI3S-2-01r35-06Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 1Easy5 marks

Calculator

The speed, v m s⁻¹, of a Formula 1 car t seconds after leaving a pit stop is modelled by
v = 12t for 0 ≤ t ≤ 4
v = 48 + 3(t − 4) for 4 < t ≤ 10.

(a)

Find the speed when t = 2.5.

(1)

(b)

Find the speed when t = 9.

(2)

(c)

Find the time at which the speed first reaches 60 m s⁻¹.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 30 m s⁻¹	✓ 1
Part (b)
M1 uses the second rule: 48 + 3 × 5	✓ 1
A1 63 m s⁻¹	✓ 1
Part (c)
M1 solves 48 + 3(t − 4) = 60 (using the second rule, since 12t ≤ 48 for the first interval)	✓ 1
A1 t = 8 s	✓ 1

Exam tip: Choose the rule that matches the interval your value lies in before substituting.

+ Add to test✓ In your test89IBMI3S-2-01r35-10Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 1Easy5 marks

Calculator

After a ward surface is left uncleaned, the number of bacteria on a sample, N, is modelled by N = 250 × 3^t/2, where t is the time in hours.

(a)

Write down the number of bacteria at the start.

(1)

(b)

Find the number of bacteria after 6 hours.

(2)

(c)

Find the time at which there are 60 750 bacteria.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 250	✓ 1
Part (b)
M1 substitutes t = 6: 250 × 3³	✓ 1
A1 6750	✓ 1
Part (c)
M1 forms 3^t/2 = 243 (= 3⁵), or uses logarithms or the GDC	✓ 1
A1 t = 10 hours	✓ 1

Exam tip: Check whether a number such as 243 is a power of the base; it can avoid logarithms completely.

+ Add to test✓ In your test90IBMI3S-2-02r35-01Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy5 marks

Calculator

The power output, P megawatts (MW), of a wind turbine is modelled by P = 0.3v − 1.5, where v m s⁻¹ is the wind speed, for 5 ≤ v ≤ 12.

(a)

Find the power output when the wind speed is 9 m s⁻¹.

(2)

(b)

Interpret the value 0.3 in this context.

(1)

(c)

Find the wind speed needed for an output of 1.8 MW.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes v = 9: 0.3 × 9 − 1.5	✓ 1
A1 1.2 MW	✓ 1
Part (b)
A1 the power output increases by 0.3 MW for each 1 m s⁻¹ increase in wind speed	✓ 1
Part (c)
M1 solves 0.3v − 1.5 = 1.8	✓ 1
A1 v = 11 m s⁻¹	✓ 1

Exam tip: Give units in an interpretation: the gradient is a rate, so state 'per' what.

+ Add to test✓ In your test91IBMI3S-2-02r35-03Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy5 marks

Calculator

A library charges a fine, F euros, for a book returned d days after its due date, as follows.
F = 0.25d for 0 ≤ d ≤ 14
F = 3.50 + 0.50(d − 14) for d > 14

(a)

Find the fine for a book returned 10 days late.

(1)

(b)

Find the fine for a book returned 20 days late.

(2)

(c)

A fine of €8.00 is paid. Find how many days late the book was returned.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €2.50	✓ 1
Part (b)
M1 uses the second rule: 3.50 + 0.50 × 6	✓ 1
A1 €6.50	✓ 1
Part (c)
M1 solves 3.50 + 0.50(d − 14) = 8 (second rule, since €8 is more than €3.50)	✓ 1
A1 d = 23	✓ 1

Exam tip: Check which interval an answer must lie in; €8.00 is above the largest fine of the first rule, so use the second rule.

+ Add to test✓ In your test92IBMI3S-2-02r35-05Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy6 marks

Calculator

After a swimming pool is treated, the concentration of chlorine, C parts per million (ppm), is modelled by C = 3.2 × 0.88^t, where t is the time in hours since treatment.

(a)

Find the concentration after 5 hours. Give your answer to 3 significant figures.

(2)

(b)

State the percentage decrease in concentration each hour.

(1)

(c)

Find the time at which the concentration falls to 1.0 ppm. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 5: 3.2 × 0.88⁵	✓ 1
A1 1.69 ppm (1.6887…)	✓ 1
Part (b)
A1 12%	✓ 1
Part (c)
M1 forms 3.2 × 0.88^t = 1, so 0.88^t = 0.3125	✓ 1
M1 uses logarithms (or GDC) to solve	✓ 1
A1 t = 9.10 hours (9.0990…)	✓ 1

Common error: A multiplier of 0.88 means a 12% decrease, not an 88% decrease.

+ Add to test✓ In your test93IBMI3S-2-02r35-09Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Easy7 marks

Calculator

A library hires out a study room. When the hourly price is €x, the daily revenue is R = 40x − 2x² euros, for 0 ≤ x ≤ 20.

(a)

Find the revenue when the hourly price is €6.

(1)

(b)

Find the hourly price that gives the greatest revenue, and state that revenue.

(3)

(c)

The library needs a revenue of at least €150. Find the greatest hourly price that satisfies this.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €168	✓ 1
Part (b)
M1 uses x = −40 ÷ (2 × −2) (or GDC maximum)	✓ 1
A1 x = 10, so price €10	✓ 1
A1 greatest revenue €200	✓ 1
Part (c)
M1 forms 40x − 2x² = 150, i.e. x² − 20x + 75 = 0	✓ 1
A1 x = 5 or x = 15	✓ 1
A1 €15	✓ 1

Exam tip: For 'at least' on a downward parabola, the allowed values lie between the two solutions.

+ Add to test✓ In your test94IBMI3S-2-04r35-01Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy4 marks

A beekeeper models the total mass of honey, H kg, harvested in a season from n hives by H = 12.5n − 5.

(a)

Interpret the value 12.5 in this model.

(1)

(b)

Find the mass of honey the model predicts from 11 hives.

(1)

(c)

Find the number of hives needed for the model to predict a harvest of 195 kg.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
R1 each additional hive increases the mass of honey harvested by 12.5 kg (accept: average yield per hive)	✓ 1
Part (b)
A1 132.5 kg	✓ 1
Part (c)
M1 12.5n − 5 = 195 (or n = (195 + 5)/12.5)	✓ 1
A1 16 hives	✓ 1

Exam tip: To find the input for a given output, set the model equal to that output and solve; do not substitute the output as if it were the input.

+ Add to test✓ In your test95IBMI3S-2-04r35-02Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

A hospital car park charges £C for a stay of t hours, where
C = 3t for 0 < t ≤ 2
C = 6 + 1.8(t − 2) for t > 2.

(a)

Find the charge for a stay of 1.5 hours.

(1)

(b)

Find the charge for a stay of 5 hours.

(2)

(c)

A visitor pays £15. Find the length of the visitor's stay.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 £4.50	✓ 1
Part (b)
M1 6 + 1.8 × (5 − 2) (uses the second rule)	✓ 1
A1 £11.40	✓ 1
Part (c)
M1 6 + 1.8(t − 2) = 15 (uses the second rule)	✓ 1
A1 7 hours	✓ 1

Common error: Check which part of a piecewise function applies before substituting; a stay of 5 hours uses the second rule, not the first.

+ Add to test✓ In your test96IBMI3S-2-04r35-03Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

The lap time, T seconds, of a racing car on lap n of a 30-lap stint is modelled by T = 0.01n² − 0.3n + 84.4, for 1 ≤ n ≤ 30. The lap time first falls as the fuel load reduces and then rises as the tyres wear.

(a)

Find the lap time on lap 1.

(1)

(b)

Find the lap number on which the model predicts the fastest lap.

(2)

(c)

Find the fastest lap time predicted by the model.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 84.11 s	✓ 1
Part (b)
M1 uses the axis of symmetry n = −(−0.3)/(2 × 0.01) (or GDC minimum)	✓ 1
A1 lap 15	✓ 1
Part (c)
M1 substitutes n = 15 into the model	✓ 1
A1 82.15 s	✓ 1

Exam tip: A quadratic with a positive n² coefficient has a minimum at its vertex, which is where the fastest lap occurs.

+ Add to test✓ In your test97IBMI3S-2-04r35-07Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

The number of forager bees in a hive t days after a hive was set up is modelled by f(t) = 5 × 1.2^t. A second hive follows the same pattern but was set up 3 days earlier, so its number of foragers is modelled by g(t) = f(t + 3).

(a)

Describe the transformation that maps the graph of y = f(t) to the graph of y = g(t).

(1)

(b)

Show that g(t) = 8.64 × 1.2^t.

(2)

(c)

Explain why g(0) is greater than f(0).

(1)

(d)

Find g(2). Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 translation 3 units to the left (vector (−3, 0))	✓ 1
Part (b)
M1 g(t) = 5 × 1.2^t+3 = 5 × 1.2³ × 1.2^t	✓ 1
A1 5 × 1.728 = 8.64, so g(t) = 8.64 × 1.2^t (AG)	✓ 1
Part (c)
R1 the second hive was set up earlier, so it has had 3 more days of growth at day 0 (accept: g(0) = f(3))	✓ 1
Part (d)
M1 8.64 × 1.2² (or 5 × 1.2⁵)	✓ 1
A1 12.4	✓ 1

Exam tip: Replacing t by t + 3 shifts a graph 3 units to the left, which is the opposite of what the plus sign may suggest.

+ Add to test✓ In your test98IBMI3S-2-03r36-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy6 marks

A light aircraft leaves with 160 litres of fuel. While cruising, the fuel remaining is modelled by F = 160 − 28t, where F is in litres and t is the time in hours after departure.

(a)

State what the value 28 represents in this model.

(1)

(b)

Find the fuel remaining after 2.5 hours.

(2)

(c)

The pilot must land with at least 25 litres in reserve. Find the greatest time the aircraft can fly before landing. Give your answer in hours correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 fuel is used at a constant rate of 28 litres per hour	✓ 1
Part (b)
M1 substitute t = 2.5 into the model	✓ 1
A1 F = 90 litres	✓ 1
Part (c)
M1 set 160 − 28t = 25 (or an equivalent inequality)	✓ 1
M1 rearrange to t = 135 ÷ 28	✓ 1
A1 t = 4.82 hours	✓ 1

Common error: Substituting F = 0 instead of the reserve value gives the time to empty, not the time to land safely.

+ Add to test✓ In your test99IBMI3S-2-03r36-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy6 marks

A farm hires a tractor. The cost C (€) of hiring it for h hours is
• C = 80 + 45h for 0 ≤ h ≤ 6
• C = 350 + 30(h − 6) for h > 6.

(a)

Find the cost of hiring the tractor for 4 hours.

(2)

(b)

Find the cost of hiring the tractor for 10 hours.

(2)

(c)

A farmer pays €590. Find how many hours the tractor was hired for.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute h = 4 into the first rule	✓ 1
A1 €260	✓ 1
Part (b)
M1 substitute h = 10 into the second rule	✓ 1
A1 €470	✓ 1
Part (c)
M1 recognise the second rule applies and set 350 + 30(h − 6) = 590	✓ 1
A1 h = 14 hours	✓ 1

Exam tip: For a piecewise model, check which interval the input belongs to before choosing the rule; here €590 exceeds C(6) = 350, so the second rule is needed.

+ Add to test✓ In your test100IBMI3S-2-03r36-03Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy8 marks

A farmer has 120 m of fencing to make a rectangular enclosure. One side of the enclosure is a straight stone wall, so fencing is needed on three sides only. The sides perpendicular to the wall each have length x metres.

(a)

Show that the area of the enclosure is A = 120x − 2x² square metres.

(2)

(b)

Find the value of x that gives the greatest area, and state this greatest area.

(3)

(c)

The farmer needs an area of at least 1600 m². Find the greatest possible value of x.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 side parallel to the wall = 120 − 2x	✓ 1
A1 A = x(120 − 2x) = 120x − 2x² (AG)	✓ 1
Part (b)
M1 use axis of symmetry x = −120 ÷ (2 × −2) or a GDC maximum	✓ 1
A1 x = 30 m	✓ 1
A1 greatest area = 1800 m²	✓ 1
Part (c)
M1 solve 120x − 2x² = 1600 (or graph A and the line 1600)	✓ 1
A1 x = 20 or x = 40	✓ 1
A1 greatest value is x = 40 m	✓ 1

Common error: Using a full four-sided perimeter for the fencing gives the wrong length for the side parallel to the wall; only three sides use fencing.

+ Add to test✓ In your test101IBMI3S-2-03r36-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy6 marks

A tuning fork produces a sound whose displacement is modelled by d = 2 sin(600πt) mm, where t is the time in seconds (angle in radians). A second sound is the same note one octave higher, which doubles the frequency, and the amplitude is reduced to 70% of the original.

(a)

Write down the period of the original sound.

(2)

(b)

Write down a model of the form d = A sin(Bt) for the second sound.

(2)

(c)

The second sound begins 0.001 s later, so its model becomes d = 1.4 sin(1200π(t − 0.001)). Find its displacement at t = 0.0015 s, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2π ÷ 600π	✓ 1
A1 1/300 s (0.00333 s)	✓ 1
Part (b)
A1 A = 1.4	✓ 1
A1 B = 1200π, so d = 1.4 sin(1200πt)	✓ 1
Part (c)
M1 substitute t = 0.0015, giving 1.4 sin(0.6π)	✓ 1
A1 d = 1.33 mm	✓ 1

Exam tip: Doubling the frequency halves the period, so the number multiplying t in the sine argument doubles; it does not stay the same.

+ Add to test✓ In your test102IBMI3S-2-02r37-02Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy5 marks

Calculator

A fishing cooperative models its weekly profit, P thousand euros, when it lands x tonnes of fish by P(x) = −0.8x² + 48x − 150, for 0 ≤ x ≤ 60.

(a)

Find the profit when 10 tonnes of fish are landed.

(1)

(b)

Find the number of tonnes that gives the greatest profit, and the value of this greatest profit.

(2)

(c)

Find the range of values of x for which the cooperative makes a profit. Give your values to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 P(10) = 250 thousand euros	✓ 1
Part (b)
A1 x = 30 tonnes	✓ 1
A1 P = 570 thousand euros	✓ 1
Part (c)
M1 solve −0.8x² + 48x − 150 = 0 (or use graph) giving 3.31 and 56.7	✓ 1
A1 3.31 < x < 56.7	✓ 1

Common error: A quadratic with a negative x² coefficient is positive between its roots, not outside them.

+ Add to test✓ In your test103IBMI3S-2-02r37-04Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy6 marks

Calculator

The number of fish in a protected area is modelled by P(t) = 800 × 1.06^t, where t is the time in years since the start of monitoring.

(a)

Find the number of fish after 5 years, to the nearest whole number.

(2)

(b)

A licence allows 200 fish to be removed, so the number of fish remaining is modelled by g(t) = P(t) − 200. Describe the transformation that maps the graph of P onto the graph of g, and find g(5) to the nearest whole number.

(2)

(c)

The graph of Q(t) = P(t + 3) models the same population when monitoring starts 3 years later. Show that Q(t) = k × P(t) and find k to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 800 × 1.06⁵	✓ 1
A1 1071	✓ 1
Part (b)
A1 translation of 200 units downwards, vector (0, −200)	✓ 1
A1 g(5) = 871	✓ 1
Part (c)
M1 800 × 1.06^t+3 = 800 × 1.06^t × 1.06³	✓ 1
A1 k = 1.19	✓ 1

Exam tip: A horizontal translation of an exponential graph is equivalent to a vertical stretch.

+ Add to test✓ In your test104IBMI3S-2-02r37-08Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Easy7 marks

Calculator

A fishing port charges a landing fee of €45 per tonne for the first 20 tonnes of fish landed by a boat, and €38 per tonne for each tonne above 20 tonnes. The fee for landing x tonnes is €C(x).

(a)

Find the fee for landing 15 tonnes.

(1)

(b)

Find the fee for landing 32 tonnes.

(2)

(c)

Find an expression for C(x) when x > 20.

(2)

(d)

A boat pays a fee of €1700. Find the mass of fish landed, in tonnes, to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €675	✓ 1
Part (b)
M1 45 × 20 + 38 × 12	✓ 1
A1 €1356	✓ 1
Part (c)
M1 900 + 38(x − 20)	✓ 1
A1 C(x) = 38x + 140	✓ 1
Part (d)
M1 38x + 140 = 1700	✓ 1
A1 41.1 tonnes	✓ 1

Exam tip: Use the second expression only when the mass is above the point where the rate changes.

+ Add to test✓ In your test105IBMI3S-2-01r38-01Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy6 marks

Calculator

A regional aircraft takes off with 12 400 kg of fuel on board. In cruise the fuel is burned at a constant rate. After 2.5 hours of flight there is 9650 kg of fuel remaining. The mass of fuel remaining, F kg, after t hours is modelled by a linear function.

(a)

Find the rate at which fuel is burned, in kg per hour.

(2)

(b)

Write down an equation for F in terms of t.

(1)

(c)

The aircraft must land with at least 2100 kg of fuel in reserve. Find the greatest flight time that meets this requirement. Give your answer in hours to 3 significant figures.

(2)

(d)

State what the value 12 400 represents in this model.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 attempt to find gradient, e.g. (9650 − 12400) ÷ 2.5	✓ 1
A1 rate = 1100 kg per hour (gradient −1100)	✓ 1
Part (b)
A1 F = 12 400 − 1100t (accept F = −1100t + 12 400; ECF from (a))	✓ 1
Part (c)
M1 set 12 400 − 1100t = 2100 (ECF from (b))	✓ 1
A1 t = 9.36 hours (9.3636…)	✓ 1
Part (d)
R1 the mass of fuel (in kg) on board at take-off (when t = 0)	✓ 1

Exam tip: Keep the unrounded gradient in your calculator until the final line, then round once to the accuracy requested.

+ Add to test✓ In your test106IBMI3S-2-01r38-02Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy7 marks

Calculator

A farm cooperative is charged for irrigation water using a two-rate tariff. For the first 400 m³ used in a month the price is €0.85 per m³. Every m³ above 400 m³ costs €1.40. The monthly cost is €C when w m³ of water are used.
The cost function is
C(w) = 0.85w for 0 ≤ w ≤ 400
C(w) = 340 + 1.40(w − 400) for w > 400.

(a)

Calculate the cost of using 250 m³ in a month.

(2)

(b)

Calculate the cost of using 640 m³ in a month.

(2)

(c)

In one month the cooperative's bill is €900. Find the volume of water used.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 0.85 × 250	✓ 1
A1 €212.50	✓ 1
Part (b)
M1 340 + 1.40 × (640 − 400)	✓ 1
A1 €676	✓ 1
Part (c)
R1 recognises that 900 > 340, so the second rule applies	✓ 1
M1 900 = 340 + 1.40(w − 400)	✓ 1
A1 800 m³	✓ 1

Common error: Using the second rate for all of the water instead of only the part above 400 m³.

+ Add to test✓ In your test107IBMI3S-2-01r38-05Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy7 marks

Calculator

A loudspeaker cone playing a pure note vibrates so that its displacement from the rest position, p millimetres, at time t seconds is modelled by
p(t) = 0.35 sin(880πt), t ≥ 0,
where the angle is in radians.

(a)

Write down the amplitude of the vibration.

(1)

(b)

Find the period of the vibration. Give your answer in seconds to 3 significant figures.

(2)

(c)

Write down the number of complete vibrations per second.

(1)

(d)

Find the first time at which the displacement is 0.20 mm. Give your answer in seconds to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.35 mm	✓ 1
Part (b)
M1 period = 2π ÷ 880π	✓ 1
A1 0.00227 s (1/440)	✓ 1
Part (c)
A1 440 (reciprocal of the period; ECF from (b))	✓ 1
Part (d)
M1 set 0.35 sin(880πt) = 0.20	✓ 1
A1 880πt = 0.609 (…) (smallest positive solution, radians)	✓ 1
A1 t = 0.000220 s (2.20 × 10⁻⁴)	✓ 1

Exam tip: Check that your calculator is in radian mode whenever the angle in a sinusoidal model has no degree sign.

+ Add to test✓ In your test108IBMI3S-2-01r38-13Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy7 marks

Calculator

A jet engine is shut down on the ground at 100 °C. The air around it is at a constant 18 °C. The engine's temperature, T °C, after t minutes is modelled by
T = 18 + 82e^−0.06t, t ≥ 0.

(a)

Show that the model gives the correct temperature at shutdown.

(1)

(b)

Find the temperature of the engine after 20 minutes. Give your answer to 3 significant figures.

(2)

(c)

Find how long it takes for the engine to cool to 30 °C. Give your answer in minutes to 3 significant figures.

(3)

(d)

State the temperature that the model predicts the engine approaches over a long time, and give a reason.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 when t = 0, T = 18 + 82e⁰ = 100	✓ 1
Part (b)
M1 T = 18 + 82e^{−0.06 × 20}	✓ 1
A1 42.7 °C	✓ 1
Part (c)
M1 82e^−0.06t = 12	✓ 1
M1 ln taken correctly, e.g. t = ln(82 ÷ 12) ÷ 0.06	✓ 1
A1 t = 32.0 minutes	✓ 1
Part (d)
R1 18 °C, as e^−0.06t → 0 so the engine reaches the temperature of the surrounding air	✓ 1

Exam tip: In a model of the form a + be⁻kt, isolate the exponential term before taking logarithms.

+ Add to test✓ In your test109IBMI3S-2-03r39-01Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Easy5 marks

A nurse sets up an intravenous drip. The volume of fluid, V mL, remaining in the bag t hours after the drip is started is modelled by V = 480 − 32t, for as long as the bag is not empty.

(a)

State what the number 32 represents in this model, including units.

(1)

(b)

Find how long the drip takes to empty the bag.

(2)

(c)

The bag is replaced when 75 mL of fluid remains. Find the time, in hours, at which the bag is replaced. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 the volume of fluid leaving the bag each hour (the drip rate), 32 mL per hour	✓ 1
Part (b)
M1 480 − 32t = 0	✓ 1
A1 t = 15 hours	✓ 1
Part (c)
M1 480 − 32t = 75	✓ 1
A1 t = 12.7 hours (12.65625…)	✓ 1

Common error: When a question asks for a time from a linear model, set the model equal to the given value and solve; do not substitute the value in for t.

+ Add to test✓ In your test110IBMI3S-2-03r39-02Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Easy7 marks

In a platform game, a character jumps from the ground. Its height above the ground is h metres, t seconds after the jump begins, modelled by h = 8t − 5t².

(a)

Find the length of time the character is in the air.

(2)

(b)

Find the maximum height reached by the character.

(2)

(c)

A coin can be collected only while the character is at least 2.4 m above the ground. Find the length of time during the jump for which the character can collect the coin.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 8t − 5t² = 0 (or factorising / GDC zero)	✓ 1
A1 t = 1.6 s	✓ 1
Part (b)
M1 use of vertex (t = 0.8 by symmetry or −b/2a) or GDC maximum	✓ 1
A1 h = 3.2 m	✓ 1
Part (c)
M1 8t − 5t² = 2.4	✓ 1
A1 t = 0.4 and t = 1.2	✓ 1
A1 duration = 1.2 − 0.4 = 0.8 s	✓ 1

Exam tip: "At least" a given height means the model is greater than or equal to that height, so two solutions of the equation give the start and end of the interval.

+ Add to test✓ In your test111IBMI3S-2-03r39-05Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Easy6 marks

A water company charges a household a fixed charge of €18 for each billing period, plus €1.45 for each cubic metre (m³) of water used up to 20 m³, plus €2.10 for each cubic metre used above 20 m³.

(a)

Find the cost of a bill for 15 m³ of water.

(1)

(b)

Find the cost of a bill for 32 m³ of water.

(2)

(c)

One household receives a bill for €96.60. Find the volume of water used. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 18 + 15 × 1.45 = €39.75	✓ 1
Part (b)
M1 18 + 20 × 1.45 + 12 × 2.10	✓ 1
A1 €72.20	✓ 1
Part (c)
M1 recognise the volume is above 20 m³ (the cost at 20 m³ is €47)	✓ 1
M1 47 + 2.10(x − 20) = 96.60	✓ 1
A1 x = 43.6 m³ (43.619…)	✓ 1

Common error: In a piecewise charge, apply the higher rate only to the volume above the threshold, not to the whole volume.

+ Add to test✓ In your test112IBMI3S-2-03r39-14Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Easy5 marks

The path of a jet of water from a fountain nozzle is modelled by h = 1.8 − 0.45(x − 2)², where h metres is the height of the water above the ground and x metres is the horizontal distance from the nozzle, for x ≥ 0.

(a)

Write down the maximum height of the water and the horizontal distance from the nozzle at which it occurs.

(2)

(b)

Find the horizontal distance from the nozzle at which the water lands on the ground.

(2)

(c)

Find the height of the water when it is 3 m from the nozzle horizontally.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum height 1.8 m	✓ 1
A1 at x = 2 m	✓ 1
Part (b)
M1 1.8 − 0.45(x − 2)² = 0 (or (x − 2)² = 4)	✓ 1
A1 x = 4 m (reject x = 0, the nozzle)	✓ 1
Part (c)
A1 1.8 − 0.45(3 − 2)² = 1.35 m	✓ 1

Exam tip: In vertex form h = k − a(x − p)² the turning point is read directly as (p, k).

+ Add to test✓ In your test113IBMI3S-2-02r40-01Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Easy5 marks

During a vaccination campaign, the total number of people vaccinated at a clinic, $N$, is modelled by $N = 120 + 45d$, where $d$ is the number of days since the clinic opened.

(a)

Interpret, in context, (i) the value 120 and (ii) the value 45.

(2)

(b)

Find the number of people vaccinated after 14 days.

(1)

(c)

Find the value of $d$ for which $N = 1000$, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 120 is the number of people already vaccinated when the clinic opened ($d = 0$)	✓ 1
A1 45 is the number of people vaccinated per day (the rate of vaccination)	✓ 1
Part (b)
A1 750 people	✓ 1
Part (c)
M1 $120 + 45d = 1000$	✓ 1
A1 $d = 19.6$	✓ 1

Exam tip: Interpret the gradient with its units: 'people per day', not just a number.

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An online shop charges for delivery according to the mass $w$ kg of the parcel:
$C(w) = 4.50$ for $0 < w \leq 2$
$C(w) = 4.50 + 1.20(w - 2)$ for $2 < w \leq 10$
$C(w) = 14.10 + 0.80(w - 10)$ for $10 < w \leq 20$
where $C$ is the charge in euros (€).

(a)

Calculate the delivery charge for a parcel of mass 6.5 kg.

(2)

(b)

A customer pays €16.50 for delivery. Find the mass of the parcel.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use the second rule: $4.50 + 1.20(6.5 - 2)$	✓ 1
A1 €9.90	✓ 1
Part (b)
R1 €16.50 is more than $C(10) = 14.10$, so use the third rule	✓ 1
M1 $14.10 + 0.80(w - 10) = 16.50$	✓ 1
A1 $w = 13$ kg	✓ 1

Exam tip: For a piecewise function, decide which rule applies by comparing the input (or the output) with the boundary values first.

+ Add to test✓ In your test115IBMI3S-2-02r40-03Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Easy6 marks

The power output $P$ kW of a rooftop solar installation is modelled by $P = -0.5t^2 + 6t$ for $0 \leq t \leq 12$, where $t$ is the number of hours after 06:00.

(a)

Find the maximum power output and the time at which it occurs.

(2)

(b)

Find the two times for which the output is exactly 10 kW.

(3)

(c)

Hence find the number of hours for which the output is greater than 10 kW.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 axis of symmetry $t = -\frac{6}{2(-0.5)} = 6$ (12:00)	✓ 1
A1 maximum output 18 kW	✓ 1
Part (b)
M1 $-0.5t^2 + 6t = 10$ (or $t^2 - 12t + 20 = 0$)	✓ 1
A1 $t = 2$ (08:00)	✓ 1
A1 $t = 10$ (16:00)	✓ 1
Part (c)
A1 8 hours	✓ 1

Exam tip: For a downward-opening quadratic, the output is above a given value between the two solutions of the equation.

+ Add to test✓ In your test116IBMI3S-2-02r40-04Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Easy6 marks

The number of new cases of an infection reported on day $d$ of an outbreak is modelled by $N = 40 \times 1.15^d$.

(a)

Find the number of cases predicted on day 10, to the nearest whole number.

(2)

(b)

Find the value of $d$ for which $N = 500$, correct to 3 significant figures.

(3)

(c)

Give one reason why this model may not be reliable for large values of $d$.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $40 \times 1.15^{10}$	✓ 1
A1 162	✓ 1
Part (b)
M1 $40 \times 1.15^d = 500$ (or $1.15^d = 12.5$)	✓ 1
M1 use of logarithms or the GDC to solve	✓ 1
A1 $d = 18.1$	✓ 1
Part (c)
R1 cases cannot grow by 15% every day indefinitely, since the susceptible population is limited and control measures will change growth	✓ 1

Exam tip: For $a \times b^d = k$, divide by $a$ first, then use logarithms or the equation solver on your calculator.

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quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

On a mountaineering expedition the air temperature at base camp (altitude 1200 m) is 14.0 °C. Higher up, the temperature falls linearly by 6.5 °C for every 1000 m climbed. Let T °C be the temperature at altitude h metres.

(a)

Find an equation for T in terms of h.

(2)

(b)

Use your model to find the temperature at the summit camp, at an altitude of 3500 m.

(2)

(c)

Find the altitude at which the model predicts a temperature of 0 °C. Give your answer to the nearest metre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient −0.0065 (or −6.5 per 1000 m) used, or a correct substitution of (1200, 14) into T = mh + c	✓ 1
A1 T = −0.0065h + 21.8 (equivalent forms such as T = 14 − 0.0065(h − 1200) accepted)	✓ 1
Part (b)
M1 substitution of h = 3500 into the model	✓ 1
A1 T = −0.95 °C (ECF from (a))	✓ 1
Part (c)
M1 setting −0.0065h + 21.8 = 0, or 14 = 0.0065(h − 1200)	✓ 1
A1 h = 3354 m (3353.8… before rounding; ECF from (a))	✓ 1

Exam tip: When a rate of change is given per 1000 m, convert it to a gradient per metre before building the equation, and check that the units of the gradient match the units of the variables.

+ Add to test✓ In your test118IBMI3S-2-01r43-05Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; 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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

A sample of cooked rice left at room temperature contains 450 bacteria per gram at t = 0. The number N of bacteria per gram is modelled by N = 450 × 2^t/25, where t is the time in minutes.

(a)

Calculate the number of bacteria per gram after 100 minutes.

(2)

(b)

Food safety guidance says the rice must be discarded once N reaches 100 000. Find the time, in minutes, at which the model predicts this. Give your answer to 4 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substituting t = 100, giving 450 × 2⁴	✓ 1
A1 7200	✓ 1
Part (b)
M1 2^t/25 = 100000/450 (= 222.2…), or 450 × 2^t/25 = 100000	✓ 1
M1 using logarithms (or a graph or solver) to solve for t	✓ 1
A1 t = 194.9 minutes	✓ 1

Exam tip: To solve for an exponent, isolate the power first, then take logarithms of both sides, or use the solver on your calculator and show the equation you entered.

+ Add to test✓ In your test119IBMI3S-2-01r43-09Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; 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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

A cheese producer's daily profit, €P, from selling x kilograms of cheese is modelled by P = −0.8x² + 24x − 60, for 0 ≤ x ≤ 30.

(a)

Find the profit when 10 kg of cheese is sold.

(1)

(b)

Find the range of values of x for which the producer makes a profit. Give the end-points to 3 significant figures.

(3)

(c)

Find the maximum daily profit.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €100	✓ 1
Part (b)
M1 solving P = 0 (formula or calculator)	✓ 1
A1 x = 2.75 and x = 27.2 (2.7526 and 27.2474 before rounding)	✓ 1
A1 profit made for 2.75 < x < 27.2	✓ 1
Part (c)
M1 vertex at x = −24/(2 × (−0.8)) = 15, or a maximum found from a graph	✓ 1
A1 maximum profit €120	✓ 1

Common error: When a quadratic model gives a profit, the interval of profitable sales lies between the two zeros; do not state the zeros as the answer to a question about where P > 0.

+ Add to test✓ In your test120IBMI3S-2-01r43-10Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

The time t minutes needed to roast a turkey of mass m kg is modelled by a linear function t = am + b. A 4.0 kg turkey needs 205 minutes and a 6.5 kg turkey needs 290 minutes.

(a)

Find the values of a and b.

(3)

(b)

Interpret the value of a in this context.

(1)

(c)

The oven can be used for at most 6 hours. Find the greatest mass of turkey that can be roasted in this time, according to the model. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient (290 − 205)/(6.5 − 4.0)	✓ 1
A1 a = 34	✓ 1
A1 b = 69	✓ 1
Part (b)
A1 each extra kilogram of mass adds 34 minutes to the roasting time	✓ 1
Part (c)
M1 34m + 69 = 360 (6 hours converted to 360 minutes)	✓ 1
A1 m = 8.56 kg (8.5588… before rounding; ECF from (a))	✓ 1

Exam tip: Check that all times are in the same unit before substituting: here 6 hours must be written as 360 minutes.

+ Add to test✓ In your test121IBMI3S-2-03r43-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy5 marks

A space probe is moving directly away from Earth. Its distance from Earth, D million km, is modelled by D = 4800 + 1.3t, where t is the number of days since the probe completed a course correction.

(a)

State what the value 1.3 represents in this model.

(1)

(b)

Find the distance of the probe from Earth when t = 200.

(2)

(c)

Find the value of t when the probe is 5500 million km from Earth. Give your answer to the nearest day.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
The probe's distance from Earth increases by 1.3 million km each day (speed of 1.3 million km per day)	✓ 1
Part (b)
M1 substitutes t = 200 into the model	✓ 1
A1 D = 5060 million km	✓ 1
Part (c)
M1 sets 5500 = 4800 + 1.3t (or 700 ÷ 1.3)	✓ 1
A1 t = 538 days (538.46…)	✓ 1

Exam tip: In a linear model the gradient is a rate of change: always give its units (here million km per day) when you interpret it.

+ Add to test✓ In your test122IBMI3S-2-03r43-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy5 marks

During a coffee roast the temperature of the beans, T °C, after t minutes is modelled by a piecewise linear function:
T = 18 + 50t for 0 ≤ t ≤ 4
T = 218 + 8(t − 4) for 4 < t ≤ 12

(a)

Find the temperature of the beans when t = 2.5.

(1)

(b)

State the gradient of each stage of the model and describe what the change in gradient means for the roast.

(2)

(c)

Find the time at which the beans reach 240 °C.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 T = 143 °C	✓ 1
Part (b)
A1 gradients 50 °C per minute (first stage) and 8 °C per minute (second stage)	✓ 1
R1 the beans heat up much more slowly after 4 minutes (rate of heating falls)	✓ 1
Part (c)
M1 uses the second stage and sets 218 + 8(t − 4) = 240	✓ 1
A1 t = 6.75 minutes	✓ 1

Common error: Use the correct piece of the function: 240 °C is above the value at t = 4, so the second rule must be used, not the first.

+ Add to test✓ In your test123IBMI3S-2-03r43-03Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy5 marks

The curve of a footbridge arch is modelled by h(x) = −0.005x² + 0.6x, where h is the height in metres of the arch above the water and x is the horizontal distance in metres from the left-hand support.

(a)

Find the width of the arch at water level.

(1)

(b)

Find the maximum height of the arch.

(2)

(c)

Two vertical cables, each of length 10 m, hang from the arch to the water. Find the horizontal distance between the two cables.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 h = 0 gives x = 0 and x = 120, so the width is 120 m	✓ 1
Part (b)
M1 uses symmetry: maximum at x = 60 (or uses the GDC or the vertex formula)	✓ 1
A1 maximum height = 18 m	✓ 1
Part (c)
M1 sets −0.005x² + 0.6x = 10 and solves (x = 20 and x = 100)	✓ 1
A1 distance = 80 m	✓ 1

Exam tip: A symmetrical parabola has its vertex halfway between its two roots, so the maximum height can be found without a GDC.

+ Add to test✓ In your test124IBMI3S-2-03r43-04Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Easy5 marks

The electrical power, P watts, produced by a probe's nuclear power source t years after launch is modelled by P = 300 × 0.992^t.

(a)

Interpret the value 0.992 in this model.

(1)

(b)

Calculate the power produced 10 years after launch.

(2)

(c)

Find the number of years after launch at which the power first falls to 200 W. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
The power output falls to 99.2% of its previous value each year (a 0.8% decrease per year)	✓ 1
Part (b)
M1 substitutes t = 10	✓ 1
A1 P = 277 W	✓ 1
Part (c)
M1 sets 300 × 0.992^t = 200 and attempts to solve (GDC or logarithms)	✓ 1
A1 t = 50.5 years	✓ 1

Common error: Do not subtract 0.992 from 1 to find the yearly percentage change and then forget to say whether it is an increase or a decrease: a base below 1 means decay.

+ Add to test✓ In your test125IBMI3S-2-03r45-01Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy5 marks

A port authority charges a container ship a berthing fee that is a linear function of the time, t hours, that the ship spends at the quay. The fee, C euros, is €1640 for 8 hours and €2300 for 14 hours.

(a)

Find the equation of the model in the form C = mt + c.

(3)

(b)

State what the value 760 represents in this context.

(1)

(c)

Use the model to find the fee for a stay of 20 hours.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient = (2300 − 1640)/(14 − 8)	✓ 1
A1 m = 110	✓ 1
A1 c = 760, so C = 110t + 760	✓ 1
Part (b)
A1 a fixed charge of €760 that is paid regardless of time (the fee for 0 hours)	✓ 1
Part (c)
A1 C = 110 × 20 + 760 = €2960 (ECF from (a))	✓ 1

Exam tip: When a question gives two data points for a linear model, write the gradient calculation out as a fraction before simplifying.

+ Add to test✓ In your test126IBMI3S-2-03r45-02Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy5 marks

During an eruption a lava bomb is thrown from a crater rim that is 60 m above a flat plain. Its height above the plain, h metres, after t seconds is modelled by
h(t) = −4.9t² + 39.2t + 60, for t ≥ 0 until the bomb lands.

(a)

Find the time at which the bomb reaches its greatest height, and that greatest height, giving the height to 3 significant figures.

(3)

(b)

Find the time at which the bomb lands on the plain, giving your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of t = −b/(2a) = −39.2/(2 × −4.9), or technology to find the maximum	✓ 1
A1 t = 4 s	✓ 1
A1 h(4) = 138.4 = 138 m	✓ 1
Part (b)
M1 set h(t) = 0 and solve using the quadratic formula or technology	✓ 1
A1 t = 9.31 s (the negative root, −1.31, is rejected)	✓ 1

Common error: Forgetting to reject the negative root: a landing time must be positive, so choose the root that lies after the launch.

+ Add to test✓ In your test127IBMI3S-2-03r45-03Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy6 marks

A theatre opens online booking for a new production. The number of people, W, on the waiting list w weeks after the announcement is modelled by
W = 45 × 1.12^w.

(a)

Write down the number of people on the waiting list at the moment of the announcement.

(1)

(b)

Find the number of people on the waiting list after 6 weeks, giving your answer to the nearest whole number.

(2)

(c)

Find the number of weeks until the waiting list first reaches 200 people, giving your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 45	✓ 1
Part (b)
M1 45 × 1.12⁶	✓ 1
A1 89	✓ 1
Part (c)
M1 45 × 1.12^w = 200	✓ 1
M1 solving by logarithms, w = ln(200/45)/ln 1.12, or by technology	✓ 1
A1 w = 13.2	✓ 1

Exam tip: To solve an equation with the unknown in an exponent, divide by the starting value first, then take logarithms or use the graphing or solver function.

+ Add to test✓ In your test128IBMI3S-2-03r45-11Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Easy5 marks

The cargo handled by a port, T million tonnes per year, is modelled by T = 2.4 × 1.06^t, where t is the number of years since 2020.

(a)

State the percentage by which the model predicts the cargo to increase each year.

(1)

(b)

Find the cargo predicted for 2030, giving your answer to 3 significant figures.

(2)

(c)

Find the cargo predicted for the year 2120 and explain why this prediction is unlikely to be reliable. Give the value to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 6 %	✓ 1
Part (b)
M1 2.4 × 1.06¹⁰	✓ 1
A1 4.30 million tonnes	✓ 1
Part (c)
A1 2.4 × 1.06¹⁰⁰ = 814 million tonnes	✓ 1
R1 an exponential model cannot continue indefinitely: the port has limited space and capacity, so such growth is unrealistic	✓ 1

Common error: Interpreting the base 1.06 as a growth of 1.06 %: a base of 1.06 means each year the value is multiplied by 1.06, which is a 6 % increase.

+ Add to test✓ In your test129IBMI3S-2-03r48-01Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy5 marks

A sprinter's coach plans a build-up block. The distance, d km, run in training week w is modelled by d = 18 + 3.5w, where w = 1 is the first week of the block.

(a)

Find the distance predicted for week 8.

(2)

(b)

Interpret the value 3.5 in this model.

(1)

(c)

Find the week in which the model predicts a distance of 60 km.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes w = 8 into the model	✓ 1
A1 46 km	✓ 1
Part (b)
A1 the weekly distance increases by 3.5 km each week	✓ 1
Part (c)
M1 60 = 18 + 3.5w	✓ 1
A1 w = 12	✓ 1

Exam tip: When asked to interpret a gradient, state the change in the output per one unit of the input, with units.

+ Add to test✓ In your test130IBMI3S-2-03r48-02Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy6 marks

A regional railway charges a fare, F pounds (£), for a journey of d km according to
F = 2.40 + 0.18d for 0 < d ≤ 50
F = 11.40 + 0.12(d − 50) for d > 50.

(a)

Calculate the fare for a journey of 30 km.

(2)

(b)

Calculate the fare for a journey of 120 km.

(2)

(c)

A passenger pays £15.00. Find the distance travelled.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes d = 30 into the first expression	✓ 1
A1 £7.80	✓ 1
Part (b)
M1 uses the second expression: 11.40 + 0.12 × 70	✓ 1
A1 £19.80	✓ 1
Part (c)
M1 recognises 15.00 > 11.40 so uses the second expression and sets 15 = 11.40 + 0.12(d − 50)	✓ 1
A1 d = 80 km	✓ 1

Common error: Using the first expression for every journey: check which interval the input (or the output) falls in before substituting.

+ Add to test✓ In your test131IBMI3S-2-03r48-06Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy4 marks

The number of passengers using a city rail network is modelled by P = 2.4 × 1.035ⁿ million per year, where n is the number of years since 2020.

(a)

Calculate the model's value of P for n = 5, giving your answer to 3 significant figures.

(2)

(b)

Find the value of n for which the model gives 3.0 million passengers. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes n = 5	✓ 1
A1 2.85 million	✓ 1
Part (b)
M1 sets 2.4 × 1.035ⁿ = 3.0	✓ 1
A1 n = 6.49	✓ 1

Common error: Writing 2.4 × 1.035 × n instead of 2.4 × 1.035ⁿ: the growth factor is raised to the power of the time.

+ Add to test✓ In your test132IBMI3S-2-03r48-12Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Easy5 marks

The roof of a rail platform consists of two identical parabolic arches. The first arch has the cross-section y = 12 − 0.04x², where x and y are in metres and y = 0 is ground level. The second arch is the image of the first after a translation of 25 m in the positive x direction.

(a)

Write down the equation of the second arch.

(2)

(b)

Find the width of one arch at ground level. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 replaces x by (x − 25)	✓ 1
A1 y = 12 − 0.04(x − 25)²	✓ 1
Part (b)
M1 sets 12 − 0.04x² = 0	✓ 1
A1 x = ±17.32…	✓ 1
A1 width 34.6 m	✓ 1

Common error: Writing (x + 25) for a translation to the right: moving the graph right by 25 replaces x with x − 25.

+ Add to test✓ In your test133IBMI3S-2-04r48-01Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; 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choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy5 marks

A beekeeper's monthly costs, €C, for running h hives are modelled by C = 85 + 12.5h.

(a)

Calculate the monthly cost of running 24 hives.

(2)

(b)

State what the value 85 represents in this context.

(1)

(c)

The beekeeper has a monthly budget of exactly €585. Find the number of hives that can be run.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes h = 24: 85 + 12.5 × 24	✓ 1
A1 €385	✓ 1
Part (b)
A1 the fixed monthly cost (for example equipment or rent), which does not depend on the number of hives	✓ 1
Part (c)
M1 585 = 85 + 12.5h	✓ 1
A1 h = 40	✓ 1

Exam tip: In a linear model y = c + mx, the constant is the value when the input is zero: say what that situation means in the context.

+ Add to test✓ In your test134IBMI3S-2-04r48-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; 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choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

A hospital pays a nurse £16.50 per hour for the first 40 hours worked in a week and £24.75 per hour for every hour after that. The weekly pay, £P, for h hours worked is
P = 16.50h for 0 ≤ h ≤ 40
P = 660 + 24.75(h − 40) for h > 40.

(a)

Calculate the pay for a week in which the nurse works 36 hours.

(2)

(b)

Calculate the pay for a week in which the nurse works 46 hours.

(2)

(c)

In one week the nurse is paid £1000. Find the number of hours worked, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes h = 36 into the first expression	✓ 1
A1 £594	✓ 1
Part (b)
M1 660 + 24.75 × 6	✓ 1
A1 £808.50	✓ 1
Part (c)
M1 recognises 1000 > 660 so uses 1000 = 660 + 24.75(h − 40)	✓ 1
A1 h = 53.7 hours	✓ 1

Common error: Applying the higher rate to all hours: the overtime rate applies only to hours beyond 40.

+ Add to test✓ In your test135IBMI3S-2-04r48-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; 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Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy4 marks

The number of bacteria in a hospital sample is modelled by N = 500 × 2^t/3, where t is the time in hours since the sample was taken.

(a)

Calculate the number of bacteria after 9 hours.

(2)

(b)

Find the time at which there are 15 000 bacteria. Give your answer in hours to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 500 × 2³	✓ 1
A1 4000	✓ 1
Part (b)
M1 sets 500 × 2^t/3 = 15 000	✓ 1
A1 14.7 hours	✓ 1

Common error: Reading the exponent t/3 as a multiplication by 3 hours: the number doubles every 3 hours, so the time is 3 times the number of doublings.

+ Add to test✓ In your test136IBMI3S-2-04r48-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Easy6 marks

A Formula 1 car starts a race with 105 kg of fuel and uses 1.65 kg per lap. The mass of fuel remaining after n laps is m kg, where m = 105 − 1.65n. The lap time, T seconds, depends on the fuel mass according to T = 83.5 + 0.04m.

(a)

Calculate the mass of fuel remaining after 20 laps.

(2)

(b)

Find the number of laps completed when the fuel mass is 39 kg.

(2)

(c)

Find the lap time when the fuel mass is 39 kg. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 105 − 1.65 × 20	✓ 1
A1 72 kg	✓ 1
Part (b)
M1 39 = 105 − 1.65n	✓ 1
A1 n = 40	✓ 1
Part (c)
M1 substitutes m = 39: 83.5 + 0.04 × 39	✓ 1
A1 85.1 seconds	✓ 1

Exam tip: When one linear model feeds into another, substitute the output of the first as the input of the second.

+ Add to test✓ In your test137IBMI-031-00-06Linear models and piecewise models Paper 1Medium7 marks

A small business makes and sells handmade candles. It has fixed monthly costs of $2400 and each candle costs $3.75 to produce. Each candle is sold for $9.50.
Let n be the number of candles made and sold in a month.

(a)

Find the least number of candles that must be sold in a month for the business to make a profit.

(3)

(b)

Find the profit made in a month when 600 candles are sold.

(2)

(c)

The business wants to break even at exactly 300 candles per month, with costs unchanged. Find the selling price per candle needed.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 equating cost and revenue: 2400 + 3.75n = 9.50n	✓ 1
A1 n = 417.39…	✓ 1
A1 418 candles	✓ 1
Part (b)
M1 9.50 × 600 − (2400 + 3.75 × 600)	✓ 1
A1 $1050.00	✓ 1
Part (c)
M1 300p = 2400 + 3.75 × 300	✓ 1
A1 p = $11.75	✓ 1

Exam tip: When a count must make a profit, round the break-even value up to the next whole number, even if the decimal part is small.

+ Add to test✓ In your test138IBMI-031-00-08Linear models and piecewise models Paper 1Medium7 marks

Two candles are lit at the same time. Candle A is initially 24 cm tall and its height decreases at a constant 1.6 cm per hour. Candle B is initially 18 cm tall and its height decreases at a constant 0.9 cm per hour.
Let t be the time in hours after the candles are lit.

(a)

Write down expressions for the heights h_A and h_B, in cm, in terms of t.

(2)

(b)

Find the time at which the candles have the same height.

(2)

(c)

Find this common height.

(1)

(d)

Find how much longer candle B burns than candle A.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 h_A = 24 − 1.6t	✓ 1
A1 h_B = 18 − 0.9t	✓ 1
Part (b)
M1 24 − 1.6t = 18 − 0.9t	✓ 1
A1 t = 8.57 hours (8.571…)	✓ 1
Part (c)
A1 10.3 cm (10.285…)	✓ 1
Part (d)
M1 burn-out times 24 ÷ 1.6 = 15 h and 18 ÷ 0.9 = 20 h	✓ 1
A1 5 hours	✓ 1

Common error: Using a rounded time such as 8.6 h to find the common height introduces error; use the full calculator value.

+ Add to test✓ In your test139IBMI-031-00-10Linear models and piecewise models Paper 1Medium5 marks

The speed of sound in air, v m s⁻¹, at temperature T °C is modelled by $v = 331 + 0.6T$.

(a)

Find the speed of sound when the temperature is 25 °C.

(1)

(b)

Find the inverse function, giving T in terms of v, and state what it allows you to calculate.

(3)

(c)

A sound is measured to travel at 340 m s⁻¹. Find the air temperature.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 346 m s⁻¹	✓ 1
Part (b)
M1 attempt to rearrange for T	✓ 1
A1 T = (v − 331)/0.6	✓ 1
R1 it gives the air temperature from a measured speed of sound	✓ 1
Part (c)
A1 15.0 °C	✓ 1

Exam tip: In context, an inverse function swaps the roles of input and output; say in words what the new input and output represent.

+ Add to test✓ In your test140IBMI-031-07-08Exponential and sinusoidal models Paper 1Medium7 marks

Ravi buys a car for 28 500 euros (EUR). Its value decreases by 14% each year.

(a)

Write down a model for the value, V EUR, of Ravi's car t years after he buys it.

(1)

(b)

Find the value of Ravi's car after 4 years. Give your answer to two decimal places.

(2)

(c)

At the same time, Sana buys a car for 32 000 EUR. The value of her car t years later is modelled by $W = 32000(0.81)^t$. Find the time at which the two cars have the same value.

(3)

(d)

Describe how the values of the two cars compare after this time.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $V = 28500(0.86)^t$	✓ 1
Part (b)
M1: $28500(0.86)^4$	✓ 1
A1: 15 589.73 EUR	✓ 1
Part (c)
M1: $28500(0.86)^t = 32000(0.81)^t$	✓ 1
M1: using GDC intersection, or $\left(\frac{0.86}{0.81}\right)^t = \frac{32000}{28500}$	✓ 1
A1: $t = 1.93$ years	✓ 1
Part (d)
R1: after 1.93 years Sana's car is worth less than Ravi's car, because her car loses value at a faster rate (19% compared with 14% per year)	✓ 1

Common error: A 14% decrease gives a multiplier of 0.86, not 0.14. Using 0.14 would wipe out 86% of the value each year.

+ Add to test✓ In your test141IBMI-031-03-r1-03Choosing and evaluating models Paper 1Medium7 marks

A footbridge over a canal has a parabolic arch. The height of the arch above the water, $h$ metres, at a horizontal distance $x$ metres from one end is modelled by $h(x) = ax^2 + bx + c$.
The arch meets the water at $x = 0$ and at $x = 40$. At $x = 10$ the arch is 7.5 m above the water.

(a)

Find the values of a, b and c.

(3)

(b)

Find the maximum height of the arch above the water.

(2)

(c)

A barge carrying a load that is 8.4 m high must pass under the arch. Find the width of the part of the canal in which the arch is at least 8.4 m above the water.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forming equations from the three points, e.g. h = ax(x − 40) with 7.5 = a(10)(−30), or solving a system on GDC	✓ 1
A1 a = −0.025	✓ 1
A1 b = 1, c = 0	✓ 1
Part (b)
M1 using the axis of symmetry x = 20 or GDC maximum	✓ 1
A1 10 m	✓ 1
Part (c)
M1 solving −0.025x² + x = 8.4 to give x = 12 and x = 28	✓ 1
A1 width = 16 m	✓ 1

Exam tip: If you know where a quadratic crosses the axis, starting from the factorised form h = ax(x − 40) leaves only one unknown to find.

+ Add to test✓ In your test142IBMI-031-02-r1-04Exponential and sinusoidal models Paper 1Medium7 marks

A new electric car is bought for $28 500. Three years later its value is $17 200.
The value of the car, V dollars, t years after it was bought is modelled by $V(t) = k\,r^t$, where k and r are constants.

(a)

Find the value of k and the value of r.

(3)

(b)

Find the predicted value of the car 7 years after it was bought. Give your answer to the nearest dollar.

(2)

(c)

The owner plans to sell the car when its value first falls below $5000. Find the time after purchase at which this occurs.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
k = 28 500 (A1)	✓ 1
Sets up 28 500 r³ = 17 200 (M1)	✓ 1
r = 0.845 (0.845073…) (A1)	✓ 1
Part (b)
Substitutes t = 7 using unrounded r: 28 500(0.845073…)⁷ (M1)	✓ 1
$8772 (A1)	✓ 1
Part (c)
Sets up 28 500(0.845073…)^t = 5000 (M1)	✓ 1
t = 10.3 years (A1)	✓ 1

Common error: Using the rounded value r = 0.845 in later parts gives $8763 in part (b) instead of $8772. Store the unrounded value of r in your calculator and use it throughout.

+ Add to test✓ In your test143IBMI-031-01-r2-04Quadratic and cubic models Paper 1Medium7 marks

A museum opens a temporary exhibition. The weekly number of visitors, N thousand, t weeks after opening is modelled by
$N(t) = -0.02t^3 + 0.6t^2 + 1.5t + 4$, for 0 ≤ t ≤ 25.

(a)

Write down the number of visitors in the week the exhibition opens.

(1)

(b)

Find the value of t when the weekly number of visitors first reaches 50 000.

(2)

(c)

Find the maximum weekly number of visitors predicted by the model and the value of t at which it occurs.

(3)

(d)

Explain why this model would not be suitable for predicting visitor numbers long after week 25.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
4000 visitors (A1)	✓ 1
Part (b)
Solves N(t) = 50, e.g. finds the intersection with y = 50 on the GDC (M1)	✓ 1
t = 8.80 weeks (A1)	✓ 1
Part (c)
Uses the GDC maximum, or solves −0.06t² + 1.2t + 1.5 = 0 (M1)	✓ 1
t = 21.2 weeks (A1)	✓ 1
Maximum ≈ 115 thousand, i.e. about 115 000 visitors (A1)	✓ 1
Part (d)
The leading term −0.02t³ dominates, so N decreases without limit and eventually becomes negative, which is impossible for visitor numbers (R1)	✓ 1

Common error: N is measured in thousands. Give 115 as '115 thousand' or '115 000 visitors', not '115 visitors'. Credit for interpretation depends on stating the units in context.

+ Add to test✓ In your test144IBMI-031-05-r1-03Linear models and piecewise models Paper 1Medium7 marks

An electricity supplier uses the following monthly tariff.
• A standing charge of $12.00 per month.
• $0.18 per kWh for the first 200 kWh used.
• $0.25 per kWh for every kWh used above 200 kWh.
The total monthly bill, in dollars, for x kWh of electricity is $C(x)$.

(a)

Find the bill for a month in which 150 kWh is used.

(1)

(b)

Write down a piecewise function for $C(x)$, stating the domain of each piece.

(3)

(c)

One household receives a bill of $103.50 for a month. Find the amount of electricity it used.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$12+0.18\times150=$ $39.00 A1	✓ 1
Part (b)
$C(x)=12+0.18x$ for $0\le x\le200$ A1 ; recognises bill at 200 kWh is $48 and adds $0.25(x-200)$ M1 ; $C(x)=0.25x-2$ (or $48+0.25(x-200)$) for $x>200$ A1	✓ 3
Part (c)
Since $103.50>48$, usage exceeds 200 kWh so the second piece applies R1 ; $0.25x-2=103.5$ M1 ; $x=422$ kWh A1	✓ 3

Exam tip: Before solving with a piecewise model, check which piece applies by evaluating the function at the boundary, here $C(200)=48$.

+ Add to test✓ In your test145IBMI-031-06-r2-03Quadratic and cubic models Paper 1Medium7 marks

An open-topped tray is made from a rectangular sheet of card measuring 30 cm by 20 cm. A square of side x cm is cut from each corner and the sides are folded up. The volume of the tray is V cm³.

(a)

Show that $V = 4x^3 - 100x^2 + 600x$.

(2)

(b)

State the possible values of x in this context.

(1)

(c)

The designer wants the tray to have a volume of 800 cm³. Find the possible values of x.

(2)

(d)

Find the maximum possible volume of the tray.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Base dimensions (30 − 2x) and (20 − 2x), so V = x(30 − 2x)(20 − 2x) M1	✓ 1
Correct expansion, e.g. x(600 − 100x + 4x²), leading to the given answer A1	✓ 1
Part (b)
0 < x < 10 A1	✓ 1
Part (c)
Solving 4x³ − 100x² + 600x = 800 with GDC (graph or solver) M1	✓ 1
x = 1.88 cm or x = 6.36 cm (16.8 rejected, as it is outside 0 < x < 10) A1	✓ 1
Part (d)
Use of GDC maximum, or V′ = 12x² − 200x + 600 = 0 giving x = 3.92 M1	✓ 1
Maximum volume = 1060 cm³ (1056.3…) A1	✓ 1

Exam tip: A cubic model can give roots that lie outside the valid domain. Check every GDC solution against the restriction you stated earlier in the question.

+ Add to test✓ In your test146IBMI-031-05-r2-04Linear models and piecewise models Paper 1Medium8 marks

A small bakery sells celebration cakes. Its monthly costs are $2400 in fixed costs plus $6.50 for the ingredients of each cake. Each cake sells for $14. Let $n$ be the number of cakes made and sold in a month.

(a)

Write down an expression for the monthly profit, $P(n)$, in dollars.

(2)

(b)

Find the number of cakes that must be sold for the bakery to break even.

(2)

(c)

Find the minimum number of cakes that must be sold for a monthly profit of at least $1750.

(2)

(d)

The bakery changes the selling price to $p$ dollars per cake so that it breaks even at 250 cakes. Assuming costs are unchanged, find $p$.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: revenue minus cost: $14n - (2400 + 6.5n)$	✓ 1
A1: $P(n) = 7.5n - 2400$	✓ 1
Part (b)
M1: $7.5n - 2400 = 0$	✓ 1
A1: $n = 320$ cakes	✓ 1
Part (c)
M1: $7.5n - 2400 \ge 1750$, giving $n \ge 553.3\ldots$	✓ 1
A1: 554 cakes (must be a whole number, rounded up)	✓ 1
Part (d)
M1: $250p = 2400 + 6.5 \times 250$	✓ 1
A1: $p = $16.10$	✓ 1

Common error: Rounding 553.3 down to 553 gives a profit just below the target; in context the number of cakes must be rounded up.

+ Add to test✓ In your test147IBMI-031-01-r1-04Quadratic and cubic models Paper 1Medium7 marks

The underside of a stone arch over a canal is modelled by $h(x) = a(x - 30)^2 + 18$, where h is the height in metres above the water and x is the horizontal distance in metres from one end of the arch. The arch meets the water at x = 0 and x = 60.

(a)

Show that a = −0.02.

(2)

(b)

A lamp is fixed to the underside of the arch at a horizontal distance of 12 m from one end. Find the height of the lamp above the water.

(2)

(c)

A barge carries a load whose top is 10 m above the water. Find the width of the part of the canal in which the top of the load can pass beneath the arch.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: substitutes (0, 0): 0 = a(0 − 30)² + 18, i.e. 900a + 18 = 0	✓ 1
A1: a = −18/900 = −0.02	✓ 1
Part (b)
M1: h(12) = −0.02(12 − 30)² + 18	✓ 1
A1: 11.5 m (11.52)	✓ 1
Part (c)
M1: sets −0.02(x − 30)² + 18 = 10	✓ 1
A1: x = 10 and x = 50	✓ 1
A1: width = 40.0 m	✓ 1

Exam tip: Vertex form a(x − h)² + k gives the maximum point directly; use the symmetry about x = 30 to check that your two solutions are equally spaced from the centre.

+ Add to test✓ In your test148IBMI-031-02-r2-04Exponential and sinusoidal models Paper 1Medium8 marks

In a northern city, the number of hours of daylight, $D$, on day $t$ of the year is modelled by
$D(t) = a\cos\big(b(t - 172)\big) + d$,
where angles are in degrees and $t = 1$ is 1 January. The longest day, day 172, has 16.4 hours of daylight and the shortest day, day 355, has 8.0 hours of daylight. The model has a period of 365 days.

(a)

Find the value of $a$ and the value of $d$.

(2)

(b)

Find the value of $b$.

(1)

(c)

Use the model to estimate the number of hours of daylight on day 100.

(2)

(d)

Find the number of days in the year on which the model predicts more than 14 hours of daylight.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
a = (16.4 − 8.0) ÷ 2 = 4.2 (A1)	✓ 1
d = (16.4 + 8.0) ÷ 2 = 12.2 (A1)	✓ 1
Part (b)
b = 360 ÷ 365 = 0.986 (0.98630…) (A1)	✓ 1
Part (c)
Substitutes t = 100: 4.2cos(0.98630 × (100 − 172)) + 12.2 (M1)	✓ 1
D = 13.6 hours (A1)	✓ 1
Part (d)
Sets up D(t) = 14 or uses graph intersections with D = 14 (M1)	✓ 1
t = 106.5 and t = 237.5 (106.479…, 237.521…) (A1)	✓ 1
Approximately 131 days (A1)	✓ 1

Common error: The amplitude is half the difference between the maximum and minimum, not the maximum value itself. Using a = 16.4 gives a model that predicts negative daylight hours.

+ Add to test✓ In your test149IBMI-031-06-r1-05Quadratic and cubic models Paper 1Medium6 marks

The entrance to a road tunnel is a parabolic arch. Take the origin at one foot of the arch on the level road. The arch meets the road again 10 m away, and its highest point is 6 m above the road. The height of the arch above the road, y metres, at a horizontal distance x metres from the origin is modelled by $y = ax^2 + bx$.

(a)

Find the values of a and b.

(3)

(b)

A lorry carries a rectangular load whose top is 4.2 m above the road. The lorry drives through the tunnel centred on the arch's line of symmetry. Find the maximum possible width of the load, giving your answer correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Forms two equations, using $(10, 0)$ and the vertex $(5, 6)$: $100a + 10b = 0$ and $25a + 5b = 6$	✓ 1
$a = -0.24$	✓ 1
$b = 2.4$	✓ 1
Part (b)
Sets $-0.24x^2 + 2.4x = 4.2$	✓ 1
$x = 2.26$ and $x = 7.74$ (to 3 s.f.)	✓ 1
Maximum width $= 7.7386 - 2.2614 = 5.48$ m	✓ 1

Exam tip: A symmetric arch has its vertex halfway between its feet. The point (5, 6) therefore gives a second equation without needing the vertex formula.

+ Add to test✓ In your test150IBMI-031-08-r2-03Choosing and evaluating models Paper 1Medium8 marks

A community podcast records its number of subscribers, $N$, at the start of month $t$. The data are:
$t=0$: 500 subscribers
$t=2$: 720 subscribers
$t=4$: 1040 subscribers
Two models are proposed, each fitted using only the data for $t=0$ and $t=4$.
Model A: $N=a+bt$
Model B: $N=500r^t$

(a)

Find the values of $a$ and $b$ for Model A.

(2)

(b)

Find the value of $r$ for Model B.

(2)

(c)

By considering the data value at $t=2$, determine which model is more appropriate. Justify your answer.

(2)

(d)

Use your chosen model to predict the number of subscribers at the start of month 10. Give your answer to the nearest whole number, using the unrounded value of any parameter you found earlier.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $a=500$	✓ 1
A1: $b=\frac{1040-500}{4}=135$	✓ 1
Part (b)
M1: $500r^4=1040$ so $r^4=2.08$	✓ 1
A1: $r=2.08^{1/4}=1.20$ (1.20093…)	✓ 1
Part (c)
A1: Model A predicts $500+135(2)=770$ and Model B predicts $500(1.20093…)^2\approx 721$	✓ 1
R1: Model B is more appropriate because its prediction is much closer to the recorded value of 720	✓ 1
Part (d)
M1: Correct substitution into their chosen model, e.g. $500(1.20093…)^{10}$ or $500(2.08)^{10/4}$	✓ 1
A1: 3120 subscribers (3119.8…)	✓ 1
Follow-through: if Model A was chosen in part (c), $500+135(10)=1850$ earns M1A1.	—

Common error: Rounding $r$ to 1.20 before raising it to the power 10 gives about 3096, which is noticeably wrong. Keep the full calculator value of $r$ (or use $2.08^{10/4}$ directly) until the final step.

+ Add to test✓ In your test151IBMI-031-06-r1-04Quadratic and cubic models Paper 1Medium7 marks

The volume of water in a reservoir, V million litres, t months after monitoring began is modelled by
$V(t) = 0.02t^3 - 0.9t^2 + 10.8t + 50$, for $0 \le t \le 24$.

(a)

Write down the volume of water when monitoring began.

(1)

(b)

Find the maximum volume of water in the reservoir during the 24 months, and the time at which it occurs.

(2)

(c)

The reservoir is classed as low when it holds less than 66 million litres. Find the total length of time during the 24 months for which the reservoir is classed as low.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$V(0) = 50$ million litres	✓ 1
Part (b)
$t = 8.29$ months (accept $15-\sqrt{45}$)	✓ 1
$V = 89.1$ million litres	✓ 1
(Note: $V(0)=50$ and $V(24)=67.28$ are both less, so the local maximum is the overall maximum.)	—
Part (c)
Attempts to solve $V(t) = 66$, e.g. by GDC or graph intersection	✓ 1
$t = 1.718$	✓ 1
$t = 20$ and $t = 23.28$	✓ 1
Low for $0 \le t < 1.718$ and $20 < t < 23.28$, so total time $= 1.718 + 3.282 = 5.00$ months (accept 5 or 5.0)	✓ 1

Common error: Forgetting the interval at the very start: the reservoir begins below 66 million litres, so the time from t = 0 to the first intersection also counts. Keep unrounded roots from your GDC when subtracting, or the total drifts to 5.02.

+ Add to test✓ In your test152IBMI-031-03-r2-04Choosing and evaluating models Paper 1Medium7 marks

At a harbour, the depth of water reaches a maximum of 8.6 m at 03:00 and the next minimum of 2.2 m at 09:15. The depth, $D$ metres, is modelled by
$D(t) = a\cos(b(t - c)) + d$,
where $t$ is the number of hours after midnight, $a, b > 0$ and $0 \le c < 6$.

(a)

Find the value of $a$ and the value of $d$.

(2)

(b)

Find the value of $b$.

(2)

(c)

Write down the value of $c$.

(1)

(d)

A fishing boat in the harbour needs a depth of at least 4.0 m to leave. It is ready to leave at 03:00. Find the latest time at which it can leave before the depth first drops below 4.0 m. Give your answer to the nearest minute.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 $a = \frac{8.6 - 2.2}{2} = 3.2$	✓ 1
A1 $d = \frac{8.6 + 2.2}{2} = 5.4$	✓ 1
Part (b)
M1 period $= 2 \times 6.25 = 12.5$ hours	✓ 1
A1 $b = \frac{2\pi}{12.5} = \frac{4\pi}{25} = 0.503$ (3 s.f.)	✓ 1
Part (c)
A1 $c = 3$	✓ 1
Part (d)
M1 solving $3.2\cos(0.503(t - 3)) + 5.4 = 4.0$ and choosing the first solution after $t = 3$, e.g. by graph intersection	✓ 1
A1 $t = 7.03$ hours, i.e. 07:02 (accept 07:01 from using $b = 0.503$)	✓ 1

Exam tip: With a cosine model, the horizontal shift is the time of a maximum, so c can be read straight from the data. Keep b unrounded in your calculator when solving for t.

+ Add to test✓ In your test153IBMI3S-2-03-04Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

The mass of caffeine in a student's bloodstream is modelled by A = 200 × 0.5^t/5.5, where A is in mg and t is the time in hours after drinking a coffee.

(a)

Find the mass of caffeine in the bloodstream after 3 hours.

(2)

(b)

The model can be written as A = 200e^−kt. Find the value of k.

(2)

(c)

Find the time taken for the mass of caffeine to fall below 30 mg.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 200 × 0.5^(3/5.5)	✓ 1
A1 137 mg	✓ 1
Part (b)
M1 e^(−k) = 0.5^(1/5.5), or k = ln 2 / 5.5	✓ 1
A1 k = 0.126	✓ 1
Part (c)
M1 200 × 0.5^(t/5.5) = 30	✓ 1
M1 taking logarithms: t = 5.5 × ln(200/30) / ln 2	✓ 1
A1 t = 15.1 hours	✓ 1

Exam tip: State what the constant 5.5 means in context (the half-life in hours) before substituting; it makes the form of the model easy to check.

+ Add to test✓ In your test154IBMI3S-2-03-07Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

An open box is made from a rectangular sheet of card measuring 30 cm by 20 cm. A square of side x cm is cut from each corner and the sides are folded up. The volume of the box is V cm³, where 0 < x < 10.

(a)

Show that V = 4x³ − 100x² + 600x.

(2)

(b)

Find the values of x for which the volume of the box is 800 cm³.

(3)

(c)

Find the value of x that gives the greatest volume.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 V = x(30 − 2x)(20 − 2x)	✓ 1
A1 expanding (30 − 2x)(20 − 2x) = 600 − 100x + 4x², so V = 4x³ − 100x² + 600x (shown)	✓ 1
Part (b)
M1 4x³ − 100x² + 600x = 800 (GDC or graph intersection)	✓ 1
A1 x = 1.88	✓ 1
A1 x = 6.36 (the third root, 16.8, is outside 0 < x < 10)	✓ 1
Part (c)
M1 dV/dx = 12x² − 200x + 600 = 0, or GDC maximum	✓ 1
A1 x = 3.92	✓ 1

Common error: Giving a root that lies outside the stated domain; always check each solution against the interval in the question.

+ Add to test✓ In your test155IBMI3S-2-03-10Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium8 marks

A cup of coffee cools in a room. Its temperature, T °C, after t minutes is modelled by T = 22 + 63e^−0.045t.

(a)

Find the initial temperature of the coffee.

(1)

(b)

Find the temperature of the coffee after 10 minutes.

(2)

(c)

Write down the temperature that the coffee approaches in the long term, and explain why.

(2)

(d)

Find the time taken for the coffee to cool to 40 °C.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 85 °C	✓ 1
Part (b)
M1 22 + 63e^(−0.45)	✓ 1
A1 62.2 °C	✓ 1
Part (c)
A1 22 °C	✓ 1
R1 as t increases, e^(−0.045t) tends to 0, so T tends to 22 (the room temperature)	✓ 1
Part (d)
M1 22 + 63e^(−0.045t) = 40	✓ 1
M1 e^(−0.045t) = 18/63 and take natural logarithms	✓ 1
A1 t = 27.8 minutes	✓ 1

Common error: Dividing the target temperature by 63 instead of first subtracting the 22 °C constant.

+ Add to test✓ In your test156IBMI3S-2-03-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

The number of hours of daylight in a northern city, D hours, on day t of the year (t = 1 is 1 January) is modelled by D = 12.1 + 4.3 sin(2π(t − 80)/365), with the angle in radians.

(a)

Find the maximum and the minimum number of hours of daylight predicted by the model.

(2)

(b)

Find the hours of daylight on day 300.

(2)

(c)

Find the number of days in the year on which the model gives more than 14 hours of daylight.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 16.4 hours	✓ 1
A1 minimum 7.8 hours	✓ 1
Part (b)
M1 substitute t = 300	✓ 1
A1 9.51 hours	✓ 1
Part (c)
M1 12.1 + 4.3 sin(2π(t − 80)/365) = 14	✓ 1
A1 t = 106.6 and t = 235.9	✓ 1
A1 235.9 − 106.6 = 129 days	✓ 1

Exam tip: For a sinusoidal inequality, find both crossing points on the graph of the model and subtract; a single solution from the calculator is not enough.

+ Add to test✓ In your test157IBMI3S-2-03b-02Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

A haulage firm charges $C for a delivery over a distance of d km, where
$C(d)=\begin{cases}15+0.8d, & 0\le d\le 50\\ 55+0.5(d-50), & d>50\end{cases}$

(a)

Find the cost of a delivery of 30 km.

(2)

(b)

Find the cost of a delivery of 120 km.

(2)

(c)

A customer pays $72.50 for a delivery. Find the distance of this delivery.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses the first rule with d = 30	✓ 1
A1 $39.00	✓ 1
Part (b)
M1 uses the second rule with d = 120	✓ 1
A1 $90.00	✓ 1
Part (c)
R1 $72.50 is greater than C(50) = $55, so the second rule applies	✓ 1
M1 sets 55 + 0.5(d − 50) = 72.5	✓ 1
A1 d = 85 km	✓ 1

Exam tip: With a piecewise model, decide which interval the given value belongs to before substituting, and check that the answer lies in that interval.

+ Add to test✓ In your test158IBMI3S-2-03b-04Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

A marine laboratory grows a plankton culture. The number of cells per mL, P, is modelled by P = P₀b^t, where t is the time in hours after the first count. At t = 0 the count is 1200 cells per mL, and at t = 3 it is 2850 cells per mL.

(a)

Find the value of b.

(2)

(b)

Interpret the value of b in this context.

(1)

(c)

Predict the count at t = 6, using the unrounded value of b.

(2)

(d)

Find the time at which the count first reaches 10 000 cells per mL.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms 1200b³ = 2850 (or b³ = 2.375)	✓ 1
A1 b = 1.33	✓ 1
Part (b)
R1 the count increases by about 33% every hour (it is multiplied by 1.33 each hour)	✓ 1
Part (c)
M1 substitutes t = 6 (for example 1200 × 2.375²)	✓ 1
A1 6770 cells per mL (accept 6640 if b = 1.33 is used)	✓ 1
Part (d)
M1 sets 1200b^t = 10 000 and solves (equation solver or logarithms)	✓ 1
A1 t = 7.35 hours	✓ 1

Exam tip: Keep unrounded values in your calculator for later parts. Rounding b early changes a prediction several steps later.

+ Add to test✓ In your test159IBMI3S-2-03b-05Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

The depth of water at the end of a harbour pier is modelled by d = c + a cos(b(t − 3)) metres, where t is the time in hours after midnight. The depth is 8.4 m at high tide at 03:00 and 3.6 m at the following low tide. The time between consecutive high tides is 12.4 hours.

(a)

Find the value of a and the value of c.

(2)

(b)

Show that b = 0.507 correct to three significant figures.

(1)

(c)

A fishing boat needs a depth of at least 5.0 m to enter the harbour. Find, for one tidal cycle centred on the high tide, the length of time during which the boat can enter.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (8.4 − 3.6) ÷ 2 = 2.4	✓ 1
A1 c = (8.4 + 3.6) ÷ 2 = 6.0	✓ 1
Part (b)
A1 b = 2π ÷ 12.4 = 0.5067… = 0.507	✓ 1
Part (c)
M1 sets 6 + 2.4cos(0.5067(t − 3)) = 5	✓ 1
A1 0.5067(t − 3) = 2.0006… (first positive solution)	✓ 1
M1 uses symmetry about the high tide, so the duration is 2(t − 3)	✓ 1
A1 7.90 hours	✓ 1

Common error: Using degrees instead of radians for the cosine. The value of b here is in radians per hour, so the calculator must be in radian mode.

+ Add to test✓ In your test160IBMI3S-2-03b-09Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

The elevation E metres of a cycling race course, at a distance d km from the start, is modelled by
\(E(d)=\begin{cases}80+15d, & 0\le d\le 6\\ 170-12(d-6), & 6

(a)

Find the elevation of the course at 8 km from the start.

(2)

(b)

Interpret the value −12 in the second rule.

(1)

(c)

Write down the greatest elevation on the course.

(1)

(d)

The course crosses the elevation of 150 m three times. Find the three distances from the start.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses the second rule, 170 − 12(8 − 6)	✓ 1
A1 146 m	✓ 1
Part (b)
R1 between 6 km and 10 km the course descends by 12 m for every kilometre	✓ 1
Part (c)
A1 222 m (at the finish, d = 15)	✓ 1
Part (d)
A1 d = 4.67 km (first rule)	✓ 1
A1 d = 7.67 km (second rule)	✓ 1
A1 d = 11.4 km (third rule)	✓ 1

Exam tip: Solve the equation in each piece separately and reject any solution that falls outside that piece's interval.

+ Add to test✓ In your test161IBMI3S-2-03b-10Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

A vet injects a sedative into a seal. The concentration of the drug in the blood, C mg/L, is modelled by C = 45e^−0.23t, where t is the time in hours after the injection.

(a)

State the concentration immediately after the injection.

(1)

(b)

Find the concentration after 6 hours.

(2)

(c)

Find the time taken for the concentration to halve.

(2)

(d)

The seal can safely be moved when the concentration is below 5 mg/L. Find the earliest time this is possible.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 45 mg/L	✓ 1
Part (b)
M1 substitutes t = 6, 45e^−0.23×6	✓ 1
A1 11.3 mg/L	✓ 1
Part (c)
M1 sets 45e^−0.23t = 22.5 (or e^−0.23t = 0.5)	✓ 1
A1 t = 3.01 hours	✓ 1
Part (d)
M1 sets 45e^−0.23t = 5	✓ 1
A1 t = 9.55 hours	✓ 1

Common error: Taking the half-life as half of the time to reach zero. Exponential decay never reaches zero, and equal times always halve the concentration.

+ Add to test✓ In your test162IBMI3S-2-03b-14Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

A warehouse records the time T minutes taken to pick an order containing n items.

n	5	10	15	20	25
T	9.8	14.1	19.5	23.9	29.4

(a)

Find the equation of the regression line of T on n.

(2)

(b)

Interpret the gradient and the intercept in this context.

(2)

(c)

The correlation coefficient is r = 0.999. Explain what this suggests about the model.

(1)

(d)

Use the model to estimate the number of items in an order that takes 60 minutes to pick.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 gradient 0.98	✓ 1
A1 intercept 4.64, so T = 0.98n + 4.64	✓ 1
Part (b)
R1 each extra item adds about 0.98 minutes to the picking time	✓ 1
R1 about 4.64 minutes of set-up time is needed even before any item is picked	✓ 1
Part (c)
R1 a very strong positive linear correlation, so a linear model fits the data very well	✓ 1
Part (d)
M1 sets 0.98n + 4.64 = 60	✓ 1
A1 n = 56.5 items (accept 56 or 57 items)	✓ 1

Exam tip: Estimating at 60 minutes is outside the data range, so mention that the estimate is an extrapolation and may be less reliable.

+ Add to test✓ In your test163IBMI3S-2-01r9-07Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

A coffee roasting business models its daily profit, P euros (€), from roasting x kg of beans in one batch by
P = −0.5x² + 40x − 150, for 0 ≤ x ≤ 100.

(a)

Find the profit when x = 20.

(2)

(b)

Find the batch size that gives the maximum profit, and the value of this maximum profit.

(3)

(c)

The business makes a loss for batches that are too large. Find the largest batch size for which the business does not make a loss. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute x = 20	✓ 1
A1 €450	✓ 1
Part (b)
M1 use the axis of symmetry x = −40 ÷ (2 × (−0.5))	✓ 1
A1 x = 40 kg	✓ 1
A1 maximum profit €650	✓ 1
Part (c)
M1 solve −0.5x² + 40x − 150 = 0	✓ 1
A1 x = 76.1 kg (other root 3.94 kg)	✓ 1

Exam tip: “Does not make a loss” means P ≥ 0; the boundary values come from solving P = 0.

+ Add to test✓ In your test164IBMI3S-2-01r9-08Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

The height, f metres, of a road surface above a reference level along the approach to a bridge is modelled by
f(x) = 0.002x³ − 0.15x² + 2.4x, for 0 ≤ x ≤ 30,
where x metres is the horizontal distance from the start of the approach.

(a)

Find the height of the road surface when x = 20.

(1)

(b)

Find the greatest height of the road surface for 0 ≤ x ≤ 20.

(2)

(c)

Find the value of x, other than x = 0, at which the road surface is at the reference level. Give your answer to 3 significant figures.

(2)

(d)

State the lowest height of the road surface over the whole interval 0 ≤ x ≤ 30.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4 m	✓ 1
Part (b)
M1 use a graphical or numerical method to locate the maximum	✓ 1
A1 11 m (at x = 10)	✓ 1
Part (c)
M1 solve f(x) = 0 (graph or factorise and use quadratic)	✓ 1
A1 x = 23.1 m	✓ 1
Part (d)
A1 −9 m, at x = 30 (an endpoint)	✓ 1

Common error: The lowest value on an interval can occur at an endpoint, not only at a turning point.

+ Add to test✓ In your test165IBMI3S-2-01r9-10Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

The altitude, h metres, of a lander above a moon’s surface is modelled by h = 0.4t² − 28t + 500, where t is the time in seconds after the descent engine starts, for 0 ≤ t ≤ 35.

(a)

Write down the altitude of the lander when the engine starts.

(1)

(b)

Find the times at which the lander is at an altitude of 100 m according to the equation, and state which time is valid for the model.

(3)

(c)

Find the lowest altitude reached by the lander during the first 35 seconds.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 500 m	✓ 1
Part (b)
M1 solve 0.4t² − 28t + 400 = 0	✓ 1
A1 t = 20 and t = 50	✓ 1
R1 t = 20 s, since 50 is outside the domain 0 ≤ t ≤ 35	✓ 1
Part (c)
M1 use the axis of symmetry t = 28 ÷ (2 × 0.4) = 35	✓ 1
A1 h = 10 m	✓ 1

Common error: A quadratic can give a solution that lies outside the domain; always check each root against the stated interval.

+ Add to test✓ In your test166IBMI3S-2-01r9-11Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

In a coffee roaster the temperature of the first batch is modelled by T(t) = 210 − 2(t − 8)² °C, for 0 ≤ t ≤ 16, where t is in minutes. A second batch starts 2 minutes later in a drum that is 15 °C hotter at every moment, so its temperature is modelled by T₂(t) = T(t − 2) + 15.

(a)

Write down the maximum temperature of the first batch and the time at which it occurs.

(2)

(b)

Describe the transformation that maps the graph of T onto the graph of T₂.

(2)

(c)

Find the length of time for which the second batch is above 200 °C. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 210 °C	✓ 1
A1 at t = 8 minutes	✓ 1
Part (b)
A1 translation 2 units in the positive t direction (right)	✓ 1
A1 and 15 units up (translation vector (2, 15))	✓ 1
Part (c)
M1 T₂(t) = 225 − 2(t − 10)² and solve T₂ = 200	✓ 1
A1 t = 6.46 and t = 13.5	✓ 1
A1 duration 7.07 minutes	✓ 1

Exam tip: A horizontal translation by a replaces t by t − a; the vertical shift is added outside the brackets.

+ Add to test✓ In your test167IBMI3S-2-01r10-05Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium6 marks

The cross-section of the floor of a volcanic crater is modelled by y = 0.01x² − 4, where x is the horizontal distance in metres from the centre line and y is the height in metres relative to a survey datum. After a collapse, the new floor is modelled by y = 0.01(x − 20)² − 10.

(a)

Describe the single transformation that maps the original graph onto the new graph.

(2)

(b)

Write down the coordinates of the lowest point of the new floor.

(2)

(c)

Calculate the height of the new floor at x = 50.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Translation (shift) 20 m to the right (horizontal)	✓ 1
and 6 m down (vertical); vector (20, −6)	✓ 1
Part (b)
x = 20	✓ 1
y = −10, so (20, −10)	✓ 1
Part (c)
Substitution 0.01(50 − 20)² − 10	✓ 1
y = −1 m	✓ 1

Common error: In y = f(x − 20) the graph moves 20 units to the right, not to the left.

+ Add to test✓ In your test168IBMI3S-2-01r10-06Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium5 marks

A port charges a ship for time spent at a berth. The charge, C euros, for h hours is given by
C = 1200h for 0 ≤ h ≤ 6
C = 7200 + 800(h − 6) for h > 6.

(a)

Find the charge for a stay of 4.5 hours.

(1)

(b)

Find the charge for a stay of 10 hours.

(2)

(c)

A ship is charged €12 000. Find the length of its stay.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
C = 1200 × 4.5 = €5400	✓ 1
Part (b)
Use of the second rule since h > 6	✓ 1
7200 + 800 × 4 = €10 400	✓ 1
Part (c)
Recognise that 12 000 > 7200 so second rule applies; 7200 + 800(h − 6) = 12 000	✓ 1
h = 12 hours	✓ 1

Exam tip: With a piecewise model, decide which rule applies by comparing the input (or the output) with the boundary value before substituting.

+ Add to test✓ In your test169IBMI3S-2-01r10-07Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium7 marks

A confetti cannon on a theatre stage fires a burst upwards from a height of 3 m above the stage floor. The height, h metres above the floor, of the burst t seconds after firing is modelled by h = 3 + 11t − 4.9t².

(a)

Write down the height of the cannon above the floor.

(1)

(b)

Find the maximum height reached by the burst. Give your answer to 3 significant figures.

(3)

(c)

Find the time at which the burst reaches the floor. Give your answer to 3 significant figures.

(2)

(d)

Hence state the domain of t for which the model is valid.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
3 m	✓ 1
Part (b)
Use of t = 11/(2 × 4.9) or GDC maximum	✓ 1
t = 1.122 s (or substitution of vertex time)	✓ 1
h = 9.17 m	✓ 1
Part (c)
Set 3 + 11t − 4.9t² = 0 and solve (GDC or formula) rejecting the negative root	✓ 1
t = 2.49 s	✓ 1
Part (d)
0 ≤ t ≤ 2.49	✓ 1

Common error: Reject the negative root of a quadratic when time is the variable; it lies outside the physical domain.

+ Add to test✓ In your test170IBMI3S-2-01r10-08Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium6 marks

The concentration of ash, C mg per cubic metre, in the air downwind of a volcano t hours after an eruption ends is modelled by C = a × b^t, where a and b are constants. At t = 0 the concentration is 80. At t = 2 the concentration is 51.2.

(a)

Find the value of a and the value of b.

(3)

(b)

Calculate the concentration at t = 7. Give your answer to 3 significant figures.

(2)

(c)

Interpret the value of b in context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
a = 80	✓ 1
80b² = 51.2 (or b² = 0.64)	✓ 1
b = 0.8	✓ 1
Part (b)
Substitution 80 × 0.8⁷	✓ 1
C = 16.8 mg per cubic metre	✓ 1
Part (c)
The concentration falls to 80% of its previous value each hour (decreases by 20% per hour)	✓ 1

Exam tip: With C = a·b^t, the value at t = 0 gives a directly, so use it first and then solve for b.

+ Add to test✓ In your test171IBMI3S-2-01r10-09Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium7 marks

The rate, f tonnes per day, at which a volcano releases sulfur dioxide varies through the day and is modelled by f = a sin(b(t − c))° + d, where t is the time in hours after midnight. The maximum rate of 940 occurs at t = 14 and the minimum rate of 340 occurs at t = 2. The pattern repeats every day.

(a)

Find the value of a and the value of d.

(2)

(b)

Show that b = 15.

(2)

(c)

Find the value of c, given that 0 < c < 24.

(1)

(d)

Calculate the rate of release at t = 17. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
a = (940 − 340) ÷ 2 = 300	✓ 1
d = (940 + 340) ÷ 2 = 640	✓ 1
Part (b)
Period is 24 hours	✓ 1
b = 360 ÷ 24 = 15	✓ 1
Part (c)
sin(15(14 − c)) = 1 so 15(14 − c) = 90, giving c = 8	✓ 1
Part (d)
Substitution 640 + 300 sin(15 × 9)°	✓ 1
f = 852 tonnes per day	✓ 1

Exam tip: The vertical shift is the midpoint of the maximum and minimum, and the amplitude is half their difference.

+ Add to test✓ In your test172IBMI3S-2-01r10-10Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium7 marks

The fuel burned by a ship each day, F tonnes, is modelled by F = kv³, where v is the speed in knots and k is a constant. At 16 knots the ship burns 38.4 tonnes per day.

(a)

Find the value of k.

(2)

(b)

Calculate the fuel burned per day at 18 knots. Give your answer to 3 significant figures.

(1)

(c)

Find the speed at which the ship burns 60 tonnes per day. Give your answer to 3 significant figures.

(2)

(d)

The ship increases its speed from 16 knots to 20 knots. Find the percentage increase in the fuel burned per day. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
38.4 = k × 16³	✓ 1
k = 0.009375	✓ 1
Part (b)
F = 54.7 tonnes (from 0.009375 × 18³ = 54.675)	✓ 1
Part (c)
Set 0.009375v³ = 60 (or v³ = 6400)	✓ 1
v = 18.6 knots	✓ 1
Part (d)
F(20) = 0.009375 × 20³ = 75 tonnes	✓ 1
Percentage increase = (75 − 38.4) ÷ 38.4 × 100 = 95.3%	✓ 1

Common error: A 25% increase in speed does not give a 25% increase in fuel; with a cubic model the fuel increases by a much larger percentage.

+ Add to test✓ In your test173IBMI3S-2-01r10-14Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium8 marks

The temperature, T °C, of a lava sample t minutes after it is collected is modelled by T = 20 + 1080e^−kt, where k is a positive constant. After 10 minutes the temperature is 560 °C.

(a)

Show that k = (ln 2)/10.

(3)

(b)

Calculate the temperature at t = 25. Give your answer to 3 significant figures.

(2)

(c)

State the temperature that the sample approaches over a long time.

(1)

(d)

Find the time at which the temperature is 120 °C. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
540 = 1080e^−10k, so e^−10k = 0.5	✓ 1
−10k = ln 0.5 = −ln 2, hence k = (ln 2)/10	✓ 1
Correct simplification to the stated result (0.0693…)	✓ 1
Part (b)
Substitution of k and t = 25	✓ 1
T = 211 °C	✓ 1
Part (c)
20 °C (the surrounding temperature, horizontal asymptote)	✓ 1
Part (d)
Set 20 + 1080e^−kt = 120 (e^−kt = 100/1080)	✓ 1
t = 34.3 minutes	✓ 1

Exam tip: For a model of the form a + b·e^(−kt), the constant a is the horizontal asymptote, so subtract it before taking logarithms.

+ Add to test✓ In your test174IBMI3S-2-02r11-03Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium7 marks

The cross-section of a stone arch at an excavation is modelled by the curve h = −0.08x² + 1.6x, where x metres is the horizontal distance from the left foot of the arch and h metres is the height above the ground. Take 0 ≤ x ≤ 20.

(a)

Find the width of the arch at ground level.

(2)

(b)

Find the maximum height of the arch.

(2)

(c)

A horizontal beam is to be fitted across the arch at a height of 6 m. Find the length of the beam.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
−0.08x² + 1.6x = 0, so x(−0.08x + 1.6) = 0	✓ 1
x = 0 and x = 20, so the width is 20 m	✓ 1
Part (b)
axis of symmetry at x = 10	✓ 1
maximum height = 8 m	✓ 1
Part (c)
−0.08x² + 1.6x = 6, so x² − 20x + 75 = 0	✓ 1
x = 5 and x = 15	✓ 1
length = 15 − 5 = 10 m	✓ 1

Exam tip: A horizontal line cuts a quadratic curve at two symmetric points; the length of the chord is the difference between the two x-values.

+ Add to test✓ In your test175IBMI3S-2-02r11-04Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium7 marks

The network traffic, D Gbps, through a data centre is modelled by D(t) = t³ − 15t² + 63t + 20 for 0 ≤ t ≤ 10, where t is the time in hours after 08:00.

(a)

Use your graphic display calculator to find the coordinates of the local maximum and the local minimum of D.

(2)

(b)

Find the total time during the 10 hours for which the traffic exceeds 100 Gbps, correct to 3 significant figures.

(3)

(c)

State the greatest value of D in the model, giving a reason.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
local maximum at (3, 101)	✓ 1
local minimum at (7, 69)	✓ 1
Part (b)
solve D(t) = 100 using the graph	✓ 1
t = 2.605, 3.423 and 8.972	✓ 1
total time = (3.423 − 2.605) + (10 − 8.972) = 1.85 hours	✓ 1
Part (c)
D(10) = 150 Gbps	✓ 1
this is greater than the local maximum value 101, so the greatest value is 150 Gbps	✓ 1

Common error: On a restricted domain the greatest value can occur at an endpoint, so compare the end values with the local maximum.

+ Add to test✓ In your test176IBMI3S-2-02r11-05Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium6 marks

An archaeologist models the percentage, P, of the original carbon-14 remaining in a bone sample after t years by P = 100 × (0.5)^t/5730.

(a)

Find the percentage remaining after 3000 years, correct to 3 significant figures.

(2)

(b)

The model can also be written as P = 100e^−kt. Find the value of k, correct to 3 significant figures.

(2)

(c)

A sample is found to contain 38% of its original carbon-14. Find its age, to the nearest 10 years. Use the unrounded value of k.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
P = 100 × 0.5^3000/5730	✓ 1
P = 69.6%	✓ 1
Part (b)
e^−5730k = 0.5, so k = ln 2 ÷ 5730	✓ 1
k = 1.21 × 10⁻⁴	✓ 1
Part (c)
100e^−kt = 38, so t = ln(100/38) ÷ k = 7998.7	✓ 1
age = 8000 years (nearest 10)	✓ 1

Exam tip: Taking natural logarithms of both sides turns an equation with the unknown in the exponent into a linear equation.

+ Add to test✓ In your test177IBMI3S-2-02r11-09Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium7 marks

A cyclist records her steady climbing speed, v km h⁻¹, on road sections of different gradient g% (g = 0 is level).

g (%)	0	2	4	6	8	10
v (km h⁻¹)	33	28	24	19	13	9

(a)

Use your graphic display calculator to find the linear regression equation for v in terms of g, with coefficients correct to 3 significant figures.

(2)

(b)

Find the quadratic regression equation v = ag² + bg + c, with coefficients correct to 3 significant figures.

(2)

(c)

On a section with gradient 12% she measures v = 3.4. Use unrounded coefficients to find the speed predicted by each model at g = 12, and state which model is closer to the measurement.

(2)

(d)

Use the linear model (unrounded coefficients) to estimate her speed at g = 7, correct to 3 significant figures.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
gradient = −2.43	✓ 1
v = −2.43g + 33.1	✓ 1
Part (b)
a = −0.0134 and b = −2.29	✓ 1
c = 33.0	✓ 1
Part (c)
linear model: 4.0 km h⁻¹; quadratic model: 3.5 km h⁻¹	✓ 1
the quadratic model is closer to 3.4, so it fits better for steep gradients	✓ 1
Part (d)
v = 16.1 km h⁻¹	✓ 1

Exam tip: Compare models by testing them against a measurement outside the original data, not only by the correlation coefficient.

+ Add to test✓ In your test178IBMI3S-2-01r12-04Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

A ball is kicked from level ground. Its height h metres at horizontal distance x metres from the kicker is modelled by h = −0.02x² + 0.72x, for 0 ≤ x ≤ 36.

(a)

Find the maximum height reached by the ball and the horizontal distance at which it occurs.

(3)

(b)

A tall fence 2.44 m high is to be passed over. The ball is higher than 2.44 m for part of its flight. Find the horizontal distance over which the ball is higher than 2.44 m. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses x = −b/(2a) = 0.72/(2 × 0.02), or symmetry about the midpoint of the roots 0 and 36, or technology	✓ 1
A1 x = 18 m	✓ 1
A1 maximum height = 6.48 m	✓ 1
Part (b)
M1 −0.02x² + 0.72x = 2.44 (or equivalent) solved	✓ 1
A1 x = 3.79 and x = 32.2 (to 3 s.f.)	✓ 1
A1 distance = 28.4 m	✓ 1

Common error: A quadratic equation set equal to a height has two solutions; an answer that asks for a 'distance over which' needs both of them.

+ Add to test✓ In your test179IBMI3S-2-01r12-06Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

A coach records the maximum bench-press load M kg of an athlete t weeks after the start of a training block.

t (weeks)	2	4	6	8
M (kg)	66	70	73	76

Two models are proposed. The linear model M = 60 + 2.1t has a sum of squared residuals of 6.60. The quadratic model is M = −0.15t² + 3.1t + 60.

(a)

Calculate the values of M predicted by the quadratic model for t = 2, 4, 6 and 8.

(2)

(b)

Calculate the sum of squared residuals for the quadratic model.

(2)

(c)

State, with a reason, which model fits the data better.

(1)

(d)

The athlete's coach uses the quadratic model to predict the load at t = 30 weeks. Calculate this prediction and comment on its reliability.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes the t values into the quadratic model	✓ 1
A1 65.6, 70.0, 73.2 and 75.2 (kg)	✓ 1
Part (b)
M1 sums the squares of (observed − predicted) for the four data points: 0.4² + 0² + 0.2² + 0.8²	✓ 1
A1 0.84	✓ 1
Part (c)
R1 the quadratic model, because its sum of squared residuals (0.84) is smaller than that of the linear model (6.60)	✓ 1
Part (d)
A1 M = 18.0 kg	✓ 1
R1 a load of 18 kg is far below the recorded values and unrealistic for a training athlete, so the model should not be used far outside the range of the data (extrapolation)	✓ 1

Exam tip: A better fit to the data in the given range does not make a model reliable outside that range. Comment on extrapolation when asked about reliability.

+ Add to test✓ In your test180IBMI3S-2-01r12-07Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

The apparent magnitude M of a variable star, measured t days after a reference time, is modelled by M(t) = a cos(bt) + d, where t is in days and b is in radians per day. The largest value of M is 5.05, the smallest is 4.15 and the period is 5.4 days.

(a)

Find the value of a and the value of d.

(2)

(b)

A second telescope starts recording 0.8 days later, so its readings follow N(t) = M(t − 0.8). Describe the transformation that maps the graph of M to the graph of N, and state the first positive time at which N is at its maximum value.

(2)

(c)

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

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (5.05 − 4.15)/2 = 0.45	✓ 1
A1 d = (5.05 + 4.15)/2 = 4.6	✓ 1
Part (b)
A1 a horizontal translation of 0.8 units in the positive t direction (to the right)	✓ 1
A1 maximum of N first occurs at t = 0.8 days	✓ 1
Part (c)
M1 b = 2π/5.4	✓ 1
A1 b = 1.16	✓ 1

Common error: The period of a cosine model is 2π divided by b. Do not divide 360 by the period when the calculator is in radian mode.

+ Add to test✓ In your test181IBMI3S-2-01r12-10Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

The concentration C mg per litre of caffeine in an athlete's blood is modelled by C = 9.0e^−0.12t, where t is the time in hours after the drink is finished.

(a)

State the initial concentration.

(1)

(b)

Calculate the concentration 6 hours after the drink.

(2)

(c)

Find the time at which the concentration first falls to 2.5 mg per litre. Give your answer to 3 significant figures.

(2)

(d)

State what happens to the concentration in the long term according to the model.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 9.0 mg per litre	✓ 1
Part (b)
M1 9.0e^{−0.12 × 6}	✓ 1
A1 4.38 mg per litre	✓ 1
Part (c)
M1 9.0e^−0.12t = 2.5	✓ 1
A1 t = 10.7 hours	✓ 1
Part (d)
R1 it approaches 0 (the horizontal asymptote C = 0) but never reaches it	✓ 1

Exam tip: Use graphing technology to solve an exponential equation if the model is given, and state the accuracy only after the final step.

+ Add to test✓ In your test182IBMI3S-2-01r12-13Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

A space probe's light sensor measures the intensity I W m⁻² of starlight at distance d AU from a star. It is modelled by I = k/d², where k is a constant. At d = 2.0 AU the intensity is 340 W m⁻².

(a)

Find the value of k.

(2)

(b)

Find the distance at which the intensity is 10 W m⁻². Give your answer to 3 significant figures.

(2)

(c)

At d = 4.0 AU the sensor reads 86.0 W m⁻². Calculate the percentage error in the model's prediction, taking the reading as the exact value. Give your answer to 3 significant figures.

(2)

(d)

State what happens to the intensity predicted by the model when the distance from the star is doubled.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 340 = k/2.0²	✓ 1
A1 k = 1360	✓ 1
Part (b)
M1 10 = 1360/d²	✓ 1
A1 d = 11.7 AU	✓ 1
Part (c)
A1 model prediction 1360/16 = 85.0 W m⁻²	✓ 1
A1 percentage error = (86.0 − 85.0)/86.0 × 100 = 1.16%	✓ 1
Part (d)
R1 it falls to one quarter of its value	✓ 1

Exam tip: For an inverse-square model, check your result by testing a doubled or halved distance. The intensity should change by a factor of 4.

+ Add to test✓ In your test183IBMI3S-2-02r17-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

A wind-farm operator models its monthly profit, P thousand euros, when n turbines are in operation by
P(n) = −0.5n² + 14n − 40, for 0 ≤ n ≤ 40, where n is a whole number.

(a)

Find the number of turbines that gives the greatest monthly profit, and state this profit in euros.

(2)

(b)

Find the values of n for which the monthly profit is positive.

(3)

(c)

Find P(40) and interpret your answer in context.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 axis of symmetry n = −14 ÷ (2 × (−0.5)) = 14 (or use of technology)	✓ 1
A1 n = 14 and profit = 58 thousand euros = €58 000	✓ 1
Part (b)
M1 solve −0.5n² + 14n − 40 = 0	✓ 1
A1 roots n = 3.23… and n = 24.7…	✓ 1
A1 4 ≤ n ≤ 24 (whole numbers only)	✓ 1
Part (c)
A1 P(40) = −280	✓ 1
A1 a monthly loss of €280 000 when all 40 turbines are in operation	✓ 1

Common error: When the variable must be a whole number, find the roots first and then choose the whole numbers that lie inside the required interval.

+ Add to test✓ In your test184IBMI3S-2-02r17-07Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

The number of people, N, in a town library t hours after it opens at 08:00 is modelled by
N(t) = −0.5t³ + 7.5t² − 24t + 40, for 0 ≤ t ≤ 10.

(a)

Find the number of people in the library 5 hours after it opens.

(1)

(b)

Find the greatest number of people in the library during opening hours and the time at which this occurs, in hours after opening.

(2)

(c)

During some of the day there are more than 60 people in the library. Find the interval of time for which this happens. Give the values of t correct to 3 significant figures.

(3)

(d)

The library aims to have at least 1 member of staff for every 12 people. Find the least number of staff needed at the busiest time.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 N(5) = 45	✓ 1
Part (b)
M1 use of technology to find the maximum of the graph for 0 ≤ t ≤ 10 (compare with the end values N(0) = 40 and N(10) = 50)	✓ 1
A1 t = 8 and N = 72	✓ 1
Part (c)
M1 solve N(t) = 60 using technology	✓ 1
A1 t = 6.17 and t = 9.51	✓ 1
A1 6.17 < t < 9.51 (the third root is negative and rejected)	✓ 1
Part (d)
A1 72 ÷ 12 = 6 staff	✓ 1

Exam tip: For a cubic on a restricted domain, the greatest value may occur at a turning point or at an end of the domain, so check both.

+ Add to test✓ In your test185IBMI3S-2-02r17-09Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium6 marks

The heating in an outdoor swimming pool is switched off at 20:00. The temperature of the water, T °C, t hours later is modelled by T = 15 + 13 × 0.96^t, for t ≥ 0.

(a)

Write down the temperature of the water when the heating is switched off, and the temperature that the model predicts in the long term.

(2)

(b)

Find the temperature of the water at 08:00 the next day. Give your answer correct to 3 significant figures.

(2)

(c)

Find how long it takes for the water to cool to 20 °C. Give your answer in hours, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 28 °C	✓ 1
A1 15 °C (as t increases, 0.96^t tends to 0)	✓ 1
Part (b)
M1 t = 12 substituted: 15 + 13 × 0.96¹²	✓ 1
A1 23.0 °C	✓ 1
Part (c)
M1 15 + 13 × 0.96^t = 20 (equation set up and solved with technology)	✓ 1
A1 t = 23.4 hours	✓ 1

Exam tip: A model of the form a × b^t + c has a horizontal asymptote at y = c; subtract c before solving for t.

+ Add to test✓ In your test186IBMI3S-2-02r17-10Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

The wind speed, v m s⁻¹, at a wind farm t hours after midnight is modelled by v = 8 + 3cos(b(t − 15)), where b is in radians and the pattern repeats every 24 hours.

(a)

Write down the greatest wind speed predicted by the model.

(1)

(b)

Show that b = ^π⁄₁₂.

(1)

(c)

Find the wind speed at 20:00. Give your answer correct to 3 significant figures.

(2)

(d)

Turbines are shut down for maintenance when the wind speed exceeds 10 m s⁻¹. Find the total time in each 24-hour period for which the wind speed exceeds 10 m s⁻¹. Give your answer in hours, correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 11 m s⁻¹	✓ 1
Part (b)
A1 b = 2π ÷ period = 2π ÷ 24 = π/12 (AG)	✓ 1
Part (c)
M1 t = 20: 8 + 3cos((π/12) × 5)	✓ 1
A1 8.78 m s⁻¹	✓ 1
Part (d)
M1 8 + 3cos((π/12)(t − 15)) = 10, solved with technology or algebra	✓ 1
A1 t = 11.79… and t = 18.2…	✓ 1
A1 duration = 6.43 hours	✓ 1

Exam tip: For y = a cos(b(t − c)) + d, the period is 2π ÷ b, the greatest value is d + a, and the greatest value occurs at t = c.

+ Add to test✓ In your test187IBMI3S-2-02r17-13Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium6 marks

A wind-turbine technician records the power output, P kW, of a turbine at four steady wind speeds, v m s⁻¹.

v (m s⁻¹): 4, 6, 8, 10
P (kW): 55, 186, 440, 860

The technician suggests a model of the form P = kv³.

(a)

Calculate P ÷ v³ for each of the four wind speeds, correct to 3 significant figures, and explain why the results support the suggested model.

(2)

(b)

Write down the model, using k = 0.86.

(1)

(c)

Use the model to estimate the power output at a wind speed of 12 m s⁻¹, to the nearest kW.

(2)

(d)

The turbine has a maximum power output of 2000 kW. Explain why the model is unlikely to be valid for very high wind speeds.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 55 ÷ 64, 186 ÷ 216, 440 ÷ 512, 860 ÷ 1000 attempted	✓ 1
A1 0.859, 0.861, 0.859, 0.860 (all four correct), which are approximately constant, so P is approximately proportional to v³	✓ 1
Part (b)
A1 P = 0.86v³	✓ 1
Part (c)
M1 0.86 × 12³	✓ 1
A1 1486 kW	✓ 1
Part (d)
R1 the model predicts power increasing without limit (for example more than 2000 kW at 14 m s⁻¹, since 0.86 × 14³ = 2360), but the turbine cannot exceed its maximum output	✓ 1

Exam tip: A table of values suggests a power model y = kx^n when y ÷ x^n is nearly constant for the right choice of n.

+ Add to test✓ In your test188IBMI3S-2-01r20-05Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium7 marks

A tidal turbine can operate only when the depth of water above it is at least 6.0 m. The depth, h metres, at the turbine site is modelled by h = 5.2 + 1.8 sin(π(t − 2)/6), where t is the time in hours after midnight, for 0 ≤ t ≤ 12. The argument of the sine is in radians.

(a)

Write down the maximum depth of water and the period of the model.

(2)

(b)

Find the depth of water at t = 7.

(2)

(c)

Find the total time, in this 12-hour period, for which the turbine can operate. Give your answer in hours to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum depth 7.0 m	✓ 1
A1 period 12 hours	✓ 1
Part (b)
M1 5.2 + 1.8 sin(5π/6)	✓ 1
A1 6.1 m	✓ 1
Part (c)
M1 equation 5.2 + 1.8 sin(π(t − 2)/6) = 6 (or GDC intersection with h = 6)	✓ 1
A1 t = 2.88 and t = 7.12 (2.8795..., 7.1205...)	✓ 1
A1 4.24 hours	✓ 1

Exam tip: Set your calculator to radians for this model; a degree-mode answer will be wrong.

+ Add to test✓ In your test189IBMI3S-2-01r20-06Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium7 marks

The concentration of a medicine in a patient's blood, C mg/L, is modelled by C = 85e^−kt, where t is the time in hours after the medicine is first given and k is a positive constant. Three hours after the medicine is given, the concentration is measured as 45.3 mg/L.

(a)

Find the value of k, giving your answer to 3 significant figures.

(3)

(b)

Using k = 0.210, find the concentration 8 hours after the medicine is given.

(2)

(c)

The medicine is no longer effective once the concentration falls below 10 mg/L. Using k = 0.210, find how long after the medicine is given this happens. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 85e^−3k = 45.3	✓ 1
M1 attempt to solve, e.g. k = ln(85/45.3)/3	✓ 1
A1 k = 0.210	✓ 1
Part (b)
M1 85e^{−0.210 × 8}	✓ 1
A1 15.8 mg/L	✓ 1
Part (c)
M1 85e^−0.210t = 10	✓ 1
A1 10.2 hours (10.19...)	✓ 1

Exam tip: Take natural logarithms of both sides to solve for an exponent; ln(ex) = x.

+ Add to test✓ In your test190IBMI3S-2-01r20-07Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium6 marks

A packaging company makes open boxes from rectangular card measuring 30 cm by 20 cm. A square of side x cm is cut from each corner and the sides are folded up to make an open box with height x cm.

(a)

Show that the volume of the box, V cm³, is given by V = 4x³ − 100x² + 600x.

(2)

(b)

State the values that x can take in this model.

(1)

(c)

Find the maximum volume of the box, giving your answer to the nearest cm³.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 length = 30 − 2x and width = 20 − 2x, so V = x(30 − 2x)(20 − 2x)	✓ 1
A1 expansion to 4x³ − 100x² + 600x (must be shown)	✓ 1
Part (b)
A1 0 < x < 10	✓ 1
Part (c)
M1 use of GDC graph or maximum function on 0 < x < 10	✓ 1
A1 x = 3.92 (3.923...)	✓ 1
A1 V = 1056 cm³ (1056.3...)	✓ 1

Exam tip: Check that your maximum lies inside the domain of the model before quoting it.

+ Add to test✓ In your test191IBMI3S-2-01r20-08Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium7 marks

The annual electricity generated by a community solar scheme, S GWh, was recorded in its first four years of operation, where t is the number of years since the scheme opened.

t	1	2	3	4
S (GWh)	10.6	13.4	17.8	23.1

Two models are proposed: Model A is S = 5t + 4 and Model B is S = 8 × 1.3^t. To compare the models, the sum of the squares of the differences between each recorded value and the value predicted by the model is calculated.

(a)

Calculate this sum of squares for Model A.

(3)

(b)

For Model B the sum of squares is 0.168. State, with a reason, which model fits the data better.

(1)

(c)

Use the better model to predict the electricity generated in year 8. Give your answer to 3 significant figures, and give one reason why this prediction may be unreliable.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 predicted values 9, 14, 19, 24 (or differences 1.6, −0.6, −1.2, −0.9)	✓ 1
M1 attempt to square the differences and add	✓ 1
A1 5.17	✓ 1
Part (b)
R1 Model B, because its sum of squares (0.168) is smaller than that of Model A (5.17)	✓ 1
Part (c)
M1 8 × 1.3⁸	✓ 1
A1 65.3 GWh	✓ 1
R1 the prediction is an extrapolation outside the range of the data (or growth cannot continue indefinitely because of limited space or resources)	✓ 1

Exam tip: A smaller sum of squared differences means a closer fit to the given data, but it does not guarantee a good prediction outside the data range.

+ Add to test✓ In your test192IBMI3S-2-01r20-09Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium7 marks

Two companies hire out electric cars for one day. Company X charges a fixed £40, which includes the first 80 miles, and £0.30 for every mile after the first 80 miles. Company Y charges a fixed £22 plus £0.25 for every mile driven. A customer drives d miles in the day.

(a)

Find the cost of hiring from Company X when d = 150.

(2)

(b)

Find the value of d, for d > 80, at which the two companies charge the same amount.

(2)

(c)

Determine the range of values of d for which Company X is cheaper than Company Y.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 40 + 0.30 × (150 − 80)	✓ 1
A1 £61	✓ 1
Part (b)
M1 40 + 0.30(d − 80) = 22 + 0.25d	✓ 1
A1 d = 120	✓ 1
Part (c)
M1 for d ≤ 80, solving 40 = 22 + 0.25d	✓ 1
A1 d = 72	✓ 1
A1 72 < d < 120	✓ 1

Exam tip: For a piecewise model, check each section of the domain separately; a crossing point is only valid if it lies in that section.

+ Add to test✓ In your test193IBMI3S-2-01r20-11Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium7 marks

The power output, P kW, of a small wind turbine during a 12-hour test is modelled by P = at² + bt + c, where t is the number of hours since the test began. The graph of the model passes through the points (0, 12), (2, 20) and (4, 24).

(a)

Find the values of a, b and c.

(4)

(b)

Find the length of time for which the power output is greater than 20 kW.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 c = 12	✓ 1
M1 forming two equations, e.g. 4a + 2b = 8 and 16a + 4b = 12	✓ 1
A1 a = −0.5	✓ 1
A1 b = 5	✓ 1
Part (b)
M1 solving −0.5t² + 5t + 12 = 20 (or graph intersection)	✓ 1
A1 t = 2 and t = 8	✓ 1
A1 6 hours	✓ 1

Exam tip: Three points on a quadratic give three equations; use the point on the vertical axis first as it gives c immediately.

+ Add to test✓ In your test194IBMI3S-2-01r25-03Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium7 marks

A mobile network offers a monthly plan, Plan A, in which the cost, €C, for using d gigabytes (GB) of data is
C(d) = 14 for 0 ≤ d ≤ 8;
C(d) = 14 + 2.25(d − 8) for 8 < d ≤ 20;
C(d) = 41 + 4(d − 20) for d > 20.

(a)

Find the cost of using 15 GB of data on Plan A.

(2)

(b)

A customer pays €50 in a month on Plan A. Find the amount of data used.

(2)

(c)

Plan B has no fixed charge and costs €2.20 per GB. For 0 < d ≤ 20, find all values of d for which Plan B is cheaper than Plan A. Give the bounds correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 14 + 2.25(15 − 8)	✓ 1
A1 €29.75	✓ 1
Part (b)
M1 41 + 4(d − 20) = 50 (third piece identified, since 50 > 41)	✓ 1
A1 d = 22.25 GB	✓ 1
Part (c)
M1 2.2d < 14 for 0 < d ≤ 8	✓ 1
A1 d < 6.36 GB	✓ 1
R1 for 8 < d ≤ 20, 2.2d > 14 + 2.25(d − 8) (e.g. at d = 8 Plan B costs €17.60 against €14, and the gap only closes at 80 GB), so Plan B is never cheaper there	✓ 1

Exam tip: With a piecewise function, first decide which piece applies by comparing the given value with the boundaries, then solve using that piece only.

+ Add to test✓ In your test195IBMI3S-2-01r25-04Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium6 marks

The number of hours of daylight, H, in a mountain village on day d of the year (where d = 1 is 1 January) is modelled by H(d) = a cos(b(d − 172)) + k, with the angle in radians. The longest day, d = 172, has 15.6 hours of daylight and the shortest day has 8.4 hours of daylight. Assume a year has 365 days.

(a)

Find the value of a and the value of k.

(2)

(b)

Show that b = 0.0172, correct to 3 significant figures.

(1)

(c)

Find the hours of daylight on day 100. Give your answer correct to 3 significant figures.

(1)

(d)

Find the first value of d at which there are 13.5 hours of daylight. Give your answer correct to 1 decimal place.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (15.6 − 8.4) ÷ 2 = 3.6	✓ 1
A1 k = (15.6 + 8.4) ÷ 2 = 12	✓ 1
Part (b)
A1 b = 2π ÷ 365 = 0.017214… = 0.0172 (period is 365 days)	✓ 1
Part (c)
A1 H(100) = 13.2 hours	✓ 1
Part (d)
M1 3.6 cos(b(d − 172)) + 12 = 13.5, solved for the smaller d (using radians)	✓ 1
A1 d = 105.7	✓ 1

Exam tip: Check that your calculator is in radian mode whenever the model is given with the angle in radians.

+ Add to test✓ In your test196IBMI3S-2-01r25-07Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium6 marks

The number of households, N, connected to a fibre broadband network in a town is modelled by N = 1200e^kt, where t is the number of months after the network launched. After 3 months, 2100 households are connected.

(a)

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

(2)

(b)

Find the number of households connected after 8 months. Use the unrounded value of k, and give your answer to the nearest whole number.

(2)

(c)

The network will be extended once 10 000 households are connected. Find how many months after launch this happens. Use the unrounded value of k and give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 2100 = 1200e^3k	✓ 1
A1 k = 0.187	✓ 1
Part (b)
M1 N = 1200e^8k (unrounded k = ln 1.75 ÷ 3)	✓ 1
A1 5337 households	✓ 1
Part (c)
M1 1200e^kt = 10 000	✓ 1
A1 t = 11.4 months	✓ 1

Exam tip: Keep the unrounded value of k in your calculator memory; using 0.187 in later parts can change the third significant figure.

+ Add to test✓ In your test197IBMI3S-2-01r25-10Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium7 marks

The boiling point of water, B °C, depends on altitude, h metres above sea level. Measurements show that B = 96.7 at h = 1000 and B = 87.1 at h = 4000. Assume B is a linear function of h.

(a)

Find an equation for B in terms of h.

(3)

(b)

Interpret, in context, (i) the gradient and (ii) the constant term of your equation.

(2)

(c)

Find the altitude at which water boils at 90 °C. Give your answer to the nearest metre.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient = (87.1 − 96.7) ÷ (4000 − 1000)	✓ 1
A1 gradient = −0.0032	✓ 1
A1 B = 99.9 − 0.0032h	✓ 1
Part (b)
A1 the boiling point falls by 0.0032 °C for each additional metre of altitude (3.2 °C per 1000 m)	✓ 1
A1 the boiling point predicted at sea level (h = 0) is 99.9 °C	✓ 1
Part (c)
M1 99.9 − 0.0032h = 90	✓ 1
A1 h = 3094 m	✓ 1

Exam tip: Interpret a gradient with its units and direction: say what changes, by how much, and per what.

+ Add to test✓ In your test198IBMI3S-2-01r25-12Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Medium6 marks

The rate of carbon dioxide production, R cm³ per minute, by a dough as it ferments is modelled by R = −0.02(T − 35)² + 9, where T °C is the temperature of the dough, for 15 ≤ T ≤ 55.

(a)

Write down the temperature at which the rate is greatest and the greatest rate.

(2)

(b)

Show that R can be written as R = −0.02T² + 1.4T − 15.5.

(2)

(c)

Find the range of temperatures for which the rate is at least 5 cm³ per minute. Give your bounds correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 T = 35 °C	✓ 1
A1 9 cm³ per minute	✓ 1
Part (b)
M1 −0.02(T² − 70T + 1225) + 9	✓ 1
A1 −0.02T² + 1.4T − 24.5 + 9 = −0.02T² + 1.4T − 15.5	✓ 1
Part (c)
M1 −0.02(T − 35)² + 9 = 5, so (T − 35)² = 200	✓ 1
A1 20.9 ≤ T ≤ 49.1	✓ 1

Exam tip: Vertex form shows the maximum immediately: the vertex of a(x − h)² + k is (h, k).

+ Add to test✓ In your test199IBMI3S-2-01r28-03Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium7 marks

A cyclist rides along a straight road away from a village. The distance d km from the village is modelled by a linear function of the time t hours after 08:00. At t = 0.5 the cyclist is 15 km from the village and at t = 2 the cyclist is 42 km from the village.

(a)

Find the gradient of the model and state its meaning.

(3)

(b)

Find the equation of the model in the form d = mt + c.

(2)

(c)

The road is 100 km long from the village. Find the time, in hours, taken from 08:00 to reach the end of the road, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (42 − 15)/(2 − 0.5)	✓ 1
A1 gradient = 18	✓ 1
A1 the cyclist's speed is 18 km h⁻¹	✓ 1
Part (b)
M1 substitute a point to find c, e.g. 15 = 18(0.5) + c	✓ 1
A1 d = 18t + 6	✓ 1
Part (c)
M1 set 18t + 6 = 100	✓ 1
A1 t = 94/18 = 5.22 hours	✓ 1

Exam tip: State the unit of a gradient as (unit of y) per (unit of x); here km per hour is km h⁻¹.

+ Add to test✓ In your test200IBMI3S-2-01r28-05Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium7 marks

Water is seeping into an excavation trench. The depth d cm of water in the trench, t hours after pumping starts, is modelled by
d(t) = −t³ + 9t² − 15t + 20, for 0 ≤ t ≤ 7.

(a)

Find the depth of water when pumping starts.

(1)

(b)

Use your graphic display calculator to find the greatest depth of water during the 7 hours, and the time at which it occurs.

(3)

(c)

Find the times, to 3 significant figures, at which the depth is 30 cm.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 20 cm	✓ 1
Part (b)
M1 use of GDC to find the local maximum of the graph on the domain	✓ 1
A1 t = 5 hours	✓ 1
A1 depth 45 cm	✓ 1
Part (c)
M1 set −t³ + 9t² − 15t + 20 = 30 and use GDC to solve	✓ 1
A1 t = 3.08	✓ 1
A1 t = 6.42	✓ 1

Exam tip: When solving with a GDC on a restricted domain, reject any solution outside the domain; here a third root is negative and is not accepted.

+ Add to test✓ In your test201IBMI3S-2-01r28-07Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium6 marks

The amount of carbon-14 in a sample from a hearth decays according to N = N₀e^−kt, where N₀ is the original amount, t is the time in years and k is a constant. The half-life of carbon-14 is 5730 years.

(a)

Show that k = 1.21 × 10⁻⁴ to 3 significant figures.

(3)

(b)

Charcoal from the hearth contains 62% of its original carbon-14. Use the unrounded value of k to find the age of the charcoal, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 set e^−5730k = 0.5	✓ 1
M1 k = ln 2 / 5730	✓ 1
A1 k = 1.209… × 10⁻⁴ = 1.21 × 10⁻⁴	✓ 1
Part (b)
M1 set 0.62 = e^−kt	✓ 1
M1 t = −ln 0.62 / k	✓ 1
A1 t = 3.95 × 10³ years	✓ 1

Common error: Do not use the rounded value of a constant in a later calculation unless told to; keep the full calculator value to protect the third significant figure.

+ Add to test✓ In your test202IBMI3S-2-01r28-10Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium6 marks

A data centre's cooling system uses E(t) = 0.5t² + 20 megawatt-hours in the first t hours after it starts up. A second building starts its cooling 3 hours later, uses 5 MWh more at every stage, and otherwise follows the same pattern. Its energy use t hours after the first system starts (for t ≥ 3) is G(t).

(a)

Write down an expression for G(t) in terms of E.

(2)

(b)

Find G(7).

(2)

(c)

Describe fully the single transformation that maps the graph of y = E(t) onto the graph of y = G(t).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 delay of 3 hours: E(t − 3)	✓ 1
A1 G(t) = E(t − 3) + 5	✓ 1
Part (b)
M1 E(4) = 0.5(16) + 20 = 28	✓ 1
A1 G(7) = 33 MWh	✓ 1
Part (c)
A1 translation	✓ 1
A1 by the vector (3, 5) (3 units right, 5 units up)	✓ 1

Common error: A translation to the right by 3 is written f(x − 3), not f(x + 3); the sign inside the bracket is opposite to the direction of the shift.

+ Add to test✓ In your test203IBMI3S-2-03r31-03Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Medium6 marks

The number of fish, N, in a managed lake is modelled by N = 4800 × 0.88^t, where t is the number of years since monitoring began.

(a)

Write down the number of fish when monitoring began.

(1)

(b)

State the percentage by which the population falls each year according to the model.

(1)

(c)

Find how long it takes for the population to fall below 1500. Give your answer in years to 3 significant figures.

(3)

(d)

Suggest one reason why the model may not be suitable for predicting the population many years after monitoring began.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4800	✓ 1
Part (b)
A1 12 %	✓ 1
Part (c)
M1 setting 4800 × 0.88^t = 1500	✓ 1
M1 attempt to solve (graphically or using logarithms)	✓ 1
A1 t = 9.10 (years)	✓ 1
Part (d)
R1 e.g. the model never reaches zero, but a real population could die out; or fishing limits/restocking could change the rate of decline; or the rate of decline is unlikely to stay constant	✓ 1

Exam tip: In exponential models of the form a × bt, the multiplier b = 0.88 means 100 % − 12 % remains each period.

+ Add to test✓ In your test204IBMI3S-2-03r31-05Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Medium6 marks

A fabric supplier charges €C for x metres of cloth, where
C(x) = 8.5x for 0 < x ≤ 40
C(x) = 340 + 6(x − 40) for x > 40.

(a)

Find the cost of 25 metres of cloth.

(1)

(b)

Find the cost of 70 metres of cloth.

(2)

(c)

A designer has a budget of €700. Find the greatest length of cloth she can buy.

(2)

(d)

Find the average cost per metre of the cloth in (c).

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €212.50	✓ 1
Part (b)
M1 use of the second rule with x = 70, e.g. 340 + 6 × 30	✓ 1
A1 €520	✓ 1
Part (c)
M1 recognising the second rule applies (700 > 340) and setting 340 + 6(x − 40) = 700	✓ 1
A1 x = 100 metres	✓ 1
Part (d)
A1 €7 per metre (ECF from (c))	✓ 1

Common error: With a piecewise function, decide which rule applies before substituting; here a cost above €340 must use the second rule.

+ Add to test✓ In your test205IBMI3S-2-03r31-07Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Medium7 marks

The annual catch of a fishery, C thousand tonnes, is recorded t years after the start of 2015. At t = 10 the recorded catch was 39.5. Two models are proposed.
Model A: C = 52.4 − 1.4t
Model B: C = 52.9 × 0.972^t

(a)

Calculate the catch predicted by each model at t = 10. Give each answer to 3 significant figures.

(2)

(b)

Calculate the percentage error in the prediction of Model A at t = 10. Give your answer to 3 significant figures.

(2)

(c)

The percentage error in the prediction of Model B at t = 10 is 0.814 %. State, with a reason, which model fits the recorded value better.

(1)

(d)

The fishery uses Model B to predict the catch at t = 40. Calculate this prediction, and give one reason why Model A would be unsuitable for such a prediction. Give the prediction to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model A: 38.4	✓ 1
A1 Model B: 39.8	✓ 1
Part (b)
M1 \|38.4 − 39.5\| / 39.5 × 100	✓ 1
A1 2.78 %	✓ 1
Part (c)
R1 Model B, since its percentage error is smaller (0.814 % < 2.78 %)	✓ 1
Part (d)
A1 17.0 (thousand tonnes)	✓ 1
R1 Model A gives a negative catch (−3.6) at t = 40, which is impossible	✓ 1

Exam tip: When a question asks which model is better, give a numerical comparison (such as percentage errors) and state the conclusion in a sentence.

+ Add to test✓ In your test206IBMI3S-2-03r31-10Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Medium6 marks

A robot launches a ball upwards. The height of the ball above the ground, h metres, is modelled by h(t) = 1.2 + 13.5t − 4.9t², where t is the time in seconds after launch.

(a)

Write down the height from which the ball is launched.

(1)

(b)

Find the greatest height reached by the ball. Give your answer to 3 significant figures.

(2)

(c)

Find the time at which the ball hits the ground. Give your answer to 3 significant figures.

(2)

(d)

Explain why the other solution of h(t) = 0 is rejected.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.2 m	✓ 1
Part (b)
M1 attempt to find the maximum (vertex at t = 13.5/9.8 or graphing)	✓ 1
A1 10.5 m	✓ 1
Part (c)
M1 setting h(t) = 0 and attempting to solve	✓ 1
A1 2.84 s	✓ 1
Part (d)
R1 it is negative (t ≈ −0.0862), which is before the launch so outside the domain	✓ 1

Common error: Setting h = 0 gives two solutions; always choose the one that makes sense in the context (here the positive time).

+ Add to test✓ In your test207IBMI3S-2-03r31-11Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Medium7 marks

The number of followers of a fashion brand's social media page is modelled by F(t) = a × b^t, where t is the number of months since the page was created. The page had 1200 followers at t = 3 and 2100 followers at t = 7.

(a)

Find the value of b. Give your answer to 4 significant figures.

(2)

(b)

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

(2)

(c)

Use the model to predict the number of followers at t = 12. Use the unrounded values of a and b and give your answer to the nearest whole number.

(2)

(d)

Interpret the value of b in this context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 dividing the two equations to obtain b⁴ = 2100/1200 = 1.75	✓ 1
A1 b = 1.150	✓ 1
Part (b)
M1 substituting into 1200 = ab³ (using unrounded b)	✓ 1
A1 a = 789	✓ 1
Part (c)
M1 substituting t = 12 (e.g. 1200 × b⁹)	✓ 1
A1 4227	✓ 1
Part (d)
A1 the number of followers grows by about 15 % each month	✓ 1

Exam tip: To find both parameters of a × bt, divide one equation by the other to eliminate a, then take the appropriate root.

+ Add to test✓ In your test208IBMI3S-2-03r31-14Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 1Medium7 marks

On a fish farm, the weight gain, Y kg, of the fish in a pond is modelled by Y(x) = ax² + bx + c, where x is the mass of feed in kg given per day. Trials give Y = 21 when x = 10, Y = 45 when x = 40 and Y = 33 when x = 70.

(a)

Write down three equations that must be satisfied by a, b and c.

(2)

(b)

Solve the equations to find a, b and c.

(2)

(c)

Find the greatest weight gain predicted by the model.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substituting the data points into the model	✓ 1
A1 100a + 10b + c = 21, 1600a + 40b + c = 45, 4900a + 70b + c = 33	✓ 1
Part (b)
M1 attempt to solve the system (e.g. calculator)	✓ 1
A1 a = −0.02, b = 1.8, c = 5	✓ 1
Part (c)
M1 x = −b/(2a) = 45 (or use of graph)	✓ 1
M1 substituting into the model	✓ 1
A1 45.5 kg	✓ 1

Exam tip: Three data points give three equations, one for each unknown; a graphing or solver tool is the quickest way to solve them in a calculator paper.

+ Add to test✓ In your test209IBMI3S-2-02r32-02Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium7 marks

Freshly roasted coffee beans are tipped into a cooling tray in a room kept at 22 °C. The temperature of the beans, T °C, after t minutes is modelled by
T(t) = 22 + 178e^−0.12t, for t ≥ 0.

(a)

Find the temperature of the beans when they are tipped into the tray.

(2)

(b)

Find the temperature of the beans after 5 minutes. Give your answer to 3 significant figures.

(2)

(c)

The beans can be packed when their temperature is 40 °C. Find the time taken to reach this temperature, giving your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 0: 22 + 178e⁰	✓ 1
A1 200 °C	✓ 1
Part (b)
M1 substitute t = 5	✓ 1
A1 120 °C (119.69…)	✓ 1
Part (c)
M1 set 22 + 178e^−0.12t = 40	✓ 1
M1 attempt to solve (e.g. e^−0.12t = 18/178, take natural logarithms, or use technology)	✓ 1
A1 19.1 minutes	✓ 1

Exam tip: Subtract the constant (the room temperature) first before taking logarithms, otherwise the exponential term is not isolated.

+ Add to test✓ In your test210IBMI3S-2-02r32-07Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium7 marks

After a thruster burn, the temperature of a sensor on a space probe is modelled by
C(t) = −45 + 85e^−0.08t, for t ≥ 0,
where C is in °C and t is the time in minutes since the burn ended.
Temperatures in °F are found from temperatures in °C using F = 1.8C + 32.

(a)

Write down the equation of the horizontal asymptote of the graph of C against t.

(1)

(b)

Describe the transformations that map the graph of C against t onto the graph of F against t.

(2)

(c)

Find an expression for F(t) in the form p + qe^−0.08t.

(2)

(d)

Find the time for the sensor temperature to fall to −20 °C. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 C = −45	✓ 1
Part (b)
A1 vertical stretch with scale factor 1.8	✓ 1
A1 vertical translation of 32 units up (stretch then translate; accept any equivalent description)	✓ 1
Part (c)
M1 F = 1.8(−45 + 85e^−0.08t) + 32	✓ 1
A1 F(t) = −49 + 153e^−0.08t	✓ 1
Part (d)
M1 set −45 + 85e^−0.08t = −20 and solve	✓ 1
A1 15.3 minutes	✓ 1

Exam tip: A conversion of the form F = mC + k stretches the graph vertically by m and then translates it by k; the asymptote moves with it.

+ Add to test✓ In your test211IBMI3S-2-02r32-09Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium7 marks

A toll bridge charges vehicles according to their mass w tonnes. The toll, T euros (€), is modelled by
T(w) = 4 for 0 < w ≤ 2,
T(w) = 4 + 1.5(w − 2) for 2 < w ≤ 6,
T(w) = 10 + 2.5(w − 6) for w > 6.

(a)

Find the toll for a vehicle of mass 4.5 tonnes.

(2)

(b)

Find the toll for a vehicle of mass 8.2 tonnes.

(2)

(c)

A lorry pays a toll of €17. Find its mass.

(2)

(d)

Show that the toll does not jump (is continuous) at w = 6.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use the second expression: 4 + 1.5(4.5 − 2)	✓ 1
A1 €7.75	✓ 1
Part (b)
M1 use the third expression: 10 + 2.5(8.2 − 6)	✓ 1
A1 €15.50	✓ 1
Part (c)
M1 set 10 + 2.5(w − 6) = 17 (the toll exceeds €10, so the third expression applies)	✓ 1
A1 8.8 tonnes	✓ 1
Part (d)
R1 second expression at w = 6 gives 4 + 1.5 × 4 = 10 and the third gives 10 + 0 = 10, so the values agree	✓ 1

Exam tip: For a piecewise function, first decide which interval the input lies in; for a given output, check which piece can produce it.

+ Add to test✓ In your test212IBMI3S-2-02r32-13Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium7 marks

The capacity of a space probe's battery, as a percentage of its original capacity, is measured every 6 months.

Time, t (months)	0	6	12	18	24
Capacity, y (%)	100	91	83	76	70

Two models are proposed.
Model A: y = 100 − 1.2t. Model B: y = 100 × 0.985^t.
For Model A, the sum of the squares of the differences between the measured and predicted values is 17.2.

(a)

Use Model B to find the predicted capacity at t = 12 and at t = 24. Give your answers to 3 significant figures.

(2)

(b)

For Model B, find the sum of the squares of the differences between the measured and predicted values for the five measurements. Use unrounded predicted values. Give your answer to 3 significant figures.

(2)

(c)

State which model fits the data better, giving a reason.

(1)

(d)

Use Model A to predict the capacity at t = 90, and explain why this shows Model A is not suitable for long-term prediction.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 12 or t = 24 into 100 × 0.985^t	✓ 1
A1 83.4 and 69.6	✓ 1
Part (b)
M1 find the five differences (measured − predicted) and square them	✓ 1
A1 0.492	✓ 1
Part (c)
R1 Model B, because its sum of squared differences (0.492) is much smaller than that of Model A (17.2)	✓ 1
Part (d)
A1 −8 %	✓ 1
R1 a capacity cannot be negative, so the linear model fails for large t	✓ 1

Exam tip: A smaller sum of squared differences means a closer fit over the data range, but a model must also behave sensibly outside that range.

+ Add to test✓ In your test213IBMI3S-2-04r32-05Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium8 marks

The rate at which water is demanded from a town's supply, D litres per second, varies during the day. It is modelled by $D = 52 + 14\sin\!\left(\dfrac{\pi}{12}(t-4)\right)$, where t is the time in hours after midnight, for $0 \le t \le 24$. The argument of the sine is in radians.

(a)

Write down the maximum and minimum demand predicted by the model.

(2)

(b)

Write down the period of the model in hours.

(1)

(c)

Find the demand at 18:00.

(2)

(d)

Find the first time after midnight at which the demand is 60 litres per second. Give your answer in hours to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 66 litres per second	✓ 1
A1 minimum 38 litres per second	✓ 1
Part (b)
A1 24 hours	✓ 1
Part (c)
M1 substitute $t = 18$	✓ 1
A1 45.0 litres per second	✓ 1
Part (d)
M1 $52 + 14\sin\!\left(\dfrac{\pi}{12}(t-4)\right) = 60$ (or GDC intersection)	✓ 1
M1 $\sin(\ldots) = \dfrac{8}{14}$, $\dfrac{\pi}{12}(t-4) = 0.6082$ (radian mode)	✓ 1
A1 $t = 6.32$, so 6.32 hours (about 06:19)	✓ 1

Common error: Set the calculator to radians when the angle is given in radians; degree mode gives a different, incorrect value.

+ Add to test✓ In your test214IBMI3S-2-04r32-07Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

In a trial of a blood pressure drug, the reduction in systolic blood pressure R (mmHg) is modelled by the linear function $R = 0.42d + 3.1$, where d is the daily dose in mg. The model was fitted using data for doses between 10 mg and 50 mg.

(a)

Interpret the value 0.42 in this context.

(1)

(b)

Use the model to estimate the reduction in blood pressure for a dose of 40 mg.

(2)

(c)

Use the model to estimate the dose needed for a reduction of 30 mmHg. Give your answer to 3 significant figures.

(2)

(d)

Comment on the reliability of your answer to part (c).

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 each additional 1 mg of daily dose is associated with an extra 0.42 mmHg reduction in blood pressure	✓ 1
Part (b)
M1 $0.42\times40 + 3.1$	✓ 1
A1 19.9 mmHg	✓ 1
Part (c)
M1 $0.42d + 3.1 = 30$	✓ 1
A1 64.0 mg	✓ 1
Part (d)
R1 the dose lies outside the 10 mg to 50 mg range of the data, so this is extrapolation and may be unreliable	✓ 1

Common error: An answer found by extrapolation outside the data range should be flagged as unreliable; interpolation within the range is more trustworthy.

+ Add to test✓ In your test215IBMI3S-2-03r34-04Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Medium7 marks

A boutique sells hand-printed scarves. The weekly profit, P pounds, when the selling price of a scarf is £x, is modelled by P(x) = −2x² + 96x − 640.

(a)

Find the selling price that gives the greatest weekly profit.

(2)

(b)

Find the range of selling prices for which the boutique makes a profit.

(3)

(c)

Find the greatest weekly profit.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 axis of symmetry x = −96/(2 × (−2)) (or graphical maximum)	✓ 1
A1 £24	✓ 1
Part (b)
M1 solving −2x² + 96x − 640 = 0	✓ 1
A1 x = 8 and x = 40	✓ 1
A1 8 < x < 40, that is, between £8 and £40	✓ 1
Part (c)
M1 substituting x = 24 into P (ECF from (a))	✓ 1
A1 £512	✓ 1

Exam tip: The roots and the vertex of a quadratic model are linked: the vertex lies halfway between the roots (24 is halfway between 8 and 40), which gives a quick check.

+ Add to test✓ In your test216IBMI3S-2-03r34-05Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Medium7 marks

The depth of water, d metres, at the end of a harbour jetty is modelled by d = a + b cos(c(t − 2)), where t is the time in hours after midnight and angles are in radians. The greatest depth, 8.1 m, occurs at t = 2. The least depth is 3.3 m. The depth pattern repeats every 12 hours.

(a)

Find the value of a and the value of b.

(2)

(b)

Find the exact value of c.

(2)

(c)

A fishing boat needs a depth of at least 4.5 m. Find the length of time, in the first 12 hours after midnight, for which the depth is less than 4.5 m.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (8.1 + 3.3)/2 = 5.7	✓ 1
A1 b = (8.1 − 3.3)/2 = 2.4	✓ 1
Part (b)
M1 period = 2π/c = 12	✓ 1
A1 c = π/6	✓ 1
Part (c)
M1 setting 5.7 + 2.4cos(π(t − 2)/6) = 4.5, so cos(π(t − 2)/6) = −0.5	✓ 1
A1 t = 6 and t = 10	✓ 1
A1 4 hours	✓ 1

Common error: Make sure the calculator is in radian mode when the model uses radians, and check both solutions of the equation within the stated interval.

+ Add to test✓ In your test217IBMI3S-2-03r34-07Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Medium6 marks

A robot's gripper follows a path with height f(x) = −0.25x² + 2x metres, where x is the horizontal distance in metres from its base, for 0 ≤ x ≤ 8. A second robot, positioned further along the line, follows the path g(x) = f(x − 3) + 1.

(a)

Describe the single transformation that maps the graph of f onto the graph of g.

(2)

(b)

The highest point of the path of f is (4, 4). Write down the highest point of the path of g.

(2)

(c)

Find g(5).

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 translation	✓ 1
A1 by the vector (3, 1), that is 3 units in the positive x-direction and 1 unit up	✓ 1
Part (b)
M1 applying the translation to (4, 4)	✓ 1
A1 (7, 5)	✓ 1
Part (c)
M1 g(5) = f(2) + 1	✓ 1
A1 4 m	✓ 1

Common error: In f(x − 3) the graph moves 3 units to the right, not to the left.

+ Add to test✓ In your test218IBMI3S-2-03r34-09Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Medium7 marks

Two companies hire out fishing boats. Company A charges €(70 + 59h) for h hours. Company B charges a fixed fee plus a constant hourly rate. Company B charges €216 for 2 hours and €396 for 6 hours.

(a)

Find the equation of the form C = mh + k for the cost C euros charged by Company B.

(3)

(b)

State what the value 126 represents.

(1)

(c)

Find the number of hours for which both companies charge the same amount.

(2)

(d)

For a 10-hour hire, find how much cheaper the cheaper company is.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient (396 − 216)/(6 − 2)	✓ 1
A1 m = 45	✓ 1
A1 k = 126, so C = 45h + 126	✓ 1
Part (b)
A1 the fixed fee (cost for zero hours) charged by Company B, €126	✓ 1
Part (c)
M1 setting 70 + 59h = 126 + 45h	✓ 1
A1 h = 4	✓ 1
Part (d)
A1 A: €660, B: €576, so B is cheaper by €84	✓ 1

Exam tip: Two different linear cost models cross at the point where they charge the same; before and after that point a different company is cheaper.

+ Add to test✓ In your test219IBMI3S-2-03r34-10Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Medium7 marks

The number of followers of a fashion brand's online page is modelled by F = 1200e^0.045t, where t is the time in months since the page was launched.

(a)

Find the number of followers after 10 months, to the nearest whole number.

(2)

(b)

Find the time taken for the number of followers to double. Give your answer to 3 significant figures.

(3)

(c)

The model can be written as F = 1200 × r^t. Find the percentage increase in followers each month, to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 1200e^{0.045 × 10}	✓ 1
A1 1882	✓ 1
Part (b)
M1 setting 1200e^0.045t = 2400 (or e^0.045t = 2)	✓ 1
M1 taking natural logarithms, 0.045t = ln 2	✓ 1
A1 t = 15.4 months	✓ 1
Part (c)
M1 r = e^0.045 = 1.0460…	✓ 1
A1 4.60 %	✓ 1

Common error: The 0.045 in the exponent is not the percentage growth rate; convert with r = e⁰.⁰⁴⁵ before reading off a percentage.

+ Add to test✓ In your test220IBMI3S-2-03r34-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 1Medium7 marks

The weekly sales of a scarf (in units) in its first six weeks are shown.

Week, w	1	2	3	4	5	6
Sales, S	120	185	230	250	248	222

A quadratic model S = aw² + bw + c is fitted to the data using quadratic regression.

(a)

Find the values of a, b and c, each to 3 significant figures.

(2)

(b)

Find the value of w at the vertex of the model, to 3 significant figures, and hence state the whole number of weeks with the greatest sales according to the model.

(2)

(c)

The model predicts negative sales for large values of w. Find the positive solution of S = 0, to 2 significant figures, and state the first week in which negative sales are predicted.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = −11.5 and b = 101	✓ 1
A1 c = 30.1	✓ 1
Part (b)
A1 w = 4.39	✓ 1
R1 the whole week is 4, since 4.39 is closer to 4 than to 5 (accept comparing S(4) and S(5))	✓ 1
Part (c)
M1 solving −11.5w² + 101w + 30.1 = 0	✓ 1
A1 w = 9.1	✓ 1
A1 week 10	✓ 1

Common error: Quadratic models turn downward without limit, so a prediction far outside the data can be impossible, as with negative sales here.

+ Add to test✓ In your test221IBMI3S-2-01r35-04Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 1Medium6 marks

Calculator

The number of bees, N, in a hive w weeks after spring inspection is modelled by N = 12 000 × 1.06^w.

(a)

Find the number of bees after 5 weeks. Give your answer to the nearest whole number.

(2)

(b)

Interpret the value 1.06 in this context.

(1)

(c)

Find the number of weeks for the population to reach 30 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes w = 5: 12 000 × 1.06⁵	✓ 1
A1 16 059 (16 058.7…)	✓ 1
Part (b)
A1 the population grows by 6% each week (multiplied by 1.06 each week)	✓ 1
Part (c)
M1 forms 12 000 × 1.06^w = 30 000, so 1.06^w = 2.5	✓ 1
M1 uses logarithms or the GDC to solve	✓ 1
A1 w = 15.7 weeks (15.725…)	✓ 1

Exam tip: Rewrite the equation so the exponential term is on its own before taking logarithms.

+ Add to test✓ In your test222IBMI3S-2-01r35-05Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 1Medium6 marks

Calculator

The number of patients, f(t), in one hospital department t hours after midnight is modelled by f(t) = 40 − 0.25(t − 10)², for 4 ≤ t ≤ 16.
A second department follows the model g(t) = f(t − 2) + 5, for 6 ≤ t ≤ 18.

(a)

Describe the single transformation that maps the graph of y = f(t) onto the graph of y = g(t).

(2)

(b)

Write down the coordinates of the vertex of the graph of y = g(t).

(1)

(c)

Find the length of time, in hours, for which g(t) is at least 40. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 translation (shift) by 2 units in the positive t direction	✓ 1
A1 and by 5 units in the positive vertical direction (vector (2, 5))	✓ 1
Part (b)
A1 (12, 45)	✓ 1
Part (c)
M1 forms g(t) = 45 − 0.25(t − 12)² = 40 (or equivalent graph intersection)	✓ 1
A1 t = 7.53… and t = 16.5… (12 ± √20)	✓ 1
A1 2√20 = 8.94 hours	✓ 1

Common error: Replacing t by t − 2 moves the graph to the right, not the left.

+ Add to test✓ In your test223IBMI3S-2-01r35-13Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 1Medium6 marks

Calculator

The mass of fuel, f kg, in a Formula 1 car is modelled as a linear function of the number of laps completed, n. After 10 laps the car carries 84.0 kg and after 30 laps it carries 56.0 kg.

(a)

Find the equation of the model in the form f = mn + c.

(3)

(b)

Interpret the value of c in this context.

(1)

(c)

The race is 58 laps long. Find the mass of fuel left at the end of the race according to the model.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient = (56 − 84) ÷ (30 − 10)	✓ 1
A1 m = −1.4	✓ 1
A1 f = 98 − 1.4n	✓ 1
Part (b)
A1 the mass of fuel (98 kg) at the start, before any laps	✓ 1
Part (c)
M1 substitutes n = 58: 98 − 1.4 × 58	✓ 1
A1 16.8 kg	✓ 1

Exam tip: A negative gradient is a rate of decrease; state the units of the rate when you interpret it.

+ Add to test✓ In your test224IBMI3S-2-02r35-02Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium6 marks

Calculator

A diver leaves a springboard at a swimming pool. Her height above the water, h metres, at time t seconds after leaving the board is modelled by h = −4.9t² + 2.4t + 1.8.

(a)

Write down the height of the springboard above the water.

(1)

(b)

Find the time at which the diver enters the water. Give your answer to 3 significant figures.

(3)

(c)

Find the greatest height reached by the diver above the water. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.8 m	✓ 1
Part (b)
M1 sets −4.9t² + 2.4t + 1.8 = 0	✓ 1
R1 rejects the negative solution (t = −0.409…) as the dive starts at t = 0	✓ 1
A1 t = 0.899 s (0.8986…)	✓ 1
Part (c)
M1 uses t = 2.4 ÷ 9.8 = 0.245… (or the maximum on the GDC)	✓ 1
A1 2.09 m (2.0939…)	✓ 1

Common error: Reject solutions that make no sense in context, such as negative times.

+ Add to test✓ In your test225IBMI3S-2-02r35-07Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium6 marks

Calculator

The power output, P MW, of a wind turbine at wind speed v m s⁻¹ is modelled by
P = 0.0025v³ for 3 ≤ v ≤ 12
P = 4.32 for 12 < v ≤ 25.

(a)

Find the power output at a wind speed of 8 m s⁻¹.

(2)

(b)

Find the wind speed at which the output is 2.7 MW. Give your answer to 3 significant figures.

(3)

(c)

Find the power output at a wind speed of 18 m s⁻¹.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes v = 8: 0.0025 × 8³	✓ 1
A1 1.28 MW	✓ 1
Part (b)
M1 forms 0.0025v³ = 2.7, so v³ = 1080	✓ 1
M1 takes the cube root (or uses the GDC)	✓ 1
A1 v = 10.3 m s⁻¹ (10.26…)	✓ 1
Part (c)
A1 4.32 MW (the constant rule applies for 12 < v ≤ 25)	✓ 1

Common error: Use the rule for the interval the wind speed lies in; 18 m s⁻¹ is not substituted into the cubic.

+ Add to test✓ In your test226IBMI3S-2-02r35-10Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium7 marks

Calculator

A wind farm's monthly output, in gigawatt-hours (GWh), was 12, 15, 19 and 24 in months m = 1, 2, 3, 4. Two models are proposed:
Model A: y = 4.2m + 7.5
Model B: y = 0.5m² + 1.5m + 10.

(a)

Calculate the sum of the squares of the differences between the recorded outputs and the values given by Model A.

(2)

(b)

State the corresponding sum for Model B.

(1)

(c)

Find the output predicted by each model for month 12.

(2)

(d)

The farm can produce at most 60 GWh per month. Which model is more suitable for predicting month 12? Justify your answer.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 calculates the differences 0.3, −0.9, −1.1, −0.3 (or 11.7, 15.9, 20.1, 24.3 for the model values)	✓ 1
A1 0.09 + 0.81 + 1.21 + 0.09 = 2.2	✓ 1
Part (b)
A1 0 (Model B gives 12, 15, 19, 24 exactly)	✓ 1
Part (c)
A1 Model A: 57.9 GWh	✓ 1
A1 Model B: 100 GWh	✓ 1
Part (d)
R1 Model A, since Model B gives 100 GWh, which is far above the maximum possible	✓ 1
R1 although Model B fits the data exactly, a better fit to existing data does not make it better for extrapolation	✓ 1

Exam tip: Fitting the data best does not make a model best; check its predictions against physical limits.

+ Add to test✓ In your test227IBMI3S-2-02r35-11Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 1Medium6 marks

Calculator

A swimming pool is drained. The volume of water in the pool, V m³, t minutes after draining begins is modelled by V = 0.02(t − 300)², for 0 ≤ t ≤ 300.

(a)

Find the volume of water at the start.

(1)

(b)

Describe the transformations that map the graph of y = x² onto the graph of V = 0.02(t − 300)².

(2)

(c)

Find the time at which 450 m³ of water remain.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.02 × (−300)² = 1800 m³	✓ 1
Part (b)
A1 translation 300 units in the positive horizontal direction	✓ 1
A1 vertical stretch with scale factor 0.02	✓ 1
Part (c)
M1 forms 0.02(t − 300)² = 450, so (t − 300)² = 22 500	✓ 1
A1 t = 150 or t = 450	✓ 1
R1 rejects t = 450 as it is outside the domain; t = 150 minutes	✓ 1

Common error: Taking a square root gives both a positive and a negative value; check both against the domain.

+ Add to test✓ In your test228IBMI3S-2-04r35-04Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

The number of patients, P, in a hospital emergency department t hours after 08:00 is modelled by
P = −0.4t³ + 6t² − 12t + 20, for 0 ≤ t ≤ 12.

(a)

Write down the number of patients at 08:00.

(1)

(b)

Find the times, in hours after 08:00, at which the model predicts there are exactly 60 patients. Give your answers to 3 significant figures where necessary.

(2)

(c)

Find the largest number of patients predicted by the model, giving your answer to the nearest whole number, and the time at which this occurs.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 20	✓ 1
Part (b)
M1 sets −0.4t³ + 6t² − 12t + 20 = 60 (or intersects the graph with P = 60)	✓ 1
A1 t = 5 and t = 11.7 (both required; reject the negative root as outside the domain)	✓ 1
Part (c)
M1 uses GDC to find the maximum of the graph on the domain (or finds where the gradient is zero)	✓ 1
A1 t = 8.87 hours after 08:00	✓ 1
A1 106 patients	✓ 1

Common error: A cubic can cross a horizontal line three times, but only solutions inside the stated domain make sense for the situation.

+ Add to test✓ In your test229IBMI3S-2-04r35-05Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

In spring, the number of bees in a colony is modelled by N = 8000 × 1.03^t, where t is the number of days since 1 April.

(a)

Find the number of bees predicted on 1 May (t = 30), to the nearest whole number.

(2)

(b)

Interpret the value 1.03 in the context of the colony.

(1)

(c)

Find the number of days after 1 April when the model predicts that the colony reaches 30 000 bees. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 8000 × 1.03³⁰	✓ 1
A1 19418	✓ 1
Part (b)
R1 the number of bees increases by 3% each day	✓ 1
Part (c)
M1 8000 × 1.03^t = 30 000 (or graph intersection)	✓ 1
M1 takes logarithms or uses a GDC solver	✓ 1
A1 44.7 days	✓ 1

Exam tip: Exponential growth of 3% per period is modelled by a base of 1.03, not 0.03 or 3.

+ Add to test✓ In your test230IBMI3S-2-04r35-09Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

A beekeeper sells honey jars at a weekly market. The weekly profit, £P, from selling x jars is modelled by P = −0.5x² + 40x − 250.

(a)

Find the two values of x for which the profit is zero. Give your answers to 3 significant figures.

(3)

(b)

Hence state the smallest whole number of jars that must be sold to make a profit.

(1)

(c)

Find the maximum weekly profit predicted by the model.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 sets −0.5x² + 40x − 250 = 0 (or uses the GDC to find the zeros)	✓ 1
A1 x = 6.83	✓ 1
A1 x = 73.2	✓ 1
Part (b)
A1 7 jars (ECF from (a))	✓ 1
Part (c)
M1 vertex at x = 40 (or GDC maximum)	✓ 1
A1 £550	✓ 1

Exam tip: When a model gives non-integer break-even values for a discrete quantity, round to a whole number in the direction that keeps the profit positive.

+ Add to test✓ In your test231IBMI3S-2-04r35-11Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

In a hospital laboratory, the viral load, V copies per millilitre, in a culture t days after infection is modelled by log₁₀ V = 2.4 + 0.35t.

(a)

Find the viral load at t = 0, to the nearest whole number.

(2)

(b)

Show that the viral load is multiplied by the same factor each day, and find this factor to 3 significant figures.

(2)

(c)

Find the number of days after which the viral load reaches 10⁶ copies per millilitre. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 V = 10^2.4	✓ 1
A1 251	✓ 1
Part (b)
M1 V = 10^{2.4 + 0.35t} = 10^2.4 × (10^0.35)^t	✓ 1
A1 factor 10^0.35 = 2.24	✓ 1
Part (c)
M1 2.4 + 0.35t = 6	✓ 1
A1 10.3 days	✓ 1

Exam tip: On a logarithmic scale, equal steps on the vertical axis mean the quantity is multiplied by a constant factor, so a straight line corresponds to exponential growth.

+ Add to test✓ In your test232IBMI3S-2-04r35-12Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

The monthly number of ambulance call-outs to a hospital was recorded for five consecutive months, with x = 0 to 4. The results were 130, 138, 153, 170, 193.
Model A is quadratic: y = 2x² + 7x + 130. Model B is linear: y = 15.5x + 123.

(a)

Calculate the sum of the squared differences between the five observed values and the values predicted by Model A.

(3)

(b)

The sum of squared differences for Model B is 114.5. State, with a reason, which model fits the data better.

(1)

(c)

Find the percentage error in the prediction of Model A when x = 4, taking the observed value as correct. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 finds the predicted values 130, 139, 152, 169, 190	✓ 1
M1 squares the differences (0, 1, 1, 1, 9) and adds	✓ 1
A1 12	✓ 1
Part (b)
R1 Model A, because its sum of squared differences (12) is smaller than 114.5	✓ 1
Part (c)
M1 \|190 − 193\| / 193 × 100	✓ 1
A1 1.55%	✓ 1

Common error: A small sum of squared differences means a better fit; do not choose the model with the larger sum.

+ Add to test✓ In your test233IBMI3S-2-03r36-05Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium8 marks

The concentration C mg kg⁻¹ of a pesticide in soil is modelled by C = 48e^−kt, where t is the time in days after application and k > 0. Six days after application the concentration is 30 mg kg⁻¹.

(a)

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

(3)

(b)

Find the half-life of the pesticide in days, correct to 3 significant figures. Use the unrounded value of k.

(2)

(c)

Crops may be planted when the concentration falls below 5 mg kg⁻¹. Find the least number of days after application before planting, correct to 3 significant figures. Use the unrounded value of k.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 set 48e^−6k = 30	✓ 1
M1 e^−6k = 0.625 and take natural logarithms (or use GDC solver)	✓ 1
A1 k = 0.0783	✓ 1
Part (b)
M1 solve e^−kt = 0.5 (or 24 = 48e^−kt)	✓ 1
A1 t = ln 2 ÷ k = 8.85 days	✓ 1
Part (c)
M1 set 48e^−kt = 5	✓ 1
M1 t = ln(9.6) ÷ k	✓ 1
A1 t = 28.9 days	✓ 1

Common error: Using the rounded k = 0.0783 in later parts can shift the third significant figure; store the full calculator value.

+ Add to test✓ In your test234IBMI3S-2-03r36-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

A rare violin is bought for €12 000. Its value is expected to increase by 6.5% each year, so its value after n years is V = 12 000 × 1.065ⁿ euros.

(a)

Find the value of the violin after 8 years, to the nearest cent.

(2)

(b)

Find the number of complete years after which the value of the violin first exceeds €25 000.

(3)

(c)

Find the time, in years correct to 3 significant figures, for the value of the violin to double.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute n = 8	✓ 1
A1 €19 859.95	✓ 1
Part (b)
M1 solve 12 000 × 1.065ⁿ = 25 000 (or use a table or graph)	✓ 1
A1 n = 11.65... (accept 11.7)	✓ 1
A1 12 years (value after 11 years is €23 989.82, after 12 years €25 549.15)	✓ 1
Part (c)
M1 solve 1.065ⁿ = 2 (or 12 000 × 1.065ⁿ = 24 000)	✓ 1
A1 n = 11.0 years	✓ 1

Exam tip: When the question says complete years, state the whole number that satisfies the condition after rounding up, not the decimal solution.

+ Add to test✓ In your test235IBMI3S-2-03r36-08Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

The temperature T °C at a small airfield is modelled by T = 12 + 7cos(π(t − 15)/12), where t is the time in hours after midnight, 0 ≤ t ≤ 24, and the angle is in radians.

(a)

Write down the maximum temperature and the time at which it occurs.

(2)

(b)

Find the minimum temperature and the time at which it occurs.

(1)

(c)

Find the length of time during the day for which the temperature is greater than 15 °C. Give your answer in hours correct to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 19 °C	✓ 1
A1 at t = 15, which is 15:00	✓ 1
Part (b)
A1 5 °C at 03:00 (t = 3)	✓ 1
Part (c)
M1 set 12 + 7cos(π(t − 15)/12) = 15 (or graph the model and the line T = 15)	✓ 1
A1 t = 10.7 and t = 19.3 (unrounded 10.6918 and 19.3082)	✓ 1
M1 subtract the two times	✓ 1
A1 8.62 hours	✓ 1

Exam tip: Check the period of a model from its stated variable: here the temperature repeats every 24 hours, so one full cycle fits the domain.

+ Add to test✓ In your test236IBMI3S-2-02r37-05Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium8 marks

Calculator

A warehouse robot raises a platform. The height of the platform above the floor is modelled by h(t) = −0.12t³ + 0.9t² + 0.5t, where h is in metres and t ≥ 0 is the time in seconds after the platform starts to move. The model is valid until the platform returns to floor level.

(a)

Find the height of the platform after 2 seconds.

(1)

(b)

Find the maximum height of the platform, and the time at which it occurs. Give your answers to three significant figures.

(3)

(c)

Find the two times at which the platform is at a height of 6 m. Give your answers to three significant figures.

(2)

(d)

Find the time at which the platform returns to floor level. Give your answer to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 h(2) = 3.64 m	✓ 1
Part (b)
M1 use of graph or solver to find the maximum (or h′(t) = 0)	✓ 1
A1 t = 5.26 s	✓ 1
A1 h = 10.1 m	✓ 1
Part (c)
M1 solve h(t) = 6	✓ 1
A1 t = 2.87 s and t = 7.09 s	✓ 1
Part (d)
M1 solve h(t) = 0 for t > 0 (e.g. −0.12t² + 0.9t + 0.5 = 0)	✓ 1
A1 8.02 s	✓ 1

Exam tip: t = 0 is also a root of the cubic, so state which root is physically meaningful.

+ Add to test✓ In your test237IBMI3S-2-02r37-07Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Medium8 marks

Calculator

The depth of water, h metres, at the entrance to a fishing harbour is modelled by h(t) = 4.2 + 1.8 sin(30(t − 2)), where t is the time in hours after midnight, 0 ≤ t ≤ 12, and the angle is in degrees.

(a)

State the maximum and minimum depth of water predicted by the model.

(2)

(b)

Find the period of the model.

(1)

(c)

Find the depth at 09:30. Give your answer to three significant figures.

(2)

(d)

A boat needs a depth of at least 3.0 m to enter the harbour. Find the times, between midnight and 12:00, when the boat can enter. Give the two times in hours after midnight to three significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 6.0 m	✓ 1
A1 minimum 2.4 m	✓ 1
Part (b)
A1 360 ÷ 30 = 12 hours	✓ 1
Part (c)
M1 substitute t = 9.5: 4.2 + 1.8 sin 225°	✓ 1
A1 2.93 m	✓ 1
Part (d)
M1 solve 4.2 + 1.8 sin(30(t − 2)) = 3.0 (or use graph)	✓ 1
A1 t = 0.606	✓ 1
A1 t = 9.39, so entry possible for 0.606 ≤ t ≤ 9.39	✓ 1

Common error: Set your calculator to degree mode before evaluating sine functions with angles in degrees.

+ Add to test✓ In your test238IBMI3S-2-01r38-03Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium7 marks

Calculator

The yield of wheat, Y tonnes per hectare, on a trial field depends on the amount of nitrogen fertiliser applied, x kg per hectare. For 0 ≤ x ≤ 250 the yield is modelled by
Y = 4.0 + 0.060x − 0.00020x².

(a)

Write down the yield predicted when no fertiliser is applied.

(1)

(b)

Find the amount of fertiliser that gives the greatest yield, and state this greatest yield.

(3)

(c)

A farmer wants a yield of at least 7.5 tonnes per hectare. Find the range of values of x that achieve this. Give your answers to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4.0 tonnes per hectare	✓ 1
Part (b)
M1 use of vertex (e.g. x = −0.060 ÷ (2 × (−0.00020))) or GDC maximum	✓ 1
A1 x = 150 kg per hectare	✓ 1
A1 Y = 8.5 tonnes per hectare	✓ 1
Part (c)
M1 set 4.0 + 0.060x − 0.00020x² = 7.5 (or sketch with line Y = 7.5)	✓ 1
A1 lower value 79.3	✓ 1
A1 upper value 221, so 79.3 ≤ x ≤ 221	✓ 1

Exam tip: When a question asks for a range, solve the equation for the boundary value(s) first and then state which side of each boundary satisfies the condition.

+ Add to test✓ In your test239IBMI3S-2-01r38-07Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium7 marks

Calculator

The function f(t) models the loudness, in decibels, of a rehearsal note t seconds after the conductor's signal. The graph of y = f(t) passes through the point (10, 62).
A second performance of the same piece starts 4 seconds later and is 6 decibels louder throughout. Its loudness is modelled by g(t) = f(t − 4) + 6.

(a)

Describe the single transformation that maps the graph of y = f(t) onto the graph of y = g(t).

(2)

(b)

Write down the coordinates of the point on the graph of y = g(t) that corresponds to (10, 62).

(2)

(c)

A third performance is played at half the speed of the original, so its loudness is modelled by h(t) = f(0.5t). Describe the transformation that maps y = f(t) onto y = h(t), and find the time at which h has the loudness of 62 decibels that f had at t = 10.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 translation 4 units in the positive t-direction (to the right)	✓ 1
A1 and 6 units in the positive y-direction (up); accept translation by vector (4, 6)	✓ 1
Part (b)
A1 t-coordinate 14	✓ 1
A1 y-coordinate 68	✓ 1
Part (c)
A1 horizontal stretch (parallel to the t-axis)	✓ 1
A1 scale factor 2 (from the y-axis)	✓ 1
A1 t = 20 seconds	✓ 1

Exam tip: For a horizontal transformation, the change to the t-value works in the opposite way to what the bracket suggests: replacing t by 0.5t doubles the t-values.

+ Add to test✓ In your test240IBMI3S-2-03r39-03Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium7 marks

The concentration of a drug in a patient's blood is modelled by C(t) = 120 × 0.85^t, where C is in mg/L and t is the time in hours after the drug is given, t ≥ 0.

(a)

Write down the concentration at the moment the drug is given, and state the percentage by which the concentration decreases each hour.

(2)

(b)

Find the concentration 4 hours after the drug is given. Give your answer to 3 significant figures.

(2)

(c)

The drug stops being effective once the concentration falls below 25 mg/L. Find the time, in hours to 3 significant figures, at which this happens.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 120 mg/L	✓ 1
A1 15% per hour	✓ 1
Part (b)
M1 120 × 0.85⁴	✓ 1
A1 62.6 mg/L	✓ 1
Part (c)
M1 120 × 0.85^t = 25 (or 0.85^t = 25/120)	✓ 1
M1 use of logarithms or GDC to solve	✓ 1
A1 t = 9.65 hours	✓ 1

Common error: A decrease of 15% per hour means the multiplier is 0.85, not 0.15.

+ Add to test✓ In your test241IBMI3S-2-03r39-04Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium7 marks

The rate of water use in a town is modelled by D(t) = 24 + 9cos(15(t − 18))°, where D is in megalitres per hour and t is the number of hours after midnight, 0 ≤ t ≤ 24. Angles are in degrees.

(a)

Write down the maximum and the minimum rate of water use.

(2)

(b)

Find the rate of water use at 08:00. Give your answer to 3 significant figures.

(2)

(c)

Find the total length of time, in hours, for which the rate of water use is greater than 30 ML/h. Give your answer to 3 significant figures; use unrounded values in your working.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 33 ML/h	✓ 1
A1 minimum 15 ML/h	✓ 1
Part (b)
M1 substitute t = 8: 24 + 9cos(−150°)	✓ 1
A1 16.2 ML/h	✓ 1
Part (c)
M1 24 + 9cos(15(t − 18)) = 30	✓ 1
A1 t = 14.79 and t = 21.21 (unrounded 14.787… and 21.213…)	✓ 1
A1 duration = 6.43 hours	✓ 1

Exam tip: Check that your calculator is in degree mode before using a model whose angle is written with a degree sign.

+ Add to test✓ In your test242IBMI3S-2-03r39-06Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium7 marks

In a role-playing game, the total experience points (XP) needed to reach level n are modelled by X(n) = n³ + 14n² + 80n, for 1 ≤ n ≤ 30.

(a)

Find the total XP needed to reach level 5.

(2)

(b)

A player has earned a total of 5000 XP. Find the highest level the player has reached.

(3)

(c)

Find the extra XP needed to go from level 12 to level 13.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute n = 5: 125 + 350 + 400	✓ 1
A1 875 XP	✓ 1
Part (b)
M1 n³ + 14n² + 80n = 5000 (or comparison of X(12) and X(13))	✓ 1
A1 n = 12.34 (or X(12) = 4704 and X(13) = 5603)	✓ 1
A1 level 12	✓ 1
Part (c)
M1 X(13) − X(12) = 5603 − 4704	✓ 1
A1 899 XP	✓ 1

Exam tip: When a model gives a non-integer solution for a quantity that must be whole, decide whether to round up or down from the context.

+ Add to test✓ In your test243IBMI3S-2-03r39-08Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium7 marks

The number of active players, N, of an online game t weeks after its release is modelled by N = ab^t, where a and b are constants. Two weeks after release there are 4159 active players and five weeks after release there are 6161 active players.

(a)

Find the value of b, correct to 3 significant figures.

(2)

(b)

Use b = 1.14 to find the value of a, correct to the nearest whole number.

(2)

(c)

Using a = 3200 and b = 1.14, find the number of weeks after release at which the model predicts 25 000 active players. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 b³ = 6161 / 4159 (dividing the two equations)	✓ 1
A1 b = 1.14	✓ 1
Part (b)
M1 a × 1.14² = 4159	✓ 1
A1 a = 3200	✓ 1
Part (c)
M1 3200 × 1.14^t = 25 000	✓ 1
M1 use of logarithms or GDC to solve	✓ 1
A1 t = 15.7 weeks	✓ 1

Common error: To find b from two data points, divide the two equations; subtracting them does not eliminate a.

+ Add to test✓ In your test244IBMI3S-2-03r39-09Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium6 marks

The height of the tide at harbour A is modelled by h = 3.2 + 1.4sin(30t)°, where h is in metres and t is the time in hours after midnight. The tide at harbour B follows the same pattern as harbour A, but each high and low tide occurs 2 hours later and the height is 0.5 m greater at every time.

(a)

Write down the maximum and minimum heights of the tide at harbour A.

(2)

(b)

Write down a model for the height of the tide at harbour B.

(2)

(c)

The first high tide at harbour A after midnight occurs when t = 3. State the time of the first high tide at harbour B and the height of the water at that time.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 4.6 m	✓ 1
A1 minimum 1.8 m	✓ 1
Part (b)
A1 horizontal translation of 2 hours: replace t by (t − 2)	✓ 1
A1 h = 3.7 + 1.4sin(30(t − 2))°	✓ 1
Part (c)
A1 t = 5 (05:00)	✓ 1
A1 5.1 m	✓ 1

Exam tip: Replacing t by (t − k) moves a graph k units to the right, so the pattern happens later, not earlier.

+ Add to test✓ In your test245IBMI3S-2-03r39-11Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium7 marks

Two companies supply bottled water to an office. Supplier 1 charges C₁ = 30 + 1.6x euros for x cubic metres of water. Supplier 2 charges a fixed amount plus a constant rate per cubic metre: it charges €53.80 for 19 m³ and €69.20 for 26 m³.

(a)

Find the equation of the cost C₂ charged by Supplier 2 in the form C₂ = mx + c.

(3)

(b)

Find the volume for which the two suppliers charge the same amount.

(2)

(c)

The office uses 45 m³. Find how much less it pays by choosing the cheaper supplier.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient (69.20 − 53.80)/(26 − 19)	✓ 1
A1 m = 2.2	✓ 1
A1 C₂ = 2.2x + 12	✓ 1
Part (b)
M1 30 + 1.6x = 2.2x + 12	✓ 1
A1 x = 30 m³	✓ 1
Part (c)
M1 C₁ = 102 and C₂ = 111 (both evaluated at 45)	✓ 1
A1 Supplier 1 is cheaper by €9	✓ 1

Exam tip: Two linear costs are equal at the intersection; the supplier with the smaller gradient becomes cheaper for larger volumes.

+ Add to test✓ In your test246IBMI3S-2-02r40-05Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium6 marks

A shop sells a gift item at a price of €$x$ each. The daily profit €$P$ is modelled by $P = -2x^2 + 120x - 900$.

(a)

Find the selling price that gives the greatest daily profit, and state this profit.

(3)

(b)

Find the range of selling prices for which the shop makes a profit.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $x = -\frac{120}{2(-2)} = 30$ (or use the GDC maximum)	✓ 1
A1 price €30	✓ 1
A1 greatest profit €900	✓ 1
Part (b)
M1 solve $-2x^2 + 120x - 900 = 0$	✓ 1
A1 $x = 8.79$	✓ 1
A1 $x = 51.2$, so $8.79 < x < 51.2$	✓ 1

Exam tip: A 'profit' range comes from the two roots of the quadratic: the profit is positive between them.

+ Add to test✓ In your test247IBMI3S-2-02r40-06Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 1Medium7 marks

The power output $P$ kW of a wind turbine for wind speeds $v$ m s⁻¹ between 4 and 12 is modelled by $P = 0.38v^3$.

(a)

Calculate the power output when the wind speed is 8 m s⁻¹, correct to 3 significant figures.

(2)

(b)

Show that, according to the model, doubling the wind speed multiplies the power output by 8.

(2)

(c)

Find the wind speed at which the power output is 400 kW, correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $0.38 \times 8^3$	✓ 1
A1 195 kW	✓ 1
Part (b)
M1 $0.38(2v)^3 = 0.38 \times 8v^3$	✓ 1
R1 $= 8 \times 0.38v^3$, which is 8 times $P$	✓ 1
Part (c)
M1 $0.38v^3 = 400$	✓ 1
M1 $v = \sqrt[3]{\frac{400}{0.38}}$	✓ 1
A1 10.2 m s⁻¹	✓ 1

Common error: To undo a cube, take a cube root: $v = \sqrt[3]{\ldots}$. Do not divide by 3.

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A retailer records its weekly advertising spend $x$ (in thousands of euros) and the weekly sales $y$ (in thousands of euros) over six weeks:
Advertising spend $x$: 1, 2, 3, 4, 5, 6
Weekly sales $y$: 12.1, 15.8, 18.4, 20.1, 21.0, 21.2

(a)

Use your GDC to find the equation of the linear regression line of $y$ on $x$. Give the coefficients to 3 significant figures.

(2)

(b)

Use your GDC to find the quadratic model $y = ax^2 + bx + c$ for these data, giving each coefficient to 3 significant figures.

(2)

(c)

The GDC gives $R^2 = 0.889$ for the linear model and $R^2 = 0.9995$ for the quadratic model. State, with a reason, which model is the better fit.

(1)

(d)

Use the better model to estimate the weekly sales when the advertising spend is €4500 per week.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 gradient 1.79	✓ 1
A1 $y = 1.79x + 11.8$	✓ 1
Part (b)
M1 attempt to use quadratic regression	✓ 1
A1 $y = -0.434x^2 + 4.83x + 7.77$	✓ 1
Part (c)
A1 quadratic, since its $R^2$ is closer to 1 (and the data level off, which a straight line cannot show)	✓ 1
Part (d)
M1 substitute $x = 4.5$ into the quadratic	✓ 1
A1 €20.7 thousand (about €20 700)	✓ 1

Exam tip: Always compare how well each model fits using a measure such as $R^2$, and look at the shape of the data too.

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A 60 kWh battery in a solar storage system is charged so that the stored energy $L$ kWh after $t$ hours is given by:
$L = 6 + 12t$ for $0 \leq t \leq 3$
$L = 42 + 5(t - 3)$ for $t > 3$, until the battery is full.

(a)

Find the stored energy after 2 hours.

(1)

(b)

State the rate of charging during the first 3 hours, with units.

(1)

(c)

Find the time at which the battery becomes full.

(3)

(d)

Calculate the average rate of charging over the whole charging time, in kWh per hour, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 30 kWh	✓ 1
Part (b)
A1 12 kWh per hour	✓ 1
Part (c)
R1 $L(3) = 42 < 60$, so the battery is full during the second rule	✓ 1
M1 $42 + 5(t - 3) = 60$	✓ 1
A1 $t = 6.6$ hours	✓ 1
Part (d)
M1 energy gained ÷ time = $\frac{60 - 6}{6.6}$	✓ 1
A1 8.18 kWh per hour	✓ 1

Exam tip: For piecewise models, check that the pieces join at the boundary: here $6 + 12(3) = 42$.

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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

A rescue flare is fired upwards from a ledge on a cliff. Its height h metres above the ledge, t seconds after firing, is modelled by
h(t) = −4.9t² + 19.6t + 5, 0 ≤ t ≤ 4.

(a)

Write down the height of the flare at the instant it is fired.

(1)

(b)

Find the maximum height of the flare above the ledge, and the time at which it is reached.

(3)

(c)

The flare is visible to the rescuers while it is higher than 15 m above the ledge. Find the length of time, in seconds, for which the flare is visible. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 5 m	✓ 1
Part (b)
M1 attempt to find the vertex (for example t = −b/(2a), or a maximum from a graph)	✓ 1
A1 t = 2 s	✓ 1
A1 maximum height = 24.6 m	✓ 1
Part (c)
M1 setting h(t) = 15, giving −4.9t² + 19.6t − 10 = 0	✓ 1
A1 t = 0.600 and t = 3.40 (0.6003 and 3.3997 before rounding)	✓ 1
A1 duration = 2.80 s (2.7994… before rounding; use unrounded roots)	✓ 1

Common error: A common slip is to give the time at which the flare returns to the ledge instead of the time spent above a given height; write the condition as an equation or inequality for h first.

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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; 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A mobile network charges a monthly fee €C that depends on the data used, d gigabytes (GB):
C(d) = 12 for 0 ≤ d ≤ 5; C(d) = 12 + 3.5(d − 5) for 5 < d ≤ 15; C(d) = 47 for d > 15.

(a)

Find the fee for a month in which 9.2 GB is used.

(2)

(b)

A customer is charged €36.25 for one month. Find the amount of data used. Give your answer to 3 significant figures.

(2)

(c)

A rival network charges €(20 + 1.2d). For 5 < d ≤ 15, find the amount of data, in GB, for which the two networks charge the same. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 using the second branch with d = 9.2, giving 12 + 3.5 × 4.2	✓ 1
A1 €26.70	✓ 1
Part (b)
M1 12 + 3.5(d − 5) = 36.25 (second branch identified, since 36.25 is between 12 and 47)	✓ 1
A1 d = 11.9 GB (11.9286… before rounding)	✓ 1
Part (c)
M1 equating 12 + 3.5(d − 5) = 20 + 1.2d	✓ 1
A1 2.3d = 25.5 (or equivalent simplification)	✓ 1
A1 d = 11.1 GB (11.087… before rounding; lies within the stated interval)	✓ 1

Exam tip: For a piecewise model, decide which branch applies before substituting; check that your answer lies inside the interval for that branch.

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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; 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A telecoms operator has f(t) = 80 × 1.12^t thousand subscribers, where t is the number of years after the start of 2020. A rival enters the market later, and its number of subscribers is modelled by g(t) = 1.5 f(t − 3) thousand, for t ≥ 3.

(a)

Describe, in terms of the graph of f, the two changes that produce the graph of g. The changes may be given in either order.

(2)

(b)

Find the number of subscribers the rival has at t = 3.

(1)

(c)

Solve g(t) = 500 to find when the rival reaches 500 thousand subscribers. Give t to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 shift 3 units to the right (a delay of 3 years)	✓ 1
A1 vertical stretch by scale factor 1.5 (about the t-axis)	✓ 1
Part (b)
A1 g(3) = 1.5 × 80 = 120 thousand	✓ 1
Part (c)
M1 120 × 1.12^t−3 = 500 (using g(t) = 120 × 1.12^t−3)	✓ 1
M1 correct method to solve (logarithms, or graph or solver)	✓ 1
A1 t = 15.6 (15.5927… before rounding)	✓ 1

Exam tip: For f(t − k) the graph moves k units to the right, which is the opposite of what the sign inside the bracket may suggest.

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quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium6 marks

A cup of coffee cools in a room. Its temperature T °C after t minutes is modelled by T = 22 + 68e^−0.045t.

(a)

Write down the initial temperature of the coffee.

(1)

(b)

State the temperature that the model predicts the coffee approaches in the long term, and explain why.

(2)

(c)

The coffee is drinkable once its temperature falls to 60 °C. Find the time taken. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 90 °C	✓ 1
Part (b)
A1 22 °C	✓ 1
R1 e^−0.045t tends to 0 as t increases (room temperature)	✓ 1
Part (c)
M1 22 + 68e^−0.045t = 60, giving e^−0.045t = 38/68	✓ 1
M1 taking natural logarithms (or using a graph or solver)	✓ 1
A1 t = 12.9 minutes (12.93… before rounding)	✓ 1

Common error: In a cooling model of the form A + Be⁻kt, subtract the constant A from both sides before taking logarithms.

+ Add to test✓ In your test254IBMI3S-2-01r43-13Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

A telephone cable hangs between two towers 120 m apart. The height y metres of the cable above the ground, at a horizontal distance x metres from the first tower, is modelled by y = 0.0025x² − 0.3x + 14, for 0 ≤ x ≤ 120.

(a)

Show that the two towers have the same cable height.

(2)

(b)

Find the minimum height of the cable above the ground, and the value of x at which it occurs.

(2)

(c)

A road runs beneath the cable. A lorry is 6.2 m tall. Find the total width of road, in metres, along which the cable is lower than 6.2 m. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 evaluating y at x = 0 and at x = 120	✓ 1
A1 both equal 14 m (y(120) = 36 − 36 + 14 = 14)	✓ 1
Part (b)
A1 x = 60	✓ 1
A1 minimum height 5 m	✓ 1
Part (c)
M1 setting 0.0025x² − 0.3x + 14 = 6.2	✓ 1
A1 x = 38.1 and x = 81.9 (38.09… and 81.91…)	✓ 1
A1 width = 43.8 m (43.818… before rounding)	✓ 1

Common error: The symmetry of a quadratic gives the vertex halfway between two points of equal height; use this as a quick check on the vertex you find.

+ Add to test✓ In your test255IBMI3S-2-03r43-05Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium6 marks

Beneath a harbour bridge the depth of water, d metres, varies with time t hours according to a model of the form d = a cos(bt) + c, where bt is in radians. At high tide, when t = 0, the depth is 5.0 m. At low tide the depth is 2.0 m. The time between consecutive high tides is 12.4 hours.

(a)

Find the values of a, b and c. Give b to 3 significant figures.

(3)

(b)

A ship can pass under the bridge only when the depth is at least 4.4 m. Around each high tide there is one continuous period during which this is possible. Find the length of this period, in hours, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = 1.5 (half the difference between 5.0 and 2.0)	✓ 1
A1 c = 3.5 (mean of 5.0 and 2.0)	✓ 1
A1 b = 2π ÷ 12.4 = 0.507	✓ 1
Part (b)
M1 sets 3.5 + 1.5 cos(bt) = 4.4 (cos(bt) = 0.6)	✓ 1
A1 first time after t = 0 is t = 1.83 (accept 1.829 to 1.830); by symmetry the period runs from −1.83 to 1.83	✓ 1
A1 length of period = 3.66 hours	✓ 1

Exam tip: A model of the form a cos(bt) + c is symmetrical about t = 0: find one solution with the GDC and double it to get a duration.

+ Add to test✓ In your test256IBMI3S-2-03r43-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium6 marks

A roaster records the percentage mass loss, L per cent, of a batch of green coffee beans after t minutes of roasting:
t (minutes): 8, 10, 12, 14, 16
L (per cent): 9.1, 11.9, 14.4, 16.6, 18.4
Two models are proposed.
Model A (linear): L = 1.17t + 0.1
Model B (quadratic): L = −0.041t² + 2.15t − 5.49
The sum of the squares of the differences between the recorded values and the values predicted by each model is 0.398 for Model A and 0.00120 for Model B.

(a)

For t = 12, calculate the value of L predicted by each model.

(2)

(b)

State which model fits the data better. Give a reason based on the information in the question.

(2)

(c)

Find the value of t at which Model B predicts the greatest mass loss. Hence explain why Model B should not be used to predict the mass loss after a long roasting time.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model A: L = 14.14 (14.1)	✓ 1
A1 Model B: L = 14.41 (14.4)	✓ 1
Part (b)
A1 Model B	✓ 1
R1 its sum of squared differences (0.00120) is much smaller than that of Model A (0.398)	✓ 1
Part (c)
A1 t = 2.15 ÷ (2 × 0.041) = 26.2 minutes (GDC or vertex formula)	✓ 1
R1 after this time Model B predicts the mass loss decreasing, which is impossible because beans cannot regain mass	✓ 1

Common error: A better fit over the data range does not mean a model is valid outside it: always check what a quadratic does beyond its vertex.

+ Add to test✓ In your test257IBMI3S-2-03r43-08Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium6 marks

A cup of coffee cools in a room at a constant 21 °C. Its temperature, T °C, after t minutes is modelled by T = 21 + 71e^−kt, where k is a positive constant. After 6 minutes the temperature is 70 °C.

(a)

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

(3)

(b)

Find the time when the coffee reaches 55 °C. 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 substitutes: 70 = 21 + 71e^−6k	✓ 1
M1 e^−6k = 49/71 and takes natural logarithms (or uses the GDC)	✓ 1
A1 k = 0.0618	✓ 1
Part (b)
M1 sets 21 + 71e^−kt = 55	✓ 1
M1 e^−kt = 34/71 and solves for t	✓ 1
A1 t = 11.9 minutes	✓ 1

Exam tip: If a later part says to use the unrounded value, store the earlier answer in the GDC memory instead of retyping the rounded value.

+ Add to test✓ In your test258IBMI3S-2-03r43-09Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 1Medium6 marks

A supplier charges C euros for x kg of green coffee beans according to
C = 8.5x for 0 ≤ x ≤ 40
C = 340 + 7.2(x − 40) for 40 < x ≤ 100
C = 772 + 6.0(x − 100) for x > 100

(a)

Find the cost of 75 kg of beans.

(2)

(b)

A café spends exactly €1000 on one order. Find the mass of beans bought.

(2)

(c)

Find the average cost per kilogram of an order of 150 kg, correct to the nearest cent.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses the second rule: 340 + 7.2 × 35	✓ 1
A1 €592	✓ 1
Part (b)
M1 uses the third rule: 772 + 6.0(x − 100) = 1000	✓ 1
A1 x = 138 kg	✓ 1
Part (c)
M1 cost = 772 + 6.0 × 50 = 1072 and divides by 150	✓ 1
A1 €7.15 per kg	✓ 1

Common error: Use the rule for the correct interval, and remember the rate applies only to the kilograms beyond the boundary, not to the whole order.

+ Add to test✓ In your test259IBMI3S-2-03r45-04Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium7 marks

The depth of water, d metres, in a harbour channel t hours after midnight is modelled by
d(t) = c + a cos(b(t − 2)), where a, b and c are positive constants and the angle is in radians.
High tide occurs at t = 2 with a depth of 10.2 m. The lowest depth is 4.6 m and the time between consecutive high tides is 12.4 hours.

(a)

Find the values of a, b and c, giving b to 3 significant figures.

(3)

(b)

A ship can use the channel only when the depth is at least 6.5 m. Find, in one tidal cycle, the total time for which the ship can use the channel. Give your answer in hours to 3 significant figures and use unrounded values throughout.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 a = (10.2 − 4.6)/2 or c = (10.2 + 4.6)/2	✓ 1
A1 a = 2.8 and c = 7.4	✓ 1
A1 b = 2π/12.4 = 0.507	✓ 1
Part (b)
M1 7.4 + 2.8 cos(b(t − 2)) = 6.5	✓ 1
M1 solving, e.g. cos(b(t − 2)) = −0.3214, giving b(t − 2) = ±1.898	✓ 1
A1 depth reaches 6.5 m at t ≈ 5.75 and t ≈ −1.75 (equivalently 10.65)	✓ 1
A1 time = 2 × 1.898/0.5067 = 7.49 hours	✓ 1

Exam tip: For the period of a sinusoidal model use b = 2π/period when working in radians, and keep the unrounded value in your calculator for later parts.

+ Add to test✓ In your test260IBMI3S-2-03r45-05Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium6 marks

A theatre sells group bookings. The cost, C euros, of a booking for n tickets is
C(n) = 18n for 1 ≤ n ≤ 10
C(n) = 12n + 60 for n > 10, where n is a whole number.

(a)

Find the cost of a booking for (i) 8 tickets, (ii) 25 tickets.

(2)

(b)

Find the average cost per ticket for a booking of 40 tickets.

(2)

(c)

A group pays €348 in total. Find the number of tickets they booked.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 (i) €144	✓ 1
A1 (ii) 12 × 25 + 60 = €360	✓ 1
Part (b)
M1 (12 × 40 + 60)/40	✓ 1
A1 €13.50	✓ 1
Part (c)
M1 recognises that 348 > 180, so uses 12n + 60 = 348	✓ 1
A1 n = 24	✓ 1

Common error: With a piecewise function, check which rule applies before substituting: a total of €348 cannot come from the first rule because that rule gives at most €180.

+ Add to test✓ In your test261IBMI3S-2-03r45-06Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium6 marks

A seismometer near a volcano records the amplitude, A micrometres, of ground tremor. For the first 8 hours of monitoring, the amplitude is modelled by
A(t) = −0.5t³ + 6t² − 18t + 40, for 0 ≤ t ≤ 8,
where t is the time in hours since monitoring began.

(a)

Solve A(t) = 30, giving each solution to 3 significant figures.

(4)

(b)

Find the total time within the first 8 hours for which the amplitude is greater than 30 micrometres. Use unrounded values from part (a) and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 sets A(t) = 30, or sketches the graph and the line A = 30	✓ 1
A1 t = 0.716	✓ 1
A1 t = 3.66	✓ 1
A1 t = 7.62	✓ 1
Part (b)
M1 identifies the intervals 0 ≤ t < 0.716 and 3.66 < t < 7.62, e.g. 0.7164 + (7.6201 − 3.6635)	✓ 1
A1 4.67 hours (accept 4.68 from rounded values)	✓ 1

Exam tip: Sketch the cubic and the horizontal line from your graphing screen so you can see how many intervals lie above the line before adding their lengths.

+ Add to test✓ In your test262IBMI3S-2-03r45-08Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium6 marks

A lighting rig is raised and lowered by a motor. For the first show, its height, h₁ metres above the stage, t seconds after the motor starts is modelled by
h₁(t) = 3.5 + 1.2 sin(0.9t), t ≥ 0, with the angle in radians.

(a)

Write down the maximum height, the minimum height and the period of the motion. Give the period to 3 significant figures.

(3)

(b)

For the second show the motor starts 2 seconds later, so the height is modelled by h₂(t) = 3.5 + 1.2 sin(0.9(t − 2)), t ≥ 2. Describe how the graph of h₂ is related to the graph of h₁.

(1)

(c)

Find the first time at which the height of the rig in the second show is 4.4 m. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 4.7 m	✓ 1
A1 minimum 2.3 m	✓ 1
A1 period = 2π/0.9 = 6.98 s	✓ 1
Part (b)
A1 a horizontal translation of 2 units to the right (no change in shape)	✓ 1
Part (c)
M1 sin(0.9(t − 2)) = 0.75, so 0.9(t − 2) = 0.848 (first positive solution)	✓ 1
A1 t = 2.94 s	✓ 1

Common error: Make sure the calculator is in radian mode when the model gives the angle in radians, and check that your answer lies in the stated domain of the model.

+ Add to test✓ In your test263IBMI3S-2-03r45-14Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Medium7 marks

The opening of a theatre's proscenium arch has height h metres at a horizontal distance x metres from the left wall, modelled by
h(x) = −0.2(x − 5)² + 7.2, for 0 ≤ x ≤ 10.

(a)

Write down the greatest height of the arch and the value of x at which it occurs.

(2)

(b)

A redesigned arch has the same shape but is moved 3 m to the right and raised by 0.8 m. Write down the equation of the redesigned arch in the form h(x) = −0.2(x − p)² + q.

(2)

(c)

A rectangular set piece 4.5 m high is carried upright through the original arch, centred at x = 5. Find the greatest width that fits beneath the arch. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 7.2 m	✓ 1
A1 x = 5	✓ 1
Part (b)
A1 p = 8	✓ 1
A1 q = 8 (so h(x) = −0.2(x − 8)² + 8)	✓ 1
Part (c)
M1 −0.2(x − 5)² + 7.2 = 4.5	✓ 1
M1 (x − 5)² = 13.5, so x = 5 ± 3.674	✓ 1
A1 width = 2√13.5 = 7.35 m	✓ 1

Exam tip: The vertex form a(x − p)² + q shows the turning point (p, q) directly, so a shift right or up changes only p or q.

+ Add to test✓ In your test264IBMI3S-2-03r48-03Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium6 marks

The reflectivity, R %, of the mirror in a small telescope falls as its coating ages. It is modelled by R = 98 × 0.992^t, where t is the time in months since the coating was applied.

(a)

Calculate the reflectivity after 24 months, giving your answer to 3 significant figures.

(2)

(b)

Find how long it takes for the reflectivity to fall to 90 %. Give your answer in months to 3 significant figures.

(3)

(c)

Explain what the value 0.992 tells you about the reflectivity.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 24: 98 × 0.992²⁴	✓ 1
A1 80.8 %	✓ 1
Part (b)
M1 sets 98 × 0.992^t = 90	✓ 1
M1 solves, e.g. by logarithms or a graphing method	✓ 1
A1 10.6 months	✓ 1
Part (c)
R1 each month the reflectivity is 99.2 % of its value the month before (it falls by 0.8 % per month)	✓ 1

Exam tip: If an unknown is in an exponent, use logarithms or the equation solver on your calculator rather than trial and improvement.

+ Add to test✓ In your test265IBMI3S-2-03r48-04Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

A goalkeeper kicks a football out of his hands. The height of the ball, h metres, at a horizontal distance x metres from the point where it is struck is modelled by
h = −0.032x² + 0.8x + 1.5, for x ≥ 0.

(a)

State the height of the ball at the moment it is struck.

(1)

(b)

Find the greatest height reached by the ball and the horizontal distance at which this occurs.

(3)

(c)

Find the horizontal distance from the kick to the point where the ball lands. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.5 m	✓ 1
Part (b)
M1 uses x = −b/(2a) or a graphical/derivative method	✓ 1
A1 x = 12.5 m	✓ 1
A1 greatest height 6.5 m	✓ 1
Part (c)
M1 sets h = 0	✓ 1
M1 solves the quadratic using the formula or a calculator, rejecting the negative root	✓ 1
A1 26.8 m	✓ 1

Common error: Giving both roots as the answer: a negative horizontal distance does not fit the context, so reject it.

+ Add to test✓ In your test266IBMI3S-2-03r48-05Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

The brightness index, B, of a variable star is modelled by B = 4.8 + 0.6 cos(bt), where t is the time in days after the star reaches its greatest index and bt is in radians. The brightness index repeats every 5.4 days.

(a)

Show that b = 1.16 correct to 3 significant figures.

(2)

(b)

Write down the maximum and minimum values of B.

(2)

(c)

Find the first time after t = 0 at which B = 4.5. Use b = 2π/5.4.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 period = 2π/b = 5.4	✓ 1
A1 b = 2π/5.4 = 1.1636… = 1.16	✓ 1
Part (b)
A1 maximum 5.4	✓ 1
A1 minimum 4.2	✓ 1
Part (c)
M1 4.8 + 0.6 cos(bt) = 4.5 so cos(bt) = −0.5	✓ 1
M1 bt = 2π/3 (first positive solution)	✓ 1
A1 t = 1.8 days	✓ 1

Exam tip: Check your calculator is in radian mode whenever the model's angle is given in radians.

+ Add to test✓ In your test267IBMI3S-2-03r48-08Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

The altitude of the Sun above the horizon at a northern observatory on one day is modelled by h = a cos(b(t − c)) + d, where h is in degrees, t is the number of hours after midnight, b = π/12 and 0 ≤ t ≤ 24. The greatest altitude is 54° at t = 12.5 and the least altitude is −18°.

(a)

Find the values of a, c and d.

(3)

(b)

The Sun is above the horizon when h > 0. Find the number of hours of daylight predicted by the model.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (54 − (−18))/2 = 36	✓ 1
A1 d = (54 + (−18))/2 = 18	✓ 1
A1 c = 12.5	✓ 1
Part (b)
M1 sets 36 cos(π/12 (t − 12.5)) + 18 = 0, so cos(π/12 (t − 12.5)) = −0.5	✓ 1
M1 π/12 (t − 12.5) = ±2π/3, so t − 12.5 = ±8	✓ 1
A1 t = 4.5 and t = 20.5	✓ 1
A1 16 hours of daylight	✓ 1

Exam tip: For a cosine model, the amplitude is half the difference between the maximum and minimum, and the midline is their mean.

+ Add to test✓ In your test268IBMI3S-2-03r48-09Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium7 marks

A cyclist takes 160 mg of caffeine before a race. The amount remaining in the body, C mg, is modelled by C = 160 × 0.5^t/5.5, where t is the time in hours after the caffeine is taken.

(a)

Calculate the amount of caffeine remaining after 8 hours, giving your answer to 3 significant figures.

(2)

(b)

Find the time at which 25 mg remains. Give your answer in hours to 3 significant figures.

(3)

(c)

Find the percentage by which the amount of caffeine falls each hour. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 8	✓ 1
A1 58.4 mg	✓ 1
Part (b)
M1 sets 160 × 0.5^t/5.5 = 25	✓ 1
M1 solves using logarithms or a graphing method	✓ 1
A1 14.7 hours	✓ 1
Part (c)
M1 hourly factor 0.5^1/5.5 = 0.8816…	✓ 1
A1 11.8 %	✓ 1

Common error: Dividing the half-life by 2 to get an hourly rate: the hourly factor is 0.5¹/⁵.⁵, not a linear fraction of the half-life.

+ Add to test✓ In your test269IBMI3S-2-03r48-10Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Medium6 marks

The table shows the scheduled journey time, T minutes, of six express services on a rail line for different distances, d km.

d (km)	40	85	120	160	210	260
T (min)	38	61	80	101	133	158

(a)

Find the equation of the least squares regression line of T on d. Give the coefficients to 3 significant figures.

(2)

(b)

Use your equation to estimate the journey time for a distance of 300 km. Give your answer to the nearest minute.

(2)

(c)

Interpret the value 14.5 in this context.

(1)

(d)

The correlation coefficient is r = 0.999. Comment on the reliability of your estimate in (b).

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 gradient 0.553	✓ 1
A1 intercept 14.5, so T = 0.553d + 14.5	✓ 1
Part (b)
M1 substitutes d = 300 into the equation from (a)	✓ 1
A1 180 minutes (ECF from (a))	✓ 1
Part (c)
A1 a fixed allowance of about 14.5 minutes per journey (for example stopping, starting and boarding), independent of distance	✓ 1
Part (d)
R1 r is very close to 1 (strong positive linear correlation), but 300 km is outside the data range 40 to 260 km, so this is extrapolation and less reliable	✓ 1

Exam tip: Always compare the requested input with the data range before commenting on reliability: interpolation is more trustworthy than extrapolation.

+ Add to test✓ In your test270IBMI3S-2-04r48-03Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; 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After an injection in a hospital ward, the concentration of a drug in a patient's blood is modelled by C = C₀e^−kt, where C is in mg L⁻¹ and t is the time in hours after the injection. At t = 0 the concentration is 48 mg L⁻¹ and at t = 3 it is 30 mg L⁻¹.

(a)

Find the value of k, giving your answer to 3 significant figures.

(3)

(b)

Find the concentration at t = 8. Use the unrounded value of k and give your answer to 3 significant figures.

(2)

(c)

A second dose is due when the concentration reaches 15 mg L⁻¹. Find the time at which this happens, in hours to 3 significant figures. Use the unrounded value of k.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 30 = 48e^−3k	✓ 1
M1 takes natural logarithms: k = ln(48/30)/3	✓ 1
A1 k = 0.157	✓ 1
Part (b)
M1 substitutes t = 8 into 48e^−kt	✓ 1
A1 13.7 mg L⁻¹	✓ 1
Part (c)
M1 sets 48e^−kt = 15 and solves using logarithms	✓ 1
A1 7.42 hours	✓ 1

Exam tip: Write the model with its unknown constant, substitute each data point, and divide the two equations to eliminate the initial value.

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choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

During a race stint, the lap time, T seconds, of a Formula 1 car is modelled by T = 0.012n² − 0.30n + 89.4, for 0 ≤ n ≤ 30, where n is the number of laps completed on a set of tyres.

(a)

Calculate the lap time predicted when n = 20.

(1)

(b)

Find the number of laps at which the model gives the shortest lap time, and find this shortest time to 3 significant figures.

(3)

(c)

The team wants lap times below 88.0 seconds. Find how many whole values of n in the model give T < 88.0.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 88.2 seconds	✓ 1
Part (b)
M1 uses n = −b/(2a) or a graphing method	✓ 1
A1 n = 12.5	✓ 1
A1 87.5 seconds	✓ 1
Part (c)
M1 solves 0.012n² − 0.30n + 89.4 = 88.0	✓ 1
A1 n = 6.21 and n = 18.8 (3 s.f.)	✓ 1
A1 n = 7, 8, …, 18, so 12 values	✓ 1

Common error: Counting the boundary values 6.21 and 18.8 as laps: only whole numbers of laps inside the interval count.

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A hospital ward's heating controller gives a temperature, T °C, of T = 21 + 1.5 sin(πt/12), where t is the number of hours after midnight and 0 ≤ t ≤ 24. The angle is in radians.

(a)

The graph of T is obtained from the graph of y = sin(πt/12) by two transformations. Describe each transformation.

(2)

(b)

The controller is reprogrammed so that the new temperature is T₂(t) = T(t − 3). State the time at which the greatest temperature now occurs, given that the original greatest temperature occurred at t = 6.

(1)

(c)

Find the times, between t = 0 and t = 24, at which T₂ = 22. Give your answers to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 vertical stretch, scale factor 1.5	✓ 1
A1 vertical translation of 21 units upward	✓ 1
Part (b)
A1 t = 9 (09:00), because the graph is translated 3 hours to the right	✓ 1
Part (c)
M1 solves 21 + 1.5 sin(π(t − 3)/12) = 22, so sin(π(t − 3)/12) = 2/3	✓ 1
A1 t = 5.79	✓ 1
A1 t = 12.2	✓ 1

Common error: Translating the wrong way: replacing t by t − 3 moves the graph 3 units to the right (later), not to the left.

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The number of bees in a colony is modelled by B = 12 000 × 1.035^w, where w is the number of weeks since the start of spring.

(a)

Calculate the number of bees after 10 weeks, giving your answer to 3 significant figures.

(2)

(b)

Interpret the value 1.035 in this context.

(1)

(c)

The beekeeper splits the colony when it first exceeds 25 000 bees. Find the value of w at which the model reaches 25 000. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes w = 10	✓ 1
A1 16900	✓ 1
Part (b)
R1 the colony grows by 3.5 % each week (it is multiplied by 1.035 each week)	✓ 1
Part (c)
M1 sets 12 000 × 1.035^w = 25 000	✓ 1
M1 solves using logarithms or a graphing method	✓ 1
A1 w = 21.3	✓ 1

Exam tip: State the growth rate as a percentage per time unit by subtracting 1 from the growth factor.

+ Add to test✓ In your test274IBMI3S-2-04r48-10Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Medium7 marks

A team records the percentage loss of grip, g %, of a set of racing tyres after n laps.

n	2	6	10	14	18	22
g	1.1	2.9	5.8	10.0	17.5	30.0

The linear regression line of g on n is g = 1.38n − 5.28, with r = 0.944.

(a)

Use a calculator to fit an exponential model of the form g = a × bⁿ. Give a and b to 3 significant figures.

(2)

(b)

Evaluate the linear model at n = 2 and use your result, together with the shape of the data, to explain which model is more suitable.

(3)

(c)

Use the exponential model, with unrounded values of a and b, to predict the percentage loss of grip after 26 laps. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = 0.988	✓ 1
A1 b = 1.17	✓ 1
Part (b)
M1 g = 1.38 × 2 − 5.28 = −2.52	✓ 1
R1 a negative loss of grip is not possible, so the linear model is unsuitable for small n	✓ 1
R1 the increases in g get larger as n increases (the data curve upward), which suits an exponential model better	✓ 1
Part (c)
M1 substitutes n = 26 into g = a × bⁿ	✓ 1
A1 64.1 %	✓ 1

Exam tip: To compare models, test them on values at both ends of the data and check whether the outputs make sense in the context, not only the value of r.

+ Add to test✓ In your test275IBMI3S-2-01r9-13Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Hard6 marks

During a planetary fly-by, the speed, v km s⁻¹, of a probe is modelled by
v = 12 + 1.5t − 0.25t² for 0 ≤ t ≤ 4
v = 14 − k(t − 4) for 4 < t ≤ 10,
where t is the time in seconds after the probe enters the planet’s influence and k is a constant. The speed at t = 10 is 8 km s⁻¹.

(a)

Find the value of k.

(2)

(b)

Find the length of time during which the speed of the probe is at least 13 km s⁻¹. Give your answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 14 − k(10 − 4) = 8	✓ 1
A1 k = 1	✓ 1
Part (b)
M1 solve 12 + 1.5t − 0.25t² = 13 for 0 ≤ t ≤ 4	✓ 1
A1 t = 0.764 s (the root 5.24 is outside this interval)	✓ 1
M1 solve 14 − (t − 4) = 13	✓ 1
A1 t = 5, so the time is 5 − 0.764 = 4.24 s	✓ 1

Exam tip: With a piecewise model, solve each rule only on its own interval and then combine the results.

+ Add to test✓ In your test276IBMI3S-2-01r9-14Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Hard7 marks

A new café models the number of customers per day by N = A × b^t, where t is the number of days since the café opened. On day t = 1 there were 120 customers and on day t = 4 there were 405 customers.

(a)

Find the value of b and the value of A.

(3)

(b)

Use the model to predict the number of customers on day t = 7. Give your answer to the nearest whole number.

(1)

(c)

Suggest one reason why the model is unlikely to be valid for large values of t.

(1)

(d)

The café can serve at most 900 customers per day. Use the model to find the value of t at which N = 900. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 Ab = 120 and Ab⁴ = 405, so b³ = 405 ÷ 120 = 3.375	✓ 1
A1 b = 1.5	✓ 1
A1 A = 80	✓ 1
Part (b)
A1 1367	✓ 1
Part (c)
R1 the café has a limited number of seats or opening hours, so the number of customers cannot grow without limit	✓ 1
Part (d)
M1 solve 80 × 1.5^t = 900	✓ 1
A1 t = 5.97	✓ 1

Common error: Do not round b or A before using them in later parts unless the question says so.

+ Add to test✓ In your test277IBMI3S-2-01r12-12Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Hard7 marks

The braking distance d metres of a train travelling at speed v km h⁻¹ is modelled by d = av² + bv, where a and b are constants. The braking distance is 150 m at 60 km h⁻¹ and 350 m at 100 km h⁻¹.

(a)

Find the values of a and b.

(3)

(b)

Find the greatest speed at which the train can travel so that its braking distance does not exceed 400 m. Give your answer to 3 significant figures.

(2)

(c)

A new braking system reduces the braking distance by 20% at every speed. Find the greatest speed at which the train can now travel with a braking distance not exceeding 400 m. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms two equations 3600a + 60b = 150 and 10000a + 100b = 350	✓ 1
M1 solves the equations simultaneously	✓ 1
A1 a = 0.025 and b = 1	✓ 1
Part (b)
M1 0.025v² + v = 400 (the negative root −148 is rejected)	✓ 1
A1 v = 108 km h⁻¹	✓ 1
Part (c)
M1 0.8(0.025v² + v) = 400, which is 0.025v² + v = 500	✓ 1
A1 v = 123 km h⁻¹	✓ 1

Common error: Reducing the speed limit by 20% when the distance is reduced by 20%. The relationship is quadratic, so the speed does not change by the same percentage.

+ Add to test✓ In your test278IBMI3S-2-01r20-14Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 1Hard7 marks

A thermal storage tank at a solar heating plant cools after the heating stops. Its temperature, T °C, after t hours is modelled by T = A + Be^−kt, where A, B and k are constants and k > 0. The temperature is 90 °C at t = 0, 74 °C at t = 4 and 62 °C at t = 8.

(a)

Show that A = 26.

(3)

(b)

Find the value of k, giving your answer to 3 significant figures.

(2)

(c)

Find the time, in hours, for the tank to cool to 34 °C. Use the unrounded value of k and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 (T − A) falls by the same factor in equal times, so (74 − A)/(90 − A) = (62 − A)/(74 − A)	✓ 1
A1 (74 − A)² = (90 − A)(62 − A), giving 5476 − 148A = 5580 − 152A	✓ 1
A1 4A = 104, so A = 26 (must be shown)	✓ 1
Part (b)
M1 B = 64 and 26 + 64e^−4k = 74 (i.e. e^−4k = 0.75)	✓ 1
A1 k = ln(4/3)/4 = 0.0719	✓ 1
Part (c)
M1 26 + 64e^−kt = 34	✓ 1
A1 t = 28.9 hours (28.91...)	✓ 1

Exam tip: In a model of the form A + Be⁻kt, the constant A is the value the quantity approaches over a long time.

+ Add to test✓ In your test279IBMI3S-2-01r28-08Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 1Hard7 marks

A cyclist pedals at a steady rate of 90 revolutions per minute. The height h metres of the right pedal above the road, t seconds after the pedal is at its mid-height and rising, is modelled by
h(t) = 0.35 + 0.17 sin(3πt), with the angle in radians.

(a)

State the greatest and least heights of the pedal above the road.

(2)

(b)

Find the height of the pedal at t = 0.1 s, to 3 significant figures.

(2)

(c)

Find the two times in the first revolution, to 3 significant figures, at which the pedal is 0.45 m above the road.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 greatest height 0.52 m	✓ 1
A1 least height 0.18 m	✓ 1
Part (b)
M1 substitute t = 0.1 (GDC in radian mode)	✓ 1
A1 0.488 m	✓ 1
Part (c)
M1 set 0.35 + 0.17 sin(3πt) = 0.45	✓ 1
A1 t = 0.0667 s	✓ 1
A1 t = 0.267 s	✓ 1

Common error: Make sure the calculator is in radian mode when the argument of the sine is in radians; degree mode gives a very different answer.

+ Add to test✓ In your test280IBMI3S-2-04r35-13Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 1Hard7 marks

A car's speed, v m s⁻¹, over a 20-second section of a racing circuit is modelled by v = 0.04t³ − 1.2t² + 9t + 60, for 0 ≤ t ≤ 20.

(a)

Find the local maximum and local minimum speeds predicted by the model for 0 < t < 20, and the times at which they occur.

(2)

(b)

Find all the times in the interval 0 ≤ t ≤ 20 at which the speed is 72 m s⁻¹. Give your answers to 3 significant figures.

(3)

(c)

Find the average rate of change of speed between the times of the local maximum and the local minimum, and state what it represents.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum speed 80 m s⁻¹ at t = 5	✓ 1
A1 minimum speed 60 m s⁻¹ at t = 15	✓ 1
Part (b)
M1 sets 0.04t³ − 1.2t² + 9t + 60 = 72 (or graph intersection with v = 72)	✓ 1
A1 any two of t = 1.69, 9.33, 19.0 (accept 18.98)	✓ 1
A1 all three values	✓ 1
Part (c)
M1 (60 − 80)/(15 − 5)	✓ 1
A1 −2 m s⁻², the average acceleration (a deceleration of 2 m s⁻²) between these times	✓ 1

Exam tip: On a cubic graph, the local maximum and minimum are found with the GDC; check that they lie inside the domain given in the question.

+ Add to test✓ In your test281IBMI3S-2-03r45-09Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 1Hard7 marks

The fuel used by a cargo ship, F tonnes per day, is modelled by F = av^b, where v is the speed in knots and a and b are positive constants. Data are collected: F = 24.0 when v = 10 and F = 98.3 when v = 16.

(a)

By taking natural logarithms, show that the graph of ln F against ln v is a straight line, and state its gradient and its vertical intercept.

(2)

(b)

Find b correct to 3 significant figures and a correct to 2 significant figures.

(3)

(c)

Using a = 0.024 and b = 3, find the speed at which the ship uses 60 tonnes per day. Give your answer to 4 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 ln F = ln a + b ln v (using laws of logarithms)	✓ 1
A1 straight line with gradient b and vertical intercept ln a	✓ 1
Part (b)
M1 gradient = (ln 98.3 − ln 24.0)/(ln 16 − ln 10)	✓ 1
A1 b = 3.00	✓ 1
A1 a = 24/10³ = 0.024	✓ 1
Part (c)
M1 0.024v³ = 60	✓ 1
A1 v = 13.57 knots	✓ 1

Exam tip: A power model y = axb becomes linear when you take logarithms of both variables, and the gradient of that line is the power b.

+ Add to test✓ In your test282IBMI3S-2-03r48-13Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 1Hard8 marks

The brightness of a fading nova is modelled by B = ae^kt, where B is in arbitrary units and t is the time in days after the nova is first observed. The brightness is 72 units when t = 1 and 24 units when t = 5.

(a)

Find the value of k, giving your answer to 3 significant figures.

(3)

(b)

Find the value of a, giving your answer to 3 significant figures. Use the unrounded value of k.

(2)

(c)

Find the time at which the model gives a brightness of 5 units. Give your answer in days to 3 significant figures, using unrounded values of a and k.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms ae^5k / ae^k = 24/72, so e^4k = 1/3	✓ 1
M1 takes natural logarithms: 4k = ln(1/3)	✓ 1
A1 k = -0.275	✓ 1
Part (b)
M1 substitutes into 72 = ae^k	✓ 1
A1 a = 94.8	✓ 1
Part (c)
M1 sets ae^kt = 5	✓ 1
A1 t = ln(5/a)/k	✓ 1
A1 10.7 days	✓ 1

Exam tip: Keep unrounded values in your calculator memory between parts, and round only the final answer.

+ Add to test✓ In your test283IBMI3S-2-04r48-13Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; 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choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; 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Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 1Hard7 marks

After a car stops in the pit lane, the temperature, T °C, of a brake disc is modelled by T = 25 + Ae^−kt, where t is the time in minutes after stopping. At t = 0 the temperature is 745 °C and at t = 4 it is 376 °C.

(a)

Find the value of A.

(1)

(b)

Find the value of k, giving your answer to 3 significant figures.

(3)

(c)

Find the time at which the disc reaches 100 °C. Use the unrounded value of k and give your answer in minutes to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 745 = 25 + A, so A = 720	✓ 1
Part (b)
M1 376 = 25 + 720e^−4k, so e^−4k = 351/720	✓ 1
M1 takes natural logarithms: k = ln(720/351)/4	✓ 1
A1 k = 0.180	✓ 1
Part (c)
M1 sets 25 + 720e^−kt = 100	✓ 1
M1 e^−kt = 75/720, so t = ln(720/75)/k	✓ 1
A1 12.6 minutes	✓ 1

Common error: Forgetting the 25 °C constant: the exponential term describes the temperature above the surroundings, so subtract 25 before taking logarithms.

+ Add to test✓ In your test284IBMI3S-2-02r37-10Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Easy9 marks

Calculator

A clothing workshop makes shirts. The cost of making x shirts is £C = 480 + 7.25x. Each shirt is sold for £15.50, so the revenue from selling x shirts is £R = 15.5x.

(a)

State what the value 480 represents in this context.

(1)

(b)

Find the profit when 120 shirts are made and sold.

(3)

(c)

Find the number of shirts, x, for which revenue equals cost. Give your answer to three significant figures.

(2)

(d)

State the smallest whole number of shirts that must be sold for the workshop to make a profit.

(1)

(e)

Find the profit when this number of shirts is made and sold.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
R1 the fixed cost (setting-up cost) of £480 that has to be paid whatever the number of shirts made	✓ 1
Part (b)
M1 R = 15.5 × 120 = £1860	✓ 1
M1 C = 480 + 7.25 × 120 = £1350	✓ 1
A1 profit £510	✓ 1
Part (c)
M1 15.5x = 480 + 7.25x	✓ 1
A1 x = 58.2	✓ 1
Part (d)
A1 59	✓ 1
Part (e)
M1 15.5 × 59 − (480 + 7.25 × 59)	✓ 1
A1 £6.75	✓ 1

Exam tip: Profit = revenue − cost; a break-even value that is not a whole number must be rounded up when only whole items can be sold.

+ Add to test✓ In your test285IBMI-031-00-05Linear models and piecewise models Paper 2Medium12 marks

In a certain country, annual income tax is calculated as follows:
• no tax is paid on the first $12 000 of income;
• 20% tax is paid on income between $12 000 and $45 000;
• 35% tax is paid on income above $45 000.
Let T(x) be the tax, in dollars, paid on an annual income of x dollars.

(a)

Find the tax paid on an income of $30 000.

(2)

(b)

Show that the tax paid on an income of $45 000 is $6600.

(1)

(c)

Write down a piecewise expression for T(x), stating the domain of each piece.

(4)

(d)

Amara paid $11 230 in tax. Find her annual income.

(3)

(e)

The effective tax rate is the total tax paid as a percentage of total income. Find the effective tax rate for an income of $80 000.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 0.20 × (30000 − 12000)	✓ 1
A1 $3600.00	✓ 1
Part (b)
M1 0.20 × (45000 − 12000) = 0.20 × 33000 = 6600	✓ 1
Part (c)
A1 T(x) = 0 for first piece	✓ 1
A1 T(x) = 0.2(x − 12000) for second piece	✓ 1
A1 T(x) = 6600 + 0.35(x − 45000) for third piece	✓ 1
A1 correct domains 0 ≤ x ≤ 12000, 12000 < x ≤ 45000, x > 45000	✓ 1
Part (d)
R1 11230 > 6600 so third piece applies	✓ 1
M1 6600 + 0.35(x − 45000) = 11230	✓ 1
A1 x = $58 228.57	✓ 1
Part (e)
M1 T(80000) = 6600 + 0.35 × 35000 = 18850 and dividing by 80000	✓ 1
A1 23.6%	✓ 1

Common error: Applying the top rate to the whole income overstates the tax; the higher rate only applies to the portion of income above each threshold.

+ Add to test✓ In your test286IBMI-031-00-07Linear models and piecewise models Paper 2Medium11 marks

A courier charges C dollars to deliver a parcel of mass m kg, where
\(C(m)=\begin{cases}6+2.5m, & 0The function C is continuous at m = 4.

(a)

Find the value of a.

(2)

(b)

Find the cost of delivering a 13 kg parcel.

(2)

(c)

A parcel costs $24.10 to deliver. Find its mass.

(2)

(d)

Find the range of C.

(2)

(e)

Jonas has two parcels, each of mass 3 kg. Find how much he saves by packing them together and sending them as one 6 kg parcel rather than sending them separately.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 6 + 2.5(4) = a + 1.8(4)	✓ 1
A1 a = 8.8	✓ 1
Part (b)
M1 8.8 + 1.8 × 13	✓ 1
A1 $32.20	✓ 1
Part (c)
M1 8.8 + 1.8m = 24.10 (second piece since 24.10 > 16)	✓ 1
A1 m = 8.5 kg	✓ 1
Part (d)
A1 lower bound 6 < C (6 not included)	✓ 1
A1 C ≤ 44.80	✓ 1
Part (e)
M1 separately: 2 × (6 + 2.5 × 3) = $27.00	✓ 1
M1 combined: 8.8 + 1.8 × 6 = $19.60	✓ 1
A1 saving $7.40	✓ 1

Exam tip: 'Continuous at m = 4' means both pieces give the same value there; set the two expressions equal at that point to find the unknown.

+ Add to test✓ In your test287IBMI-031-07-07Exponential and sinusoidal models Paper 2Medium10 marks

In a northern city, the number of hours of daylight varies through the year. The maximum of 16.4 hours occurs on day 172 and the minimum of 8.0 hours occurs half a year later. The number of hours of daylight, D, on day t is modelled by
$D(t) = a\cos\big(b(t - 172)\big) + d$,
where a, b, d > 0, a year has 365 days and the angle is in degrees.

(a)

Find the values of a and d.

(2)

(b)

Find the value of b.

(1)

(c)

Find the number of hours of daylight on day 60.

(2)

(d)

Find the number of days in the year on which there are more than 14 hours of daylight.

(4)

(e)

Suggest one reason why the model might not give exactly the right number of daylight hours on a particular day.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $a = \frac{16.4 - 8.0}{2} = 4.2$	✓ 1
A1: $d = \frac{16.4 + 8.0}{2} = 12.2$	✓ 1
Part (b)
A1: $b = \frac{360}{365} = 0.986$ (0.98630…)	✓ 1
Part (c)
M1: $4.2\cos\big(0.98630\ldots(60 - 172)\big) + 12.2$	✓ 1
A1: 10.7 hours	✓ 1
Part (d)
M1: setting $D(t) = 14$, i.e. $\cos\big(b(t-172)\big) = \frac{1.8}{4.2}$, or sketching $D$ with the line $y = 14$	✓ 1
A1: $t = 106.5$ (106.48…)	✓ 1
A1: $t = 237.5$ (237.52…)	✓ 1
A1: $237.5 - 106.5 \approx 131$ days (accept 131.0; counting days 107 to 237 inclusive also gives 131)	✓ 1
Part (e)
R1: a valid reason, e.g. the real variation is not exactly sinusoidal, the year is not exactly 365 days (leap years), or terrain/atmospheric effects	✓ 1

Common error: In part (d), a frequent error is giving one of the intersection days instead of the length of the interval between them. Read what the question is asking for.

+ Add to test✓ In your test288IBMI-031-07-10Exponential and sinusoidal models Paper 2Medium10 marks

On a summer day in a desert town, the maximum temperature of 31 °C occurs at 15:00 and the minimum temperature of 19 °C occurs at 03:00. The temperature, T °C, t hours after midnight is modelled by
$T(t) = a\sin\big(b(t - c)\big) + d$,
where a, b, d > 0, 0 ≤ c < 24 and the angle is in degrees. Assume the pattern repeats every 24 hours.

(a)

Find the values of a and d.

(2)

(b)

Find the value of b.

(2)

(c)

Find the value of c.

(2)

(d)

Find the first time after midnight at which the temperature reaches 28 °C.

(2)

(e)

Find the number of hours during the day for which the temperature is above 28 °C.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $a = 6$	✓ 1
A1: $d = 25$	✓ 1
Part (b)
M1: period = 24 hours, so $b = \frac{360}{24}$	✓ 1
A1: $b = 15$	✓ 1
Part (c)
M1: the temperature is 25 °C and rising halfway between 03:00 and 15:00	✓ 1
A1: $c = 9$	✓ 1
Part (d)
M1: $6\sin\big(15(t - 9)\big) + 25 = 28$ or graph intersection	✓ 1
A1: $t = 11$, i.e. 11:00	✓ 1
Part (e)
M1: second solution $t = 19$, or use of symmetry about $t = 15$	✓ 1
A1: $19 - 11 = 8$ hours	✓ 1

Exam tip: For a sine model, the horizontal shift c is the time when the curve crosses the mean value going upwards, which is halfway between a minimum and the next maximum.

+ Add to test✓ In your test289IBMI-031-02-r1-03Exponential and sinusoidal models Paper 2Medium12 marks

On a particular summer day in a coastal city, the air temperature reached a maximum of 27 °C at 15:00 and a minimum of 13 °C at 03:00.
The temperature, T °C, is modelled by
$T(t) = a\cos\big(b(t - c)\big) + d$,
where t is the number of hours after midnight and a, b, c, d are positive constants, with $0 < c < 24$. Assume the temperature pattern repeats every 24 hours.

(a)

Find the value of a and the value of d.

(2)

(b)

Find the value of b.

(2)

(c)

Write down the value of c.

(1)

(d)

Use the model to estimate the temperature at 10:00.

(2)

(e)

Outdoor workers must take extra breaks whenever the temperature is above 24 °C. Find the total length of time during this day for which the temperature is above 24 °C.

(4)

(f)

State one reason why this model may not give reliable predictions of the temperature one week later.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
a = (27 − 13)/2 = 7 (A1)	✓ 1
d = (27 + 13)/2 = 20 (A1)	✓ 1
Part (b)
Uses period = 24, so b = 2π/24 (M1)	✓ 1
b = π/12 (= 0.262) (A1)	✓ 1
Part (c)
c = 15 (since the cosine curve has its maximum when t − c = 0) (A1)	✓ 1
Part (d)
Substitutes t = 10: 7cos(π(10 − 15)/12) + 20 (M1)	✓ 1
= 21.8 °C (A1)	✓ 1
Part (e)
Sets up 7cos(π(t − 15)/12) + 20 = 24, e.g. graph intersection (M1)	✓ 1
t = 11.3 (A1)	✓ 1
t = 18.7 (A1)	✓ 1
Total time = 18.677 − 11.323 = 7.35 hours (A1)	✓ 1
Part (f)
Valid reason in context, e.g. the model assumes every day has the same maximum and minimum, but weather conditions (cloud, wind, rain) change from day to day (R1)	✓ 1

Exam tip: For a cosine model, the horizontal shift c is the time of a maximum; for a sine model it would be the time the curve crosses its principal axis going up. Choose whichever matches the information given to avoid extra working.

+ Add to test✓ In your test290IBMI-031-05-r1-06Linear models and piecewise models Paper 2Medium9 marks

A cyclist rides along a straight route. Her distance from the start, d km, t minutes after setting off is modelled by
\(d(t)=\begin{cases}0.35t, & 0\le t\le 20\\ 7, & 20The route is 15.4 km long and ends at time T.

(a)

Find the distance travelled after 12 minutes.

(1)

(b)

Interpret the second piece of the function in context.

(2)

(c)

Find the value of T.

(2)

(d)

Find the average speed for the whole journey, in km h⁻¹.

(2)

(e)

Find the time at which the cyclist is 11 km from the start.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$d(12)=0.35\times12=4.2$ km A1	✓ 1
Part (b)
Distance stays constant at 7 km R1 ; so the cyclist stops (rests) for 10 minutes, from $t=20$ to $t=30$ R1	✓ 2
Part (c)
$7+0.28(T-30)=15.4$ M1 ; $T=60$ A1	✓ 2
Part (d)
$\frac{15.4}{60}$ km min⁻¹ or 15.4 km in 1 hour M1 ; 15.4 km h⁻¹ A1	✓ 2
Part (e)
Uses third piece: $7+0.28(t-30)=11$ M1 ; $t=44.3$ minutes (44.285…) A1	✓ 2

Exam tip: Decide which piece applies by comparing the target value with the values at the boundaries: 11 km is more than 7 km, so only the third piece can give it.

+ Add to test✓ In your test291IBMI-031-06-r2-04Quadratic and cubic models Paper 2Medium13 marks

The temperature inside a greenhouse, T °C, t hours after 06:00 is modelled by
$T(t) = at^3 + bt^2 + ct + d$, for $0 \leq t \leq 16$.
The following readings were taken: T(0) = 12, T(2) = 12.8, T(6) = 26.4 and T(10) = 40.

(a)

Write down the value of d, and write down three equations in a, b and c.

(3)

(b)

Find the values of a, b and c.

(2)

(c)

Find the maximum temperature in the greenhouse and the time of day at which it occurs.

(3)

(d)

Find T(20) and comment on the use of the model at this time.

(2)

(e)

Find the length of time for which the temperature is above 30 °C.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
d = 12 A1	✓ 1
Substituting at least one data point into the cubic M1	✓ 1
8a + 4b + 2c = 0.8, 216a + 36b + 6c = 14.4, 1000a + 100b + 10c = 28 (or equivalent forms) A1	✓ 1
Part (b)
a = −0.05 and b = 0.9 A1	✓ 1
c = −1.2 A1	✓ 1
Part (c)
Use of GDC maximum, or T′(t) = −0.15t² + 1.8t − 1.2 = 0 M1	✓ 1
Maximum temperature = 41.2 °C A1	✓ 1
t = 11.3 hours, so at about 17:17 (accept 17:18) A1	✓ 1
Part (d)
T(20) = −52 °C A1	✓ 1
This is unrealistic, since t = 20 lies outside the domain of the model, so the model should not be used to extrapolate R1	✓ 1
Part (e)
Setting T(t) = 30, or using the intersection with y = 30 on a GDC M1	✓ 1
t = 6.86 and t = 14.7 A1	✓ 1
Length of time = 7.84 hours A1	✓ 1

Exam tip: For time-of-day answers, convert the decimal part of the hours into minutes (0.29 h ≈ 17 min) and add the result to the starting time.

+ Add to test✓ In your test292IBMI-031-05-r2-05Linear models and piecewise models Paper 2Medium12 marks

Amira travels from home to work. Her distance from home, $d$ km, $t$ minutes after leaving is modelled by
$d(t) = \begin{cases} 0.08t, & 0 \le t \le 10 \\ 0.8, & 10 < t \le 16 \\ 0.8 + 0.5(t - 16), & 16 < t \le 36 \end{cases}$
She walks to a bus stop, waits, and then travels by bus along a straight road.

(a)

(i) Interpret the value 0.08 in context and convert it to km h⁻¹.
(ii) Explain why $d$ is constant for $10 < t \le 16$.

(3)

(b)

Find Amira's distance from home 30 minutes after leaving.

(2)

(c)

Find the time at which Amira is 5 km from home.

(2)

(d)

Her brother leaves the same house 8 minutes after Amira and cycles along the same route at a constant 0.3 km per minute, so his distance from home is $f(t) = 0.3(t - 8)$. Find the time at which he passes the bus stop, and state what Amira is doing at that moment.

(2)

(e)

Find the time at which Amira's bus catches up with her brother, and how far from home this happens.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: (i) Amira's walking speed is 0.08 km per minute	✓ 1
A1: $0.08 \times 60 = 4.8$ km h⁻¹	✓ 1
R1: (ii) she is waiting at the bus stop, so her distance from home does not change	✓ 1
Part (b)
M1: uses third piece: $0.8 + 0.5(30 - 16)$	✓ 1
A1: $d = 7.8$ km	✓ 1
Part (c)
M1: $0.8 + 0.5(t - 16) = 5$	✓ 1
A1: $t = 24.4$ minutes	✓ 1
Part (d)
A1: $0.3(t - 8) = 0.8 \Rightarrow t = 10.7$ minutes (10.666…)	✓ 1
R1: $10 < 10.7 \le 16$, so Amira is waiting at the bus stop	✓ 1
Part (e)
M1: $0.3(t - 8) = 0.8 + 0.5(t - 16)$	✓ 1
A1: $t = 24$ minutes (within $16 < t \le 36$)	✓ 1
A1: $d = 0.3(24 - 8) = 4.8$ km	✓ 1

Exam tip: After solving for an intersection, confirm the solution lies in the domain of the piece you used; otherwise it is not valid in context.

+ Add to test✓ In your test293IBMI-031-08-r2-04Choosing and evaluating models Paper 2Medium11 marks

The depth of water at a harbour entrance varies with the tides. On a particular day, high tide occurs at 03:00 with a depth of 8.6 m, and the next low tide occurs 6.2 hours later with a depth of 2.2 m.
The depth, $D$ metres, $t$ hours after midnight is modelled by
$D(t)=a\cos\big(b(t-3)\big)+d$, where $a,b,d>0$ and $b$ is measured in degrees per hour.

(a)

Find the value of $a$ and the value of $d$.

(2)

(b)

Find the value of $b$.

(2)

(c)

Use the model to find the depth of water at 14:00.

(2)

(d)

A fishing boat needs a depth of at least 4.0 m to pass through the entrance. Find the length of time between $t=3$ and $t=15.4$ for which the boat cannot pass.

(4)

(e)

Suggest one reason why the harbour master should not rely on this model for planning several weeks ahead.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: $a=\frac{8.6-2.2}{2}=3.2$	✓ 1
A1: $d=\frac{8.6+2.2}{2}=5.4$	✓ 1
Part (b)
M1: period $=2\times6.2=12.4$ hours	✓ 1
A1: $b=\frac{360}{12.4}=29.0$ (29.032…)	✓ 1
Part (c)
M1: substituting $t=14$, $3.2\cos(29.032\times11)+5.4$	✓ 1
A1: $D=7.83$ m	✓ 1
Part (d)
M1: setting $3.2\cos(b(t-3))+5.4=4$ or sketching on GDC with line $D=4$	✓ 1
A1: $t=6.99$	✓ 1
A1: $t=11.4$ (11.406…)	✓ 1
A1: duration $=4.41$ hours	✓ 1
Part (e)
R1: e.g. tidal heights vary over the month (spring/neap tides) so the amplitude is not constant; weather such as storm surges alters depth; small errors in the period accumulate over many cycles	✓ 1

Common error: Check your GDC is in degree mode when $b$ is in degrees per hour — using radian mode gives a wildly wrong depth with no warning.

+ Add to test✓ In your test294IBMI-031-08-r1-03Choosing and evaluating models Paper 2Medium12 marks

A jet of water in a garden fountain leaves a nozzle and its height $h$ metres above ground level, $t$ seconds after leaving the nozzle, is modelled by
$h(t) = at^2 + bt + c$.
Measurements show that $h = 1.2$ when $t = 0$, $h = 4.45$ when $t = 0.5$ and $h = 5.2$ when $t = 1$.

(a)

Write down the value of $c$ and two further equations in $a$ and $b$.

(3)

(b)

Find the value of $a$ and the value of $b$.

(2)

(c)

Find the maximum height of the water and the time at which it occurs.

(3)

(d)

Find the time at which the model predicts the water reaches ground level.

(2)

(e)

Video footage shows the water actually reaches ground level after 2.05 seconds. Calculate the percentage error in the model's prediction.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$c = 1.2$ A1	✓ 1
$0.25a + 0.5b + 1.2 = 4.45$ A1	✓ 1
$a + b + 1.2 = 5.2$ A1	✓ 1
Part (b)
Attempt to solve simultaneously (e.g. GDC) M1	✓ 1
$a = -5$, $b = 9$ A1	✓ 1
Part (c)
Uses $t = -\frac{b}{2a}$ or GDC maximum M1	✓ 1
$t = 0.9$ s A1	✓ 1
$h = 5.25$ m A1	✓ 1
Part (d)
Solves $-5t^2 + 9t + 1.2 = 0$ M1	✓ 1
$t = 1.92$ s (1.9247…; reject negative root) A1	✓ 1
Part (e)
$\frac{\|1.9247 - 2.05\|}{2.05} \times 100$ M1	✓ 1
6.11% A1	✓ 1

Exam tip: Carry the unrounded root 1.9247… into the percentage error; using 1.92 gives 6.34%, which may lose the final accuracy mark.

+ Add to test✓ In your test295IBMI-031-01-r1-05Quadratic and cubic models Paper 2Medium10 marks

A workshop makes x handmade lamps per week, where 0 ≤ x ≤ 70. The weekly cost, in dollars, of making x lamps is modelled by
$C(x) = 0.02x^3 - 1.5x^2 + 60x + 400$.
Each lamp is sold for $45 and every lamp made is sold.

(a)

Interpret the value 400 in this context.

(1)

(b)

Show that the weekly profit, P dollars, is $P(x) = -0.02x^3 + 1.5x^2 - 15x - 400$.

(2)

(c)

Find the values of x for which the workshop breaks even.

(2)

(d)

Hence state the numbers of lamps the workshop should make each week in order to make a profit.

(1)

(e)

Find the number of lamps that gives the maximum weekly profit, and the value of this maximum profit.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: the workshop has fixed weekly costs of $400, paid even if no lamps are made	✓ 1
Part (b)
M1: uses P = revenue − cost: 45x − (0.02x³ − 1.5x² + 60x + 400)	✓ 1
A1: simplifies correctly to the given expression	✓ 1
Part (c)
A1: x = 32.0 (31.97)	✓ 1
A1: x = 54.5 (54.51)	✓ 1
Part (d)
A1: 32 ≤ x ≤ 54, with x a whole number	✓ 1
Part (e)
M1: uses GDC maximum or sets $P'(x) = -0.06x^2 + 3x - 15 = 0$	✓ 1
A1: x = 44.4 (44.36)	✓ 1
R1: lamps must be a whole number, so compares P(44) = 140.32 and P(45) = 140.00	✓ 1
A1: 44 lamps, maximum profit $140.32	✓ 1

Common error: Rounding 44.36 to 44 without checking is not enough: for a discrete quantity, compare the profit at the whole numbers either side of the turning point.

+ Add to test✓ In your test296IBMI-031-01-r1-03Quadratic and cubic models Paper 2Medium10 marks

An open-topped box is made from a rectangular sheet of card measuring 40 cm by 25 cm. A square of side x cm is cut from each corner and the sides are folded up.
The volume of the box, V cm³, is given by $V(x) = x(40 - 2x)(25 - 2x)$.

(a)

State the set of possible values of x for this model, giving a reason for the upper limit.

(2)

(b)

Find the volume of the box when x = 3.

(2)

(c)

The box must hold at least 2000 cm³. Find the range of values of x for which this is possible.

(3)

(d)

Find the value of x that maximises the volume, and the maximum volume.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1: x > 0	✓ 1
A1: x < 12.5, because the width 25 − 2x must be positive (this is more restrictive than 40 − 2x > 0)	✓ 1
Part (b)
M1: 3 × 34 × 19	✓ 1
A1: V = 1938 cm³ (1940 cm³ to 3 s.f.)	✓ 1
Part (c)
M1: solves V(x) = 2000, e.g. intersection of graphs on GDC	✓ 1
A1: x = 3.20	✓ 1
A1: x = 7.01, so 3.20 ≤ x ≤ 7.01	✓ 1
Part (d)
M1: uses GDC maximum, or sets $V'(x) = 12x^2 - 260x + 1000 = 0$	✓ 1
A1: x = 5 (cm)	✓ 1
A1: maximum volume = 2250 cm³	✓ 1

Common error: Setting V′(x) = 0 gives x = 5 and x ≈ 16.7. The second turning point lies outside the domain 0 < x < 12.5 and has no physical meaning. Always check your solutions against the domain of the model.

+ Add to test✓ In your test297IBMI-031-06-r2-05Quadratic and cubic models Paper 2Medium10 marks

A concert hall manager models the number of tickets sold, n, when the ticket price is p dollars by $n = 1200 - 40p$. The hall has fixed costs of 3500 dollars per concert, plus a cost of 4 dollars for each ticket sold.

(a)

Write down an expression for the revenue, R dollars, in terms of p.

(1)

(b)

Find the ticket price that maximises revenue, and the maximum revenue.

(2)

(c)

Show that the profit, P dollars, is given by $P = -40p^2 + 1360p - 8300$.

(2)

(d)

Find the range of ticket prices, to the nearest cent, for which the concert makes a profit.

(3)

(e)

Find the maximum profit and the ticket price at which it occurs.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
R = p(1200 − 40p) or R = 1200p − 40p² A1	✓ 1
Part (b)
p = 15 dollars A1	✓ 1
Maximum revenue = 9000 dollars A1	✓ 1
Part (c)
P = 1200p − 40p² − 3500 − 4(1200 − 40p) M1	✓ 1
Correct expansion, e.g. −40p² + 1200p + 160p − 3500 − 4800, leading to the given answer A1	✓ 1
Part (d)
Setting P = 0, or P > 0, and solving, e.g. with a GDC M1	✓ 1
p = 7.972… and p = 26.027… A1	✓ 1
7.98 ≤ p ≤ 26.02 (dollars), with the prices rounded towards the inside of the interval so that the profit stays positive; accept the equivalent form 7.97 < p < 26.03 R1	✓ 1
Part (e)
p = 17 dollars A1	✓ 1
Maximum profit = 3260 dollars A1	✓ 1

Common error: Rounding a break-even price to the nearest cent in the usual way can give a price outside the profitable range. Round towards the inside of the interval.

+ Add to test✓ In your test298IBMI-031-03-r2-03Choosing and evaluating models Paper 2Medium12 marks

A new online video channel records its number of subscribers, $S$, $t$ months after launch.

$t$ (months)	0	4	8
$S$	1500	2400	3900

Two models are proposed, each passing through the points for $t = 0$ and $t = 8$:
Model A: $L(t) = pt + q$
Model B: $E(t) = 1500b^t$

(a)

Find the equation of Model A.

(2)

(b)

Show that $b = 1.13$, correct to 3 significant figures.

(2)

(c)

By evaluating both models at $t = 4$, determine which model fits the data better. Justify your answer.

(3)

(d)

At $t = 12$ the channel has 6100 subscribers. Use Model B to predict the number of subscribers at $t = 12$, and find the percentage error in this prediction.

(3)

(e)

Explain why Model B is unlikely to be suitable for predicting the number of subscribers several years after launch.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient $\frac{3900-1500}{8} = 300$	✓ 1
A1 $L(t) = 300t + 1500$	✓ 1
Part (b)
M1 $1500b^8 = 3900$ so $b^8 = 2.6$	✓ 1
A1 $b = 2.6^{1/8} = 1.12686...$, which is 1.13 to 3 s.f.	✓ 1
Part (c)
A1 $L(4) = 2700$	✓ 1
A1 $E(4) = 2420$ (accept 2418.7, or 2450 from using $b = 1.13$)	✓ 1
R1 Model B is better because its value is much closer to the observed 2400 than 2700 is	✓ 1
Part (d)
A1 $E(12) = 6290$ (accept 6288.6; using $b = 1.13$ gives 6502, which earns this mark only as follow-through)	✓ 1
M1 $\frac{\|E(12) - 6100\|}{6100} \times 100$ with their $E(12)$	✓ 1
A1 3.09% (accept answers from 3.08% to 3.11% from a rounded $E(12)$; follow-through 6.59% from $b = 1.13$)	✓ 1
Part (e)
R1 the exponential model increases without limit	✓ 1
R1 the number of potential subscribers is limited, so growth is likely to slow down (level off) as the market saturates	✓ 1

Common error: Using the rounded value b = 1.13 for t = 12 gives about 6500 instead of 6290. Store the unrounded value of b (2.6¹/⁸) and use it in later parts.

+ Add to test✓ In your test299IBMI-031-05-r2-03Linear models and piecewise models Paper 2Medium11 marks

A water company charges households for each quarter according to the volume of water used, $x$ m³. The charge, $C$ dollars, is modelled by
$C(x) = \begin{cases} 0.95x, & 0 \le x \le 20 \\ 19 + 1.60(x - 20), & 20 < x \le 50 \\ 67 + 2.40(x - 50), & x > 50 \end{cases}$

(a)

Find the charge for a household that uses 14 m³.

(2)

(b)

Find the charge for a household that uses 63 m³.

(2)

(c)

Show that the expressions for the second and third pieces give the same value when $x = 50$.

(2)

(d)

A household receives a bill of 54.36 dollars. Find the volume of water it used.

(3)

(e)

The company considers replacing this tariff with a single rate of $k$ dollars per m³. The rate is chosen so that a household using 63 m³ pays the same as under the current model. Find $k$.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: uses first piece: $0.95 \times 14$	✓ 1
A1: 13.30 dollars	✓ 1
Part (b)
M1: uses third piece: $67 + 2.40(63 - 50)$	✓ 1
A1: 98.20 dollars	✓ 1
Part (c)
M1: second expression at $x = 50$: $19 + 1.60 \times 30 = 67$	✓ 1
A1: third expression at $x = 50$: $67 + 2.40 \times 0 = 67$, so both give 67	✓ 1
Part (d)
R1: $19 < 54.36 \le 67$, so the second piece applies	✓ 1
M1: $19 + 1.60(x - 20) = 54.36$	✓ 1
A1: $x = 42.1$ m³	✓ 1
Part (e)
M1: $63k = 98.20$	✓ 1
A1: $k = 1.56$ dollars per m³ (3 s.f.)	✓ 1

Exam tip: Before solving a piecewise equation, evaluate the function at the boundaries to find which interval the given output lies in.

+ Add to test✓ In your test300IBMI-031-01-r2-03Quadratic and cubic models Paper 2Medium13 marks

The fuel consumption of a car, F litres per 100 km, when it travels at a constant speed of v km h⁻¹ is modelled by
$F(v) = av^2 + bv + c$, for 30 ≤ v ≤ 130.
Tests give the following results: at 40 km h⁻¹ the consumption is 7.6 litres per 100 km; at 80 km h⁻¹ it is 6.0 litres per 100 km; at 120 km h⁻¹ it is 8.0 litres per 100 km.

(a)

Write down a system of three equations in a, b and c.

(2)

(b)

Find the values of a, b and c.

(2)

(c)

Find the speed at which the fuel consumption is lowest, and state this lowest consumption.

(3)

(d)

Find the range of speeds for which the fuel consumption is at most 7 litres per 100 km.

(3)

(e)

A driver makes a 250 km journey at a constant 100 km h⁻¹. Fuel costs 1.62 USD per litre. Find the cost of the fuel used, in USD, giving your answer to two decimal places.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Substitutes at least one data pair into the model (M1)	✓ 1
1600a + 40b + c = 7.6; 6400a + 80b + c = 6.0; 14400a + 120b + c = 8.0 (A1)	✓ 1
Part (b)
a = 0.001125 (A1)	✓ 1
b = −0.175 and c = 12.8 (A1)	✓ 1
Part (c)
Uses $v = -\frac{b}{2a}$ or the GDC minimum (M1)	✓ 1
v = 77.8 km h⁻¹ (A1)	✓ 1
F = 5.99 litres per 100 km (A1)	✓ 1
Part (d)
Sets F(v) = 7, e.g. finds where the graph meets y = 7 (M1)	✓ 1
v = 47.9 and v = 108 (accept 107.7) (A1)	✓ 1
47.9 ≤ v ≤ 108 km h⁻¹ (A1)	✓ 1
Part (e)
F(100) = 6.55 litres per 100 km (M1)	✓ 1
Fuel used = 6.55 × 2.5 = 16.375 litres (A1)	✓ 1
Cost = 16.375 × 1.62 = 26.53 USD (A1)	✓ 1

Exam tip: Keep the exact coefficients (0.001125, −0.175, 12.8) stored in your GDC for every later part. Rounding a coefficient early can change the final cost in the second decimal place.

+ Add to test✓ In your test301IBMI-031-03-r1-04Choosing and evaluating models Paper 2Medium12 marks

The depth of water, $D$ metres, at a harbour entrance is recorded on a particular day. High tide gives a maximum depth of 7.8 m at 04:00. The next low tide gives a minimum depth of 1.4 m at 10:12.
The depth is modelled by $D(t) = a\cos(b(t - c)) + d$, where $t$ is the time in hours after midnight and $a, b, c > 0$.

(a)

Find the value of a and the value of d.

(2)

(b)

Find the value of b.

(2)

(c)

Write down a possible value of c.

(1)

(d)

Use the model to find the depth of water at 13:00.

(2)

(e)

A ferry can only use the harbour entrance when the depth is at least 5.5 m. Find the total length of time between 04:00 and 16:24 on this day during which the ferry can use the entrance.

(4)

(f)

Suggest one reason why this model may not predict the depth accurately on later days.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (7.8 − 1.4)/2 = 3.2	✓ 1
A1 d = (7.8 + 1.4)/2 = 4.6	✓ 1
Part (b)
M1 period = 2 × 6.2 = 12.4 hours	✓ 1
A1 b = 2π/12.4 = 0.507 (0.506707…)	✓ 1
Part (c)
A1 c = 4, the time of the maximum (accept any equivalent value 4 + 12.4k, e.g. 16.4)	✓ 1
Part (d)
M1 substituting t = 13 into their model	✓ 1
A1 D = 4.12 m	✓ 1
Part (e)
M1 setting up D(t) = 5.5 or D(t) ≥ 5.5	✓ 1
A1 t = 6.537… and t = 13.86…	✓ 1
M1 adding the intervals at each end, e.g. (6.537 − 4) + (16.4 − 13.863)	✓ 1
A1 5.07 hours (accept 5.08 from rounded intermediate values)	✓ 1
Part (f)
R1 e.g. the tidal period and range vary over the month (spring and neap tides), or weather such as wind or storms changes the water level	✓ 1

Common error: Using a period of 6.2 hours. The time from a maximum to the next minimum is only half a period.

+ Add to test✓ In your test302IBMI-031-06-r1-03Quadratic and cubic models Paper 2Medium11 marks

A small business sells scented candles. The weekly profit, $P, when each candle is sold at a price of $x is modelled by $P(x) = ax^2 + bx + c$.
Market research gives the following information:
• at a price of $5, the weekly profit is $1200;
• at a price of $10, the weekly profit is $2700;
• at a price of $20, the weekly profit is $1200.

(a)

Write down a system of three equations in a, b and c.

(2)

(b)

Find the values of a, b and c.

(2)

(c)

Interpret the value of c in context.

(1)

(d)

Find the price that maximises weekly profit and the maximum weekly profit.

(3)

(e)

Find the range of prices for which the business makes a positive profit.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Any two correct equations from $25a + 5b + c = 1200$, $100a + 10b + c = 2700$, $400a + 20b + c = 1200$	✓ 1
All three equations correct	✓ 1
Part (b)
Solves the system using a GDC or by elimination, e.g. $b = -25a$	✓ 1
$a = -30$, $b = 750$, $c = -1800$	✓ 1
Part (c)
At a price of $0 (candles given away) the business makes a weekly loss of $1800, i.e. its fixed weekly costs are $1800	✓ 1
Part (d)
Uses the symmetry of the data, $x = -\frac{b}{2a}$ or a GDC	✓ 1
Price = $12.50	✓ 1
Maximum weekly profit = $2887.50	✓ 1
Part (e)
Solves $-30x^2 + 750x - 1800 = 0$ or uses the graph's x-intercepts	✓ 1
$x = 2.69$	✓ 1
$x = 22.31$, so $2.69 < x < 22.31$	✓ 1

Exam tip: Two of the data points share the same profit, so the axis of symmetry lies midway between their prices. This gives a quick check on your vertex.

+ Add to test✓ In your test303IBMI-031-02-r1-05Exponential and sinusoidal models Paper 2Medium9 marks

A cup of tea is placed on a desk in a room. The temperature of the tea, H °C, t minutes after it is placed on the desk, is modelled by
$H(t) = 20 + 75e^{-kt}$, $t \ge 0$,
where k is a positive constant. After 4 minutes the temperature of the tea is 70 °C.

(a)

Write down the initial temperature of the tea.

(1)

(b)

Find the value of k.

(2)

(c)

Find the temperature of the tea after 10 minutes.

(2)

(d)

Write down the temperature that the tea approaches in the long term, and interpret this value in context.

(2)

(e)

The tea is considered too cool to enjoy once its temperature drops below 60 °C. Find the time after which this happens.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
95 °C (A1)	✓ 1
Part (b)
Sets up 20 + 75e^−4k = 70, or equivalent, e.g. e^−4k = 2/3 (M1)	✓ 1
k = 0.101 (0.101366…; exact form ln(1.5)/4) (A1)	✓ 1
Part (c)
Substitutes t = 10 with their k: 20 + 75e^−10k (M1)	✓ 1
= 47.2 °C (unrounded k). Accept 47.3 °C from k = 0.101 (A1)	✓ 1
Part (d)
20 °C (A1)	✓ 1
This is the temperature of the room; the tea cools towards room temperature (R1)	✓ 1
Part (e)
Sets up 20 + 75e^−kt = 60 with their k (or the inequality H < 60) (M1)	✓ 1
t = 6.20 minutes (unrounded k). Accept 6.22 minutes from k = 0.101 (A1)	✓ 1

Exam tip: Keep the unrounded value of k in your calculator memory for later parts. Rounding k early can change the third significant figure of later answers.

+ Add to test✓ In your test304IBMI-031-08-r1-04Choosing and evaluating models Paper 2Medium10 marks

At a harbour entrance on one particular day, high tide gives a water depth of 8.6 m at 03:00 and the following low tide gives a depth of 2.2 m at 09:15.
The depth $D$ metres, $t$ hours after midnight, is modelled by
$D(t) = p\cos\big(q(t - r)\big) + s$, where $p > 0$, $q$ is measured in degrees per hour and $0 \le r < 12$.

(a)

Find the values of $p$, $q$, $r$ and $s$.

(5)

(b)

Use the model to estimate the depth of water at 12:00.

(2)

(c)

A cargo boat can only pass the entrance when the depth is at least 4 m. Find the length of time, around the 09:15 low tide, during which the boat cannot pass.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
$p = \frac{8.6 - 2.2}{2} = 3.2$ A1	✓ 1
$s = \frac{8.6 + 2.2}{2} = 5.4$ A1	✓ 1
Period $= 2 \times 6.25 = 12.5$ hours M1	✓ 1
$q = \frac{360}{12.5} = 28.8$ A1	✓ 1
$r = 3$ (maximum of the cosine at the 03:00 high tide) A1	✓ 1
Part (b)
Substitutes $t = 12$: $3.2\cos(28.8 \times 9) + 5.4$ M1	✓ 1
$D = 4.80$ m A1	✓ 1
Part (c)
Solves $3.2\cos(28.8(t-3)) + 5.4 = 4$ (e.g. by GDC intersection) M1	✓ 1
$t = 7.03$ and $t = 11.5$ (7.0257…, 11.474…) A1	✓ 1
Duration $= 4.45$ hours A1	✓ 1

Common error: Leaving the calculator in radian mode when $q$ is in degrees gives a completely wrong depth, so check the mode before substituting.

+ Add to test✓ In your test305IBMI-031-02-r2-03Exponential and sinusoidal models Paper 2Medium11 marks

A cup of coffee is left to cool in a room. Its temperature, $T$ °C, $t$ minutes after it is poured is modelled by
$T(t) = 20 + 70e^{-kt}$, $t \ge 0$,
where $k$ is a positive constant. After 5 minutes the temperature of the coffee is 62 °C.

(a)

Write down the temperature of the coffee when it is poured.

(1)

(b)

Find the value of $k$.

(2)

(c)

Find the temperature of the coffee 12 minutes after it is poured.

(2)

(d)

The coffee is considered pleasant to drink once its temperature is 45 °C or lower. Find the time after pouring at which the coffee first becomes pleasant to drink.

(2)

(e)

Write down the equation of the horizontal asymptote of the graph of $T$ and interpret it in context.

(2)

(f)

A student claims that, according to the model, the coffee has reached exactly 20 °C after one hour. Use the model to comment on this claim.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
90 °C (A1)	✓ 1
Part (b)
Sets up 20 + 70e^−5k = 62 (M1)	✓ 1
k = 0.102 (0.102165…) (A1)	✓ 1
Part (c)
Substitutes t = 12 with their k: 20 + 70e^−12k (M1)	✓ 1
T = 40.5 °C (40.54…); accept 40.6 °C from k = 0.102 (A1)	✓ 1
Part (d)
Sets up 20 + 70e^−kt = 45 (M1)	✓ 1
t = 10.1 minutes (10.08…) (A1)	✓ 1
Part (e)
T = 20 (A1)	✓ 1
The coffee cools towards room temperature, 20 °C, but never falls below it (R1)	✓ 1
Part (f)
T(60) = 20.15 °C (20.152…) (A1)	✓ 1
The temperature is very close to 20 °C. However, e^−kt > 0 for all t, so the model never reaches exactly 20 °C and the claim is incorrect (R1)	✓ 1

Exam tip: Carry the unrounded value of k (0.102165…) into later parts. Using 0.102 throughout can shift answers in the third significant figure.

+ Add to test✓ In your test306IBMI3S-2-03-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Medium13 marks

A bike-share company models the number of rentals per day, n, at a price of $p per rental by n = 600 − 40p.

(a)

Find the number of rentals per day when p = 5, and hence show that the daily revenue is R = 600p − 40p².

(3)

(b)

The daily cost is $(1000 + 3n). Show that the daily profit is P = −40p² + 720p − 2800.

(3)

(c)

Find the price that maximises the daily profit, and the maximum daily profit.

(3)

(d)

Find the range of prices for which the company makes a profit.

(3)

(e)

State one restriction on p for the model n = 600 − 40p to be valid.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 n = 400	✓ 1
M1 R = p × n = p(600 − 40p)	✓ 1
A1 R = 600p − 40p² (shown)	✓ 1
Part (b)
M1 cost = 1000 + 3(600 − 40p) = 2800 − 120p	✓ 1
M1 P = R − cost	✓ 1
A1 P = −40p² + 720p − 2800 (shown)	✓ 1
Part (c)
M1 vertex p = −720 / (2 × −40), or GDC maximum	✓ 1
A1 p = $9	✓ 1
A1 maximum profit $440	✓ 1
Part (d)
M1 solve −40p² + 720p − 2800 = 0 (or > 0)	✓ 1
A1 p = 5.68 and p = 12.3	✓ 1
A1 5.68 < p < 12.3	✓ 1
Part (e)
R1 p ≤ 15 (or p < 15), since otherwise the number of rentals would be negative	✓ 1

Exam tip: Use the structure of the question: profit is revenue minus cost, so build the profit function in stages and check it against the value you found for p = 5.

+ Add to test✓ In your test307IBMI3S-2-03-09Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Medium10 marks

The area covered by an invasive weed on a pond, A m², is measured every two days.

t (days): 0, 2, 4, 6
A (m²): 12.0, 16.8, 23.5, 32.9
The pond has a surface area of 120 m².

(a)

Show, using the data, that an exponential model is more suitable than a linear model.

(3)

(b)

The weed area is modelled by A = 12b^t. Use the data to find b.

(2)

(c)

Use the model, with the unrounded value of b, to predict the area after 9 days.

(2)

(d)

Find the time at which the model first predicts that the weed covers the whole pond, and comment on the model beyond this time.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 differences 4.8, 6.7, 9.4 are not constant, so not linear	✓ 1
M1 ratios 16.8/12 = 1.40, 23.5/16.8 = 1.40, 32.9/23.5 = 1.40	✓ 1
R1 an approximately constant ratio indicates exponential growth	✓ 1
Part (b)
M1 b² = 1.4	✓ 1
A1 b = 1.18	✓ 1
Part (c)
M1 12 × 1.4^(9/2)	✓ 1
A1 54.5 m²	✓ 1
Part (d)
M1 12 × 1.4^(t/2) = 120	✓ 1
A1 t = 13.7 days	✓ 1
R1 the weed cannot cover more than the pond, so the exponential model is not valid beyond this time	✓ 1

Exam tip: To compare models from a table, test both the differences (linear) and the ratios (exponential) at equal steps of the independent variable.

+ Add to test✓ In your test308IBMI3S-2-03b-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Medium13 marks

The daily profit $P of a freight depot that dispatches n pallets in a day is modelled by P = −0.6n² + 84n − 2250.

(a)

Find the number of pallets that gives the maximum profit, and the maximum profit.

(3)

(b)

Find the numbers of pallets for which the depot makes a profit.

(3)

(c)

Find the numbers of pallets for which the profit is at least $500.

(3)

(d)

Show that the model gives an unrealistic result for 150 pallets, and suggest a reason.

(2)

(e)

The depot can dispatch at most 90 pallets a day. Find the profit when n = 90.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 uses n = −84 ÷ (2 × (−0.6)) (or technology)	✓ 1
A1 n = 70	✓ 1
A1 maximum profit $690	✓ 1
Part (b)
A1 roots n = 36.1 and n = 104 (103.9…)	✓ 1
A1 profit when 36.1 < n < 103.9	✓ 1
A1 whole numbers of pallets 37 ≤ n ≤ 103	✓ 1
Part (c)
M1 solves −0.6n² + 84n − 2250 = 500	✓ 1
A1 n = 52.2 and n = 87.8	✓ 1
A1 53 ≤ n ≤ 87 (whole numbers)	✓ 1
Part (d)
A1 P(150) = −3150, a loss that keeps growing as n increases	✓ 1
R1 for example the depot has a fixed capacity, so the model cannot hold for large n	✓ 1
Part (e)
M1 substitutes n = 90	✓ 1
A1 $450.00	✓ 1

Exam tip: For inequalities, a quick sketch of the parabola shows whether the answer lies between the roots or outside them.

+ Add to test✓ In your test309IBMI3S-2-01r9-05Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 2Medium14 marks

The temperature, T °C, of coffee beans in a roasting drum t minutes after loading is modelled by
T = 25 + 30t for 0 ≤ t < 4
T = 145 + 12(t − 4) for 4 ≤ t ≤ 10.
The roast ends at t = 10.

(a)

Find the temperature of the beans when t = 2.5.

(1)

(b)

Show that the model gives the same temperature at t = 4 from both rules.

(3)

(c)

Show that, for 4 ≤ t ≤ 10, the model can be written as T = 12t + 97.

(2)

(d)

Find the time at which the beans reach 190 °C.

(2)

(e)

The beans reach a stage called “first crack” at 196 °C. Find the temperature at the end of the roast, and hence the time spent above 196 °C.

(4)

(f)

Suggest one reason why the second rule cannot be used for large values of t.

(1)

(g)

Express the time found in part (e) as a percentage of the total roast time.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 100 °C	✓ 1
Part (b)
M1 first rule: 25 + 30 × 4	✓ 1
A1 = 145	✓ 1
R1 second rule gives 145 + 12(0) = 145, so the two rules agree (the model is continuous)	✓ 1
Part (c)
M1 expand 145 + 12t − 48	✓ 1
A1 T = 12t + 97 (AG)	✓ 1
Part (d)
M1 use second rule, 12t + 97 = 190	✓ 1
A1 t = 7.75 minutes	✓ 1
Part (e)
A1 T(10) = 217 °C	✓ 1
M1 solve 12t + 97 = 196	✓ 1
A1 t = 8.25 minutes	✓ 1
A1 time above 196 °C = 10 − 8.25 = 1.75 minutes	✓ 1
Part (f)
R1 the temperature would rise without limit (beans would burn), so the model only applies up to the end of the roast	✓ 1
Part (g)
A1 17.5%	✓ 1

Exam tip: To show two pieces of a piecewise model meet, evaluate both rules at the boundary value and compare.

+ Add to test✓ In your test310IBMI3S-2-01r9-06Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 2Medium14 marks

A footbridge has a parabolic arch. The height, h metres, of the arch above the water at a horizontal distance x metres from the left pier is modelled by h = −0.02x² + 1.2x, for 0 ≤ x ≤ 60.

(a)

Find the height of the arch when x = 10.

(1)

(b)

Find the width of the arch at water level.

(2)

(c)

Find the maximum height of the arch and the value of x at which it occurs.

(3)

(d)

A boat needs headroom of at least 14 m. Find the width of the arch over which the height is at least 14 m. Give your answer to 3 significant figures.

(3)

(e)

A pontoon 4 m wide carries a cabin 4.6 m high. Its centre line is at x = 5, so its edges are at x = 3 and x = 7. Determine whether the pontoon can pass under the arch.

(2)

(f)

The pontoon is moved so that its centre line is at x = c. Find the smallest value of c for which the cabin just clears the arch. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 10 m	✓ 1
Part (b)
M1 solve −0.02x² + 1.2x = 0	✓ 1
A1 x = 0 and x = 60, so the width is 60 m	✓ 1
Part (c)
M1 use symmetry or the axis of symmetry x = −1.2 ÷ (2 × (−0.02))	✓ 1
A1 x = 30	✓ 1
A1 h = 18 m	✓ 1
Part (d)
M1 solve −0.02x² + 1.2x = 14	✓ 1
A1 x = 15.86… and x = 44.14…	✓ 1
A1 width = 28.3 m	✓ 1
Part (e)
M1 evaluate the arch height at the nearer edge, x = 3	✓ 1
A1 h(3) = 3.42 m, which is less than 4.6 m, so it cannot pass	✓ 1
Part (f)
M1 the left edge is at x = c − 2; solve −0.02x² + 1.2x = 4.6	✓ 1
A1 x = 4.116… (smaller root)	✓ 1
A1 c = 6.12 m	✓ 1

Common error: A quadratic equation set equal to a height usually has two roots; choose the one that suits the situation described.

+ Add to test✓ In your test311IBMI3S-2-01r9-09Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 2Medium12 marks

The charge level, C %, of the battery in a space probe is modelled by C = 100 × 0.955^t, where t is the time in years since launch. An engineer proposes a different model, L = 100 − 3.9t, where L % is the charge level.

(a)

Use the exponential model to find the charge level after 4 years. Give your answer to 3 significant figures.

(1)

(b)

Use the linear model to find the charge level after 20 years.

(1)

(c)

Use the exponential model to find the charge level after 20 years. Give your answer to 3 significant figures.

(1)

(d)

A measurement taken 20 years after launch shows a charge level of 40 %. State which model is more suitable, giving a reason.

(2)

(e)

Use the exponential model to find the time at which the charge level falls to 50 %. Give your answer to 3 significant figures.

(2)

(f)

Interpret the values 100 and 0.955 in the exponential model.

(2)

(g)

The probe can operate only while its charge level is at least 30 %. Use the exponential model to find the number of complete years for which it can operate.

(2)

(h)

Find the time at which the linear model predicts a charge level of 0. Give your answer to 3 significant figures.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 83.2 %	✓ 1
Part (b)
A1 22 %	✓ 1
Part (c)
A1 39.8 %	✓ 1
Part (d)
A1 the exponential model	✓ 1
R1 its prediction (39.8 %) is much closer to 40 % than the linear prediction (22 %)	✓ 1
Part (e)
M1 solve 100 × 0.955^t = 50	✓ 1
A1 t = 15.1 years	✓ 1
Part (f)
A1 100 is the initial (launch) charge level, 100 %	✓ 1
A1 0.955 means the charge level is multiplied by 0.955 each year (it falls by 4.5 % per year)	✓ 1
Part (g)
M1 solve 100 × 0.955^t = 30, giving t = 26.1…	✓ 1
A1 26 complete years	✓ 1
Part (h)
A1 t = 25.6 years	✓ 1

Exam tip: To compare models, evaluate both at a value that is far from the data used to build them; this is where they disagree most.

+ Add to test✓ In your test312IBMI3S-2-01r12-05Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium12 marks

The number of passengers N on board a commuter train is modelled by N(t) = −2t³ + 30t² − 96t + 120, for 0 ≤ t ≤ 10, where t is the time in hours after 06:00.

(a)

Calculate the number of passengers on board at 06:00 and at 16:00.

(2)

(b)

Use technology to find the greatest number of passengers on board during the period 06:00 to 16:00, and the time at which this occurs.

(2)

(c)

Use technology to find the least number of passengers on board during the period, and the time at which this occurs.

(2)

(d)

Find the time, to the nearest minute, at which the number of passengers on board first reaches 150.

(4)

(e)

The railway company uses the model to predict the number of passengers at 18:00. Find the model's value for N at 18:00 and explain why the model should not be used at this time.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 N(0) = 120	✓ 1
A1 N(10) = 160	✓ 1
Part (b)
A1 maximum is 248 passengers	✓ 1
A1 at t = 8, that is 14:00	✓ 1
Part (c)
A1 minimum is 32 passengers (the value at t = 10 is 160 and at t = 0 is 120, so this is the least)	✓ 1
A1 at t = 2, that is 08:00	✓ 1
Part (d)
M1 sets −2t³ + 30t² − 96t + 120 = 150 and solves (graph intersection or technology)	✓ 1
A1 t = 5.19 hours (the only solution in the domain)	✓ 1
M1 converts to a clock time by adding to 06:00	✓ 1
A1 11:11	✓ 1
Part (e)
A1 N(12) = −168	✓ 1
R1 the number of passengers cannot be negative, so the model is not valid outside 0 ≤ t ≤ 10	✓ 1

Common error: Reading only the end points of a domain. A turning point inside the domain can give a greater or smaller value than either end.

+ Add to test✓ In your test313IBMI3S-2-01r12-11Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium12 marks

The length of daylight D hours in a northern town on day d of the year (d = 1 is 1 January) is modelled by D(d) = 12.2 + 5.8 sin(2π(d − 80)/365), where the angle is in radians.

(a)

Write down the longest and the shortest length of daylight predicted by the model.

(2)

(b)

Find the day of the year, to the nearest whole day, on which the longest daylight occurs.

(2)

(c)

Calculate the length of daylight on day 45. Give your answer to 3 significant figures.

(2)

(d)

Find the number of days in the year for which the model predicts more than 15 hours of daylight. Give your answer to 3 significant figures.

(3)

(e)

A measurement on day 355 gives 6.7 hours of daylight. Find the model's value for day 355 and calculate the percentage error of the model, taking 6.7 hours as the exact value. Use the unrounded model value in your calculation and give the percentage error to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 longest 18.0 hours	✓ 1
A1 shortest 6.4 hours	✓ 1
Part (b)
M1 2π(d − 80)/365 = π/2, or technology	✓ 1
A1 d = 171	✓ 1
Part (c)
M1 substitutes d = 45	✓ 1
A1 8.91 hours	✓ 1
Part (d)
M1 sets 12.2 + 5.8 sin(2π(d − 80)/365) = 15	✓ 1
M1 finds both solutions, about d = 109.3 and d = 233.2 (or uses a sketch of the graph)	✓ 1
A1 about 124 days	✓ 1
Part (e)
A1 D(355) = 6.40 hours	✓ 1
M1 percentage error = \|6.7 − 6.401\|/6.7 × 100	✓ 1
A1 4.46%	✓ 1

Common error: Putting a calculator in degree mode when the model uses radians. Check the mode by testing a value you can predict, such as sin(π/2) = 1.

+ Add to test✓ In your test314IBMI3S-2-02r17-05Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Medium10 marks

A swimmer completes a 1500 m open-water swim. The distance swum, d metres, after t seconds is modelled by
d = 1.6t for 0 ≤ t ≤ 600
d = 1.2t + c for 600 < t ≤ T,
where T is the time at which the swimmer finishes.

(a)

Find the distance swum after 600 s.

(1)

(b)

The two pieces of the model give the same distance at t = 600. Find the value of c.

(2)

(c)

Find T, the time taken to finish the swim, in seconds.

(2)

(d)

Find the swimmer's average speed over the whole swim, in m s⁻¹, correct to 3 significant figures.

(2)

(e)

A second swimmer swims the whole 1500 m at a constant speed of 1.35 m s⁻¹. Find by how many seconds the first swimmer finishes ahead of the second swimmer, correct to 3 significant figures. Use the unrounded time for the second swimmer.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.6 × 600 = 960 m	✓ 1
Part (b)
M1 960 = 1.2 × 600 + c (ECF from (a))	✓ 1
A1 c = 240	✓ 1
Part (c)
M1 1.2T + 240 = 1500	✓ 1
A1 T = 1050 s	✓ 1
Part (d)
M1 average speed = total distance ÷ total time = 1500 ÷ 1050 (ECF from (c))	✓ 1
A1 1.43 m s⁻¹	✓ 1
Part (e)
M1 time for second swimmer = 1500 ÷ 1.35 = 1111.1… s	✓ 1
M1 difference = 1111.1… − 1050 (ECF from (c))	✓ 1
A1 61.1 s	✓ 1

Exam tip: A piecewise model should join continuously at its boundary, so substitute the boundary value into both pieces to find an unknown constant.

+ Add to test✓ In your test315IBMI3S-2-02r17-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Medium9 marks

A library records the number of e-books borrowed in each of five years, where t is the number of years since 2020.

t: 0, 1, 2, 3, 4
Loans, L: 1200, 1580, 2010, 2570, 3300

Two models are proposed:
Model A: L = 1200 × 1.28^t
Model B: L = 540t + 950

(a)

Find the number of loans predicted by each model when t = 4. Give your answers to the nearest whole number.

(2)

(b)

By calculating ratios of loans in successive years, and differences in loans in successive years, explain which type of model the data suggest.

(2)

(c)

For the five data points, the sum of the differences between the actual loans and the model values, ignoring signs, is 220 for Model A and 550 for Model B. State which model fits the data better and give a reason.

(2)

(d)

Use the better model to estimate the number of loans when t = 6. Give your answer correct to 3 significant figures.

(2)

(e)

Comment on the reliability of your estimate in part (d).

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model A: 3221	✓ 1
A1 Model B: 3110	✓ 1
Part (b)
M1 ratios 1.32, 1.27, 1.28, 1.28 are approximately constant (and differences 380, 430, 560, 730 are not constant)	✓ 1
A1 so the data suggest an exponential model rather than a linear model	✓ 1
Part (c)
M1 compare 220 and 550 (the smaller total difference)	✓ 1
A1 Model A fits better, which agrees with part (b)	✓ 1
Part (d)
M1 1200 × 1.28⁶	✓ 1
A1 5280	✓ 1
Part (e)
R1 the estimate is an extrapolation beyond the range of the data, and growth cannot continue to increase at the same percentage rate indefinitely (for example it is limited by the number of readers or the budget)	✓ 1

Exam tip: To judge a model, compare it with the data in more than one way: check the pattern in the data, the size of the differences, and whether predictions outside the data are sensible.

+ Add to test✓ In your test316IBMI3S-2-01r25-08Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 2Medium12 marks

A pot of soup is left to cool in a kitchen. Its temperature, T °C, is measured t minutes after it is removed from the heat.
t (min): 0, 10, 20, 30, 40
T (°C): 85, 66, 53, 43, 37
Two models are proposed:
Model A (linear): T = 85 − 1.2t
Model B (exponential): T = 20 + 65e^−0.035t

(a)

Complete a table of the temperatures predicted by Model A and by Model B at t = 10, 20, 30 and 40, giving each to 1 decimal place.

(2)

(b)

For each model, find the sum of the squares of the differences between the measured and predicted temperatures at the five times. Use the unrounded predictions from Model B.

(4)

(c)

State, with a reason, which model fits the data better.

(1)

(d)

Use each model to predict the temperature of the soup at t = 70, and comment on the validity of each prediction. The kitchen temperature is 20 °C.

(3)

(e)

Use Model B to find the time at which the soup has cooled to 60 °C. Give your answer correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model A: 73.0, 61.0, 49.0, 37.0	✓ 1
A1 Model B: 65.8, 52.3, 42.7, 36.0	✓ 1
Part (b)
M1 differences (measured − predicted) for Model A: 0, −7, −8, −6, 0	✓ 1
A1 sum of squares for Model A = 149	✓ 1
M1 corresponding differences for Model B found, then squared and summed	✓ 1
A1 sum of squares for Model B = 1.57	✓ 1
Part (c)
R1 Model B, since its sum of squares is much smaller	✓ 1
Part (d)
A1 Model A gives 1 °C	✓ 1
A1 Model B gives 25.6 °C	✓ 1
R1 the Model A prediction is impossible because soup in a 20 °C kitchen cannot cool below 20 °C, whereas Model B approaches 20 °C	✓ 1
Part (e)
M1 20 + 65e^−0.035t = 60	✓ 1
A1 t = 13.9 minutes	✓ 1

Exam tip: When asked to compare models, use a numerical measure of fit and then also check whether the model's long-term behaviour makes sense in context.

+ Add to test✓ In your test317IBMI3S-2-01r28-13Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Medium13 marks

The height h metres above sea level of a cycling route, at a distance x km from the start, is modelled by
h = 50 + 30x for 0 ≤ x ≤ 4
h = 170 − 15(x − 4) for 4 < x ≤ 10
h = 80 + 25(x − 10) for 10 < x ≤ 14.

(a)

Find the height at x = 2.5.

(1)

(b)

Find the height at x = 7.

(2)

(c)

Write down the gradient of the third section of the route, and state what it means.

(2)

(d)

Show that the model gives the same height at x = 4 and at x = 10 whichever of the two neighbouring rules is used.

(2)

(e)

Find all the distances at which the height is 150 m. Give answers to 3 significant figures.

(4)

(f)

Find the total height gained on the two climbing sections of the route.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 125 m	✓ 1
Part (b)
M1 170 − 15(3)	✓ 1
A1 125 m	✓ 1
Part (c)
A1 gradient 25	✓ 1
A1 the route climbs 25 m for every 1 km travelled along it	✓ 1
Part (d)
M1 first and second rules at x = 4: 170 and 170	✓ 1
A1 second and third rules at x = 10: 80 and 80	✓ 1
Part (e)
M1 solve 150 = each rule within its own interval	✓ 1
A1 x = 3.33	✓ 1
A1 x = 5.33	✓ 1
A1 x = 12.8	✓ 1
Part (f)
M1 (170 − 50) + (180 − 80)	✓ 1
A1 220 m	✓ 1

Exam tip: For a piecewise function, solve against each rule separately and keep only solutions that lie inside that rule's interval.

+ Add to test✓ In your test318IBMI3S-2-01r28-14Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Medium13 marks

On an excavation, the number N of artefacts found per square metre at depth d metres is recorded.
d: 0.5, 1.0, 1.5, 2.0, 2.5
N: 41, 33, 24, 19, 10

Two models are proposed: Model A: N = −15d + 49 and Model B: N = 55e^−0.6d.

(a)

Find the value of N predicted by each model at d = 1.5. Give answers to 3 significant figures.

(2)

(b)

Model A gives predictions 41.5, 34, 26.5, 19, 11.5. Find the sum of the squares of the differences between the data and Model A.

(3)

(c)

Model B gives predictions 40.7, 30.2, 22.4, 16.6, 12.3 (3 significant figures). Find the sum of the squares of the differences between the data and Model B, using these values.

(3)

(d)

State which model fits the data better, giving a reason.

(1)

(e)

Find the largest depth, to 3 significant figures, for which Model A gives a non-negative number of artefacts.

(2)

(f)

Use Model B to estimate N at d = 4 to 3 significant figures, and explain why this gives Model B an advantage for deep layers.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model A: 26.5	✓ 1
A1 Model B: 22.4	✓ 1
Part (b)
M1 differences −0.5, −1, −2.5, 0, −1.5	✓ 1
M1 sum of squares 0.25 + 1 + 6.25 + 0 + 2.25	✓ 1
A1 9.75	✓ 1
Part (c)
M1 differences 0.3, 2.8, 1.6, 2.4, −2.3	✓ 1
M1 sum of squares 0.09 + 7.84 + 2.56 + 5.76 + 5.29	✓ 1
A1 21.5	✓ 1
Part (d)
R1 Model A, because its sum of squared differences (9.75) is smaller than that of Model B	✓ 1
Part (e)
M1 set −15d + 49 = 0	✓ 1
A1 d = 3.27 m	✓ 1
Part (f)
A1 N = 4.99	✓ 1
R1 Model B stays positive for all depths, whereas Model A would give a negative (impossible) count beyond 3.27 m	✓ 1

Common error: A better fit inside the data range does not make a model suitable outside it; always check that extrapolated values make sense in context.

+ Add to test✓ In your test319IBMI3S-2-04r32-06Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium10 marks

The number of players, P thousand, who are online at the same time in a new video game is modelled by $P = -t^{3} + 9t^{2} - 15t + 30$, where t is the time in weeks after the launch of the game, for $0 \le t \le 7$.

(a)

Find the number of players online at the same time 3 weeks after launch.

(2)

(b)

Find the maximum number of players predicted by the model and the week in which it occurs.

(2)

(c)

Find the two times when the number of players online is 50 thousand. Give your answers to 3 significant figures.

(3)

(d)

A designer uses the model to predict the number of players 10 weeks after launch. Find the value of P given by the model when $t = 10$ and explain why the model is not suitable for this prediction.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $P(3) = -27 + 81 - 45 + 30$	✓ 1
A1 39 thousand (39 000 players)	✓ 1
Part (b)
M1 GDC maximum of the graph (or $P'(t)=0$ giving $t=5$)	✓ 1
A1 $t = 5$ weeks, maximum 55 thousand	✓ 1
Part (c)
M1 $-t^{3}+9t^{2}-15t+30 = 50$ (or GDC intersections with $y = 50$)	✓ 1
A1 $t = 4.00$ weeks	✓ 1
A1 $t = 5.85$ weeks	✓ 1
Part (d)
M1 $P(10) = -1000 + 900 - 150 + 30$	✓ 1
A1 −220 thousand, which is negative	✓ 1
R1 a number of players cannot be negative, so the model should not be used outside $0 \le t \le 7$	✓ 1

Exam tip: Use the GDC to solve a cubic equation: plot both sides as graphs and use the intersection tool, then check each solution lies inside the stated domain.

+ Add to test✓ In your test320IBMI3S-2-04r32-09Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium9 marks

The number of subscribers to an online game is modelled by $N = 4200\times1.12^{t}$, where t is the number of months after the game's launch.

(a)

Find the number of subscribers after 6 months. Give your answer to the nearest 10.

(2)

(b)

State the percentage by which the number of subscribers increases each month, according to the model.

(1)

(c)

Find the time taken for the number of subscribers to reach 20 000. Give your answer to 3 significant figures.

(3)

(d)

The developer wants the number of subscribers to reach 20 000 after exactly 12 months, still starting from 4200 and growing by the same percentage each month. Find the required monthly percentage increase. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $4200\times1.12^{6}$	✓ 1
A1 8290 so 8290 subscribers	✓ 1
Part (b)
A1 12%	✓ 1
Part (c)
M1 $4200\times1.12^{t} = 20000$ (or GDC intersection)	✓ 1
M1 take logarithms: $t = \dfrac{\ln(20000/4200)}{\ln1.12}$	✓ 1
A1 13.8 months	✓ 1
Part (d)
M1 $4200\times r^{12} = 20000$	✓ 1
M1 $r = \left(\dfrac{20000}{4200}\right)^{1/12} = 1.1389$	✓ 1
A1 13.9%	✓ 1

Common error: A monthly growth factor of 1.139 is an increase of 13.9%, not 113.9%: subtract 1 before converting to a percentage.

+ Add to test✓ In your test321IBMI3S-2-04r32-10Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium9 marks

The depth L metres of water in a reservoir is recorded every 3 days during a dry spell. t is the number of days since the first reading.

t	0	3	6	9
L	80.0	70.8	62.6	55.4

(a)

Calculate the average rate of change of depth between $t = 0$ and $t = 3$, and between $t = 6$ and $t = 9$.

(2)

(b)

Hence explain why a linear model is not suitable for these data.

(1)

(c)

Calculate the ratio of each depth to the previous depth. Hence explain why an exponential model is suitable.

(2)

(d)

The data are modelled by $L = 80\times k^{t}$. Use the ratio 0.885 for a 3-day interval to find the value of k to 3 significant figures.

(2)

(e)

Use the model $L = 80\times0.960^{t}$ to predict the depth at $t = 12$. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 $\dfrac{70.8-80.0}{3} = -3.07$ m per day	✓ 1
A1 $\dfrac{55.4-62.6}{3} = -2.40$ m per day	✓ 1
Part (b)
R1 the rate of change is not constant (it becomes smaller in magnitude), but a linear model has constant gradient	✓ 1
Part (c)
M1 $\dfrac{70.8}{80.0} = 0.885$, $\dfrac{62.6}{70.8} = 0.884$, $\dfrac{55.4}{62.6} = 0.885$	✓ 1
R1 the ratios are (approximately) constant, which is characteristic of exponential decay	✓ 1
Part (d)
M1 $k^{3} = 0.885$	✓ 1
A1 $k = 0.960$	✓ 1
Part (e)
M1 $80\times0.960^{12}$	✓ 1
A1 49.0 m	✓ 1

Exam tip: To test for an exponential model, check whether equal steps in the input give a constant ratio between outputs; for a linear model check for a constant difference.

+ Add to test✓ In your test322IBMI3S-2-04r32-11Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium9 marks

A water company charges each household a bill, B euros, every quarter. The charge is €18 plus €2.10 for each cubic metre (m³) used, for the first 40 m³, plus €3.45 for each cubic metre used above 40 m³.

(a)

Find the bill for a household that uses 35 m³.

(2)

(b)

Find the bill for a household that uses 62 m³.

(2)

(c)

Show that, for a household using V m³ where $V > 40$, the bill is given by $B = 3.45V - 36$.

(2)

(d)

One household's bill is €205.50. Find the amount of water used by this household.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $18 + 2.10\times35$	✓ 1
A1 €91.50	✓ 1
Part (b)
M1 $18 + 2.10\times40 + 3.45\times22$	✓ 1
A1 €177.90	✓ 1
Part (c)
M1 $B = 18 + 84 + 3.45(V-40)$	✓ 1
A1 $B = 102 + 3.45V - 138 = 3.45V - 36$ (answer given, correct working shown)	✓ 1
Part (d)
M1 note that €205.50 exceeds €102 (the bill at 40 m³), so use $3.45V - 36 = 205.50$	✓ 1
M1 $V = \dfrac{241.5}{3.45}$	✓ 1
A1 70 m³	✓ 1

Common error: The higher rate applies only to the water above 40 m³, not to the whole amount used.

+ Add to test✓ In your test323IBMI3S-2-03r34-14Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 2Medium12 marks

The height, A metres, of a survey drone t seconds after take-off is modelled by
A = 12t for 0 ≤ t ≤ 10
A = 120 for 10 < t ≤ 25
A = 120 − 0.192(t − 25)² for 25 < t ≤ 50

(a)

Find the height of the drone at t = 7 and at t = 18.

(2)

(b)

Find the time at which the drone reaches a height of 100 m. Give your answer to 3 significant figures.

(2)

(c)

Find the height of the drone at t = 40.

(2)

(d)

Show that the model is continuous at t = 10 and at t = 25.

(3)

(e)

Find the time during the descent at which the drone's height is 50 m. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 A(7) = 84 m	✓ 1
A1 A(18) = 120 m	✓ 1
Part (b)
M1 12t = 100 (first piece)	✓ 1
A1 t = 8.33 s	✓ 1
Part (c)
M1 120 − 0.192 × 15²	✓ 1
A1 76.8 m	✓ 1
Part (d)
M1 at t = 10: first piece 12 × 10 = 120, second piece 120	✓ 1
M1 at t = 25: second piece 120, third piece 120 − 0.192 × 0 = 120	✓ 1
R1 the values from adjacent pieces agree at both joins, so the model is continuous	✓ 1
Part (e)
M1 setting 120 − 0.192(t − 25)² = 50	✓ 1
M1 (t − 25)² = 70/0.192 and taking the positive square root, since t > 25	✓ 1
A1 t = 44.1 s	✓ 1

Common error: A quadratic equation has two roots; reject the one that lies outside the interval for which that piece of the model applies.

+ Add to test✓ In your test324IBMI3S-2-01r35-03Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 2Medium13 marks

Calculator

[Set your calculator to radians.] During a Formula 1 race, the surface temperature of a front tyre, T °C, is modelled by
T = 95 + 18 sin(π/45 (t − 10)),
where t is the time in seconds since the car crossed the start line and the temperature pattern repeats every lap.

(a)

Find the time taken for one complete cycle of the model.

(2)

(b)

Find the temperature of the tyre when t = 25. Give your answer to 3 significant figures.

(3)

(c)

Find the first time, t ≥ 0, at which the temperature is 100 °C. Give your answer to 3 significant figures.

(3)

(d)

Find the maximum temperature of the tyre and the first time at which it occurs.

(2)

(e)

In one cycle, find the length of time for which the temperature is at least 100 °C. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 period = 2π ÷ (π/45)	✓ 1
A1 90 seconds	✓ 1
Part (b)
M1 substitutes t = 25	✓ 1
A1 95 + 18 sin(π/3) = 110.588…	✓ 1
A1 111 °C	✓ 1
Part (c)
M1 forms 95 + 18 sin(π/45 (t − 10)) = 100, so sin(π/45 (t − 10)) = 5/18	✓ 1
M1 uses inverse sine (or GDC) to obtain π/45 (t − 10) = 0.2815	✓ 1
A1 t = 14.0 seconds (14.03…)	✓ 1
Part (d)
A1 maximum temperature 113 °C (95 + 18)	✓ 1
A1 first occurs at t = 32.5 seconds (when π/45 (t − 10) = π/2)	✓ 1
Part (e)
M1 finds the two solutions of the equation in one cycle (t = 14.03… and t = 50.97…) or uses sinθ ≥ 5/18 for θ between 0.2815 and π − 0.2815	✓ 1
A1 time = (π − 2 × 0.2815) × 45/π = 36.94…	✓ 1
A1 36.9 seconds	✓ 1

Common error: A sine equation has a second solution in each cycle; for a time interval above a value you need both.

+ Add to test✓ In your test325IBMI3S-2-01r35-07Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 2Medium12 marks

Calculator

The number of occupied beds, B, in a hospital ward is modelled by B = t³ − 15t² + 63t + 20, where t is the number of weeks since 1 November, for 0 ≤ t ≤ 10.

(a)

Find the number of occupied beds when t = 5.

(2)

(b)

Use your graphic display calculator to find the coordinates of the local maximum and the local minimum of the graph of B for 0 ≤ t ≤ 10.

(4)

(c)

The ward has 95 beds. Find the total time, in weeks, for which the model predicts more than 95 beds are needed during the 10 weeks. Give your answer to 3 significant figures.

(4)

(d)

Calculate the value of B predicted by the model when t = 12 and comment on its validity.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 5: 125 − 375 + 315 + 20	✓ 1
A1 85	✓ 1
Part (b)
A1 local maximum at t = 3	✓ 1
A1 B = 101	✓ 1
A1 local minimum at t = 7	✓ 1
A1 B = 69	✓ 1
Part (c)
M1 forms t³ − 15t² + 63t + 20 = 95 (or draws y = 95 on the graph)	✓ 1
A1 solutions t = 2.07…, 4.11…, 8.82…	✓ 1
M1 adds the two intervals (4.107… − 2.069…) + (10 − 8.823…)	✓ 1
A1 3.21 weeks	✓ 1
Part (d)
A1 B = 344	✓ 1
R1 not valid: the model gives a number of beds far above any realistic ward size (and is outside the domain 0 ≤ t ≤ 10)	✓ 1

Common error: A cubic can equal a value three times; check every solution lies inside the stated domain and count each interval.

+ Add to test✓ In your test326IBMI3S-2-01r35-11Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 2Medium12 marks

Calculator

A Formula 1 team models the lap time, L seconds, on lap n of a set of tyres by L = an² + bn + c. The recorded lap times are 90.34 s on lap 3, 90.04 s on lap 6 and 90.20 s on lap 10.

(a)

Find the values of a, b and c.

(4)

(b)

Find the lap number on which the model gives the shortest lap time, and that shortest lap time.

(3)

(c)

Find the value of n with n > 7 for which L = 90.30. Give your answer to 3 significant figures.

(3)

(d)

Find the lap time predicted by the model for lap 25 and comment on its reliability.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms three equations, e.g. 9a + 3b + c = 90.34, 36a + 6b + c = 90.04, 100a + 10b + c = 90.20	✓ 1
A1 a = 0.02	✓ 1
A1 b = −0.28	✓ 1
A1 c = 91	✓ 1
Part (b)
M1 uses n = −b ÷ 2a (or the minimum on the GDC)	✓ 1
A1 lap 7	✓ 1
A1 90.02 s	✓ 1
Part (c)
M1 forms 0.02n² − 0.28n + 91 = 90.30, i.e. n² − 14n + 35 = 0	✓ 1
A1 n = 3.26… or 10.74… (7 ± √14)	✓ 1
A1 n = 10.7 (the root with n > 7)	✓ 1
Part (d)
A1 L = 96.5 s	✓ 1
R1 unreliable: lap 25 is far outside the recorded laps (extrapolation), and tyres would normally have been changed	✓ 1

Exam tip: For a full set of coefficients from three data points, write the three equations and use the GDC simultaneous-equation solver.

+ Add to test✓ In your test327IBMI3S-2-02r35-04Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Medium12 marks

Calculator

[Set your calculator to radians.] The hub of a wind turbine is 90 m above the ground and each blade is 45 m long. The turbine rotates 15 times per minute. The height above the ground of the tip of one blade, h metres, at time t seconds is modelled by h = 90 − 45 cos(bt), where the tip is at its lowest point when t = 0.

(a)

Write down the greatest height of the blade tip above the ground.

(1)

(b)

Write down the least height of the blade tip above the ground.

(1)

(c)

Find the value of b.

(2)

(d)

Find the height of the blade tip when t = 1.3. Give your answer to 3 significant figures.

(2)

(e)

Find the first time at which the tip is 100 m above the ground. Give your answer to 3 significant figures.

(3)

(f)

In one rotation, find the percentage of the time for which the tip is at least 100 m above the ground. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 135 m	✓ 1
Part (b)
A1 45 m	✓ 1
Part (c)
M1 period = 60 ÷ 15 = 4 s, so b = 2π ÷ 4	✓ 1
A1 b = π/2 (1.57)	✓ 1
Part (d)
M1 substitutes into 90 − 45 cos(π/2 × 1.3)	✓ 1
A1 110 m (110.4…)	✓ 1
Part (e)
M1 forms 90 − 45 cos(πt/2) = 100, so cos(πt/2) = −10/45	✓ 1
M1 uses inverse cosine (or GDC): πt/2 = 1.7954…	✓ 1
A1 t = 1.14 s (1.1427…)	✓ 1
Part (f)
M1 uses symmetry: the tip is above 100 m between t = 1.1427… and t = 4 − 1.1427… = 2.8573…	✓ 1
A1 time above 100 m = 1.715… s	✓ 1
A1 42.9%	✓ 1

Common error: Use the unrounded first time when finding the interval; subtracting rounded values can change the third significant figure.

+ Add to test✓ In your test328IBMI3S-2-02r35-08Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Medium12 marks

Calculator

[Set your calculator to radians.] A wave pool produces waves so that the water depth at a marker, D metres, at time t seconds is modelled by D = a sin(bt) + d, where a, b > 0. The depth varies between 1.05 m and 1.75 m, and one complete wave takes 8 seconds. When t = 0 the depth is at its middle value and rising.

(a)

Find the values of a and d.

(2)

(b)

Find the value of b, as a multiple of π.

(2)

(c)

Find the depth when t = 1.5. Give your answer to 3 significant figures.

(2)

(d)

Find the first time at which the depth is 1.6 m. Give your answer to 3 significant figures.

(3)

(e)

Part of the pool is closed to children while the depth exceeds 1.55 m. In one wave, find the length of time for which this happens. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (1.75 − 1.05) ÷ 2 = 0.35	✓ 1
A1 d = (1.75 + 1.05) ÷ 2 = 1.4	✓ 1
Part (b)
M1 b = 2π ÷ 8	✓ 1
A1 b = π/4	✓ 1
Part (c)
M1 substitutes t = 1.5: 1.4 + 0.35 sin(1.5π/4)	✓ 1
A1 1.72 m (1.7234…)	✓ 1
Part (d)
M1 forms 0.35 sin(πt/4) + 1.4 = 1.6, so sin(πt/4) = 0.2/0.35	✓ 1
M1 uses inverse sine: πt/4 = 0.6082…	✓ 1
A1 t = 0.774 s (0.7744…)	✓ 1
Part (e)
M1 forms sin(πt/4) = 0.15/0.35 and finds both solutions in one wave: πt/4 = 0.4429… and π − 0.4429…	✓ 1
A1 t = 0.564… and t = 3.44…	✓ 1
A1 2.87 s (2.872…)	✓ 1

Common error: Make sure your calculator is in radian mode when the argument of the sine is a multiple of π.

+ Add to test✓ In your test329IBMI3S-2-04r35-08Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Transformations of graphs and logarithmic scales Paper 2Medium12 marks

During a race, a driver's lap time, L seconds, on lap n is modelled by
L = 84 + 0.06n for 1 ≤ n ≤ 20 (first set of tyres)
L = 85.2 + 0.03(n − 21) for 21 ≤ n ≤ 50 (second set of tyres).
The driver makes one pit stop at the end of lap 20, which adds 22 s to the total race time. The race is 50 laps long.

(a)

Find the lap time on lap 10 and on lap 20.

(2)

(b)

Find the lap time on lap 40.

(1)

(c)

Find the total time taken for laps 1 to 20.

(3)

(d)

Find the total time taken for laps 21 to 50.

(3)

(e)

Find the total race time, including the pit stop, in minutes. Give your answer to 3 significant figures. Use the unrounded values from the earlier parts.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 L(10) = 84.6 s	✓ 1
A1 L(20) = 85.2 s	✓ 1
Part (b)
A1 85.77 s	✓ 1
Part (c)
M1 identifies an arithmetic sequence with first term 84.06, common difference 0.06 and 20 terms	✓ 1
M1 S₂₀ = 20/2 × (84.06 + 85.2)	✓ 1
A1 1692.6 s	✓ 1
Part (d)
M1 identifies first term 85.2, common difference 0.03 and 30 terms	✓ 1
M1 S₃₀ = 30/2 × (2 × 85.2 + 29 × 0.03)	✓ 1
A1 2569.05 s	✓ 1
Part (e)
M1 1692.6 + 2569.05 + 22 = 4283.65 s (ECF from (c) and (d))	✓ 1
M1 divides by 60	✓ 1
A1 71.4 minutes	✓ 1

Common error: The pit stop time is added once to the total; it should not be added to every lap or left out of the race time.

+ Add to test✓ In your test330IBMI3S-2-03r36-04Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Medium10 marks

The height h metres of a hot-air balloon above the ground is modelled by h(t) = −2t³ + 30t², for 0 ≤ t ≤ 15, where t is the time in minutes after take-off.

(a)

Find the height of the balloon after 4 minutes.

(2)

(b)

Find the time at which the balloon lands.

(2)

(c)

Find the maximum height reached, and the time at which it occurs.

(3)

(d)

Find the length of time for which the balloon is higher than 800 m. Give your answer in minutes correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 4	✓ 1
A1 352 m	✓ 1
Part (b)
M1 solve h(t) = 0 for t > 0 (e.g. −2t²(t − 15) = 0)	✓ 1
A1 t = 15 minutes	✓ 1
Part (c)
M1 use a GDC maximum or solve h′(t) = 0	✓ 1
A1 t = 10 minutes	✓ 1
A1 maximum height 1000 m	✓ 1
Part (d)
M1 solve h(t) = 800 (graph intersections)	✓ 1
A1 t = 7.13 and t = 12.4 (unrounded 7.1286 and 12.3975)	✓ 1
A1 duration = 5.27 minutes (use unrounded values)	✓ 1

Exam tip: A cubic can cross a horizontal line twice within the domain; write both intersection times before subtracting.

+ Add to test✓ In your test331IBMI3S-2-03r36-07Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Medium9 marks

The displacement d millimetres of a loudspeaker cone playing a steady note is modelled by d = 0.8 sin(880πt), where t is the time in seconds and the angle is in radians.

(a)

Write down the amplitude of the motion.

(1)

(b)

Find the period of the motion in seconds, correct to 3 significant figures, and hence the frequency of the note.

(3)

(c)

Find the displacement when t = 0.0004 s. Give your answer correct to 3 significant figures.

(2)

(d)

Find the first time after t = 0 at which the displacement reaches 0.5 mm. Give your answer in seconds correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.8 mm	✓ 1
Part (b)
M1 period = 2π ÷ 880π	✓ 1
A1 period = 0.00227 s	✓ 1
A1 frequency = 440 Hz	✓ 1
Part (c)
M1 substitute t = 0.0004 (radian mode)	✓ 1
A1 d = 0.715 mm	✓ 1
Part (d)
M1 set 0.8 sin(880πt) = 0.5	✓ 1
M1 880πt = sin⁻¹(0.625) (smallest positive solution)	✓ 1
A1 t = 0.000244 s	✓ 1

Common error: Leaving a GDC in degree mode makes every sine value wrong when the model uses radians.

+ Add to test✓ In your test332IBMI3S-2-03r36-10Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Medium9 marks

The table shows the fuel burn rate y (kg per minute) of a regional jet at different take-off masses x (tonnes).
Mass x (tonnes): 40, 45, 50, 55, 60, 65
Rate y (kg per minute): 38.2, 41.5, 45.1, 47.9, 51.8, 54.6

(a)

Find the equation of the least-squares regression line of y on x, with coefficients correct to 3 significant figures.

(3)

(b)

Write down the Pearson correlation coefficient r, and describe the correlation.

(2)

(c)

Interpret the gradient of the regression line in context.

(1)

(d)

Use your equation to estimate the burn rate for a take-off mass of 52 tonnes, correct to 3 significant figures.

(2)

(e)

A student uses the line to estimate the burn rate for a mass of 120 tonnes. Give one reason why this estimate may be unreliable.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use regression on a GDC	✓ 1
A1 gradient 0.661	✓ 1
A1 y = 0.661x + 11.8	✓ 1
Part (b)
A1 r = 0.999	✓ 1
A1 very strong positive linear correlation	✓ 1
Part (c)
A1 each extra tonne of take-off mass increases the burn rate by about 0.661 kg per minute	✓ 1
Part (d)
M1 substitute x = 52	✓ 1
A1 46.2 kg per minute	✓ 1
Part (e)
R1 120 tonnes is far outside the data range 40–65 tonnes (extrapolation), so the linear trend may not continue	✓ 1

Common error: A value of r close to 1 supports using the line within the data range only; it does not make predictions outside the range reliable.

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During take-off and climb, the height h metres of an aircraft above the runway is modelled, for 0 ≤ t ≤ 20, where t is the time in minutes after the start of the climb, by
• h = 60t² for 0 ≤ t ≤ 5
• h = 1500 + 600(t − 5) for 5 < t ≤ 15
• h = 7500 for 15 < t ≤ 20.

(a)

Find the height of the aircraft when t = 3.

(2)

(b)

Show that the model gives the same height from the first two rules at t = 5.

(1)

(c)

Find the average rate of climb, in metres per minute, between t = 0 and t = 15.

(2)

(d)

State the rate of climb for 5 < t ≤ 15, giving a reason for your answer.

(2)

(e)

Describe what happens to the aircraft for 15 < t ≤ 20.

(1)

(f)

Find the time at which the aircraft is at a height of 7000 m. Give your answer in minutes correct to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 3 into 60t²	✓ 1
A1 540 m	✓ 1
Part (b)
A1 60 × 25 = 1500 and 1500 + 600 × 0 = 1500, so the rules agree (AG)	✓ 1
Part (c)
M1 (7500 − 0) ÷ 15	✓ 1
A1 500 m per minute	✓ 1
Part (d)
M1 the second rule is linear, so the rate is its gradient	✓ 1
A1 600 metres per minute (constant)	✓ 1
Part (e)
A1 the height stays constant at 7500 m (level cruise)	✓ 1
Part (f)
M1 recognise the second rule applies and set 1500 + 600(t − 5) = 7000	✓ 1
M1 t − 5 = 5500 ÷ 600	✓ 1
A1 t = 14.2 minutes	✓ 1

Exam tip: For a piecewise function, pick the rule by checking which interval the given height or time belongs to before solving.

+ Add to test✓ In your test334IBMI3S-2-02r37-01Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Medium10 marks

Calculator

A delivery robot starts a shift with its battery 100% charged. For the first 2 hours the charge, C%, decreases at a constant rate of 12 percentage points per hour. After that, the robot switches to an economy mode and the charge decreases at a constant rate of 6.5 percentage points per hour. Let t be the time in hours since the start of the shift.

(a)

Write down an expression for C in terms of t for 0 ≤ t ≤ 2.

(1)

(b)

Find an expression for C in terms of t for t > 2.

(2)

(c)

The robot returns to its charging station when the charge reaches 20%. Find the time at which this happens, in hours to three significant figures.

(2)

(d)

Find the mean rate at which the charge decreases over the whole period from the start of the shift until the robot returns to the station. Give your answer in percentage points per hour to three significant figures. Use the unrounded time from part (c).

(2)

(e)

A second robot has charge modelled by C = 96 − 8t for all t ≥ 0. Find the time at which the two robots have equal charge. Give your answer in hours to three significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 C = 100 − 12t	✓ 1
Part (b)
M1 C(2) = 76 and gradient −6.5, e.g. C = 76 − 6.5(t − 2)	✓ 1
A1 C = 89 − 6.5t	✓ 1
Part (c)
M1 89 − 6.5t = 20	✓ 1
A1 10.6 hours	✓ 1
Part (d)
M1 (100 − 20) ÷ 10.615…	✓ 1
A1 7.54 percentage points per hour	✓ 1
Part (e)
M1 recognise that the equal charge occurs for t > 2, so use 89 − 6.5t	✓ 1
M1 96 − 8t = 89 − 6.5t	✓ 1
A1 t = 4.67 hours	✓ 1

Exam tip: For a piecewise model, check that the solution you find lies inside the interval for the expression you used.

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Calculator

The positioning error, E mm, of a robot arm is measured at five speeds, v m s⁻¹.

v	1	2	3	4	5
E	0.3	0.6	1.0	1.7	2.4

Two models are proposed: model A is E = 0.42v + 0.1 and model B is E = 0.1v² + 0.2. For each model, a residual is (observed value of E) − (value predicted by the model).

(a)

Find the value of E predicted by model A when v = 3 and when v = 5.

(2)

(b)

Find the sum of the squares of the residuals for model A.

(3)

(c)

Find the sum of the squares of the residuals for model B.

(3)

(d)

State which model fits the data better, giving a reason.

(1)

(e)

Use model B to estimate the error at 8 m s⁻¹ and comment on the reliability of the estimate.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1.36	✓ 1
A1 2.2	✓ 1
Part (b)
M1 residuals −0.22, −0.34, −0.36, −0.08, 0.2	✓ 1
M1 squares added	✓ 1
A1 0.34	✓ 1
Part (c)
M1 residuals 0, 0, −0.1, −0.1, −0.3	✓ 1
M1 squares added	✓ 1
A1 0.11	✓ 1
Part (d)
R1 model B, as it has the smaller sum of squared residuals	✓ 1
Part (e)
A1 E = 6.6 mm	✓ 1
R1 this is extrapolation far outside the data range (1 to 5), so the estimate may be unreliable	✓ 1

Exam tip: Show the residual for each data point in a table so a single slip is easy to find.

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Calculator

A fish crate with no lid is made from a rectangular sheet of plastic measuring 60 cm by 40 cm. Squares of side x cm are cut from each corner, and the sides are folded up to make an open box of volume V cm³.

(a)

Show that V = 4x³ − 200x² + 2400x.

(2)

(b)

State the range of possible values of x.

(1)

(c)

The crate must hold at least 5000 cm³. Find the range of values of x for which V ≥ 5000. Give your answers to three significant figures.

(3)

(d)

Find the value of x that gives the maximum volume, and find this maximum volume. Give your answers to three significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 V = x(60 − 2x)(40 − 2x) (length, width and height identified)	✓ 1
A1 expand to 4x³ − 200x² + 2400x	✓ 1
Part (b)
A1 0 < x < 20	✓ 1
Part (c)
M1 solve 4x³ − 200x² + 2400x = 5000 (graph or solver)	✓ 1
A1 x = 2.63 and x = 14.4	✓ 1
A1 2.63 ≤ x ≤ 14.4	✓ 1
Part (d)
M1 use of graph or solver to find the maximum (or V′(x) = 0)	✓ 1
A1 x = 7.85	✓ 1
A1 V = 8450 cm³	✓ 1

Common error: The width 40 − 2x must stay positive, which limits x to less than 20.

+ Add to test✓ In your test337IBMI3S-2-01r38-06Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Medium14 marks

Calculator

The number of hours of daylight, D, on day t of the year at a farm is modelled by
D(t) = a sin(b(t − c)) + d, 0 ≤ t ≤ 365,
where the angle is in radians. The model has a maximum of 15.4 hours on day 172, and a minimum of 8.6 hours. One full cycle takes 365 days.

(a)

Find the value of a and the value of d.

(3)

(b)

Show that b = 0.0172 correct to 3 significant figures.

(2)

(c)

The maximum occurs when t = 172. Find the value of c.

(3)

(d)

Use the model to find the days of the year on which there are more than 14 hours of daylight. Use b = 2π ÷ 365 and c = 80.75. Give the first and last day as whole numbers of days.

(4)

(e)

Hence find the number of days of the year on which there are more than 14 hours of daylight.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 a = (15.4 − 8.6) ÷ 2 or d = (15.4 + 8.6) ÷ 2	✓ 1
A1 a = 3.4	✓ 1
A1 d = 12	✓ 1
Part (b)
M1 b = 2π ÷ 365	✓ 1
A1 0.017214… = 0.0172 (answer must be seen to more than 3 s.f. or as 2π/365)	✓ 1
Part (c)
M1 b(172 − c) = π ÷ 2 (sine equal to 1) or quarter of a period 365 ÷ 4 = 91.25	✓ 1
A1 172 − c = 91.25	✓ 1
A1 c = 80.75	✓ 1
Part (d)
M1 set 3.4 sin(b(t − 80.75)) + 12 = 14	✓ 1
A1 boundary values t = 117.28… and t = 226.72…	✓ 1
A1 first day 118	✓ 1
A1 last day 226	✓ 1
Part (e)
M1 226 − 118 + 1 (count both end days)	✓ 1
A1 109 days	✓ 1

Common error: When counting whole days between two boundary days, forgetting to add 1 (226 − 118 gives 108, not 109).

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Calculator

The number of mould spores, N thousand, on a sample of stored grain is counted t days after storage begins.
t: 0, 2, 4, 6, 8
N: 118, 245, 480, 990, 1960
A researcher suspects the model N = Ae^kt, where A and k are constants, and plots ln N against t.
In this question use the unrounded values from your calculator until your final answers.

(a)

Find the value of ln N when t = 8. Give your answer to 3 significant figures.

(1)

(b)

Use linear regression to find the equation of the regression line of ln N on t, in the form ln N = mt + c. Give m and c to 3 significant figures.

(2)

(c)

Hence find the values of A and k in the model N = Ae^kt. Give both to 3 significant figures.

(3)

(d)

Use the model to predict the number of spores, in thousands, after 12 days. Give your answer to 3 significant figures.

(2)

(e)

The correlation coefficient between ln N and t is 0.9999. State, with a reason, whether the model N = Ae^kt is suitable for the data.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 7.58	✓ 1
Part (b)
A1 m = 0.351	✓ 1
A1 c = 4.78	✓ 1
Part (c)
M1 N = e^{mt + c} or A = e^c	✓ 1
A1 A = 119 (accept 119.3 from unrounded c)	✓ 1
A1 k = 0.351	✓ 1
Part (d)
M1 N = 119e^{0.351 × 12} (or using unrounded values)	✓ 1
A1 8030	✓ 1
Part (e)
R1 suitable, since the very strong linear relationship between ln N and t means N grows exponentially with t	✓ 1

Exam tip: A straight line on a graph of ln N against t corresponds to exponential growth in N: the gradient is k and the intercept is ln A.

+ Add to test✓ In your test339IBMI3S-2-01r38-10Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Medium9 marks

Calculator

The number of times a new song is streamed in month t after its release is modelled by the cubic function
S(t) = −0.04t³ + 0.7t² − 1.2t + 3, t ≥ 0,
where S is in millions of streams. The model was fitted using data for the first 12 months.

(a)

Calculate the number of streams the model predicts in month 6. Give your answer to 3 significant figures.

(2)

(b)

Find the positive value of t greater than 12 for which S(t) = 0. Give your answer to 3 significant figures.

(2)

(c)

Explain why the model should not be used for values of t greater than 15.9.

(2)

(d)

Find the greatest number of streams in a month predicted by the model for 0 ≤ t ≤ 12, in millions to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 6	✓ 1
A1 12.4 million	✓ 1
Part (b)
M1 solve −0.04t³ + 0.7t² − 1.2t + 3 = 0 (GDC)	✓ 1
A1 t = 15.9	✓ 1
Part (c)
R1 the model gives negative values of S for t > 15.9, but a number of streams cannot be negative	✓ 1
R1 these values are outside the data range used to fit the model (extrapolation)	✓ 1
Part (d)
M1 use GDC maximum or solve S′(t) = 0	✓ 1
A1 maximum at t = 10.7	✓ 1
A1 21.3 million	✓ 1

Common error: Treating the turning point of a cubic as the greatest value on the whole domain without checking the end points of the given interval.

+ Add to test✓ In your test340IBMI3S-2-03r39-12Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Medium12 marks

The concentration, C mg/L, of an antibiotic in a patient's blood is measured t hours after an injection.
t (hours): 1, 2, 3, 4, 5, 6
C (mg/L): 82, 66, 53, 43, 34, 28
An exponential model of the form C = a × b^t is fitted to the data.

(a)

Use your graphic display calculator to find the values of a and b, each correct to 3 significant figures.

(3)

(b)

Interpret the value of b in this context.

(1)

(c)

Use unrounded values from (a) to predict the concentration 8 hours after the injection. Give your answer to 3 significant figures.

(2)

(d)

Find the time at which the model predicts the concentration to be 10 mg/L. Use unrounded values from (a) and give your answer to 3 significant figures.

(3)

(e)

A linear model C = mt + k is also fitted to the data and has gradient about −10.7. Give one reason why the exponential model is more suitable for predicting the concentration at large values of t.

(1)

(f)

Find the time taken for the concentration to halve, according to the exponential model. Use the unrounded value of b and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of exponential regression	✓ 1
A1 a = 102	✓ 1
A1 b = 0.805	✓ 1
Part (b)
R1 each hour the concentration is about 80.5% of its value in the previous hour (it decreases by about 19.5% each hour)	✓ 1
Part (c)
M1 substitute t = 8 into the model	✓ 1
A1 18.0 mg/L	✓ 1
Part (d)
M1 ab^t = 10	✓ 1
M1 use of logarithms or GDC to solve	✓ 1
A1 t = 10.7 hours	✓ 1
Part (e)
R1 the linear model predicts a negative concentration for large t, whereas the exponential model stays positive and approaches zero	✓ 1
Part (f)
M1 b^t = 0.5 (or ln 0.5 / ln b)	✓ 1
A1 3.20 hours	✓ 1

Exam tip: Use unrounded regression values stored in your calculator for later parts; rounding a and b early can change the third significant figure of a prediction.

+ Add to test✓ In your test341IBMI3S-2-02r40-08Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Medium13 marks

A tidal energy turbine sits in water whose depth $D$ metres at time $t$ hours after midnight is modelled by $D = 8.5 + 3.2\sin(0.507t)$, where the angle is in radians. The turbine operates only when the depth is at least 10 m.

(a)

Write down the maximum and minimum depth of water predicted by the model.

(2)

(b)

Find the period of the model, correct to 3 significant figures.

(2)

(c)

Find the depth at 04:00, correct to 3 significant figures.

(2)

(d)

Find the two times in the first cycle when the depth is exactly 10 m, correct to 3 significant figures.

(4)

(e)

Hence find the length of time in the first cycle for which the turbine operates, correct to 3 significant figures.

(1)

(f)

The pattern repeats every period. Find the total time the turbine operates during the first 24 hours, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 11.7 m	✓ 1
A1 minimum 5.3 m	✓ 1
Part (b)
M1 $\frac{2\pi}{0.507}$	✓ 1
A1 12.4 hours	✓ 1
Part (c)
M1 substitute $t = 4$ (calculator in radian mode)	✓ 1
A1 11.4 m	✓ 1
Part (d)
M1 $8.5 + 3.2\sin(0.507t) = 10$	✓ 1
M1 $\sin(0.507t) = 0.46875$	✓ 1
A1 $t = 0.962$	✓ 1
A1 $t = 5.23$	✓ 1
Part (e)
A1 $5.23 - 0.962 = 4.27$ hours	✓ 1
Part (f)
M1 two full operating periods occur in 24 hours (the third starts after $t = 24$), so total is $2 \times 4.27$	✓ 1
A1 8.54 hours	✓ 1

Exam tip: Set your calculator to radians when the angle is not in degrees, and use the second solution $\pi - \theta$ to find the other time.

+ Add to test✓ In your test342IBMI3S-2-02r40-09Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Medium12 marks

After a patient takes a dose of a medicine, the concentration $C$ mg L⁻¹ in the blood at $t$ hours is modelled by $C = 12e^{-0.21t}$ for $t \geq 0$.

(a)

Write down the concentration immediately after the dose.

(1)

(b)

Find the concentration after 6 hours, correct to 3 significant figures.

(2)

(c)

Find the time at which the concentration falls to 2 mg L⁻¹, correct to 3 significant figures.

(3)

(d)

Find the half-life of the medicine, correct to 3 significant figures.

(2)

(e)

When the concentration has just fallen to 2 mg L⁻¹, a second dose raises the concentration by 12 mg L⁻¹ and it then decays at the same rate. Find how long after this second dose the concentration again falls to 2 mg L⁻¹, correct to 3 significant figures.

(3)

(f)

State the value that $C$ approaches as $t$ becomes large, and interpret it in context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 12 mg L⁻¹	✓ 1
Part (b)
M1 $12e^{-0.21 \times 6}$	✓ 1
A1 3.40 mg L⁻¹	✓ 1
Part (c)
M1 $12e^{-0.21t} = 2$ (or $e^{-0.21t} = \frac{1}{6}$)	✓ 1
M1 take natural logs: $t = \frac{\ln 6}{0.21}$	✓ 1
A1 8.53 hours	✓ 1
Part (d)
M1 $e^{-0.21t} = \frac{1}{2}$	✓ 1
A1 3.30 hours	✓ 1
Part (e)
A1 concentration just after second dose is $2 + 12 = 14$ mg L⁻¹	✓ 1
M1 $14e^{-0.21t} = 2$	✓ 1
A1 $t = \frac{\ln 7}{0.21} = 9.27$ hours	✓ 1
Part (f)
R1 $C \to 0$: the medicine is gradually eliminated from the blood but the model never reaches exactly zero	✓ 1

Exam tip: A half-life does not depend on the starting amount: solve $e^{-kt} = \frac{1}{2}$.

+ Add to test✓ In your test343IBMI3S-2-02r40-11Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Medium12 marks

A retailer makes an open box from a rectangular sheet of card 40 cm by 30 cm. A square of side $x$ cm is cut from each corner, and the sides are folded up. The volume of the box is $V$ cm³.

(a)

Show that $V = 4x^3 - 140x^2 + 1200x$.

(2)

(b)

Write down the possible values of $x$ for this model.

(1)

(c)

Use your GDC to find the maximum volume and the value of $x$ at which it occurs, correct to 3 significant figures.

(2)

(d)

Find the two values of $x$ for which the volume is 2500 cm³, correct to 3 significant figures.

(3)

(e)

Hence find the range of values of $x$ for which the volume is at least 2500 cm³, and the width of this range, correct to 3 significant figures.

(2)

(f)

When the volume is a maximum, find the percentage of the original sheet's area that is cut away as waste, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 box dimensions are $(40 - 2x)$, $(30 - 2x)$ and $x$, so $V = x(40 - 2x)(30 - 2x)$	✓ 1
A1 expanding $(40-2x)(30-2x) = 1200 - 140x + 4x^2$, so $V = 4x^3 - 140x^2 + 1200x$	✓ 1
Part (b)
A1 $0 < x < 15$	✓ 1
Part (c)
A1 $x = 5.66$	✓ 1
A1 maximum volume 3030 cm³	✓ 1
Part (d)
M1 solve $4x^3 - 140x^2 + 1200x = 2500$ using the GDC	✓ 1
A1 $x = 3.11$	✓ 1
A1 $x = 8.63$ (reject 23.3, which is outside the domain)	✓ 1
Part (e)
A1 $3.11 \leq x \leq 8.63$	✓ 1
A1 width 5.52 cm	✓ 1
Part (f)
M1 four squares: $\frac{4 \times 5.657^2}{1200} \times 100$	✓ 1
A1 10.7%	✓ 1

Exam tip: Use the context to restrict the domain: lengths must be positive, so reject solutions outside $0 < x < 15$.

+ Add to test✓ In your test344IBMI3S-2-01r43-06Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; 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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium13 marks

The temperature T °C at a mountain hut is modelled by T = a sin(b(t − c)) + d, where t is the time in hours after midnight, with a > 0, b > 0 and 0 ≤ c < 24. The temperature repeats every 24 hours. The lowest temperature is −3 °C at 03:00 and the highest temperature is 11 °C at 15:00.

(a)

Find the values of a, b, c and d.

(4)

(b)

Find the temperature at 18:00. Give your answer to 3 significant figures.

(2)

(c)

Find the total time in one day, in hours, for which the temperature is above 0 °C. Give your answer to 3 significant figures.

(3)

(d)

A second hut has temperature T_B(t) = T(t − 1.5) − 2, where T is the model above. Describe the change from the graph of T to the graph of T_B.

(2)

(e)

Find the maximum temperature at the second hut and the time of day at which it occurs.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = 7 (half of 11 − (−3))	✓ 1
A1 d = 4 (midpoint of the extreme values)	✓ 1
A1 b = 2π/24 = π/12 (0.262)	✓ 1
A1 c = 9 (the model passes through its midline, rising, at 09:00)	✓ 1
Part (b)
M1 substituting t = 18 into the model	✓ 1
A1 T = 8.95 °C (8.9497… before rounding; ECF from (a))	✓ 1
Part (c)
M1 solving 7 sin(π(t − 9)/12) + 4 = 0 (graph or solver)	✓ 1
A1 the temperature is below 0 °C from t = 23.3 to t = 30.7 (that is, until 06:41 the next day); unrounded values 23.3233 and 30.6767	✓ 1
A1 time above 0 °C = 24 − 7.35 = 16.6 hours (16.6467… before rounding)	✓ 1
Part (d)
A1 shift of 1.5 hours later (translation 1.5 units to the right)	✓ 1
A1 shift down by 2 °C (translation 2 units down)	✓ 1
Part (e)
A1 maximum temperature 9 °C (11 − 2)	✓ 1
A1 at t = 16.5, that is 16:30	✓ 1

Common error: In a sin(b(t − c)) + d, the period is 2π/b; using b = 24 (the period itself) is a frequent slip.

+ Add to test✓ In your test345IBMI3S-2-01r43-08Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; 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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; 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Atmospheric pressure P kPa at altitude h metres is measured at a research station. Two models are proposed:
Model A (linear): P = 101.3 − 0.0083h
Model B (exponential): P = 101.3e^−0.000125h
The measured pressures are 54.0 kPa at 5000 m and 37.1 kPa at 8000 m.

(a)

Calculate the pressure predicted by each model at h = 5000.

(2)

(b)

Calculate the percentage error at 5000 m for each model, using the unrounded predictions. Give your answers to 3 significant figures.

(3)

(c)

Find the pressure predicted by each model at 8000 m, and state which model is closer to the measured value of 37.1 kPa there.

(2)

(d)

Model A predicts a pressure of zero at a certain altitude. Find this altitude, to 3 significant figures, and explain why this makes Model A unsuitable for very high altitudes.

(2)

(e)

Use Model B to find the altitude at which the pressure is half of the sea-level pressure. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model A: 59.8 kPa	✓ 1
A1 Model B: 54.2 kPa (54.222… before rounding)	✓ 1
Part (b)
M1 percentage error = \|model − measured\| / measured × 100	✓ 1
A1 Model A: 10.7%	✓ 1
A1 Model B: 0.411%	✓ 1
Part (c)
A1 Model A: 34.9 kPa, Model B: 37.3 kPa (37.266… before rounding)	✓ 1
A1 Model B is closer (difference about 0.17 kPa compared with 2.2 kPa)	✓ 1
Part (d)
A1 h = 101.3 / 0.0083 = 12 200 m (12 205 m before rounding)	✓ 1
R1 pressure would then become negative above this altitude, which is impossible, whereas Model B stays positive	✓ 1
Part (e)
M1 101.3e^−0.000125h = 50.65 (or e^−0.000125h = 0.5)	✓ 1
M1 taking natural logarithms: −0.000125h = ln 0.5	✓ 1
A1 h = 5550 m (5545.2… before rounding)	✓ 1

Exam tip: Percentage error is measured relative to the accepted (measured) value, not relative to the model's prediction; write the formula before substituting.

+ Add to test✓ In your test346IBMI3S-2-03r43-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Medium12 marks

A road bridge has a parabolic arch. Let x metres be the horizontal distance from the left-hand support and h metres the height of the arch above the road. The arch meets the road at x = 0 and x = 80, and its greatest height above the road is 16 m, at x = 40. The arch is modelled by a quadratic function h = ax² + bx + c.

(a)

Show that h = −0.01x² + 0.8x.

(3)

(b)

A sign is fixed to the arch at the point where the arch is 12 m above the road. Find the possible distances of the sign from the left-hand support.

(2)

(c)

A lorry is 4.2 m high. It can pass under the arch only where the arch is at least 4.2 m above the road. Find the distance from the left-hand support to the nearest point at which the arch is exactly 4.2 m above the road. Give your answer to 3 significant figures.

(3)

(d)

A surveyor measures the height of the arch at x = 10 as 6.8 m. Calculate the percentage error in the height predicted by the model at this point, taking the surveyor's value as the exact value. Give your answer to 3 significant figures.

(2)

(e)

A cable is attached to the arch at a point 25 m from the right-hand support. Find the height of the arch above the road at this point.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 c = 0 because h = 0 when x = 0	✓ 1
M1 forms two equations from (40, 16) and (80, 0): 1600a + 40b = 16 and 6400a + 80b = 0	✓ 1
A1 solves to obtain a = −0.01 and b = 0.8	✓ 1
Part (b)
M1 sets −0.01x² + 0.8x = 12	✓ 1
A1 x = 20 or x = 60	✓ 1
Part (c)
M1 sets −0.01x² + 0.8x = 4.2	✓ 1
M1 solves (GDC or quadratic formula): x = 5.65 or x = 74.35	✓ 1
A1 x = 5.65 m	✓ 1
Part (d)
M1 model value h(10) = 7 and \|7 − 6.8\| ÷ 6.8 × 100	✓ 1
A1 2.94%	✓ 1
Part (e)
M1 the point is at x = 80 − 25 = 55 from the left-hand support	✓ 1
A1 h = 13.75 m	✓ 1

Exam tip: For a 'show that' question write every step, including the equations you form: the given answer cannot be assumed.

+ Add to test✓ In your test347IBMI3S-2-03r43-13Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Medium12 marks

A small roastery roasts n kg of coffee each month. Its monthly costs are modelled by C = 1800 + 3.2n euros.

(a)

Interpret the numbers 1800 and 3.2 in this model.

(2)

(b)

Coffee is sold for €9.50 per kg, so the revenue is R = 9.5n. Find the smallest whole number of kilograms that must be sold in a month to make a profit.

(3)

(c)

Find the profit for a month in which 500 kg is sold.

(1)

(d)

Selling more than 600 kg in a month requires a discount: the price for kilograms beyond the first 600 kg is €8.00 per kg. Show that the profit for 900 kg is €3420.

(3)

(e)

Find the smallest whole number of kilograms that must be sold in a month to make a profit of at least €5000.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 €1800 is the fixed monthly cost (cost when nothing is roasted)	✓ 1
A1 €3.20 is the extra cost for each additional kilogram roasted	✓ 1
Part (b)
M1 sets 9.5n = 1800 + 3.2n (or 6.3n = 1800)	✓ 1
A1 n = 285.7…	✓ 1
A1 286 kg (must round up to a whole number of kilograms)	✓ 1
Part (c)
A1 €1350	✓ 1
Part (d)
M1 revenue = 9.5 × 600 + 8.0 × 300 = 5700 + 2400 = 8100	✓ 1
M1 cost = 1800 + 3.2 × 900 = 4680	✓ 1
A1 profit = 8100 − 4680 = 3420	✓ 1
Part (e)
R1 for 600 kg or fewer the profit is at most 6.3 × 600 − 1800 = 1980, so more than 600 kg are needed	✓ 1
M1 5700 + 8.0(n − 600) − (1800 + 3.2n) = 5000, i.e. 4.8n − 900 = 5000	✓ 1
A1 n = 1229.2…, so 1230 kg	✓ 1

Exam tip: In a piecewise model, check which interval the answer lies in before solving; if the answer lands outside the interval you used, it is not valid.

+ Add to test✓ In your test348IBMI3S-2-03r48-14Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Medium13 marks

The average number of passengers per day on a commuter line is modelled by N = 1800 + 12t + 300 sin(πt/6), where t is the time in months since 1 January and the angle is in radians.

(a)

Calculate the number of passengers per day predicted for 1 January.

(1)

(b)

Calculate the number of passengers per day predicted at t = 9.

(2)

(c)

State the period, in months, of the seasonal variation in the model.

(1)

(d)

For 0 ≤ t ≤ 12, find the greatest value of N predicted by the model. Give your answer to the nearest whole number.

(3)

(e)

Find the first time at which the model predicts more than 2200 passengers per day. Give your answer in months to 3 significant figures.

(3)

(f)

The operator uses the model to predict the number of passengers per day at t = 14. Calculate this prediction to the nearest whole number and give one reason why it may be unreliable.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1800	✓ 1
Part (b)
M1 substitutes t = 9: 1800 + 108 + 300 sin(1.5π)	✓ 1
A1 1608	✓ 1
Part (c)
A1 12 months	✓ 1
Part (d)
M1 uses a graphing method to find the maximum on the interval	✓ 1
A1 t = 3.15 months	✓ 1
A1 N = 2137	✓ 1
Part (e)
M1 sets 1800 + 12t + 300 sin(πt/6) = 2200	✓ 1
M1 uses a graphing method or equation solver and chooses the first solution after the maximum found in (d)	✓ 1
A1 t = 13.7 months	✓ 1
Part (f)
M1 substitutes t = 14: 1800 + 168 + 300 sin(14π/6)	✓ 1
A1 2228	✓ 1
R1 the model assumes the linear growth of 12 passengers per day each month continues unchanged, which may not hold so far into the future (extrapolation)	✓ 1

Exam tip: For a model that adds a trend to a sinusoid, sketch the graph on your calculator and read off turning points and intersections rather than trying to solve by hand.

+ Add to test✓ In your test349IBMI3S-2-04r48-14Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Medium13 marks

The number of patients arriving at a hospital emergency department per hour is modelled by A = 22 + 14 sin(π(t − 8)/12), where t is the number of hours after midnight and the angle is in radians.

(a)

Write down the greatest and least number of arrivals per hour predicted by the model.

(2)

(b)

State the period of the model in hours.

(1)

(c)

Calculate the number of arrivals per hour predicted at t = 19, to 3 significant figures.

(2)

(d)

Find the two times in the first 24 hours at which the model gives 30 arrivals per hour. Give your answers to 3 significant figures.

(3)

(e)

At t = 10, 31 arrivals were recorded. Compare this with the model's value.

(2)

(f)

Each doctor can see 4 patients per hour. Find the number of hours in a day for which the model predicts that more than 7 doctors are needed. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 greatest 36	✓ 1
A1 least 8	✓ 1
Part (b)
A1 24 hours	✓ 1
Part (c)
M1 substitutes t = 19: 22 + 14 sin(11π/12)	✓ 1
A1 25.6	✓ 1
Part (d)
M1 solves 22 + 14 sin(π(t − 8)/12) = 30, so sin(π(t − 8)/12) = 8/14	✓ 1
A1 t = 10.3	✓ 1
A1 t = 17.7	✓ 1
Part (e)
M1 A(10) = 22 + 14 sin(π/6) = 29	✓ 1
R1 the model is 2 below the recorded value, so it slightly underestimates arrivals at this time	✓ 1
Part (f)
M1 more than 7 doctors needed when A > 28	✓ 1
M1 solves 22 + 14 sin(π(t − 8)/12) = 28 and finds the two times	✓ 1
A1 8.62 hours	✓ 1

Exam tip: To find the length of time a sinusoidal model stays above a value, find both crossing times and subtract.

+ Add to test✓ In your test350IBMI-031-00-11Linear models and piecewise models Paper 2Hard14 marks

A water company offers two pricing plans for x cubic metres (m³) of water used per quarter.
Plan A: a fixed charge of $18, plus $1.40 per m³ for the first 20 m³, $2.10 per m³ for the next 30 m³, and $3.20 per m³ for all water above 50 m³.
Plan B: a fixed charge of $45, plus $2.05 per m³ for all water used.

(a)

Find the Plan A cost for 35 m³.

(2)

(b)

Find the Plan A cost, A(x), as a piecewise linear function, giving each piece in the form mx + c with its domain.

(4)

(c)

Write down the Plan B cost, B(x).

(1)

(d)

Find the set of values of x for which Plan B is cheaper than Plan A.

(5)

(e)

A household uses 120 m³ in one quarter. Find how much it saves by choosing Plan B.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 18 + 20 × 1.40 + 15 × 2.10	✓ 1
A1 $77.50	✓ 1
Part (b)
A1 A(x) = 1.4x + 18 for 0 ≤ x ≤ 20	✓ 1
M1 A(20) = 46 used to form 46 + 2.1(x − 20), or equivalent	✓ 1
A1 A(x) = 2.1x + 4 for 20 < x ≤ 50	✓ 1
A1 A(x) = 3.2x − 51 for x > 50	✓ 1
Part (c)
A1 B(x) = 45 + 2.05x	✓ 1
Part (d)
M1 comparing B with each piece of A	✓ 1
R1 for 0 ≤ x ≤ 50, A(x) < B(x) (e.g. B − A = 27 + 0.65x > 0 and B − A = 41 − 0.05x > 0 on these domains)	✓ 1
M1 3.2x − 51 = 45 + 2.05x	✓ 1
A1 x = 83.478… (= 96/1.15)	✓ 1
A1 Plan B cheaper for x > 83.478… m³ (accept x > 83.5 to 3 s.f.)	✓ 1
Part (e)
M1 A(120) = 333 and B(120) = 291	✓ 1
A1 $42.00	✓ 1

Common error: Solving A(x) = B(x) using only one piece can give a solution outside that piece's domain. Always check that the solution lies in the interval you used.

+ Add to test✓ In your test351IBMI-031-07-11Exponential and sinusoidal models Paper 2Hard10 marks

After a filter is installed at a factory, the concentration of a pollutant in a nearby river is monitored. The concentration, C mg L⁻¹, t weeks after installation is modelled by
$C(t) = ab^t + c$,
where a > 0 and 0 < b < 1. The measurements are C(0) = 85, C(2) = 61 and C(4) = 45.

(a)

Show that $b^2 = \frac{2}{3}$.

(3)

(b)

Hence find the values of a, b and c.

(3)

(c)

The river is considered safe for fishing when the concentration falls below 20 mg L⁻¹. Find the time taken for the river to become safe.

(3)

(d)

Interpret the value of c in context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: forming two equations from differences, $a(1 - b^2) = 24$ and $ab^2(1 - b^2) = 16$	✓ 1
M1: dividing to eliminate a: $b^2 = \frac{16}{24}$	✓ 1
A1: $b^2 = \frac{2}{3}$ shown	✓ 1
Part (b)
A1: $b = 0.816$ (0.81649…)	✓ 1
A1: $a = \frac{24}{1/3} = 72$	✓ 1
A1: $c = 85 - 72 = 13$	✓ 1
Part (c)
M1: $72(0.81649\ldots)^t + 13 = 20$	✓ 1
M1: $(0.81649\ldots)^t = \frac{7}{72}$ or GDC intersection	✓ 1
A1: 11.5 weeks	✓ 1
Part (d)
R1: 13 mg L⁻¹ is the long-term (background) concentration that the pollutant level approaches; the model predicts it will never fall below this	✓ 1

Exam tip: With equally spaced data for a model ab^t + c, subtracting consecutive values removes c, and dividing the differences removes a.

+ Add to test✓ In your test352IBMI-031-07-12Exponential and sinusoidal models Paper 2Hard12 marks

A conservation team introduces 400 deer into a nature reserve that can support at most 12 000 deer. The number of deer, P, t years after introduction is modelled by
$P(t) = \frac{L}{1 + Ce^{-kt}}$,
where L, C and k are positive constants and L = 12 000. After 3 years there are 1500 deer.

(a)

Find the value of C.

(2)

(b)

Find the value of k.

(3)

(c)

The population grows fastest when it reaches half of its carrying capacity. Find the time at which this occurs.

(3)

(d)

Find the time taken for the population to reach 90% of the carrying capacity.

(3)

(e)

Interpret the value of L in context.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $\frac{12000}{1 + C} = 400$	✓ 1
A1: $C = 29$	✓ 1
Part (b)
M1: $\frac{12000}{1 + 29e^{-3k}} = 1500$	✓ 1
M1: $e^{-3k} = \frac{7}{29}$ or GDC solver	✓ 1
A1: $k = 0.474$ (0.47379…)	✓ 1
Part (c)
M1: $P = 6000$	✓ 1
M1: $29e^{-kt} = 1$ or GDC solver with their k	✓ 1
A1: 7.11 years	✓ 1
Part (d)
M1: $P = 10800$	✓ 1
M1: $29e^{-kt} = \frac{1}{9}$ or GDC solver	✓ 1
A1: 11.7 years	✓ 1
Part (e)
R1: 12 000 is the maximum number of deer the reserve can support, or the value the population approaches over time; accept either statement	✓ 1

Common error: Substituting t = 0 gives P = L/(1 + C), not L/C. Forgetting the 1 in the denominator gives C = 30, which spoils every later part.

+ Add to test✓ In your test353IBMI-031-03-r2-06Choosing and evaluating models Paper 2Hard13 marks

A cup of coffee is placed in a room kept at 21 °C. Its temperature is measured at three times.

$t$ (minutes)	0	5	10
Temperature (°C)	88	70	57

Model 1 for the temperature is $T(t) = 21 + Be^{-kt}$, where $B, k > 0$.

(a)

Find the value of $B$, and interpret the value 21 in the context of the model.

(2)

(b)

Show that $k = 0.0626$, correct to 3 significant figures.

(3)

(c)

Model 2 is a quadratic, $Q(t) = pt^2 + qt + 88$, that passes through all three data points. Find the value of $p$ and the value of $q$.

(2)

(d)

Find the minimum value of $Q(t)$. Hence explain why Model 2 is unsuitable for large values of $t$.

(3)

(e)

The coffee is considered cold when its temperature reaches 45 °C. Use Model 1 to find the time at which this occurs, and explain why Model 2 cannot answer this question.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 $88 = 21 + B$, so $B = 67$	✓ 1
R1 21 °C is the temperature the coffee approaches in the long term, which is room temperature	✓ 1
Part (b)
M1 $70 = 21 + 67e^{-5k}$	✓ 1
M1 $e^{-5k} = \frac{49}{67}$, so $k = \frac{1}{5}\ln\frac{67}{49}$	✓ 1
A1 $k = 0.062574...$, which is 0.0626 to 3 s.f.	✓ 1
Part (c)
A1 $p = 0.1$	✓ 1
A1 $q = -4.1$	✓ 1
Part (d)
M1 minimum at $t = \frac{4.1}{0.2} = 20.5$	✓ 1
A1 minimum value 46.0 °C	✓ 1
R1 after 20.5 minutes Model 2 predicts the coffee warming up again, and it never approaches room temperature, which is unrealistic	✓ 1
Part (e)
M1 solving $21 + 67e^{-0.06257t} = 45$	✓ 1
A1 $t = 16.4$ minutes	✓ 1
R1 the minimum of $Q$ is 46.0 °C, which is above 45 °C, so Model 2 never reaches 45 °C	✓ 1

Exam tip: A model can fit every data point exactly and still be a poor choice. Check its long-term behaviour against the physical situation.

+ Add to test✓ In your test354IBMI-031-08-r2-06Choosing and evaluating models Paper 2Hard13 marks

A cup of coffee is poured at time $t=0$ minutes in a room kept at a constant 21 °C. Its temperature is 90 °C when poured and 62 °C after 10 minutes. A later measurement at $t=30$ minutes gives 32 °C.

(a)

A student fits a linear model through the readings at $t=0$ and $t=10$. Use this model to predict the temperature at $t=30$, and explain why the linear model is unsuitable for longer times.

(3)

(b)

A second model is $T=21+69e^{-kt}$. Using the reading at $t=10$, show that $k=0.0521$ to 3 significant figures.

(3)

(c)

Use the exponential model to predict the temperature at $t=30$ and find the percentage error compared with the measured value.

(3)

(d)

State the equation of the horizontal asymptote of the exponential model and interpret it in context.

(2)

(e)

Use the exponential model to find the time taken for the coffee to cool to 50 °C.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: gradient $=\frac{62-90}{10}=-2.8$, so $T=90-2.8t$	✓ 1
A1: $T(30)=6$ °C	✓ 1
R1: this is below room temperature (21 °C), which is impossible for coffee cooling in the room / the model eventually predicts negative temperatures	✓ 1
Part (b)
M1: substituting $62=21+69e^{-10k}$	✓ 1
M1: $e^{-10k}=\frac{41}{69}$ and taking logs, $k=\frac{1}{10}\ln\frac{69}{41}$	✓ 1
A1: $k=0.052053…=0.0521$	✓ 1
Part (c)
A1: $T(30)=35.5$ °C (35.476…)	✓ 1
M1: $\frac{\|35.476-32\|}{32}\times100$	✓ 1
A1: 10.9%	✓ 1
Part (d)
A1: $T=21$	✓ 1
R1: the coffee's temperature approaches (but never falls below) room temperature as time increases	✓ 1
Part (e)
M1: solving $21+69e^{-0.05205t}=50$, e.g. $t=\frac{\ln(69/29)}{k}$	✓ 1
A1: $t=16.7$ minutes (accept 16.6 from using $k=0.0521$)	✓ 1

Exam tip: When evaluating a model, check its long-term behaviour against the context: a model with the right asymptote can be judged more appropriate even if it is not perfect at every data point.

+ Add to test✓ In your test355IBMI-031-03-r1-06Choosing and evaluating models Paper 2Hard13 marks

The population, $P$ thousand, of a town $t$ years after the start of 2000 is given in the table.

t: 0, 5, 10
P: 12.0, 15.6, 20.4

Two models are proposed.
Model A: $P = 12r^t$
Model B: $P = at^2 + bt + 12$

(a)

Using the data for t = 10, find the value of r for Model A. Hence show that Model A gives a population that rounds to 15.6 thousand when t = 5.

(3)

(b)

Using the data for t = 5 and t = 10, find the values of a and b for Model B.

(3)

(c)

In 2020 (t = 20) the population was found to be 33.1 thousand. Determine which model fits this value better, justifying your answer.

(4)

(d)

Use Model B to find the value of t when the population first reaches 50 thousand.

(2)

(e)

Give one reason why Model B might not be suitable for predicting the population far into the future.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 12r¹⁰ = 20.4, so r¹⁰ = 1.7	✓ 1
A1 r = 1.7^0.1 = 1.0545 (1.05 to 3 s.f.)	✓ 1
A1 using the unrounded r: 12 × 1.0545⁵ = 12√1.7 = 15.646, which rounds to 15.6	✓ 1
Part (b)
M1 forming two equations: 25a + 5b = 3.6 and 100a + 10b = 8.4	✓ 1
A1 a = 0.024	✓ 1
A1 b = 0.6	✓ 1
Part (c)
M1 substituting t = 20 into both models	✓ 1
A1 Model A: 34.7 thousand (34.68)	✓ 1
A1 Model B: 33.6 thousand	✓ 1
R1 Model B is better, as its prediction is closer to 33.1 (error 0.5 thousand compared with about 1.6 thousand)	✓ 1
Part (d)
M1 solving 0.024t² + 0.6t + 12 = 50	✓ 1
A1 t = 29.2 years (positive root)	✓ 1
Part (e)
R1 e.g. the quadratic increases without limit, but limited housing, land or resources would restrict growth; or the model would give unrealistic values outside the data range	✓ 1

Common error: In a 'show that' step, keep the unrounded value of r in your calculator. Using r = 1.05 gives 15.3 and fails to show the result.

+ Add to test✓ In your test356IBMI-031-05-r2-06Linear models and piecewise models Paper 2Hard13 marks

A traveller exchanges pounds for euros at an airport kiosk and later uses a card that converts euros into US dollars. The kiosk converts $x$ pounds into euros according to $f(x) = 1.16(x - 3)$, $x > 3$, which includes a fixed £3 fee. The card converts $y$ euros into US dollars according to $g(y) = 1.08y - 2$, which includes a fixed fee of 2 US dollars.

(a)

Find $(g \circ f)(x)$, giving your answer in the form $ax + b$.

(3)

(b)

Find the number of US dollars the traveller obtains from £250, and interpret what $(g \circ f)(x)$ represents.

(2)

(c)

Find $(g \circ f)^{-1}(x)$ and hence find the number of pounds needed to obtain 500 US dollars.

(4)

(d)

A second kiosk converts $x$ pounds into euros according to $k(x) = 1.20x - 9$. Find the amount of pounds for which both kiosks give the same number of euros, and state the amounts for which the second kiosk gives more euros.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1: $g(f(x)) = 1.08 \times 1.16(x - 3) - 2$	✓ 1
A1: $a = 1.2528$	✓ 1
A1: $b = -5.7584$, so $(g \circ f)(x) = 1.2528x - 5.7584$	✓ 1
Part (b)
A1: $(g \circ f)(250) = 307.44$ US dollars	✓ 1
R1: $(g \circ f)(x)$ gives the number of US dollars received from $x$ pounds after both conversions and both fees	✓ 1
Part (c)
M1: rearranges $y = 1.2528x - 5.7584$ for $x$ (or swaps variables first)	✓ 1
A1: $(g \circ f)^{-1}(x) = \frac{x + 5.7584}{1.2528}$	✓ 1
M1: substitutes 500	✓ 1
A1: £403.70	✓ 1
Part (d)
M1: sets $1.16(x - 3) = 1.20x - 9$	✓ 1
M1: simplifies to $0.04x = 5.52$	✓ 1
A1: $x = 138$ pounds	✓ 1
R1: the second kiosk gives more euros when more than £138 is exchanged ($x > 138$), because its gradient is larger	✓ 1

Common error: $(g \circ f)(x)$ means apply $f$ first, then $g$. Reversing the order applies the dollar conversion to pounds and gives a different function.

+ Add to test✓ In your test357IBMI-031-01-r2-06Quadratic and cubic models Paper 2Hard12 marks

A small company makes handmade guitars. When it makes and sells x guitars per week, the selling price of each guitar is (80 − 0.4x) hundred dollars. The weekly cost, in hundreds of dollars, is modelled by
$C(x) = 0.002x^3 - 0.3x^2 + 25x + 600$, where 600 represents fixed costs.

(a)

Show that the weekly profit, P hundred dollars, is given by $P(x) = -0.002x^3 - 0.1x^2 + 55x - 600$.

(2)

(b)

Find the number of guitars the company should make each week to maximise its profit. Justify your answer.

(3)

(c)

Write down the maximum weekly profit in dollars.

(1)

(d)

Find the selling price of each guitar at this level of production.

(1)

(e)

Find the set of whole-number values of x for which the company makes a profit.

(3)

(f)

The fixed costs increase from 600 to F hundred dollars. All other parts of the model are unchanged. Find the largest value of F, to the nearest whole number, for which the company can still avoid making a loss.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Revenue = x(80 − 0.4x) = 80x − 0.4x², and uses P = R − C (M1)	✓ 1
Simplifies correctly to −0.002x³ − 0.1x² + 55x − 600 (A1) (AG)	✓ 1
Part (b)
GDC maximum, or solves P′(x) = 0, giving x = 80.5 (80.516…) (M1)	✓ 1
Because x must be a whole number, compares P(80) = 2136 with P(81) = 2136.018 (R1)	✓ 1
81 guitars (A1)	✓ 1
Part (c)
2136.018 hundred dollars, so 213 601.80 dollars (accept 213 602 dollars) (A1)	✓ 1
Part (d)
80 − 0.4 × 81 = 47.6 hundred dollars, so 4760 dollars (A1)	✓ 1
Part (e)
Solves P(x) = 0 (M1)	✓ 1
x = 11.2 and x = 136.0 (135.99…) (A1)	✓ 1
12 ≤ x ≤ 135, where x is an integer (A1)	✓ 1
Part (f)
Recognises that a change in fixed costs lowers the whole profit curve by the same amount, so the maximum still occurs at x = 81, and that the company needs F − 600 ≤ 2136.018 (M1)	✓ 1
F = 2736 (hundred dollars) (A1)	✓ 1

Exam tip: When the variable counts whole items, a turning point at x = 80.5 is not the final answer. Evaluate the model at the integers on either side, and give the one with the larger profit as your justification.

+ Add to test✓ In your test358IBMI3S-2-03-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard13 marks

A car is bought new. A dealer says its value is V = V₀e^−kt dollars after t years, with k > 0. The car is worth $24 000 after 2 years and $17 500 after 5 years.

(a)

Find the value of k and the value of V₀.

(4)

(b)

Use the exponential model to find the value of the car after 8 years.

(2)

(c)

Find the time for the car to lose half of its initial value.

(2)

(d)

A second model is linear and passes through the same two data points. Find its equation, and the value of the car after 8 years.

(3)

(e)

Find when the linear model gives a value of zero, and suggest which model is more realistic for long times.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 24000 = V0 e^(−2k) and 17500 = V0 e^(−5k)	✓ 1
M1 dividing: e^(3k) = 24000/17500	✓ 1
A1 k = 0.105	✓ 1
A1 V0 = 29 600	✓ 1
Part (b)
M1 substitute t = 8 (unrounded k and V0)	✓ 1
A1 $12 800	✓ 1
Part (c)
M1 e^(−kt) = 0.5, so t = ln 2 / k	✓ 1
A1 t = 6.58 years	✓ 1
Part (d)
M1 gradient (17500 − 24000)/3 = −2167	✓ 1
A1 V = 28 300 − 2170t	✓ 1
A1 $11 000	✓ 1
Part (e)
A1 t = 13.1 years	✓ 1
R1 the exponential model never reaches zero, so it is more realistic for old cars	✓ 1

Exam tip: When fitting an exponential through two points, divide the two equations: this eliminates the unknown initial value in one step.

+ Add to test✓ In your test359IBMI3S-2-03-13Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard11 marks

A footbridge has a parabolic arch. Taking the ground at the left foot of the arch as the origin, the height of the arch is h metres at a horizontal distance x metres from the origin. The arch meets the ground at x = 0 and x = 40, and its greatest height is 8 m.

(a)

Show that h = −0.02x² + 0.8x.

(3)

(b)

A pedestrian group carries a banner 3.5 m high. Find the width of the part of the arch at which the banner can pass.

(2)

(c)

A truck 4.0 m wide drives so that its centre line is under the highest point of the arch. Find the maximum height of the truck.

(2)

(d)

A rectangular screen of width w m is placed on the ground, centred under the highest point of the arch, with its top corners touching the arch. Find the greatest possible area of the screen.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 h = ax(x − 40)	✓ 1
M1 h(20) = 8, so −400a = 8	✓ 1
A1 a = −0.02, so h = −0.02x² + 0.8x (shown)	✓ 1
Part (b)
M1 −0.02x² + 0.8x = 3.5	✓ 1
A1 x = 5 and x = 35, so the width is 30 m	✓ 1
Part (c)
M1 h(18) or h(22)	✓ 1
A1 7.92 m	✓ 1
Part (d)
M1 height at x = 20 + w/2 is 8 − 0.005w²	✓ 1
M1 area A = w(8 − 0.005w²)	✓ 1
A1 w = 23.1 m (GDC maximum)	✓ 1
A1 A = 123 m²	✓ 1

Common error: Using the whole span as the width available; a tall object only fits where the arch is at least as high as the object.

+ Add to test✓ In your test360IBMI3S-2-03-14Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard10 marks

The tip of a wind turbine blade of length 40 m is at height H metres above the ground at time t seconds. The centre of the rotor is 90 m above the ground. The rotor turns at 15 revolutions per minute, and at t = 0 the tip is at its lowest point.

(a)

Show that H = 90 − 40cos(bt) with b = π/2.

(3)

(b)

Find the first time at which the tip is at a height of 100 m.

(2)

(c)

Find the time in each revolution for which the tip is above 110 m.

(2)

(d)

A second blade is 120° ahead of the first, so its tip has height 90 − 40cos(πt/2 + 2π/3). Find the first time t > 0 at which both tips are at the same height, and that height.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 midline 90 and amplitude 40 with a reflected cosine, since the tip starts at the lowest point	✓ 1
A1 period = 60/15 = 4 s	✓ 1
A1 b = 2π/4 = π/2 (shown)	✓ 1
Part (b)
M1 90 − 40cos(πt/2) = 100	✓ 1
A1 t = 1.16 s	✓ 1
Part (c)
M1 boundaries t = 4/3 and t = 8/3 from cos(πt/2) = −0.5	✓ 1
A1 1.33 s	✓ 1
Part (d)
M1 equate the two height expressions (GDC intersection)	✓ 1
A1 t = 1.33 s	✓ 1
A1 height 110 m	✓ 1

Exam tip: When a rotating point starts at the bottom, a negative cosine model fits directly and avoids a phase shift.

+ Add to test✓ In your test361IBMI3S-2-03b-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard14 marks

A logistics company makes open-topped boxes from a rectangular metal sheet 60 cm by 40 cm. A square of side x cm is cut from each corner and the sides are folded up to form a cuboid of volume V cm³.

(a)

Show that V = 4x³ − 200x² + 2400x.

(2)

(b)

State the possible values of x.

(1)

(c)

Find the maximum volume of the box and the value of x at which it occurs.

(3)

(d)

The company needs boxes of volume at least 8000 cm³. Find the possible values of x.

(3)

(e)

Find the two values of x that give a box of volume 5000 cm³.

(2)

(f)

The area of metal cut away is 4x² cm². Using x = 2.629 and x = 14.437, calculate how much more metal is cut away for the larger value than for the smaller value.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 V = x(60 − 2x)(40 − 2x)	✓ 1
A1 expands to 4x³ − 200x² + 2400x	✓ 1
Part (b)
A1 0 < x < 20	✓ 1
Part (c)
M1 uses a graph or table to find the maximum for 0 < x < 20	✓ 1
A1 x = 7.85 cm	✓ 1
A1 V = 8450 cm³	✓ 1
Part (d)
M1 solves V = 8000	✓ 1
A1 x = 5.86 and x = 10 (the third root, 34.1, is outside the domain)	✓ 1
A1 5.86 ≤ x ≤ 10	✓ 1
Part (e)
A1 x = 2.63	✓ 1
A1 x = 14.4	✓ 1
Part (f)
M1 area removed for the larger value 4 × 14.437² = 833.7	✓ 1
M1 area removed for the smaller value 4 × 2.629² = 27.6	✓ 1
A1 806 cm² more	✓ 1

Exam tip: After solving a cubic equation, check each root against the domain stated or implied in the context. A root outside it is not a valid answer.

+ Add to test✓ In your test362IBMI3S-2-03b-07Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard13 marks

The diameter d cm of a coral colony is measured each year for five years.

Year t	0	1	2	3	4	5
d (cm)	4.0	5.1	6.6	8.4	10.9	14.0

(a)

Use technology to find the linear regression equation d = mt + c.

(2)

(b)

Use technology to find the exponential regression equation d = a × b^t.

(2)

(c)

Write down the value of R² for each model.

(2)

(d)

Give two reasons why the exponential model is more suitable than the linear model for these data.

(2)

(e)

Use d = 3.98 × 1.29^t to predict the diameter at t = 7.

(2)

(f)

Comment on the reliability of the prediction in part (e).

(1)

(g)

Use the model d = 3.98 × 1.29^t to find when the diameter is predicted to exceed 30 cm.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 m = 1.98	✓ 1
A1 c = 3.22	✓ 1
Part (b)
A1 a = 3.98	✓ 1
A1 b = 1.29	✓ 1
Part (c)
A1 linear 0.967	✓ 1
A1 exponential 0.9999 (accept 1.00)	✓ 1
Part (d)
R1 its R² value is much closer to 1	✓ 1
R1 the yearly differences (1.1, 1.5, 1.8, 2.5, 3.1) are not constant, whereas the yearly ratios are all close to 1.28 (between 1.27 and 1.30)	✓ 1
Part (e)
M1 substitutes t = 7	✓ 1
A1 d = 23.7 cm	✓ 1
Part (f)
R1 t = 7 lies outside the data range 0 to 5, so this is extrapolation and may be unreliable	✓ 1
Part (g)
M1 sets 3.98 × 1.29^t = 30	✓ 1
A1 t = 7.93 years	✓ 1

Common error: Choosing a model on R² alone. Also check the pattern in the data and whether the model still makes sense outside the range measured.

+ Add to test✓ In your test363IBMI3S-2-03b-13Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard15 marks

At a coastal station the concentration of carbon dioxide, C ppm, is modelled by C = 412 + 2.4t + 3.5 sin(2πt), where t is the time in years after 1 January 2020. A second model proposes E = 412 × 1.0058^t.

(a)

State the period of the seasonal term 3.5 sin(2πt).

(1)

(b)

Find C when t = 0.25.

(2)

(c)

Find C when t = 0.75, and comment on how it compares with the value in part (b).

(2)

(d)

Find the first time at which C reaches 420 ppm.

(3)

(e)

Show that the two models agree to within 0.1 ppm when t = 5.

(2)

(f)

Give two ways the two models differ in the behaviour they predict.

(2)

(g)

Calculate the percentage difference between E and C when t = 30, relative to C.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1 year	✓ 1
Part (b)
M1 uses sin(π ÷ 2) = 1 or substitutes t = 0.25	✓ 1
A1 416.1 ppm	✓ 1
Part (c)
A1 410.3 ppm	✓ 1
R1 the value falls between t = 0.25 and t = 0.75 even though the long-term trend is rising: the seasonal swing outweighs the trend within a year	✓ 1
Part (d)
M1 sets 412 + 2.4t + 3.5 sin(2πt) = 420	✓ 1
M1 uses a graph or solver to find the first intersection	✓ 1
A1 t = 2.15 years	✓ 1
Part (e)
A1 C(5) = 424.0 (the sine term is 0)	✓ 1
A1 E(5) = 424.09, a difference of 0.09 ppm	✓ 1
Part (f)
R1 only C shows seasonal rises and falls	✓ 1
R1 C grows by a fixed amount per year, whereas E grows by a fixed percentage and so speeds up	✓ 1
Part (g)
M1 E(30) = 490.06…	✓ 1
M1 C(30) = 484.0 (sine term is 0)	✓ 1
A1 1.25%	✓ 1

Common error: Rounding intermediate values such as E(30) to three significant figures before finding a small percentage difference. Keep full calculator values until the end.

+ Add to test✓ In your test364IBMI3S-2-01r10-11Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Hard13 marks

Volcanologists measure the height, H metres, of a growing lava dome t days after monitoring began. The measurements are:

t (days)	0	2	4	6	8
H (m)	10.0	13.9	19.2	28.6	38.5

Two models are proposed:
Model A: H = 10 + 3.6t
Model B: H = 0.3t² + 1.2t + 10

(a)

Calculate the height at t = 6 predicted by Model A and by Model B.

(2)

(b)

For each model, calculate the sum of the absolute differences between the predicted and measured heights at the five times given. Hence state which model fits the data better.

(4)

(c)

Explain, using the data, why a linear model is unsuitable for the dome's growth.

(1)

(d)

Use Model B to predict the height at t = 12, and give one reason why this prediction may be unreliable.

(3)

(e)

Use Model B to find the time at which the dome reaches a height of 50 m. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
Model A: 10 + 3.6 × 6 = 31.6 m	✓ 1
Model B: 0.3 × 36 + 1.2 × 6 + 10 = 28.0 m	✓ 1
Part (b)
Differences for Model A at the five times: 0, 3.3, 5.2, 3.0, 0.3	✓ 1
Sum for Model A = 11.8	✓ 1
Sum for Model B = 1.6 (differences 0, 0.3, 0.4, 0.6, 0.3)	✓ 1
Model B fits better since its total difference is smaller	✓ 1
Part (c)
The increase in height per 2 days grows (3.9, 5.3, 9.4, 9.9), so the rate of growth is not constant	✓ 1
Part (d)
Substitution of t = 12 (0.3 × 144 + 1.2 × 12 + 10)	✓ 1
H = 67.6 m	✓ 1
t = 12 lies outside the range of the data (extrapolation); growth may change or the dome may collapse	✓ 1
Part (e)
Set 0.3t² + 1.2t + 10 = 50 (0.3t² + 1.2t − 40 = 0)	✓ 1
Use of GDC or quadratic formula, rejecting the negative root	✓ 1
t = 9.72 days	✓ 1

Exam tip: When comparing models, give a quantitative measure of fit for each and then state the conclusion in words.

+ Add to test✓ In your test365IBMI3S-2-01r10-12Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Hard13 marks

The depth of water, d metres, in a shipping channel is modelled by d = 7.5 + 3.5 cos(30t)°, where t is the time in hours after high water, for 0 ≤ t ≤ 12. Use your calculator in degree mode. A ship can use the channel only when the depth is at least 1.0 m more than the ship's draught (the depth of the ship below the waterline).

(a)

Write down the depth at high water and at low water, the time between successive high waters, and the depth at t = 3.

(4)

(b)

A ship has a draught of 8.5 m. Find the two times in the interval 0 ≤ t ≤ 12 at which the depth is exactly 9.5 m. Give your answers to 3 significant figures.

(3)

(c)

Hence find the length of time, in hours, for which this ship can use the channel around one high water. Give your answer to 3 significant figures.

(2)

(d)

State one reason why the model may not predict depths accurately in practice.

(1)

(e)

Another ship has a draught of 9.1 m. Find the length of time, in hours, for which this ship can use the channel around one high water. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
High water depth 11.0 m	✓ 1
Low water depth 4.0 m (7.5 − 3.5)	✓ 1
Time between high waters = 360 ÷ 30 = 12 hours	✓ 1
d = 7.5 + 3.5 cos 90° = 7.5 m at t = 3	✓ 1
Part (b)
Set 7.5 + 3.5 cos(30t) = 9.5	✓ 1
t = 1.84 (GDC or inverse cosine)	✓ 1
t = 10.16 (symmetry about t = 6 or second intersection)	✓ 1
Part (c)
Window runs from t = 10.16 (before high water) to t = 1.84 (after), i.e. 2 × 1.838…	✓ 1
Length = 3.68 hours	✓ 1
Part (d)
Real tides vary from day to day / are affected by weather, or the 12-hour period is only approximate	✓ 1
Part (e)
Required depth 10.1 m: 7.5 + 3.5 cos(30t) = 10.1	✓ 1
30t = cos⁻¹(2.6/3.5) = 42.0…°, so t = 1.40… hours	✓ 1
Length = 2 × 1.40… = 2.80 hours	✓ 1

Exam tip: Solve the equality first, then decide from the shape of the graph whether the required times lie between or outside the two solutions.

+ Add to test✓ In your test366IBMI3S-2-02r11-08Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 2Hard12 marks

At an archaeological dig site, the number of hours of daylight on day d of the year (d = 1 is 1 January) is modelled by D(d) = 12.2 + 2.8 sin(2π(d − 80) ÷ 365), where the angle is in radians.

(a)

Write down the greatest and least number of daylight hours predicted by the model.

(2)

(b)

On day 200 the observed daylight is 14.9 hours. Find the model's prediction for day 200, and the percentage error in this prediction relative to the observed value, correct to 3 significant figures.

(2)

(c)

Find the day number on which the model predicts the greatest daylight.

(2)

(d)

At a second site the greatest and least daylight hours are 16.0 and 8.0, and the greatest occurs on the same day as at the first site. Write down a model of the same form for the second site.

(2)

(e)

Using the original model, find the number of days in a year on which the daylight exceeds 13.5 hours, to the nearest whole number.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
greatest = 12.2 + 2.8 = 15.0 hours	✓ 1
least = 12.2 − 2.8 = 9.4 hours	✓ 1
Part (b)
D(200) = 14.66 hours	✓ 1
percentage error = (14.9 − 14.66) ÷ 14.9 × 100 = 1.58%	✓ 1
Part (c)
2π(d − 80) ÷ 365 = π/2	✓ 1
d = 171.25, so day 171	✓ 1
Part (d)
principal axis 12 and amplitude 4	✓ 1
D₂(d) = 12 + 4 sin(2π(d − 80) ÷ 365)	✓ 1
Part (e)
12.2 + 2.8 sin θ = 13.5, so sin θ = 1.3/2.8	✓ 1
θ = 0.4828 (first value)	✓ 1
second value θ = π − 0.4828 = 2.6588	✓ 1
number of days = (2.6588 − 0.4828) ÷ (2π) × 365 = 126.4, so 126 days	✓ 1

Common error: The answers to sin θ = k come in pairs, θ and π − θ, and the interval between them is the time the model stays above that level.

+ Add to test✓ In your test367IBMI3S-2-02r11-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 2Hard12 marks

A cycle workshop builds x bicycles per week and sells them all. The weekly revenue is R(x) = x(120 − 0.4x) euros and the weekly cost is C(x) = 1500 + 31x euros. The profit is P(x) = R(x) − C(x).

(a)

Show that P(x) = −0.4x² + 89x − 1500.

(3)

(b)

Find the numbers of bicycles per week for which the workshop makes a profit.

(2)

(c)

The workshop can only build whole numbers of bicycles. Find the number built per week that gives the greatest profit, and the greatest profit.

(3)

(d)

Find the selling price per bicycle, in euros, when 111 bicycles are built per week.

(2)

(e)

Find the value of x at which the price per bicycle given by the model becomes zero, and explain what this implies about the domain of the model.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
R(x) = 120x − 0.4x²	✓ 1
P = 120x − 0.4x² − 1500 − 31x	✓ 1
P = −0.4x² + 89x − 1500	✓ 1
Part (b)
solve −0.4x² + 89x − 1500 = 0: x = 18.37 and x = 204.13	✓ 1
profit when 18.37 < x < 204.13, so 19 to 204 bicycles (inclusive)	✓ 1
Part (c)
vertex at x = 89 ÷ (2 × 0.4) = 111.25	✓ 1
P(111) = 3450.60 and P(112) = 3450.40	✓ 1
111 bicycles, maximum profit €3450.60	✓ 1
Part (d)
price = R(111) ÷ 111 = 120 − 0.4 × 111	✓ 1
= €75.60	✓ 1
Part (e)
120 − 0.4x = 0, so x = 300	✓ 1
the model is only valid for x < 300 as the price cannot be zero or negative	✓ 1

Exam tip: When the unknown must be a whole number, evaluate the function at the whole numbers either side of the vertex rather than rounding the vertex.

+ Add to test✓ In your test368IBMI3S-2-01r12-08Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard14 marks

A train travels between two stations. Its distance s metres from the first station, t seconds after departure, is modelled by:
s = 0.2t² for 0 ≤ t ≤ 40
s = 320 + 16(t − 40) for 40 < t ≤ 160
s = 2240 + 16(t − 160) − 0.1(t − 160)² for 160 < t ≤ 240

(a)

Calculate the distance travelled when t = 25.

(1)

(b)

Calculate the average speed of the train during the first 40 seconds.

(2)

(c)

Show that the first two expressions give the same distance at t = 40.

(2)

(d)

Find the time at which the train has travelled 1000 m.

(2)

(e)

The train brakes during the last stage and stops at t = 240. Show that the greatest distance given by the third expression is 2880 m.

(3)

(f)

Find the length of time, to 3 significant figures, for which the train travels the final 200 m of its journey.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 s = 125 m	✓ 1
Part (b)
M1 uses s(40)/40 = 320/40	✓ 1
A1 8 m s⁻¹	✓ 1
Part (c)
M1 first expression: 0.2 × 40² = 320	✓ 1
A1 second expression: 320 + 16(0) = 320, so they agree	✓ 1
Part (d)
M1 320 + 16(t − 40) = 1000 (1000 lies between 320 and 2240, so the second expression applies)	✓ 1
A1 t = 82.5 s	✓ 1
Part (e)
M1 uses the vertex of the quadratic in u = t − 160: u = 16/(2 × 0.1) = 80, or technology	✓ 1
A1 t = 240	✓ 1
A1 s = 2240 + 1280 − 640 = 2880 m	✓ 1
Part (f)
M1 2240 + 16u − 0.1u² = 2680	✓ 1
M1 solves, choosing the root less than 80	✓ 1
A1 u = 35.3, so t = 195 s when 200 m remain	✓ 1
A1 time = 240 − 195.28 = 44.7 s (accept 44.7 from u = 80 − 35.3 = 44.7)	✓ 1

Exam tip: For a piecewise model, decide which expression applies by comparing the given value of t or s with the break points before substituting.

+ Add to test✓ In your test369IBMI3S-2-01r12-14Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard13 marks

The cross-section of a railway tunnel is modelled by y = 7.2 − 0.2x², where x metres is the horizontal distance from the centre line and y metres is the height above the floor y = 0.

(a)

Write down the maximum height of the tunnel.

(1)

(b)

Find the width of the tunnel at floor level.

(2)

(c)

A train has a rectangular cross-section 3.4 m wide and 5.6 m high and runs centred on the centre line. Show that the train fits in the tunnel.

(3)

(d)

Two identical trains, each 3.4 m wide, run side by side with a 1.0 m gap between them, symmetrically about the centre line. Find the greatest height the trains can have. Give your answer to 3 significant figures.

(3)

(e)

A rectangular container of width 2x sits on the floor, symmetric about the centre line, with its top corners on the curve. Find the greatest possible cross-sectional area of the container. Give your answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 7.2 m	✓ 1
Part (b)
M1 7.2 − 0.2x² = 0, so x = ±6	✓ 1
A1 width = 12 m	✓ 1
Part (c)
M1 top corners at x = ±1.7	✓ 1
A1 tunnel height at x = 1.7 is 7.2 − 0.2 × 1.7² = 6.622 m	✓ 1
R1 6.622 > 5.6, so the train fits	✓ 1
Part (d)
M1 outer edges at x = ±(0.5 + 3.4) = ±3.9	✓ 1
M1 y = 7.2 − 0.2 × 3.9²	✓ 1
A1 4.16 m	✓ 1
Part (e)
M1 area A = 2x(7.2 − 0.2x²) = 14.4x − 0.4x³	✓ 1
M1 finds the maximum using technology or differentiation	✓ 1
A1 x = 3.46 m	✓ 1
A1 maximum area = 33.3 m²	✓ 1

Exam tip: For a 'greatest value' question, write the quantity as a function of one variable first, then use technology to find the maximum within the allowed domain.

+ Add to test✓ In your test370IBMI3S-2-02r17-08Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard10 marks

An overhead power cable at a wind farm hangs between two pylons that are 120 m apart. The cable is attached to the top of each pylon, 45 m above level ground, and its lowest point is 30 m above the ground, midway between the pylons. Let x metres be the horizontal distance from the first pylon and h metres the height of the cable above the ground. The cable is modelled by h = a(x − 60)² + 30, for 0 ≤ x ≤ 120.

(a)

Find the value of a.

(2)

(b)

Find the height of the cable at a horizontal distance of 20 m from the first pylon, correct to 3 significant figures.

(2)

(c)

A tall vehicle of height 36 m must stay clear of the cable. Find the total length of the horizontal section of the road, in metres, along which the cable is lower than 36 m. Give your answer correct to 3 significant figures.

(3)

(d)

The cable is tightened so that the new model is h = (1/300)(x − 60)² + k, with the cable still attached 45 m above the ground at both pylons. Find the value of k and state what it represents.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 45 = a(0 − 60)² + 30	✓ 1
A1 a = 15/3600 = 1/240 (= 0.00417)	✓ 1
Part (b)
M1 h = (1/240)(20 − 60)² + 30	✓ 1
A1 36.7 m	✓ 1
Part (c)
M1 (1/240)(x − 60)² + 30 = 36	✓ 1
M1 (x − 60)² = 1440, so x = 60 ± 37.9… (22.05 and 97.95)	✓ 1
A1 75.9 m	✓ 1
Part (d)
M1 45 = (1/300)(0 − 60)² + k	✓ 1
A1 k = 45 − 12 = 33	✓ 1
A1 the new height of the lowest point of the cable above the ground, 33 m	✓ 1

Exam tip: In vertex form y = a(x − p)² + q, the vertex is at (p, q): use it to read off the turning point and substitute one other point to find a.

+ Add to test✓ In your test371IBMI3S-2-02r17-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard12 marks

In a wave pool, the height, h metres, of the water above the pool floor at a fixed marker t seconds after the waves start is modelled by h = acos(b(t − c)) + d, where a, b, c and d are positive constants and b is in radians. The height reaches a maximum of 2.35 m at t = 1.5 s, and the next minimum of 1.15 m occurs at t = 4.5 s.

(a)

Find the value of a and the value of d.

(2)

(b)

Find the values of b and c.

(3)

(c)

Find the first time after t = 0 at which the height is 2.0 m. Give your answer in seconds, correct to 3 significant figures.

(3)

(d)

In each cycle, the height is above 2.2 m for part of the time. Find the length of this time, in seconds, correct to 3 significant figures.

(3)

(e)

Find the percentage of each cycle for which the height is above 2.2 m, correct to 3 significant figures. Use the unrounded answer from part (d).

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (2.35 − 1.15) ÷ 2 = 0.6	✓ 1
A1 d = (2.35 + 1.15) ÷ 2 = 1.75	✓ 1
Part (b)
M1 half a period = 4.5 − 1.5 = 3, so the period is 6 s	✓ 1
A1 b = 2π ÷ 6 = π/3 (= 1.05)	✓ 1
A1 c = 1.5 (the maximum of a cosine curve occurs at t = c)	✓ 1
Part (c)
M1 1.75 + 0.6cos((π/3)(t − 1.5)) = 2.0, so cos((π/3)(t − 1.5)) = 0.4167…	✓ 1
M1 (π/3)(t − 1.5) = −1.141… (negative value chosen for the earliest time)	✓ 1
A1 t = 0.410 s	✓ 1
Part (d)
M1 1.75 + 0.6cos((π/3)(t − 1.5)) = 2.2, so cos(…) = 0.75	✓ 1
M1 half-width of the interval = acos(0.75) ÷ (π/3) = 0.690…, so the interval is twice this	✓ 1
A1 1.38 s	✓ 1
Part (e)
A1 (1.380… ÷ 6) × 100 = 23.0%	✓ 1

Exam tip: A cosine model reaches its maximum at t = c, and the graph is symmetrical about this value, so each interval above a given height is centred on a maximum.

+ Add to test✓ In your test372IBMI3S-2-02r17-14Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard12 marks

The tip of a wind-turbine blade moves in a vertical circle. The hub is 80 m above the ground and the blade is 45 m long. The blade makes one complete rotation every 4 seconds. At time t = 0 the tip is at its lowest point. The height of the tip above the ground, in metres, is modelled by h(t) = 80 − 45cos(bt), where b is in radians.

(a)

Write down the least and greatest heights of the tip above the ground.

(2)

(b)

Find the value of b, and hence find the height of the tip at t = 1.3 s. Give your answer correct to 3 significant figures.

(3)

(c)

The tip of the blade on a second turbine is at its lowest point at t = 1, with the same rotation period and dimensions. Its height is modelled by h₂(t) = h(t − 1). Describe the transformation that maps the graph of h onto the graph of h₂.

(2)

(d)

Find the values of t, for 0 ≤ t ≤ 4, at which the two tips are at the same height.

(3)

(e)

Find the common height of the two tips at the earlier of these two times, correct to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 least height = 35 m	✓ 1
A1 greatest height = 125 m	✓ 1
Part (b)
M1 period = 4 = 2π ÷ b	✓ 1
A1 b = π/2	✓ 1
A1 h(1.3) = 80 − 45cos(1.3π/2) = 100 m	✓ 1
Part (c)
A1 a translation (horizontal shift) of 1 unit	✓ 1
A1 in the positive t direction, that is, by the vector (1, 0)	✓ 1
Part (d)
M1 h(t) = h₂(t), so cos(πt/2) = cos(π(t − 1)/2)	✓ 1
M1 cos(πt/2 − π/2) = sin(πt/2), so tan(πt/2) = 1 (or solving the equation graphically)	✓ 1
A1 t = 0.5 and t = 2.5	✓ 1
Part (e)
M1 h(0.5) = 80 − 45cos(π/4)	✓ 1
A1 48.2 m	✓ 1

Common error: A horizontal shift of a periodic graph changes when the pattern happens, not its range: the greatest and least values stay the same.

+ Add to test✓ In your test373IBMI3S-2-01r20-13Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 2Hard12 marks

A manufacturer of heat pumps models its annual profit by P(x) = −x³ + 9x² + 24x − 40, for 0 ≤ x ≤ 12, where P is the profit in thousands of euros and x is the number of heat pumps produced, in hundreds.

(a)

Find P(0) and interpret this value in context.

(2)

(b)

Find the production level that maximises the profit, and the maximum profit in euros.

(3)

(c)

Find the range of production levels, in hundreds of heat pumps, for which the manufacturer makes a profit. Give the end points to 3 significant figures.

(4)

(d)

Find the value of P(12) and state what this represents.

(2)

(e)

Suggest one reason why profit might fall at high production levels.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 P(0) = −40	✓ 1
A1 with no production there is a loss of €40 000 (fixed costs)	✓ 1
Part (b)
M1 use of GDC to find maximum on 0 ≤ x ≤ 12	✓ 1
A1 x = 7.12 hundred, i.e. approximately 712 heat pumps	✓ 1
A1 maximum profit €226 000 (226.2 thousand)	✓ 1
Part (c)
M1 solve P(x) = 0 (or use GDC graph)	✓ 1
A1 x = 1.20	✓ 1
A1 x = 10.9	✓ 1
A1 1.20 < x < 10.9	✓ 1
Part (d)
M1 P(12) = −12³ + 9 × 12² + 24 × 12 − 40	✓ 1
A1 −184, a loss of €184 000	✓ 1
Part (e)
R1 e.g. higher costs from overtime or extra machinery, or the market becomes saturated so prices must be cut	✓ 1

Exam tip: A GDC solver or graph is the quickest route for a cubic; sketch the graph so you can see which roots lie in the domain.

+ Add to test✓ In your test374IBMI3S-2-01r25-11Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 2Hard15 marks

Two telecoms companies, X and Y, compete in a region with 60 000 households that may take broadband. t months after Y enters the market, the number of subscribers of each is modelled by:
Company X: N_X = 4500 + 380t
Company Y: N_Y = 2000 × 1.16^t

(a)

State the number of subscribers of each company when Y enters the market, and the percentage by which the number of subscribers of Y increases each month.

(2)

(b)

Find the number of subscribers of each company when t = 12, to the nearest whole number, and the difference between them.

(3)

(c)

Find the value of t at which the two companies have equal numbers of subscribers, correct to 3 significant figures, and the number of subscribers each then has, to the nearest hundred.

(4)

(d)

Using the model for Y, find when Y alone would have 60 000 subscribers. Give your answer correct to 3 significant figures.

(3)

(e)

Find the number of subscribers the model predicts for Y at t = 36, to the nearest thousand, and explain why the model is unlikely to remain valid for this long.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 X: 4500 and Y: 2000	✓ 1
A1 16% per month	✓ 1
Part (b)
A1 N_X(12) = 9060	✓ 1
A1 N_Y(12) = 11872	✓ 1
A1 Y has 2812 more subscribers	✓ 1
Part (c)
M1 4500 + 380t = 2000 × 1.16^t (equation or graphs intersection with a positive t value)	✓ 1
A1 t = 9.40 months	✓ 1
M1 substitution of the unrounded t into either model	✓ 1
A1 about 8100 subscribers each	✓ 1
Part (d)
M1 2000 × 1.16^t = 60 000 (1.16^t = 30)	✓ 1
A1 t = 22.9 months	✓ 1
A1 during the 23rd month after Y enters the market (or: after about 23 months)	✓ 1
Part (e)
A1 N_Y(36) ≈ 418000	✓ 1
R1 this exceeds the 60 000 households in the region	✓ 1
R1 the growth cannot stay at a fixed 16% per month as the market becomes saturated, so a model whose growth slows is needed	✓ 1

Exam tip: To compare a linear and an exponential model, use the intersection tool on your calculator, and always check that the solution is inside the context's domain.

+ Add to test✓ In your test375IBMI3S-2-01r25-14Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models Paper 2Hard12 marks

Meltwater flows from a glacier in a daily cycle. The flow rate, Q litres per second, in a mountain stream is modelled by Q(t) = a cos(b(t − 16)) + d, where t is the number of hours after midnight (0 ≤ t < 24) and the angle is in radians. The flow is greatest at 16:00, at 38 litres per second, and least at 04:00, at 14 litres per second.

(a)

Find the values of a, b and d.

(3)

(b)

Find the flow rate at 13:30. Give your answer correct to 3 significant figures.

(2)

(c)

Find the times of day between which the flow exceeds 30 litres per second. Give the times to the nearest minute, and state the length of this period in hours correct to 3 significant figures.

(4)

(d)

Crossing the stream is safe only while the flow is below 20 litres per second. Find the length of time in each 24-hour cycle for which crossing is safe.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (38 − 14) ÷ 2 = 12	✓ 1
A1 d = (38 + 14) ÷ 2 = 26	✓ 1
A1 b = 2π ÷ 24 = π/12 (0.262)	✓ 1
Part (b)
M1 Q(13.5) = 12 cos(π/12 × (13.5 − 16)) + 26	✓ 1
A1 35.5 litres per second	✓ 1
Part (c)
M1 12 cos(π/12 × (t − 16)) + 26 = 30 (cos(π/12 × (t − 16)) = 1/3)	✓ 1
A1 t = 11.30, i.e. about 11:18	✓ 1
A1 t = 20.70, i.e. about 20:42	✓ 1
A1 length of period = 9.40 hours	✓ 1
Part (d)
M1 12 cos(π/12 × (t − 16)) + 26 = 20 (cos = −0.5), giving t = 8 and t = 24 (equivalently 0)	✓ 1
A1 flow is below 20 between 00:00 and 08:00 (by symmetry, since Q(0) = 20 and the minimum is at 04:00)	✓ 1
A1 8 hours	✓ 1

Exam tip: Convert decimal hours into hours and minutes at the end: 0.3 hours is 18 minutes, not 30.

+ Add to test✓ In your test376IBMI3S-2-01r28-12Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Hard12 marks

The number of requests y (millions per day) received by a new online service in week x after launch is recorded.
x: 1, 2, 3, 4, 5, 6
y: 3.1, 4.6, 6.9, 10.5, 15.6, 23.2
It is suggested that y = a × b^x. To test this, a student plots log₁₀y against x.

(a)

Find log₁₀y when x = 1 and when x = 6, to 3 significant figures.

(2)

(b)

The regression line of log₁₀y on x is log₁₀y = mx + c. Use your graphic display calculator to find m and c to 3 significant figures.

(2)

(c)

Hence show that y = a × b^x and find a and b to 3 significant figures, using the unrounded values of m and c.

(3)

(d)

Use the unrounded model to predict the number of requests per day, in millions, in week 9. Give your answer to 3 significant figures.

(2)

(e)

Interpret the value of b in this context.

(1)

(f)

Find the value of x at which the model predicts 100 million requests per day. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 0.491	✓ 1
A1 1.37	✓ 1
Part (b)
M1 regression applied to the points (x, log₁₀y)	✓ 1
A1 m = 0.176 and c = 0.314	✓ 1
Part (c)
M1 y = 10^{mx + c} = 10^c × (10^m)^x	✓ 1
A1 a = 2.06	✓ 1
A1 b = 1.50	✓ 1
Part (d)
M1 substitute x = 9	✓ 1
A1 78.4 million	✓ 1
Part (e)
R1 the number of requests is multiplied by about 1.50 each week, i.e. increases by about 50% per week	✓ 1
Part (f)
M1 set 100 = 10^{mx + c}, i.e. mx + c = 2	✓ 1
A1 x = 9.60 weeks	✓ 1

Exam tip: Taking log₁₀ of y = a × bx gives log₁₀y = log₁₀a + x log₁₀b, a straight line with gradient log₁₀b.

+ Add to test✓ In your test377IBMI3S-2-03r31-04Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 2Hard12 marks

The height of the water, h metres, in a harbour is modelled by h(t) = a cos(bt) + d, where t is the time in hours after a high tide and the angle is in radians. At high tide the height is 5.6 m, at low tide it is 0.8 m, and the time between consecutive high tides is 12.4 hours.

(a)

Find the value of a and the value of d.

(3)

(b)

Find the value of b. Give your answer to 3 significant figures, and use the unrounded value in later parts.

(2)

(c)

Find the height of the water 4 hours after a high tide. Give your answer to 3 significant figures.

(2)

(d)

A ship can enter the harbour only when the height of the water is greater than 4 m. Find the total length of time, in hours, during the interval 0 ≤ t ≤ 12.4 for which the ship can enter. Give your answer to 3 significant figures.

(5)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 a = (5.6 − 0.8)/2 or d = (5.6 + 0.8)/2	✓ 1
A1 a = 2.4	✓ 1
A1 d = 3.2	✓ 1
Part (b)
M1 period = 2π/b = 12.4 (or equivalent)	✓ 1
A1 b = 0.507	✓ 1
Part (c)
M1 substituting t = 4 into the model (calculator in radian mode)	✓ 1
A1 h = 2.14 m	✓ 1
Part (d)
M1 setting 3.2 + 2.4 cos(bt) = 4	✓ 1
A1 first crossing t = 2.43	✓ 1
M1 using symmetry or the graph to find the second crossing (t = 12.4 − 2.43 = 9.97)	✓ 1
M1 total time = 2.43 + (12.4 − 9.97), i.e. the time before the first and after the second crossing	✓ 1
A1 4.86 hours	✓ 1

Exam tip: Check that your calculator is in radian mode whenever the model uses a period such as 2π/b; a degree-mode answer will not match the context.

+ Add to test✓ In your test378IBMI3S-2-03r31-06Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models Paper 2Hard12 marks

A robotics club makes open boxes for small parts from a rectangular sheet of plastic 60 cm by 40 cm. A square of side x cm is cut from each corner of the sheet and the sides are folded up to make a box with no lid.

(a)

Show that the volume of the box, V cm³, is given by V = 4x³ − 200x² + 2400x.

(3)

(b)

State the possible values of x in this model.

(1)

(c)

Find the value of x that gives the greatest volume, and the greatest volume. Give both answers to 3 significant figures.

(3)

(d)

The club wants a box with volume 4500 cm³. Find the two values of x in the domain that give this volume.

(3)

(e)

When the box is made with the value of x from (c), the four cut-out squares are wasted. Find the total area of plastic wasted. Use the unrounded value of x from (c) and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 length 60 − 2x and width 40 − 2x and height x	✓ 1
M1 V = x(60 − 2x)(40 − 2x)	✓ 1
A1 expanding to 4x³ − 200x² + 2400x (AG)	✓ 1
Part (b)
A1 0 < x < 20	✓ 1
Part (c)
M1 attempt to find the maximum from a graph of V for 0 < x < 20	✓ 1
A1 x = 7.85 cm	✓ 1
A1 V = 8450 cm³	✓ 1
Part (d)
M1 setting 4x³ − 200x² + 2400x = 4500 (or intersecting graphs)	✓ 1
A1 x = 2.29	✓ 1
A1 x = 15.0	✓ 1
Part (e)
M1 area wasted = 4x²	✓ 1
A1 246 cm²	✓ 1

Exam tip: State the domain from the physical situation: every dimension of the box must stay positive, so 40 − 2x > 0 gives x < 20.

+ Add to test✓ In your test379IBMI3S-2-02r32-05Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Hard15 marks

A probe orbits a moon. Its altitude above the moon's surface, h km, t minutes after it passes its highest point, is modelled by
h(t) = a cos(bt) + d, where the angle is in radians.
The highest altitude is 1800 km, the lowest altitude is 400 km, and one complete orbit takes 90 minutes.

(a)

Find the values of a and d.

(2)

(b)

Find the value of b, giving your answer to 3 significant figures.

(2)

(c)

Find the altitude of the probe when t = 20. Use the unrounded value of b. Give your answer to 3 significant figures.

(2)

(d)

The probe can transmit to the surface only when its altitude is above 1500 km. During the first orbit (0 ≤ t ≤ 90), find the two intervals of time for which this is true. Give your answers to 3 significant figures.

(3)

(e)

Hence find the time in one orbit for which the probe can transmit. Use the unrounded value from (d). Give your answer to 3 significant figures.

(2)

(f)

Find the percentage of each orbit for which the probe can transmit. Give your answer to 3 significant figures.

(2)

(g)

The probe makes 16 complete orbits in 24 hours. Find the total time in 24 hours, in minutes, for which it can transmit. Use the unrounded value from (e). Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = 700 (half the difference of 1800 and 400)	✓ 1
A1 d = 1100 (the mean of 1800 and 400)	✓ 1
Part (b)
M1 2π/b = 90	✓ 1
A1 b = 0.0698	✓ 1
Part (c)
M1 substitute t = 20 into 1100 + 700cos(bt)	✓ 1
A1 1220 km	✓ 1
Part (d)
M1 set 1100 + 700cos(bt) > 1500 (solve cos(bt) = 4/7 or use technology)	✓ 1
A1 0 ≤ t < 13.8	✓ 1
A1 76.2 < t ≤ 90	✓ 1
Part (e)
M1 add the two intervals (2 × 13.7875…, or 13.79… + (90 − 76.21…))	✓ 1
A1 27.6 minutes	✓ 1
Part (f)
M1 27.575… ÷ 90 × 100 (ECF from (e))	✓ 1
A1 30.6 %	✓ 1
Part (g)
M1 16 × 27.575…	✓ 1
A1 441 minutes	✓ 1

Exam tip: For a cosine model, the amplitude is half of (maximum − minimum) and the mean level is the average of the two; check that h(0) gives the stated maximum.

+ Add to test✓ In your test380IBMI3S-2-02r32-12Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Hard13 marks

A steel girder is 10 m long. When it is loaded, its vertical offset from a straight line, d millimetres, at a distance x metres from the left end is modelled by
d(x) = 0.004x(x − 6)(x − 10), for 0 ≤ x ≤ 10.
A positive value of d means the girder is above the straight line.

(a)

Write down the values of x at which the girder lies on the straight line.

(2)

(b)

Find the offset at x = 3.

(2)

(c)

Find the greatest offset above the straight line, and the value of x at which it occurs. Give your answers to 3 significant figures.

(3)

(d)

Find the greatest offset below the straight line, and the value of x at which it occurs. Give your answers to 3 significant figures.

(2)

(e)

The girder must be rejected if the offset is more than 0.1 mm above the straight line at any point. Find the values of x between 0 and 6 for which d(x) > 0.1. Give your answers to 3 significant figures where appropriate.

(3)

(f)

State the greatest distance, in mm, between the girder and the straight line at any point along its length. Give your answer to 3 significant figures.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 x = 6	✓ 1
A1 x = 0 and x = 10	✓ 1
Part (b)
M1 substitute x = 3: 0.004 × 3 × (−3) × (−7)	✓ 1
A1 0.252 mm	✓ 1
Part (c)
M1 use technology to find the local maximum of d(x) for 0 < x < 6	✓ 1
A1 x = 2.43 m	✓ 1
A1 d = 0.263 mm	✓ 1
Part (d)
A1 x = 8.24 m	✓ 1
A1 d = −0.130 mm (offset of 0.130 mm below the line)	✓ 1
Part (e)
M1 set d(x) = 0.1 and solve (technology)	✓ 1
A1 x = 0.475	✓ 1
A1 x = 5, so 0.475 < x < 5	✓ 1
Part (f)
A1 0.263 mm, since \|−0.130\| < 0.263 (ECF from (c) and (d))	✓ 1

Exam tip: A cubic can have a local maximum and a local minimum; use technology to find each and compare their absolute values when asked for the greatest distance from a line.

+ Add to test✓ In your test381IBMI3S-2-02r32-14Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Hard12 marks

During a coffee roast, the percentage of mass lost by the beans, y %, is recorded at different times t minutes after the roast begins.

t (min)	4	6	8	10	12	14
y (%)	1.8	4.6	8.9	13.1	16.2	18.0

Two models are proposed.
Model L: y = 1.71t − 5.0. Model Q: y = −0.044t² + 2.5t − 8.0.

(a)

Use Model L to find the predicted values of y when t = 8 and when t = 14.

(3)

(b)

For Model L, find the sum of the squares of the differences between the measured and predicted values for all six measurements. Give your answer to 3 significant figures.

(2)

(c)

For Model Q, the sum of the squares of the differences is 1.68. State which model fits the measurements better and give a reason.

(2)

(d)

Find the values of y predicted by Model L and by Model Q when t = 20.

(2)

(e)

Find the value of t at which Model Q has its greatest value, giving your answer to 3 significant figures. Explain why Model Q should not be used for times after this.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 8 or t = 14 into 1.71t − 5.0	✓ 1
A1 8.68	✓ 1
A1 18.94	✓ 1
Part (b)
M1 find the six differences (measured − predicted) and square them	✓ 1
A1 2.83	✓ 1
Part (c)
A1 Model Q	✓ 1
R1 its sum of squared differences (1.68) is smaller than that of Model L (2.83)	✓ 1
Part (d)
A1 Model L: 29.2 %	✓ 1
A1 Model Q: 24.4 %	✓ 1
Part (e)
M1 vertex of the quadratic: t = −2.5/(2 × (−0.044)) (or use technology)	✓ 1
A1 t = 28.4 minutes	✓ 1
R1 after this time Model Q predicts the percentage of mass lost decreases, but beans cannot regain lost mass	✓ 1

Exam tip: A model that fits well inside the range of the data can still give impossible predictions outside it, so always check extrapolated values against the context.

+ Add to test✓ In your test382IBMI3S-2-04r32-12Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard13 marks

A games company sells a new game for p euros per download. The number of downloads is modelled by $N = 5000 - 40p$. Each download costs the company €1.50 in server fees, and there is a fixed cost of €20 000. The profit, P euros, is the total income from downloads minus all costs.

(a)

Show that $P = -40p^{2} + 5060p - 27500$.

(3)

(b)

Find the price that gives the greatest profit.

(2)

(c)

Find the greatest profit and the number of downloads at this price.

(2)

(d)

Find the range of prices for which the company makes a profit. Give your answers to the nearest cent.

(3)

(e)

The company decides that the price must not be more than €50. Find the greatest profit the company can make under this condition, and justify your answer.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $P = pN - 1.5N - 20000 = (p-1.5)(5000-40p) - 20000$	✓ 1
A1 expansion $5000p - 40p^{2} - 7500 + 60p - 20000$	✓ 1
A1 $-40p^{2} + 5060p - 27500$ (answer given)	✓ 1
Part (b)
M1 $p = -\dfrac{5060}{2\times(-40)}$ or GDC maximum	✓ 1
A1 €63.25	✓ 1
Part (c)
A1 €132522.50 (greatest profit; 63.25 substituted into $P$)	✓ 1
A1 2470 downloads	✓ 1
Part (d)
M1 solve $P > 0$ (or $P = 0$)	✓ 1
A1 lower limit €5.69	✓ 1
A1 upper limit €120.81	✓ 1
Part (e)
R1 the vertex (€63.25) lies above €50 and the graph is increasing for $p < 63.25$, so the greatest profit is at $p = 50$	✓ 1
M1 $P(50) = -40\times50^{2} + 5060\times50 - 27500$	✓ 1
A1 €125500	✓ 1

Exam tip: When a restriction removes the vertex from the allowed domain, the maximum of a quadratic occurs at the end of the domain nearest the vertex.

+ Add to test✓ In your test383IBMI3S-2-04r32-14Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard12 marks

In an online game, the experience points (XP) needed to reach level L are modelled by $X = 100\times1.5^{L}$. A player makes a graph of $\log_{10}X$ against L, and the points lie on a straight line.

(a)

Calculate the value of $\log_{10}X$ for level 8. Give your answer to 3 significant figures.

(2)

(b)

Find the gradient and the vertical intercept of the line. Give the gradient to 3 significant figures.

(3)

(c)

Find the lowest level that needs more than 1 million XP.

(3)

(d)

A second game uses $X = 250\times1.3^{L}$. Find the value of L at which the two games need the same XP. Give your answer to 3 significant figures.

(3)

(e)

State which game needs more XP for large values of L, giving a reason.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $\log_{10}(100\times1.5^{8}) = \log_{10}(2562.89\ldots)$	✓ 1
A1 3.41	✓ 1
Part (b)
M1 $\log_{10}X = \log_{10}100 + L\log_{10}1.5$	✓ 1
A1 gradient $\log_{10}1.5 = 0.176$	✓ 1
A1 vertical intercept 2	✓ 1
Part (c)
M1 $100\times1.5^{L} > 10^{6}$, so $1.5^{L} > 10^{4}$	✓ 1
M1 $L > \dfrac{\ln 10^{4}}{\ln 1.5}$ (= 22.7) or GDC table/graph	✓ 1
A1 level 23 (not 22.7, as levels are whole numbers)	✓ 1
Part (d)
M1 $100\times1.5^{L} = 250\times1.3^{L}$ (or GDC intersection)	✓ 1
M1 $\left(\dfrac{1.5}{1.3}\right)^{L} = 2.5$ and take logarithms	✓ 1
A1 $L = 6.40$	✓ 1
Part (e)
R1 the first game, because its growth factor 1.5 is larger than 1.3, so it eventually overtakes the second	✓ 1

Exam tip: On a log scale, an exponential model becomes a straight line: the gradient is the logarithm of the growth factor and the intercept is the logarithm of the starting value.

+ Add to test✓ In your test384IBMI3S-2-03r34-06Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 2Hard13 marks

A delivery robot's battery charge, C %, is recorded t hours after it leaves its dock.

t (hours)	0	1	2	3	4	5
C (%)	100	82	67	55	45	37

Two models are proposed: a linear model C = mt + k and an exponential model C = a × b^t. Use the regression facility of your calculator (use the stored, unrounded values in later calculations).

(a)

Find the equation of the linear regression line, giving m and k to 3 significant figures.

(2)

(b)

Find the exponential regression model, giving a to 3 significant figures and b to 3 significant figures.

(2)

(c)

Interpret the value of b in the exponential model.

(1)

(d)

State the value of r² for each model, to 3 significant figures.

(2)

(e)

Use each model to predict the charge at t = 8, to the nearest whole number, and hence explain which model is more suitable for predicting beyond the data.

(3)

(f)

The robot must return to its dock when the charge reaches 30 %. Use the exponential model to find the time at which this occurs. Give your answer to 2 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 m = −12.5	✓ 1
A1 k = 95.6	✓ 1
Part (b)
A1 a = 100 (accept 99.95…, 100)	✓ 1
A1 b = 0.819	✓ 1
Part (c)
A1 the charge falls to about 81.9 % of its previous value each hour (a decrease of about 18.1 % per hour)	✓ 1
Part (d)
A1 linear: r² = 0.979	✓ 1
A1 exponential: r² = 1.00 (accept 0.99999…)	✓ 1
Part (e)
A1 linear: −4 %	✓ 1
A1 exponential: 20 %	✓ 1
R1 the linear model gives a negative charge, which is impossible, so the exponential model (with a higher r²) is more suitable	✓ 1
Part (f)
M1 setting 99.95… × 0.819…^t = 30	✓ 1
M1 solving using logarithms or a graph / solver	✓ 1
A1 t = 6.0 hours	✓ 1

Exam tip: A high r² shows a good fit within the data only; always check whether a model's predictions make sense outside the data range.

+ Add to test✓ In your test385IBMI3S-2-03r34-08Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 2Hard12 marks

A fish-processing plant makes open crates from rectangular metal sheets measuring 60 cm by 40 cm. A square of side x cm is cut from each corner and the sides are folded up to form a crate of volume V cm³.

(a)

Show that V = 4x³ − 200x² + 2400x.

(2)

(b)

State the values of x for which the model is valid.

(1)

(c)

Use your calculator to find the value of x that gives the greatest volume, and the greatest volume. Give both answers to 3 significant figures.

(3)

(d)

The plant requires a volume of at least 6000 cm³. Find the range of values of x that satisfy this requirement, giving your answers to 3 significant figures.

(3)

(e)

The cutting machine can only produce values of x that are multiples of 0.5 cm. Find the value of x that gives the greatest volume and state that volume.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 base dimensions (60 − 2x) and (40 − 2x), height x, so V = x(60 − 2x)(40 − 2x)	✓ 1
A1 expansion (2400 − 200x + 4x²) × x leading to the given result	✓ 1
Part (b)
A1 0 < x < 20	✓ 1
Part (c)
M1 graph or derivative used to locate the maximum within the domain	✓ 1
A1 x = 7.85 cm	✓ 1
A1 V = 8450 cm³	✓ 1
Part (d)
M1 solving V = 6000 (graph or solver)	✓ 1
A1 x = 3.40 and x = 13.2	✓ 1
A1 3.40 ≤ x ≤ 13.2	✓ 1
Part (e)
M1 comparing volumes at the multiples of 0.5 nearest to 7.85, that is x = 7.5 and x = 8	✓ 1
A1 V(7.5) = 8437.5 and V(8) = 8448	✓ 1
A1 x = 8 cm, V = 8448 cm³	✓ 1

Exam tip: In an optimisation problem, the best value allowed by a restriction is not always the nearest one to the unrestricted maximum, so compare the volumes on both sides.

+ Add to test✓ In your test386IBMI3S-2-03r34-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models Paper 2Hard12 marks

A wave-making robot in a test tank produces water-level readings modelled by h(t) = a + b cos(c(t − 1)), where h is the height in metres above the tank floor, t is the time in seconds and angles are in radians. The greatest height is 1.9 m, the least height is 0.7 m, and the first maximum occurs at t = 1. The pattern repeats every 8 seconds.

(a)

Find the values of a, b and c.

(3)

(b)

Find the height of the water at t = 6.5. Give your answer to 3 significant figures.

(2)

(c)

Find the first time after t = 0 at which the height is 1.0 m. Give your answer to 3 significant figures.

(2)

(d)

In each cycle, find the length of time for which the height is greater than 1.0 m. Give your answer to 3 significant figures.

(3)

(e)

The controller is reprogrammed so that the new height is h₂(t) = h(t − 2) + 0.2. Find the greatest value of h₂ and the first time after t = 0 at which it occurs.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = 1.3	✓ 1
A1 b = 0.6	✓ 1
A1 c = 2π/8 = π/4	✓ 1
Part (b)
M1 1.3 + 0.6cos(π × 5.5/4)	✓ 1
A1 1.07 m	✓ 1
Part (c)
M1 solving 1.3 + 0.6cos(π(t − 1)/4) = 1.0, that is cos(π(t − 1)/4) = −0.5	✓ 1
A1 t = 3.67 s	✓ 1
Part (d)
M1 second solution in the same cycle, t = 1 + 16/3 = 6.33	✓ 1
M1 time below 1.0 m is 6.33 − 3.67 = 2.67 s, so time above is 8 − 2.67	✓ 1
A1 5.33 s	✓ 1
Part (e)
A1 greatest value 2.1 m (translation of 0.2 upward)	✓ 1
A1 first at t = 3 s (translation of 2 to the right)	✓ 1

Exam tip: For an equation of the form cos(x) = −0.5 within one cycle there are two solutions symmetric about the maximum or minimum; use symmetry to find the second.

+ Add to test✓ In your test387IBMI3S-2-01r35-08Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 2Hard13 marks

Calculator

A beekeeper fills a feeder with sugar syrup. The mass of syrup remaining, M kg, is modelled by M = M₀ e^−kt, where t is the number of days after the feeder was filled and M₀ and k are positive constants.
The feeder holds 3.20 kg after 2 days and 1.70 kg after 5 days.
In parts (b) to (e), use the unrounded values from your calculator.

(a)

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

(3)

(b)

Find the value of M₀. Give your answer to 3 significant figures.

(2)

(c)

Find the percentage of syrup, relative to the amount at the start of a day, that the feeder loses each day. Give your answer to 3 significant figures.

(2)

(d)

Find the time taken for the mass of syrup to halve. Give your answer to 2 significant figures.

(2)

(e)

Find the time at which 0.50 kg of syrup remains. Give your answer to 3 significant figures.

(3)

(f)

Explain why the model cannot predict the moment the feeder becomes completely empty.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 forms two equations M₀e^−2k = 3.2 and M₀e^−5k = 1.7	✓ 1
M1 divides to get e^3k = 3.2/1.7 (or e^−3k = 1.7/3.2)	✓ 1
A1 k = 0.211 (0.2108…)	✓ 1
Part (b)
M1 substitutes into M₀ = 3.2e^2k	✓ 1
A1 M₀ = 4.88 kg	✓ 1
Part (c)
M1 1 − e^−k	✓ 1
A1 19.0%	✓ 1
Part (d)
M1 e^−kt = 0.5, so t = ln 2 ÷ k	✓ 1
A1 3.3 days	✓ 1
Part (e)
M1 forms M₀e^−kt = 0.5	✓ 1
M1 takes natural logarithms: t = ln(M₀ ÷ 0.5) ÷ k	✓ 1
A1 t = 10.8 days (10.80…)	✓ 1
Part (f)
R1 M = M₀e^−kt is always positive (approaches 0 but never reaches it)	✓ 1

Exam tip: Keep unrounded values in your calculator memory; rounding k early can change the third significant figure of later answers.

+ Add to test✓ In your test388IBMI3S-2-01r35-12Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 2Hard13 marks

Calculator

[Set your calculator to radians.] The number of bees leaving a hive each minute is modelled by B(t) = a sin(b(t − c)) + d, where t is the time in hours after midnight and a, b > 0. The greatest value of B is 78 and the least value is 46. The pattern repeats every 24 hours.

(a)

Find the values of a, d and b. Give b as a multiple of π.

(3)

(b)

The greatest value occurs at 15:00 (t = 15). Find a possible value of c with 0 ≤ c < 24.

(2)

(c)

Use the model with c = 9 to find the two times in one day (0 ≤ t ≤ 24) at which B = 70.

(4)

(d)

Hence find the number of hours in one day for which B is at least 70.

(2)

(e)

Find the value of B at midnight (t = 0). Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 a = (78 − 46)/2 = 16	✓ 1
A1 d = (78 + 46)/2 = 62	✓ 1
A1 b = 2π/24 = π/12	✓ 1
Part (b)
M1 sets π/12 (15 − c) = π/2, so 15 − c = 6	✓ 1
A1 c = 9	✓ 1
Part (c)
M1 forms 16 sin(π/12 (t − 9)) + 62 = 70	✓ 1
A1 sin(π/12 (t − 9)) = 0.5	✓ 1
M1 uses π/12 (t − 9) = π/6 or 5π/6 (or the GDC)	✓ 1
A1 t = 11 and t = 19	✓ 1
Part (d)
M1 19 − 11 (ECF from (c))	✓ 1
A1 8 hours	✓ 1
Part (e)
M1 substitutes t = 0: 16 sin(−3π/4) + 62	✓ 1
A1 50.7 (50.686…)	✓ 1

Common error: Sine equations have two solutions in each cycle; finding only the first misses the end of the interval.

+ Add to test✓ In your test389IBMI3S-2-01r35-14Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Exponential and sinusoidal models Paper 2Hard13 marks

Calculator

A hospital warms a unit of donated blood in a room at constant temperature. The temperature of the blood, T °C, after t minutes is modelled by T = 22 − Ae^−kt, where A and k are positive constants. The blood is at 4.0 °C when t = 0 and at 13.9 °C when t = 10.
In parts (c) to (e), use k = 0.08.

(a)

Find the value of A.

(2)

(b)

Find the value of k. Give your answer to 2 significant figures.

(3)

(c)

State the value that T approaches as t becomes large and say what it represents.

(2)

(d)

Find the time taken for the blood to reach 10 °C. Give your answer to 3 significant figures.

(3)

(e)

The blood can be used when its temperature is between 10 °C and 18 °C. Find the length of time, in minutes, for which it is in this range. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 0: 22 − A = 4	✓ 1
A1 A = 18	✓ 1
Part (b)
M1 forms 13.9 = 22 − 18e^−10k	✓ 1
A1 e^−10k = 0.45	✓ 1
A1 k = 0.080 (0.0798…)	✓ 1
Part (c)
A1 22 °C	✓ 1
A1 the room temperature	✓ 1
Part (d)
M1 forms 22 − 18e^−0.08t = 10, so e^−0.08t = 2/3	✓ 1
M1 takes logarithms: t = ln 1.5 ÷ 0.08	✓ 1
A1 t = 5.07 minutes	✓ 1
Part (e)
M1 forms 22 − 18e^−0.08t = 18 and solves: t = ln 4.5 ÷ 0.08 = 18.8…	✓ 1
A1 uses the difference between the two times: 18.80… − 5.07…	✓ 1
A1 13.7 minutes	✓ 1

Common error: Subtracting the constant before taking logarithms is essential: 22 − 18e⁻⁰.⁰⁸t = 10 must be rearranged to e⁻⁰.⁰⁸t = 12/18 first.

+ Add to test✓ In your test390IBMI3S-2-02r35-13Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard13 marks

Calculator

A library records the number of new members, M, joining in month t after a promotion starts, for 0 ≤ t ≤ 16. A quadratic model is M = a(t − 8)² + k. The greatest number of members occurs when t = 8, there were 500 new members when t = 0, and 548 when t = 4.

(a)

Find the values of a and k.

(4)

(b)

Write M in the form M = −t² + bt + c.

(2)

(c)

Find the values of t for which M = 550. Give your answers to 3 significant figures.

(3)

(d)

Find the values of t for which M > 500.

(2)

(e)

Find the length of time, in months, for which M ≥ 550. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 0: 64a + k = 500	✓ 1
M1 substitutes t = 4: 16a + k = 548	✓ 1
A1 a = −1	✓ 1
A1 k = 564	✓ 1
Part (b)
M1 expands −(t − 8)² + 564 (ECF from (a))	✓ 1
A1 M = −t² + 16t + 500	✓ 1
Part (c)
M1 forms −(t − 8)² + 564 = 550, so (t − 8)² = 14	✓ 1
A1 t = 8 − √14 = 4.26	✓ 1
A1 t = 8 + √14 = 11.7	✓ 1
Part (d)
M1 solves −t² + 16t > 0 (equivalently t(16 − t) > 0)	✓ 1
A1 0 < t < 16	✓ 1
Part (e)
M1 difference between the two solutions in (c): (8 + √14) − (8 − √14)	✓ 1
A1 2√14 = 7.48 months	✓ 1

Exam tip: Vertex form gives the greatest value directly; two pieces of information about the curve give two equations for two unknowns.

+ Add to test✓ In your test391IBMI3S-2-02r35-14Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models Paper 2Hard15 marks

Calculator

The installed capacity, C MW, of a wind-power company is modelled by C = A × b^t, where t is the number of years after 1 January 2020 and A, b are positive constants. The capacity was 72 MW when t = 2 and 124.4 MW when t = 5.
In parts (b) to (f), use the exact values b = 1.2 and A = 50 where needed.

(a)

Find the value of b. Give your answer to 2 significant figures.

(3)

(b)

Show that A = 50.

(2)

(c)

Interpret the value of b in this context.

(1)

(d)

Find the time for the capacity to reach 400 MW. Give your answer to 3 significant figures.

(3)

(e)

Find the capacity predicted by the exponential model when t = 10. Give your answer to 3 significant figures.

(2)

(f)

A linear model C = mt + n is fitted through the same two data points (2, 72) and (5, 124.4). Find the capacity it predicts when t = 10. Give your answer to 3 significant figures.

(3)

(g)

The actual capacity when t = 10 was 295 MW. State which model is more suitable and give a reason.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 divides: b³ = 124.4 ÷ 72	✓ 1
A1 b³ = 1.7278…	✓ 1
A1 b = 1.2	✓ 1
Part (b)
M1 substitutes t = 2: A × 1.2² = 72	✓ 1
A1 A = 72 ÷ 1.44 = 50	✓ 1
Part (c)
A1 capacity increases by 20% each year	✓ 1
Part (d)
M1 forms 50 × 1.2^t = 400, so 1.2^t = 8	✓ 1
M1 uses logarithms: t = ln 8 ÷ ln 1.2	✓ 1
A1 t = 11.4 years (11.405…)	✓ 1
Part (e)
M1 substitutes t = 10: 50 × 1.2¹⁰	✓ 1
A1 310 MW (309.6…)	✓ 1
Part (f)
M1 gradient = (124.4 − 72) ÷ (5 − 2) = 17.46…	✓ 1
M1 equation C = 72 + 17.46…(t − 2) evaluated at t = 10	✓ 1
A1 212 MW (211.7…)	✓ 1
Part (g)
R1 the exponential model, as its prediction (310 MW) is much closer to 295 MW than the linear prediction (212 MW)	✓ 1

Exam tip: When you are told to use given values in later parts, use them exactly rather than your own rounded answers.

+ Add to test✓ In your test392IBMI3S-2-03r36-09Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard14 marks

A study of wheat yield uses the model Y = ax² + bx + c, where Y is the yield in tonnes per hectare and x is the mass of fertiliser applied in kg per hectare, for 0 ≤ x ≤ 120. The measured yields are
• 3.0 tonnes per hectare with no fertiliser
• 5.8 tonnes per hectare with 40 kg per hectare
• 6.6 tonnes per hectare with 80 kg per hectare.

(a)

Find the values of a, b and c.

(5)

(b)

Find the amount of fertiliser that gives the greatest yield, and state this yield.

(3)

(c)

The grain sells for €210 per tonne and the fertiliser costs €1.20 per kg. Show that the profit per hectare, from the grain and fertiliser only, is P = 630 + 18.75x − 0.13125x² euros.

(2)

(d)

Find the maximum profit per hectare, giving your answer to the nearest cent, and state the amount of fertiliser needed to achieve it (to 3 significant figures).

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 c = 3.0	✓ 1
M1 form two equations, e.g. 1600a + 40b = 2.8 and 6400a + 80b = 3.6	✓ 1
M1 solve the system (GDC or elimination)	✓ 1
A1 a = −0.000625	✓ 1
A1 b = 0.095	✓ 1
Part (b)
M1 use x = −b ÷ 2a or a GDC maximum	✓ 1
A1 x = 76 kg per hectare	✓ 1
A1 Y = 6.61 tonnes per hectare	✓ 1
Part (c)
M1 P = 210Y − 1.2x	✓ 1
A1 expand 210(3 + 0.095x − 0.000625x²) − 1.2x = 630 + 18.75x − 0.13125x² (AG)	✓ 1
Part (d)
M1 use x = −18.75 ÷ (2 × −0.13125) or a GDC maximum	✓ 1
A1 x = 71.4 kg per hectare	✓ 1
M1 substitute the unrounded x into P	✓ 1
A1 maximum profit €1299.64	✓ 1

Exam tip: The fertiliser amount that maximises profit is lower than the amount that maximises yield, because extra fertiliser has a cost; say so when comparing the two.

+ Add to test✓ In your test393IBMI3S-2-03r36-12Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard12 marks

A farmer counts aphids on a sample of crop plants. The number N of aphids is recorded t days after counting begins.
t (days): 0, 2, 4, 6, 8
N (aphids): 40, 61, 89, 133, 198

(a)

Find the linear regression equation of N on t, and the value of r to 3 significant figures.

(3)

(b)

Use the linear model to find the predicted number of aphids at t = 0 and compare this with the data. Hence give one reason why a linear model is not suitable.

(2)

(c)

The farmer proposes the model N = ab^t. Use exponential regression to find a and b correct to 3 significant figures.

(3)

(d)

Interpret the value of b in context.

(1)

(e)

Use the exponential model to predict the number of aphids at t = 10, correct to 3 significant figures. Use the unrounded regression values.

(2)

(f)

Give one reason why the exponential model may not be valid for large values of t.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use regression on a GDC	✓ 1
A1 N = 19.4t + 26.6	✓ 1
A1 r = 0.975	✓ 1
Part (b)
A1 predicted value 26.6 (about 27), compared with 40 recorded	✓ 1
R1 the linear model under-estimates at the start (and the end), so the pattern is curved; the counts increase faster each day	✓ 1
Part (c)
M1 use exponential regression on a GDC	✓ 1
A1 a = 40.4	✓ 1
A1 b = 1.22	✓ 1
Part (d)
A1 the number of aphids increases by about 22% each day (multiplied by 1.22 per day)	✓ 1
Part (e)
M1 substitute t = 10	✓ 1
A1 295	✓ 1
Part (f)
R1 the model has no upper limit, but food supply, space or pest control will restrict real growth	✓ 1

Common error: Choosing a model from the value of r alone is risky; check how well each model fits the whole pattern of the data.

+ Add to test✓ In your test394IBMI3S-2-03r36-14Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard12 marks

Grain is stored in a silo and cools after harvest. Its temperature T °C, t hours after it enters the silo, is modelled by T = 12 + Ae^−kt, where A and k are positive constants. When the grain enters the silo its temperature is 40 °C, and after 3 hours it is 30 °C.

(a)

Find the value of A.

(2)

(b)

Find the value of k, correct to 3 significant figures.

(3)

(c)

Find the temperature of the grain after 8 hours, correct to 3 significant figures. Use the unrounded value of k.

(2)

(d)

As t becomes very large, state the value that T approaches, and give one interpretation of this value.

(2)

(e)

Find the time taken for the grain to cool to 15 °C. Give your answer in hours correct to 3 significant figures. Use the unrounded value of k.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 0 into the model	✓ 1
A1 A = 28	✓ 1
Part (b)
M1 28e^−3k = 18	✓ 1
M1 take natural logarithms or use a GDC solver	✓ 1
A1 k = 0.147	✓ 1
Part (c)
M1 substitute t = 8	✓ 1
A1 20.6 °C	✓ 1
Part (d)
A1 T approaches 12 °C	✓ 1
R1 this is the temperature of the surroundings (room temperature in the silo)	✓ 1
Part (e)
M1 28e^−kt = 3	✓ 1
M1 t = ln(28 ÷ 3) ÷ k	✓ 1
A1 t = 15.2 hours	✓ 1

Common error: The model approaches 12 °C but never reaches it; to find when the grain hits a target temperature, solve the equation rather than assuming the limit.

+ Add to test✓ In your test395IBMI3S-2-02r37-12Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Hard11 marks

Calculator

After a robot's motor is switched off, its temperature, T °C, t minutes later is modelled by T = 22 + Be^−kt, where B and k are positive constants. The temperature is 80 °C at t = 0 and 60 °C at t = 5.

(a)

Find the value of B.

(2)

(b)

Find the value of k, to three significant figures.

(3)

(c)

Find the temperature of the motor at t = 15. Use the unrounded value of k and give your answer to three significant figures.

(2)

(d)

According to the model, state the temperature the motor approaches as t becomes very large, and suggest what this temperature represents.

(2)

(e)

Find the time at which the temperature of the motor is 30 °C. Use the unrounded value of k and give your answer to three significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 22 + B = 80	✓ 1
A1 B = 58	✓ 1
Part (b)
M1 22 + 58e^−5k = 60	✓ 1
M1 e^−5k = 38/58 and take logarithms	✓ 1
A1 k = 0.0846	✓ 1
Part (c)
M1 22 + 58e^−15k (or 22 + 58 × (38/58)³)	✓ 1
A1 38.3 °C	✓ 1
Part (d)
A1 22 °C	✓ 1
R1 the surrounding (room) temperature	✓ 1
Part (e)
M1 22 + 58e^−kt = 30, so e^−kt = 8/58	✓ 1
A1 23.4 minutes	✓ 1

Common error: Subtract the horizontal asymptote (22) before taking logarithms in a model of the form a + be⁻kt.

+ Add to test✓ In your test396IBMI3S-2-02r37-13Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Hard13 marks

Calculator

A fashion brand records its monthly online sales, S thousand units, in month t after launching a new range.

t	1	2	3	4	5	6
S	4.1	5.0	6.3	7.6	9.4	11.5

Two models are proposed: the linear model L(t) = 1.4t + 1.8 and the exponential model E(t) = 3.4 × 1.22^t. A residual is (observed value) − (model value).

(a)

Calculate the sum of the squares of the residuals for the linear model.

(3)

(b)

Calculate the sum of the squares of the residuals for the exponential model. Give your answer to three significant figures.

(3)

(c)

State which model fits the data better, giving a reason.

(1)

(d)

Using the exponential model, find the value of t for which the predicted sales are 20 thousand units, to three significant figures. Hence state the first month in which the model predicts sales above 20 thousand units.

(3)

(e)

The brand can produce at most 30 thousand units each month. Find the sales predicted by each model in month 12, to three significant figures, and comment on which prediction is more realistic.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 residuals 0.9, 0.4, 0.3, 0.2, 0.6, 1.3	✓ 1
M1 squares added	✓ 1
A1 3.15	✓ 1
Part (b)
M1 residuals −0.048, −0.061, 0.126, 0.068, 0.211, 0.289	✓ 1
M1 squares added	✓ 1
A1 0.155	✓ 1
Part (c)
R1 the exponential model, as its sum of squared residuals is much smaller	✓ 1
Part (d)
M1 3.4 × 1.22^t = 20	✓ 1
A1 t = 8.91	✓ 1
A1 month 9	✓ 1
Part (e)
A1 L(12) = 18.6	✓ 1
A1 E(12) = 37.0	✓ 1
R1 E predicts sales above the capacity of 30 thousand, so it is unrealistic that far ahead (growth cannot continue at a fixed rate); the linear prediction is within capacity	✓ 1

Exam tip: A small sum of squared residuals only supports a model inside the range of the data; test predictions against real limits before trusting them further out.

+ Add to test✓ In your test397IBMI3S-2-02r37-14Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models Paper 2Hard12 marks

Calculator

A fishery records its monthly catch, C tonnes, in month t of the year (t = 1 is January).

t	1	3	6	9	12
C	14	21	27	19	13

Two models are proposed: the sinusoidal model S(t) = 20 + 8 sin(30(t − 3)), with the angle in degrees, and the quadratic model Q(t) = −0.6t² + 7.6t + 8. A residual is (observed value) − (model value).

(a)

Write down the maximum value, the minimum value and the period of the sinusoidal model.

(2)

(b)

Find the sum of the squares of the residuals for the sinusoidal model. Give your answer to three significant figures.

(3)

(c)

Find the sum of the squares of the residuals for the quadratic model. Give your answer to three significant figures.

(3)

(d)

State which model fits the data better, giving a reason.

(1)

(e)

Both models are used to predict the catch in month 18 (June of the following year). Find each prediction, and explain why only one of them is sensible.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 28 and minimum 12	✓ 1
A1 period 360 ÷ 30 = 12 months	✓ 1
Part (b)
M1 values 13.07, 20, 28, 20, 12 and residuals 0.93, 1, −1, −1, 1	✓ 1
M1 squares added	✓ 1
A1 4.86	✓ 1
Part (c)
M1 values 15, 25.4, 32, 27.8, 12.8 and residuals −1, −4.4, −5, −8.8, 0.2	✓ 1
M1 squares added	✓ 1
A1 123	✓ 1
Part (d)
R1 the sinusoidal model, since its sum of squared residuals is much smaller	✓ 1
Part (e)
A1 Q(18) = −49.6 tonnes	✓ 1
R1 a negative catch is impossible, so the quadratic model fails here	✓ 1
A1 S(18) = 28 tonnes, which is sensible as the catch repeats seasonally each year	✓ 1

Exam tip: Seasonal data repeat, so a periodic model is usually the better choice outside the observed months.

+ Add to test✓ In your test398IBMI3S-2-01r38-09Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Hard14 marks

Calculator

The annual number of passengers, P million, using a regional airport is recorded in each of seven years. Let x be the number of years after 2015.
  x: 0, 1, 2, 3, 4, 5, 6
  P: 2.1, 2.6, 3.1, 3.9, 4.7, 5.8, 7.1
Two models are proposed:
  Model A: P = 0.82x + 1.72
  Model B: P = 2.10 × 1.22^x

(a)

For x = 3, calculate the value of P predicted by each model. Give your answers to 2 decimal places.

(2)

(b)

For each model, calculate the sum of the squares of the differences between the predicted and the observed values of P for the seven years. Give your answers to 3 significant figures.

(3)

(c)

Hence state which model fits the data better. Give one other observation about the data that supports your choice.

(2)

(d)

Use the better model to predict the number of passengers in 2024. Give your answer to 3 significant figures.

(2)

(e)

The terminal can handle at most 11 million passengers per year. According to Model B, find the value of x at which P reaches 11. Give your answer to 3 significant figures.

(3)

(f)

Hence state the first year in which Model B predicts that the terminal limit is exceeded, and give one reason why this prediction may be unreliable.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model A: 4.18	✓ 1
A1 Model B: 3.81	✓ 1
Part (b)
M1 attempt to sum (predicted − observed)² over all seven points	✓ 1
A1 Model A: 0.596	✓ 1
A1 Model B: 0.0582	✓ 1
Part (c)
R1 Model B, as its sum of squared differences is smaller	✓ 1
R1 the yearly increases in P are not constant (0.5, 0.5, 0.8, 0.8, 1.1, 1.3), they get larger, so growth is faster than linear	✓ 1
Part (d)
M1 substitute x = 9 into Model B (ECF from (c))	✓ 1
A1 12.6 million	✓ 1
Part (e)
M1 2.10 × 1.22^x = 11	✓ 1
M1 correct use of logarithms or GDC solve	✓ 1
A1 x = 8.33	✓ 1
Part (f)
A1 2024 (the first whole year after x = 8.33)	✓ 1
R1 it is an extrapolation beyond the data range, and growth may not stay exponential (accept capacity or demand limits)	✓ 1

Exam tip: Compare models using a measure of fit, such as the sum of squared differences, as well as the shape of the data, and always comment on extrapolation.

+ Add to test✓ In your test399IBMI3S-2-01r38-14Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Hard15 marks

Calculator

A promoter sells tickets for a concert. At a ticket price of €40, 800 tickets are sold. At a price of €56, 400 tickets are sold. The number of tickets sold, N, is modelled as a linear function of the ticket price, €p.

(a)

Find an expression for N in terms of p.

(3)

(b)

Hence show that the ticket revenue, €R, is given by R = 1800p − 25p².

(2)

(c)

The venue has 1200 seats. Find the lowest ticket price that can be set.

(2)

(d)

The promoter pays €6000 to hire the venue plus €8 for each ticket sold. Show that the profit, €P, is P = −25p² + 2000p − 20 400.

(2)

(e)

Find the ticket price that gives the greatest profit, and the value of this greatest profit.

(3)

(f)

The promoter wants a profit of at least €15 000. Find the range of ticket prices that achieves this, giving your answers to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 gradient = (400 − 800) ÷ (56 − 40)	✓ 1
A1 gradient = −25	✓ 1
A1 N = 1800 − 25p	✓ 1
Part (b)
M1 R = p × N	✓ 1
A1 R = 1800p − 25p² (AG)	✓ 1
Part (c)
M1 1800 − 25p ≤ 1200	✓ 1
A1 p ≥ €24	✓ 1
Part (d)
M1 P = R − (6000 + 8N) or (p − 8)N − 6000	✓ 1
A1 expands correctly to −25p² + 2000p − 20 400 (AG)	✓ 1
Part (e)
M1 vertex p = −2000 ÷ (2 × (−25)) or GDC maximum	✓ 1
A1 p = €40	✓ 1
A1 €19 600	✓ 1
Part (f)
M1 −25p² + 2000p − 20 400 = 15 000	✓ 1
A1 lower boundary 26.4	✓ 1
A1 upper boundary 53.6, so 26.4 ≤ p ≤ 53.6 (both values exceed 24)	✓ 1

Exam tip: A 'show that' for a profit function should show revenue and cost as separate expressions before expanding.

+ Add to test✓ In your test400IBMI3S-2-03r39-07Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Hard14 marks

The volume of water, V million m³, stored in a reservoir is measured at the start of each month for eight months. Time t is the number of months since the start of the first measurement period.
t (months): 1, 2, 3, 4, 5, 6, 7, 8
V (million m³): 14.2, 17.1, 19.3, 20.9, 21.8, 22.2, 21.9, 21.0
A linear model fitted to the data has coefficient of determination R² = 0.704. A quadratic model V = at² + bt + c is also fitted to the data.

(a)

Use your graphic display calculator to find the values of a, b and c, each correct to 3 significant figures.

(3)

(b)

The quadratic model has R² = 0.9999. Explain, with reference to the data and to both R² values, why the quadratic model is more suitable than the linear model.

(2)

(c)

Use the quadratic model, with unrounded values of a, b and c, to find the maximum volume stored and the time at which it occurs. Give each answer to 3 significant figures.

(3)

(d)

Use the quadratic model to predict the volume stored at t = 10. Give your answer to 3 significant figures.

(2)

(e)

The authorities plan to restrict water use once the volume falls below 12 million m³. According to the quadratic model, find the time at which this happens. Give your answer to 3 significant figures.

(3)

(f)

State one reason why the quadratic model should not be used to predict the volume for large values of t.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 use of quadratic regression	✓ 1
A1 a = -0.313, b = 3.78	✓ 1
A1 c = 10.8	✓ 1
Part (b)
A1 the quadratic R² is much closer to 1 than 0.704, so it fits the data much better	✓ 1
R1 the data rise and then begin to fall, which a straight line cannot represent	✓ 1
Part (c)
M1 use of vertex t = −b/2a or GDC maximum	✓ 1
A1 t = 6.04 months	✓ 1
A1 V = 22.2 million m³	✓ 1
Part (d)
M1 substitute t = 10 into the model (unrounded values)	✓ 1
A1 17.3 million m³	✓ 1
Part (e)
M1 at² + bt + c = 12	✓ 1
M1 use of GDC or quadratic formula to solve	✓ 1
A1 t = 11.7 months (reject the root t = 0.339 as it is before the volume reaches its maximum)	✓ 1
Part (f)
R1 the model eventually predicts a negative volume (the volume would be negative for large t), which is impossible	✓ 1

Exam tip: A high R² shows a good fit within the data range only; always comment on how sensible a prediction is outside it.

+ Add to test✓ In your test401IBMI3S-2-03r39-10Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Hard13 marks

After a patient takes a tablet, the concentration of a drug in the blood is modelled by C(t) = 40te^−0.5t, where C is in mg/L and t is the time in hours after the tablet is taken, t ≥ 0.

(a)

Find the concentration 1 hour after the tablet is taken. Give your answer to 3 significant figures.

(2)

(b)

Use your graphic display calculator to find the maximum concentration and the time at which it occurs. Give each answer to 3 significant figures.

(3)

(c)

The drug is effective while the concentration is above 20 mg/L. Find the two times at which the concentration is exactly 20 mg/L. Give each answer to 3 significant figures.

(3)

(d)

Hence find the length of time, in hours, for which the drug is effective. Use unrounded values from (c) and give your answer to 3 significant figures.

(2)

(e)

Describe what happens to the concentration as t becomes very large.

(1)

(f)

A second tablet is taken 10 hours after the first. Find the concentration of the drug that remains from the first tablet at that moment. Give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitute t = 1: 40e^−0.5	✓ 1
A1 24.3 mg/L	✓ 1
Part (b)
M1 use of GDC to find the maximum of C(t)	✓ 1
A1 t = 2.00 hours	✓ 1
A1 C = 29.4 mg/L	✓ 1
Part (c)
M1 40te^−0.5t = 20 (or intersection of the graph with y = 20)	✓ 1
A1 t = 0.71 hours	✓ 1
A1 t = 4.31 hours	✓ 1
Part (d)
M1 subtract the two times from (c)	✓ 1
A1 3.59 hours (ECF from (c))	✓ 1
Part (e)
A1 the concentration approaches zero (the graph has a horizontal asymptote at C = 0)	✓ 1
Part (f)
M1 substitute t = 10: 40 × 10 × e⁻⁵	✓ 1
A1 2.70 mg/L	✓ 1

Common error: A model that equals a target value twice has two solutions; give both and say which is the start and which is the end of the interval.

+ Add to test✓ In your test402IBMI3S-2-03r39-13Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Hard14 marks

An online shop sells a game accessory. When the price is €p, the number of accessories sold each week is modelled by n = 640 − 8p, for 0 ≤ p ≤ 80. Each accessory costs the shop €12 to buy.

(a)

Show that the weekly revenue, €R, is given by R = 640p − 8p².

(1)

(b)

Find the price that gives the greatest weekly revenue, and the value of that revenue.

(3)

(c)

Show that the weekly profit, €P, can be written as P = −8p² + 736p − 7680, and find the price that gives the greatest profit, and that profit.

(4)

(d)

Find the two prices at which the shop makes zero profit.

(2)

(e)

The shop wants a weekly profit of at least €8000. Find the range of prices that achieves this. Give your prices to 3 significant figures.

(3)

(f)

Explain why the model for n is not valid when p is greater than 80.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 R = p × n = p(640 − 8p) = 640p − 8p²	✓ 1
Part (b)
M1 vertex p = −640/(2 × −8) or GDC maximum	✓ 1
A1 p = €40	✓ 1
A1 R = €12 800	✓ 1
Part (c)
M1 profit = revenue − cost = (p − 12)(640 − 8p)	✓ 1
A1 expansion to −8p² + 736p − 7680	✓ 1
A1 p = €46	✓ 1
A1 P = €9248	✓ 1
Part (d)
M1 (p − 12)(640 − 8p) = 0 or solving −8p² + 736p − 7680 = 0	✓ 1
A1 p = €12 and p = €80	✓ 1
Part (e)
M1 −8p² + 736p − 7680 = 8000 (or p² − 92p + 1960 = 0)	✓ 1
A1 p = 33.5 and p = 58.5	✓ 1
A1 33.5 ≤ p ≤ 58.5	✓ 1
Part (f)
R1 the number sold would be negative, which is impossible	✓ 1

Common error: Profit is revenue minus total cost, so the cost per item must be multiplied by the number sold, not by the price.

+ Add to test✓ In your test403IBMI3S-2-02r40-12Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Hard14 marks

The electricity demand $D$ kW of a store at $t$ hours after midnight is modelled by $D = a\cos(b(t - c)) + d$ with $a > 0$ and $b > 0$. The demand has a maximum of 84 kW at 14:00 and a minimum of 36 kW at 02:00, and the pattern repeats every 24 hours.

(a)

Find the values of $a$ and $d$.

(2)

(b)

Find the value of $b$, in radians per hour, correct to 3 significant figures, and write down the value of $c$.

(3)

(c)

Find the demand at 09:00, correct to 3 significant figures.

(2)

(d)

Find the times between 06:00 and 22:00 when the demand is 75 kW, correct to 3 significant figures.

(4)

(e)

Hence find the number of hours in each day for which the demand exceeds 75 kW, correct to 3 significant figures (use unrounded values).

(1)

(f)

The same model can be written $D = 24\sin(b(t - k)) + 60$ with $b = \frac{\pi}{12}$. Find the smallest positive value of $k$.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 $a = \frac{84 - 36}{2} = 24$	✓ 1
A1 $d = \frac{84 + 36}{2} = 60$	✓ 1
Part (b)
M1 $b = \frac{2\pi}{24}$	✓ 1
A1 $b = 0.262$ (exact value $\frac{\pi}{12}$)	✓ 1
A1 $c = 14$	✓ 1
Part (c)
M1 $60 + 24\cos\left(\frac{\pi}{12}(9 - 14)\right)$	✓ 1
A1 66.2 kW	✓ 1
Part (d)
M1 $24\cos\left(\frac{\pi}{12}(t - 14)\right) + 60 = 75$	✓ 1
M1 $\cos\left(\frac{\pi}{12}(t - 14)\right) = 0.625$ and inverse cosine	✓ 1
A1 $t = 10.6$ (about 10:35)	✓ 1
A1 $t = 17.4$ (about 17:25)	✓ 1
Part (e)
A1 6.84 hours	✓ 1
Part (f)
M1 $\cos\theta = \sin\left(\theta + \frac{\pi}{2}\right)$, so a quarter period (6 hours) must be subtracted from 14	✓ 1
A1 $k = 8$	✓ 1

Exam tip: A cosine graph is a sine graph shifted by a quarter of a period, so $c$ values differ by 6 hours here.

+ Add to test✓ In your test404IBMI3S-2-02r40-13Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Hard14 marks

The number of rooftop solar installations completed in a region in year $t$ (where $t$ is the number of years since 2018) is modelled by $N = N_0 b^t$. Data show that $N = 1450$ when $t = 2$ and $N = 3100$ when $t = 5$.

(a)

Find the value of $b$, correct to 3 significant figures.

(3)

(b)

Find the value of $N_0$, correct to 3 significant figures. Use the unrounded value of $b$.

(2)

(c)

Use the model to predict the number of installations when $t = 8$, correct to 3 significant figures. Use unrounded values.

(2)

(d)

A linear model $N = mt + k$ is fitted through the same two data points. Find $m$ and $k$.

(2)

(e)

In year $t = 8$, the actual number of installations was 6400. Calculate the percentage error, using the unrounded predictions, for (i) the exponential model (ii) the linear model, each correct to 3 significant figures, and state which model is more accurate.

(4)

(f)

Give one reason why the exponential model may be less reliable for predictions far into the future.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $N_0 b^2 = 1450$ and $N_0 b^5 = 3100$	✓ 1
M1 divide: $b^3 = \frac{3100}{1450}$	✓ 1
A1 $b = 1.29$	✓ 1
Part (b)
M1 $N_0 = \frac{1450}{b^2}$	✓ 1
A1 $N_0 = 874$	✓ 1
Part (c)
M1 $N = 3100 \times b^3$ (or $874 \times b^8$)	✓ 1
A1 6630	✓ 1
Part (d)
M1 gradient $= \frac{3100 - 1450}{5 - 2} = 550$	✓ 1
A1 $N = 550t + 350$	✓ 1
Part (e)
M1 linear prediction $550 \times 8 + 350 = 4750$ and use of $\frac{\|\text{prediction} - \text{actual}\|}{\text{actual}} \times 100$	✓ 1
A1 exponential model: 3.56%	✓ 1
A1 linear model: 25.8%	✓ 1
R1 the exponential model is more accurate	✓ 1
Part (f)
R1 installations cannot keep growing by about 29% each year indefinitely, since the number of suitable roofs is limited	✓ 1

Exam tip: Percentage error compares the difference with the actual value, not with the predicted value.

+ Add to test✓ In your test405IBMI3S-2-02r40-14Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models Paper 2Hard15 marks

The number of new daily cases of an infection, $C$, on day $t$ of an outbreak is modelled by the quadratic function $C = at^2 + bt + c$. The reported values are $C = 40$ when $t = 0$, $C = 215$ when $t = 5$ and $C = 240$ when $t = 10$.

(a)

Write down the value of $c$ and form two equations in $a$ and $b$.

(2)

(b)

Solve the equations to find $a$ and $b$.

(3)

(c)

Find the day on which the model predicts the greatest number of new cases, and the number of cases on that day, each correct to 3 significant figures.

(3)

(d)

Find the period during which the model predicts more than 100 new cases per day, correct to 3 significant figures for each end and for the length of the period.

(4)

(e)

Find the day on which the model predicts that the number of new cases reaches zero, correct to 3 significant figures.

(2)

(f)

Explain why the model cannot be used for $t > 17.4$.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 $c = 40$	✓ 1
A1 $25a + 5b = 175$ and $100a + 10b = 200$ (equivalent forms accepted)	✓ 1
Part (b)
M1 attempt to solve the simultaneous equations (by hand or GDC)	✓ 1
A1 $a = -3$	✓ 1
A1 $b = 50$	✓ 1
Part (c)
M1 $t = -\frac{b}{2a} = \frac{50}{6}$	✓ 1
A1 $t = 8.33$	✓ 1
A1 $C = 248$	✓ 1
Part (d)
M1 $-3t^2 + 50t + 40 = 100$	✓ 1
A1 $t = 1.30$	✓ 1
A1 $t = 15.4$	✓ 1
A1 length $15.4 - 1.30 = 14.1$ days	✓ 1
Part (e)
M1 solve $-3t^2 + 50t + 40 = 0$ and take the positive root	✓ 1
A1 $t = 17.4$	✓ 1
Part (f)
R1 for $t > 17.4$ the model gives negative numbers of cases, which is impossible	✓ 1

Exam tip: A quadratic model is only valid on the domain where its outputs make sense in context.

+ Add to test✓ In your test406IBMI3S-2-01r43-04Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; 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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard14 marks

A baker models the height h cm of dough in a tin t minutes after it is placed in a warm proving cabinet by
h(t) = 4 + 0.30t + 0.012t² − 0.0004t³, 0 ≤ t ≤ 40.

(a)

State the height of the dough at t = 0.

(1)

(b)

Find the height of the dough after 20 minutes.

(2)

(c)

Using your graphic display calculator, find the maximum height reached by the dough during the 40 minutes, and the time at which it occurs. Give both answers to 3 significant figures.

(3)

(d)

The tin is 12.5 cm tall. The dough spills over the top whenever h(t) > 12.5. Find the length of time, in minutes, for which the model predicts the dough is above the rim. Give your answer to 3 significant figures.

(3)

(e)

Find h(60) and use it to explain why the model should not be used for t = 60.

(2)

(f)

A measurement at t = 35 gives a height of 11.8 cm. Calculate the percentage error in the model's prediction at t = 35. Give your answer to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4 cm	✓ 1
Part (b)
M1 substituting t = 20	✓ 1
A1 11.6 cm	✓ 1
Part (c)
M1 graph or solver used to locate the maximum of h on the interval	✓ 1
A1 t = 28.7 minutes	✓ 1
A1 maximum height = 13.0 cm (13.04 before rounding)	✓ 1
Part (d)
M1 solving h(t) = 12.5	✓ 1
A1 t = 23.6 and t = 33.4 (23.5707 and 33.4120 before rounding)	✓ 1
A1 duration = 9.84 minutes (use unrounded roots)	✓ 1
Part (e)
A1 h(60) = −21.2	✓ 1
R1 a negative height is impossible, so the model is not valid outside the stated interval	✓ 1
Part (f)
A1 model value h(35) = 12.05	✓ 1
M1 percentage error = \|12.05 − 11.8\| / 11.8 × 100	✓ 1
A1 2.12%	✓ 1

Common error: When a cubic is only valid on a stated interval, solutions outside that interval must be rejected; an equation such as h = 8 can have roots that are not physically meaningful.

+ Add to test✓ In your test407IBMI3S-2-01r43-12Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; 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Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; 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choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard13 marks

The monthly data traffic D (in petabytes) carried by a regional network is recorded at the end of each year for six years, where t is the number of years since the first record.
t: 0, 1, 2, 3, 4, 5
D: 12, 15, 19, 24, 31, 39
Two models are proposed: Model L: D = 5.5t + 10 and Model E: D = 12 × 1.26^t.

(a)

Calculate the values predicted by each model at t = 3 and t = 5.

(2)

(b)

For each model, calculate the sum of the squares of the differences between the model's predictions and the six recorded values. Give each answer to 2 decimal places and use unrounded predictions.

(4)

(c)

State, with a reason, which model fits the data better.

(1)

(d)

Find the values predicted by each model at t = 10.

(2)

(e)

The network can carry at most 80 petabytes per month. Use Model E to find when this capacity is first reached. Give your answer to 3 significant figures.

(3)

(f)

Give one reason why Model E might not be reliable for predictions beyond t = 10.

(1)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 Model L: 26.5 and 37.5	✓ 1
A1 Model E: 24.0 and 38.1 (24.0045 and 38.1096 before rounding)	✓ 1
Part (b)
M1 differences (recorded − predicted) found for all six values, for at least one model	✓ 1
M1 differences squared and summed for a model	✓ 1
A1 Model L: 17.75	✓ 1
A1 Model E: 1.38 (1.3789… before rounding)	✓ 1
Part (c)
R1 Model E, because its sum of squared differences is much smaller (1.38 compared with 17.75)	✓ 1
Part (d)
A1 Model L: 65	✓ 1
A1 Model E: 121 (121.03… before rounding)	✓ 1
Part (e)
M1 12 × 1.26^t = 80	✓ 1
M1 correct method to solve (logarithms, or graph or solver)	✓ 1
A1 t = 8.21 years (8.2087… before rounding)	✓ 1
Part (f)
R1 for example: growth cannot continue exponentially forever, because of limited users, devices or capacity; or the model is extrapolated far beyond the data range	✓ 1

Exam tip: A smaller total of squared differences means a closer fit to the recorded data, but a good fit within the data range does not guarantee a good prediction far outside it.

+ Add to test✓ In your test408IBMI3S-2-01r43-14Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard13 marks

The number of hours of daylight D in an Alpine village on day d of the year (where d = 1 is 1 January) is modelled by
D = 12.2 + 3.4 sin(2π(d − 80)/365), 1 ≤ d ≤ 365.

(a)

Find the longest and shortest daylight, in hours, according to the model.

(2)

(b)

Find the day of the year, to the nearest whole number, on which the daylight is longest.

(2)

(c)

Find the daylight, in hours, on day 300. Give your answer to 3 significant figures.

(2)

(d)

On day 355 the measured daylight was 8.6 hours. Find the percentage error in the model's prediction for day 355, to 3 significant figures.

(3)

(e)

Find the number of days in the year on which the model predicts more than 14 hours of daylight. Give your answer to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 longest 15.6 hours (12.2 + 3.4)	✓ 1
A1 shortest 8.8 hours (12.2 − 3.4)	✓ 1
Part (b)
M1 2π(d − 80)/365 = π/2, or a maximum found from a graph	✓ 1
A1 d = 171 (171.25 before rounding)	✓ 1
Part (c)
M1 substituting d = 300 (angle in radians)	✓ 1
A1 10.2 hours (10.154… before rounding)	✓ 1
Part (d)
A1 model value 8.80 hours (8.8008… before rounding)	✓ 1
M1 \|8.8008 − 8.6\| / 8.6 × 100	✓ 1
A1 2.33%	✓ 1
Part (e)
M1 solving 12.2 + 3.4 sin(2π(d − 80)/365) = 14	✓ 1
A1 d = 112.4 and d = 230.1 (112.41 and 230.09 before rounding)	✓ 1
M1 subtracting the two solutions	✓ 1
A1 118 days (117.68… before rounding)	✓ 1

Exam tip: Set your calculator to radians for models such as this one, where the angle is written as a multiple of π; a quick check of the value at the first or last day of the year will show if the mode is wrong.

+ Add to test✓ In your test409IBMI3S-2-03r43-10Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Hard12 marks

The charge remaining in a probe's backup battery, B per cent of full charge, is measured at the end of selected years t after launch:
t (years): 0, 2, 4, 6, 8
B (per cent): 100.0, 87.4, 76.1, 66.2, 58.3
Two models are proposed:
Model L (linear): B = 100 − 5.3t
Model E (exponential): B = 100 × 0.934^t

(a)

Calculate the value of B predicted by Model L when t = 4, and hence the difference between this prediction and the measured value.

(2)

(b)

Calculate the value of B predicted by Model E when t = 4.

(1)

(c)

Find the sum of the squares of the differences between the measured values and the values predicted by Model L, for the five measurements. Give your answer to 3 significant figures.

(3)

(d)

For Model E the sum of the squares of the differences is 0.212. State, with a reason, which model fits the data better.

(1)

(e)

Use Model E to find when the charge first falls below 30%. Give your answer in years to 3 significant figures.

(3)

(f)

Model L is used to estimate the same time. Find the difference between the two estimates. Use unrounded values throughout and give your answer to 3 significant figures.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 B = 78.8	✓ 1
A1 difference = 78.8 − 76.1 = 2.7 (accept −2.7)	✓ 1
Part (b)
A1 B = 76.1	✓ 1
Part (c)
M1 finds the predicted values 100, 89.4, 78.8, 68.2, 57.6	✓ 1
M1 squares the differences (0, 2.0², 2.7², 2.0², 0.7²) and adds	✓ 1
A1 15.8	✓ 1
Part (d)
R1 Model E, because its sum of squared differences (0.212) is smaller than that of Model L (15.8)	✓ 1
Part (e)
M1 sets 100 × 0.934^t = 30	✓ 1
M1 attempts to solve (GDC or logarithms: t = ln 0.3 ÷ ln 0.934)	✓ 1
A1 t = 17.6 years	✓ 1
Part (f)
M1 Model L: 100 − 5.3t = 30 gives t = 13.2 (13.207…)	✓ 1
A1 difference = 4.43 years	✓ 1

Exam tip: When asked to compare models, use the numerical evidence given (here the sum of squared differences) and say which is smaller; do not simply say that one model looks better.

+ Add to test✓ In your test410IBMI3S-2-03r43-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Choosing and evaluating models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models Paper 2Hard13 marks

The temperature in a coffee-roasting room, T °C, t hours after midnight, is modelled by T = 22 + 5 sin(π(t − 9) ÷ 12) for 0 ≤ t ≤ 24, with the angle in radians.

(a)

Write down the maximum and minimum temperatures predicted by the model, and the times at which they occur.

(3)

(b)

State the period of the model.

(1)

(c)

Find the temperature at 11:30. Give your answer to 3 significant figures.

(2)

(d)

Find the first time after midnight at which the temperature reaches 24 °C. Give your answer in hours and minutes, to the nearest minute.

(3)

(e)

Roasting is not possible while the temperature is above 24 °C. Find the length of time each day for which the temperature is above 24 °C. Give your answer in hours to 3 significant figures.

(4)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 maximum 27 °C and minimum 17 °C	✓ 1
A1 maximum when t = 15 (15:00)	✓ 1
A1 minimum when t = 3 (03:00)	✓ 1
Part (b)
A1 24 hours	✓ 1
Part (c)
M1 substitutes t = 11.5	✓ 1
A1 T = 25.0 °C	✓ 1
Part (d)
M1 sets 22 + 5 sin(π(t − 9) ÷ 12) = 24 (sin = 0.4)	✓ 1
A1 t = 10.57… hours (GDC)	✓ 1
A1 10:34	✓ 1
Part (e)
M1 recognises a second solution of sin = 0.4 in the same cycle (e.g. angle π − 0.4115… or reading it from the GDC graph)	✓ 1
A1 second time t = 19.43… hours	✓ 1
M1 subtracts: 19.43… − 10.57…	✓ 1
A1 8.86 hours	✓ 1

Common error: Set the GDC to radians for this model, and remember that a sine equation has a second solution in each cycle: the temperature is above 24 °C between the two solutions, not after the first one.

+ Add to test✓ In your test411IBMI3S-2-03r45-13Quadratic and cubic models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Exponential and sinusoidal models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models Paper 2Hard13 marks

A sample of lava is exposed to air at 24 °C at time t = 0 hours, when its temperature is 1120 °C. Its temperature, T °C, is modelled by T = 24 + Ae^−kt, where A and k are constants and k > 0. After 2 hours the temperature is 640 °C.

(a)

Show that A = 1096.

(1)

(b)

Find the value of k, giving your answer to 3 significant figures.

(3)

(c)

Find the temperature of the lava after 5 hours, giving your answer to 3 significant figures. Use the unrounded value of k.

(2)

(d)

Find the time taken for the lava to cool to 100 °C. Use the unrounded value of k and give your answer to 3 significant figures.

(3)

(e)

Show that the time for the difference between the lava temperature and the air temperature to halve is the same at every stage, and find this time to 3 significant figures.

(2)

(f)

A researcher claims that the lava will reach exactly 24 °C after 30 hours. Find the temperature the model gives at 30 hours, to 3 significant figures, and comment on the claim. Use the unrounded value of k.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 1120 = 24 + Ae⁰, so A = 1096	✓ 1
Part (b)
M1 640 = 24 + 1096e^−2k	✓ 1
M1 e^−2k = 616/1096, then takes natural logarithms	✓ 1
A1 k = 0.288	✓ 1
Part (c)
M1 T = 24 + 1096e^−5k	✓ 1
A1 284 °C	✓ 1
Part (d)
M1 100 = 24 + 1096e^−kt	✓ 1
M1 e^−kt = 76/1096 and takes logarithms	✓ 1
A1 t = 9.26 hours (accept 9.27 from rounded k)	✓ 1
Part (e)
M1 e^−kt = 1/2, so the halving time is ln 2/k, which does not depend on the starting difference	✓ 1
A1 2.41 hours	✓ 1
Part (f)
A1 24.2 °C	✓ 1
R1 the temperature gets very close to 24 °C but, according to the model, never reaches it, since 1096e^−kt is always positive	✓ 1

Common error: Subtracting the air temperature too late: set up the equation as 24 + Ae⁻kt and isolate the exponential term before taking logarithms.

+ Add to test✓ In your test412IBMI3S-2-03r48-07Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard13 marks

During the first 8 minutes after leaving a station, the speed of a train, v km h⁻¹, is modelled by
v = −0.6t³ + 5.5t² + 8t, for 0 ≤ t ≤ 8,
where t is the time in minutes after the train starts moving.

(a)

Calculate the speed of the train after 2 minutes.

(2)

(b)

Find the maximum speed predicted by the model during the first 8 minutes, and the time at which it occurs. Give both answers to 3 significant figures.

(3)

(c)

The speed limit on the first section of track is 110 km h⁻¹. Find the two times at which the model gives a speed of exactly 110 km h⁻¹. Give your answers to 3 significant figures.

(3)

(d)

Hence find, to 3 significant figures, how long the train is travelling faster than 110 km h⁻¹. Use the unrounded times from (c).

(1)

(e)

The timetable says the train is stopped at the next station at t = 8. Use the model to explain why it is not suitable at t = 8.

(2)

(f)

Find the positive value of t, to 3 significant figures, at which the model gives v = 0, and state what this means for the use of the model beyond this time.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes t = 2	✓ 1
A1 33.2 km h⁻¹	✓ 1
Part (b)
M1 uses a graphing method or solves dv/dt = 0 (or an equivalent calculator method)	✓ 1
A1 t = 6.77 minutes	✓ 1
A1 v = 120 km h⁻¹	✓ 1
Part (c)
M1 sets −0.6t³ + 5.5t² + 8t = 110	✓ 1
A1 t = 5.46 minutes	✓ 1
A1 t = 7.94 minutes	✓ 1
Part (d)
A1 2.47 minutes	✓ 1
Part (e)
M1 v(8) = 108.8 km h⁻¹	✓ 1
R1 the model gives a high speed instead of the train being stationary, so it does not fit the end of the section	✓ 1
Part (f)
M1 solves −0.6t² + 5.5t + 8 = 0 (after dividing by t)	✓ 1
A1 t = 10.4 minutes (a negative speed would follow, which is not possible)	✓ 1

Exam tip: State the domain limits from the question when you evaluate a model: results outside the stated interval are extrapolation.

+ Add to test✓ In your test413IBMI3S-2-03r48-11Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Linear models and piecewise models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Quadratic and cubic models; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Quadratic and cubic models Paper 2Hard15 marks

A rail operator runs a weekend heritage train. When the ticket price is £p, the number of tickets sold per weekend is modelled by n = 4200 − 150p, for 0 ≤ p ≤ 28. Treat p and n as continuous.

(a)

Calculate the number of tickets sold when the price is £12.

(1)

(b)

Show that the ticket revenue, £R, is given by R = 4200p − 150p².

(2)

(c)

Find the price that gives the greatest revenue, and state this greatest revenue.

(3)

(d)

The weekend costs are £9000 plus £6 for each ticket sold. Show that the profit, £P, is given by P = −150p² + 5100p − 34 200.

(3)

(e)

Find the ticket price that gives the greatest profit, and state this greatest profit.

(3)

(f)

Find the range of prices for which the operator makes a profit. Give your answers to the nearest penny.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
A1 4200 − 150 × 12 = 2400	✓ 1
Part (b)
M1 R = p × n = p(4200 − 150p)	✓ 1
A1 expands to 4200p − 150p² (AG)	✓ 1
Part (c)
M1 uses p = −4200/(2 × (−150)) or a graphing method	✓ 1
A1 £14	✓ 1
A1 revenue £29 400	✓ 1
Part (d)
M1 cost = 9000 + 6(4200 − 150p) = 34 200 − 900p	✓ 1
M1 P = R − cost	✓ 1
A1 −150p² + 4200p − 34 200 + 900p = −150p² + 5100p − 34 200 (AG)	✓ 1
Part (e)
M1 uses p = −5100/(2 × (−150)) or a graphing method	✓ 1
A1 £17	✓ 1
A1 profit £9150	✓ 1
Part (f)
M1 sets P > 0 (or P = 0)	✓ 1
A1 lower limit £9.19	✓ 1
A1 upper limit £24.81, so £9.19 < p < £24.81	✓ 1

Common error: Assuming the price that maximises revenue also maximises profit: costs that depend on sales change the position of the vertex.

+ Add to test✓ In your test414IBMI3S-2-04r48-07Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; 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choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard13 marks

The output power, P kW, of a Formula 1 engine is modelled by P = −1.6r³ + 25r² − 30r, for 4 ≤ r ≤ 12, where r is the engine speed in thousands of revolutions per minute.

(a)

Calculate the power when r = 8.

(2)

(b)

Find the greatest power predicted by the model and the engine speed at which it occurs. Give both answers to 3 significant figures.

(3)

(c)

Find the range of engine speeds for which the power is at least 550 kW. Give your answers to 3 significant figures.

(3)

(d)

A student suggests using the model for r = 1. Show that this is not sensible.

(2)

(e)

A driver changes gear so that the engine speed falls from 11.5 to 9.5 thousand revolutions per minute. Find the increase in power, to 3 significant figures.

(3)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 substitutes r = 8	✓ 1
A1 540.8 kW	✓ 1
Part (b)
M1 uses a graphing method or solves dP/dr = 0	✓ 1
A1 r = 9.78	✓ 1
A1 P = 601 kW	✓ 1
Part (c)
M1 sets P = 550	✓ 1
A1 r = 8.15	✓ 1
A1 r = 11.2, so 8.15 ≤ r ≤ 11.2	✓ 1
Part (d)
M1 substituting r = 1 gives −1.6 + 25 − 30 = −6.6 kW	✓ 1
R1 a negative power is not possible, so the model cannot be used for engine speeds this low	✓ 1
Part (e)
M1 P(11.5) = 527.85 and P(9.5) = 599.45	✓ 1
M1 finds the difference P(9.5) − P(11.5)	✓ 1
A1 71.6 kW	✓ 1

Exam tip: Check the endpoints of the stated domain: a cubic model can give impossible values (such as negative power) just outside the interval it was fitted to.

+ Add to test✓ In your test415IBMI3S-2-04r48-11Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; 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choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Transformations of graphs and logarithmic scales; Linear models and piecewise models; quadratic and cubic models; exponential and sinusoidal models; choosing and evaluating models; Choosing and evaluating models; Quadratic and cubic models; Exponential and sinusoidal models; Quadratic and cubic models; Choosing and evaluating models; Exponential and sinusoidal models; Linear models and piecewise models; Quadratic and cubic models; Linear models and piecewise models; Quadratic and cubic models; Choosing and evaluating models; Transformations of graphs and logarithmic scales; Exponential and sinusoidal models; Transformations of graphs and logarithmic scales Paper 2Hard14 marks

A beekeeper fences a rectangular apiary using 80 m of fencing. A straight hedge forms the fourth side, so fencing is needed on only three sides. The sides of the rectangle at right angles to the hedge each have length x metres.

(a)

Show that the area of the apiary, A m², is A = 80x − 2x².

(2)

(b)

Write down the set of possible values of x, giving a reason.

(2)

(c)

Find the value of x that gives the greatest area, and find this greatest area.

(3)

(d)

Each hive needs 6 m² of ground space. Find the greatest number of hives that fit in the apiary of greatest area.

(2)

(e)

The local council requires an area of at least 700 m². Find the range of values of x that satisfy this, to 3 significant figures.

(3)

(f)

Hence find the greatest possible length of the side parallel to the hedge when the area is at least 700 m². Give your answer to 3 significant figures, using the unrounded lower value of x.

(2)

Show mark scheme

Marking point	Mark	Notes
Part (a)
M1 length parallel to the hedge = 80 − 2x	✓ 1
A1 A = x(80 − 2x) = 80x − 2x² (AG)	✓ 1
Part (b)
A1 0 < x < 40	✓ 1
R1 both side lengths must be positive: x > 0 and 80 − 2x > 0	✓ 1
Part (c)
M1 uses x = −80/(2 × (−2)) or a graphing method	✓ 1
A1 x = 20	✓ 1
A1 A = 800 m²	✓ 1
Part (d)
M1 800 ÷ 6 = 133.3…	✓ 1
A1 133 hives	✓ 1
Part (e)
M1 solves 80x − 2x² = 700	✓ 1
A1 x = 12.9 and x = 27.1	✓ 1
A1 12.9 ≤ x ≤ 27.1	✓ 1
Part (f)
M1 greatest length occurs for the smallest x: 80 − 2x	✓ 1
A1 54.1 m	✓ 1

Common error: Giving the length as 80 − x or 80 − 4x: only three sides are fenced, so two sides of length x and one side of the remaining length use up the 80 m.

Marking point	Mark	Notes
Part (a)
A1: \(T(0) = 20 + 70 = 90\) °C	✓ 1
Part (b)
M1: substituting \(t = 10\): \(20 + 70(0.92)^{10}\)	✓ 1
A1: 50.4 °C	✓ 1
Part (c)
M1: setting \(20 + 70(0.92)^t = 40\) or using a graph / solver	✓ 1
A1: \(t = 15.0\) minutes	✓ 1
Part (d)
R1: 20 °C is the room temperature; the coffee approaches this temperature over time but never cools below it	✓ 1

Marking point	Mark	Notes
Part (a)
A1: maximum height \(26 + 24 = 50\) m	✓ 1
A1: minimum height \(26 - 24 = 2\) m	✓ 1
Part (b)
M1: period \(= \frac{360}{12}\)	✓ 1
A1: 30 minutes	✓ 1
Part (c)
M1: \(h(8) = 26 - 24\cos(96°)\)	✓ 1
A1: 28.5 m	✓ 1

Marking point	Mark	Notes
Part (a)
M1: \(5000(1.035)^t = 8000\) or \((1.035)^t = 1.6\)	✓ 1
A1: \(t = 13.66\ldots\)	✓ 1
A1: 14 complete years	✓ 1
Part (b)
M1: \(5000(1.035)^6\)	✓ 1
A1: 6146.28 USD	✓ 1

Marking point	Mark	Notes
Part (a)
A1: \(a = \frac{8.8 - 2.4}{2} = 3.2\)	✓ 1
Part (b)
A1: \(c = \frac{8.8 + 2.4}{2} = 5.6\)	✓ 1
Part (c)
M1: \(b = \frac{360}{12.4}\)	✓ 1
A1: \(b = 29.0\) (29.032…)	✓ 1
Part (d)
M1: \(d(5) = 3.2\sin(29.032\ldots \times 5) + 5.6\)	✓ 1
A1: 7.43 m (7.428…)	✓ 1

Marking point	Mark	Notes
Part (a)
Base dimensions \((30-2x)\) and \((20-2x)\), height \(x\), so \(V = x(30-2x)(20-2x)\) (M1)	✓ 1
Expands correctly: \(x(600 - 100x + 4x^2) = 4x^3 - 100x^2 + 600x\) (A1) (AG)	✓ 1
Part (b)
0 < x < 10, because the 20 cm side requires 20 − 2x > 0 (A1)	✓ 1
Part (c)
Uses the GDC maximum on the graph of V, or solves V′(x) = 12x² − 200x + 600 = 0 (M1)	✓ 1
x = 3.92 cm (A1)	✓ 1
V = 1060 cm³ (1056.3…) (A1)	✓ 1

Marking point	Mark	Notes
Part (a)
M1 attempt at gradient: \(m = \frac{18.20-9.80}{10-4}\)	✓ 1
A1 \(m = 1.40\)	✓ 1
A1 \(c = 9.80 - 4(1.40) = 4.20\)	✓ 1
Part (b)
R1 \(c\) is the fixed booking fee of $4.20, charged before any distance is travelled	✓ 1
Part (c)
A1 $C = 1.40(25) + 4.20 = $39.20	✓ 1

Marking point	Mark	Notes
Part (a)
\(m=\frac{13.9-18.4}{5-2}\) M1 ; \(m=-1.5\) A1	✓ 2
Part (b)
\(18.4=-1.5(2)+c \Rightarrow c=21.4\) A1	✓ 1
Part (c)
The height of the candle decreases by 1.5 cm every hour R1	✓ 1
Part (d)
Sets \(-1.5t+21.4=0\) M1 ; \(t=14.3\) hours (14.266…) A1	✓ 2

Marking point	Mark	Notes
Part (a)
M1: substitutes \(d = 12.4\): \(C = 3.50 + 1.85 \times 12.4\)	✓ 1
A1: \(C = $26.44\)	✓ 1
Part (b)
A1: the fare increases by $1.85 for each additional kilometre travelled (cost per km)	✓ 1
Part (c)
M1: sets up \(3.50 + 1.85d = 41.30\)	✓ 1
A1: \(d = 20.4\) km (20.432…)	✓ 1

Marking point	Mark	Notes
Part (a)
M1: \(10\log_{10}\left(\frac{3.2\times10^{-5}}{1.0\times10^{-12}}\right)\)	✓ 1
A1: 75.1 dB (75.05...)	✓ 1
Part (b)
M1: \(\frac{I}{I_0}=10^{9.2}\)	✓ 1
A1: \(I=1.58\times10^{-3}\) W m⁻² (1.5849...×10⁻³)	✓ 1
Part (c)
A1: \(\frac{1.585\times10^{-3}}{3.2\times10^{-5}}=49.5\) times. Also accept \(10^{(92-75.05)/10}\approx49.5\). Accept 49.0 to 49.6 where it follows from rounded earlier answers.	✓ 1

Marking point	Mark	Notes
Part (a)
Uses a vertex method, e.g. \(t = -\frac{14.7}{2(-4.9)}\), or the GDC maximum	✓ 1
\(t = 1.5\) s	✓ 1
Maximum height = 12.8 m (12.825)	✓ 1
Part (b)
Sets \(h(t) = 0\) and solves, e.g. by GDC or the quadratic formula	✓ 1
\(t = 3.12\) s, rejecting the negative root	✓ 1
Part (c)
The ball is 1.8 m above the ground at the moment it is released (at \(t = 0\))	✓ 1

Marking point	Mark	Notes
Part (a)
\(C(d)=1.85d+3.50\): coefficient of d is 1.85 A1 ; constant term is 3.50 A1	✓ 2
Part (b)
\(C(12.4)=3.50+1.85\times12.4=26.44\), so the cost is USD 26.44 A1	✓ 1
Part (c)
Sets \(1.85d+3.50=40\) (or \(\le 40\)) M1 ; \(d=19.7\) km (19.729…), also accept 19.72 km, but not a value rounded up to 19.73 km A1	✓ 2

Marking point	Mark	Notes
Part (a)
A1: price = 7.00 dollars (from \(x=\frac{-560}{2(-40)}\) or using GDC)	✓ 1
A1: maximum profit = 960.00 dollars	✓ 1
Part (b)
M1: attempt to solve \(-40x^2+560x-1000=0\) (e.g. GDC or quadratic formula)	✓ 1
A1: \(x=2.10\) dollars and \(x=11.90\) dollars	✓ 1
Part (c)
R1: e.g. the model predicts an ever-increasing loss, whereas in reality almost no smoothies would be sold, so the loss would be limited to roughly the fixed costs (or the owner would stop selling); OR such prices are likely outside the range of data used to build the model	✓ 1

Marking point	Mark	Notes
Part (a)
M1: substitutes correctly, \(L = 10\log_{10}(3.2\times10^{-5}/10^{-12}) = 10\log_{10}(3.2\times10^{7})\)	✓ 1
A1: L = 75.1 dB (accept 75.05)	✓ 1
Part (b)
M1: rearranges to \(I = 10^{-12}\times10^{9.2}\)	✓ 1
A1: I = 1.58×10⁻³ W m⁻² (accept 1.585×10⁻³)	✓ 1
Part (c)
M1: recognises that an increase of 15 dB multiplies the intensity by \(10^{15/10}\)	✓ 1
A1: factor = 31.6	✓ 1

Marking point	Mark	Notes
Part (a)
Substitutes \(t = 4\): \(32000 \times 0.85^4\) M1	✓ 1
16704.20 dollars A1	✓ 1
Part (b)
Each year the car keeps 85% of its value from the previous year, so it loses 15% of its value per year. R1	✓ 1
Part (c)
Sets up \(32000 \times 0.85^{t} = 5000\) and solves by GDC or logarithms M1	✓ 1
\(t = 11.4\) years A1	✓ 1
Part (d)
Any valid reason, e.g. a very old car may be scrapped, so its value is only its scrap value; or a rare model may gain value as a classic; or the rate of depreciation is unlikely to stay at 15% per year for 30 years. R1	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(6 + 0.8\times14\)	✓ 1
A1 €17.20	✓ 1
Part (b)
M1 \(12 + 0.5\times50\)	✓ 1
A1 €37.00	✓ 1
Part (c)
M1 recognise that €30 is above the fee at 20 hours (€22), so use \(12 + 0.5h = 30\)	✓ 1
A1 36 hours	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(80\times0.85^{3}\)	✓ 1
A1 49.1 mg per litre	✓ 1
Part (b)
M1 \(80\times0.85^{t} = 40\) or \(0.85^{t} = 0.5\)	✓ 1
M1 take logarithms or use a GDC: \(t = \dfrac{\ln0.5}{\ln0.85}\)	✓ 1
A1 4.27 hours	✓ 1

Marking point	Mark	Notes
Part (a)
M1 use the second rule: \(4.50 + 1.20(6.5 - 2)\)	✓ 1
A1 €9.90	✓ 1
Part (b)
R1 €16.50 is more than \(C(10) = 14.10\), so use the third rule	✓ 1
M1 \(14.10 + 0.80(w - 10) = 16.50\)	✓ 1
A1 \(w = 13\) kg	✓ 1

Marking point	Mark	Notes
Part (a)
M1 axis of symmetry \(t = -\frac{6}{2(-0.5)} = 6\) (12:00)	✓ 1
A1 maximum output 18 kW	✓ 1
Part (b)
M1 \(-0.5t^2 + 6t = 10\) (or \(t^2 - 12t + 20 = 0\))	✓ 1
A1 \(t = 2\) (08:00)	✓ 1
A1 \(t = 10\) (16:00)	✓ 1
Part (c)
A1 8 hours	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(40 \times 1.15^{10}\)	✓ 1
A1 162	✓ 1
Part (b)
M1 \(40 \times 1.15^d = 500\) (or \(1.15^d = 12.5\))	✓ 1
M1 use of logarithms or the GDC to solve	✓ 1
A1 \(d = 18.1\)	✓ 1
Part (c)
R1 cases cannot grow by 15% every day indefinitely, since the susceptible population is limited and control measures will change growth	✓ 1

Marking point	Mark	Notes
Part (a)
A1: \(V = 28500(0.86)^t\)	✓ 1
Part (b)
M1: \(28500(0.86)^4\)	✓ 1
A1: 15 589.73 EUR	✓ 1
Part (c)
M1: \(28500(0.86)^t = 32000(0.81)^t\)	✓ 1
M1: using GDC intersection, or \(\left(\frac{0.86}{0.81}\right)^t = \frac{32000}{28500}\)	✓ 1
A1: \(t = 1.93\) years	✓ 1
Part (d)
R1: after 1.93 years Sana's car is worth less than Ravi's car, because her car loses value at a faster rate (19% compared with 14% per year)	✓ 1

Marking point	Mark	Notes
Part (a)
\(12+0.18\times150=\) $39.00 A1	✓ 1
Part (b)
\(C(x)=12+0.18x\) for \(0\le x\le200\) A1 ; recognises bill at 200 kWh is $48 and adds $0.25(x-200)$ M1 ; $C(x)=0.25x-2$ (or $48+0.25(x-200)$) for $x>200$ A1	✓ 3
Part (c)
Since \(103.50>48\), usage exceeds 200 kWh so the second piece applies R1 ; \(0.25x-2=103.5\) M1 ; \(x=422\) kWh A1	✓ 3

Marking point	Mark	Notes
Part (a)
M1: revenue minus cost: \(14n - (2400 + 6.5n)\)	✓ 1
A1: \(P(n) = 7.5n - 2400\)	✓ 1
Part (b)
M1: \(7.5n - 2400 = 0\)	✓ 1
A1: \(n = 320\) cakes	✓ 1
Part (c)
M1: \(7.5n - 2400 \ge 1750\), giving \(n \ge 553.3\ldots\)	✓ 1
A1: 554 cakes (must be a whole number, rounded up)	✓ 1
Part (d)
M1: \(250p = 2400 + 6.5 \times 250\)	✓ 1
A1: \(p = $16.10\)	✓ 1

Marking point	Mark	Notes
Part (a)
Forms two equations, using \((10, 0)\) and the vertex \((5, 6)\): \(100a + 10b = 0\) and \(25a + 5b = 6\)	✓ 1
\(a = -0.24\)	✓ 1
\(b = 2.4\)	✓ 1
Part (b)
Sets \(-0.24x^2 + 2.4x = 4.2\)	✓ 1
\(x = 2.26\) and \(x = 7.74\) (to 3 s.f.)	✓ 1
Maximum width \(= 7.7386 - 2.2614 = 5.48\) m	✓ 1

Marking point	Mark	Notes
Part (a)
A1: \(a=500\)	✓ 1
A1: \(b=\frac{1040-500}{4}=135\)	✓ 1
Part (b)
M1: \(500r^4=1040\) so \(r^4=2.08\)	✓ 1
A1: \(r=2.08^{1/4}=1.20\) (1.20093…)	✓ 1
Part (c)
A1: Model A predicts \(500+135(2)=770\) and Model B predicts \(500(1.20093…)^2\approx 721\)	✓ 1
R1: Model B is more appropriate because its prediction is much closer to the recorded value of 720	✓ 1
Part (d)
M1: Correct substitution into their chosen model, e.g. \(500(1.20093…)^{10}\) or \(500(2.08)^{10/4}\)	✓ 1
A1: 3120 subscribers (3119.8…)	✓ 1
Follow-through: if Model A was chosen in part (c), \(500+135(10)=1850\) earns M1A1.	—

Marking point	Mark	Notes
Part (a)
\(V(0) = 50\) million litres	✓ 1
Part (b)
\(t = 8.29\) months (accept \(15-\sqrt{45}\))	✓ 1
\(V = 89.1\) million litres	✓ 1
(Note: \(V(0)=50\) and \(V(24)=67.28\) are both less, so the local maximum is the overall maximum.)	—
Part (c)
Attempts to solve \(V(t) = 66\), e.g. by GDC or graph intersection	✓ 1
\(t = 1.718\)	✓ 1
\(t = 20\) and \(t = 23.28\)	✓ 1
Low for \(0 \le t < 1.718\) and \(20 < t < 23.28\), so total time \(= 1.718 + 3.282 = 5.00\) months (accept 5 or 5.0)	✓ 1

Marking point	Mark	Notes
Part (a)
A1 \(a = \frac{8.6 - 2.2}{2} = 3.2\)	✓ 1
A1 \(d = \frac{8.6 + 2.2}{2} = 5.4\)	✓ 1
Part (b)
M1 period \(= 2 \times 6.25 = 12.5\) hours	✓ 1
A1 \(b = \frac{2\pi}{12.5} = \frac{4\pi}{25} = 0.503\) (3 s.f.)	✓ 1
Part (c)
A1 \(c = 3\)	✓ 1
Part (d)
M1 solving \(3.2\cos(0.503(t - 3)) + 5.4 = 4.0\) and choosing the first solution after \(t = 3\), e.g. by graph intersection	✓ 1
A1 \(t = 7.03\) hours, i.e. 07:02 (accept 07:01 from using \(b = 0.503\))	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(x = -\frac{120}{2(-2)} = 30\) (or use the GDC maximum)	✓ 1
A1 price €30	✓ 1
A1 greatest profit €900	✓ 1
Part (b)
M1 solve \(-2x^2 + 120x - 900 = 0\)	✓ 1
A1 \(x = 8.79\)	✓ 1
A1 \(x = 51.2\), so \(8.79 < x < 51.2\)	✓ 1

Marking point	Mark	Notes
Part (a)
M1 \(0.38 \times 8^3\)	✓ 1
A1 195 kW	✓ 1
Part (b)
M1 \(0.38(2v)^3 = 0.38 \times 8v^3\)	✓ 1
R1 \(= 8 \times 0.38v^3\), which is 8 times \(P\)	✓ 1
Part (c)
M1 \(0.38v^3 = 400\)	✓ 1
M1 \(v = \sqrt[3]{\frac{400}{0.38}}\)	✓ 1
A1 10.2 m s⁻¹	✓ 1

Marking point	Mark	Notes
Part (a)
A1: \(a = \frac{16.4 - 8.0}{2} = 4.2\)	✓ 1
A1: \(d = \frac{16.4 + 8.0}{2} = 12.2\)	✓ 1
Part (b)
A1: \(b = \frac{360}{365} = 0.986\) (0.98630…)	✓ 1
Part (c)
M1: \(4.2\cos\big(0.98630\ldots(60 - 172)\big) + 12.2\)	✓ 1
A1: 10.7 hours	✓ 1
Part (d)
M1: setting \(D(t) = 14\), i.e. \(\cos\big(b(t-172)\big) = \frac{1.8}{4.2}\), or sketching \(D\) with the line \(y = 14\)	✓ 1
A1: \(t = 106.5\) (106.48…)	✓ 1
A1: \(t = 237.5\) (237.52…)	✓ 1
A1: \(237.5 - 106.5 \approx 131\) days (accept 131.0; counting days 107 to 237 inclusive also gives 131)	✓ 1
Part (e)
R1: a valid reason, e.g. the real variation is not exactly sinusoidal, the year is not exactly 365 days (leap years), or terrain/atmospheric effects	✓ 1

Marking point	Mark	Notes
Part (a)
A1: \(a = 6\)	✓ 1
A1: \(d = 25\)	✓ 1
Part (b)
M1: period = 24 hours, so \(b = \frac{360}{24}\)	✓ 1
A1: \(b = 15\)	✓ 1
Part (c)
M1: the temperature is 25 °C and rising halfway between 03:00 and 15:00	✓ 1
A1: \(c = 9\)	✓ 1
Part (d)
M1: \(6\sin\big(15(t - 9)\big) + 25 = 28\) or graph intersection	✓ 1
A1: \(t = 11\), i.e. 11:00	✓ 1
Part (e)
M1: second solution \(t = 19\), or use of symmetry about \(t = 15\)	✓ 1
A1: \(19 - 11 = 8\) hours	✓ 1