Number and algebra

Find \(ab\).

Find \(\dfrac{a}{b}\).

Write \(0.000\,407\) in the form \(a\times10^{k}\), where \(1\le a\lt 10\) and \(k\in\mathbb{Z}\).

Write \(52.3\times10^{4}\) in the form \(a\times10^{k}\), where \(1\le a\lt 10\) and \(k\in\mathbb{Z}\).

Write the following numbers in order of size, starting with the smallest.
\(3.1\times10^{-2}\qquad 0.0298\qquad 31.2\times10^{-3}\qquad 2.99\times10^{-2}\)

Find \(a+b\).

Find \(a-10b\).

Find \(2a^{2}\).

Find the common difference, \(d\).

Find the first term, \(u_{1}\).

Find the sum of the first 20 terms of the sequence.

Write down the common difference.

Find an expression for \(u_{n}\) in the form \(an+b\), where \(a,b\in\mathbb{Z}\).

Find the value of \(n\) for which \(u_{n}=-121\).

Determine whether \(-100\) is a term of the sequence.

Find the interest paid each year.

Find the annual rate of interest.

Find the number of complete years after which the value of the account first exceeds $6000.

Find the amount by which the value decreases each year.

Find the value of the machine after 10 years.

Find the number of complete years after which the value of the machine is first less than 10% of the price paid for it.

Write down the common ratio.

Find \(u_{8}\).

Given that \(2187=3^{7}\), show that \(u_{n}=3^{8-n}\).

Hence find the value of \(n\) for which \(u_{n}=\dfrac{1}{81}\).

An amount of €5000 is invested at an interest rate of 10% per year, compounded annually. Find the value of the investment after 2 years.

A boat is bought for €20 000. It depreciates by 25% each year. Find the value of the boat after 2 years.

Hence find the total percentage decrease in the value of the boat over the 2 years.

Simplify the expression, giving your answer in the form \(ax^{p}y^{q}\), where \(a,p,q\in\mathbb{Z}\).

Hence find the value of the expression when \(x=2\) and \(y=2\).

Write down the value of \(\log_{10}0.001\).

Write down the value of \(\ln\dfrac{1}{\mathrm{e}^{2}}\).

Solve \(10^{2x-1}=1000\).

Write the equation \(10^{-2}=0.01\) in logarithmic form.

Given that \(\ln y=3\), write down the value of \(y\) as a power of \(\mathrm{e}\).

Given that \(\log_{a}81=4\), where \(a\gt 0\), \(a\neq1\), find the value of \(a\).

Explain why there is no real value of \(x\) for which \(10^{x}=-4\).

Write \(\dfrac{8^{-2}\times4^{3}}{2^{-5}}\) in the form \(2^{k}\), where \(k\in\mathbb{Z}\).

Find the value of \(n\) for which \(5^{n}\times25^{-1}=\dfrac{1}{125}\).

Show that \(8^{2}-7^{2}=8+7\).

Show that \((n+1)^{2}-n^{2}\equiv 2n+1\).

Hence write 41 as the difference of the squares of two consecutive integers.

By substituting \(x=1\), show that the student’s statement is not an identity.

Find all values of \(x\) for which \((x+2)^{2}=x^{2}+4\).

Show that \((x+2)^{2}-(x-2)^{2}\equiv8x\).

Check your result in part (c) by substituting \(x=5\).

Find the value of \(32^{\frac{3}{5}}\).

Find the value of \(\left(\dfrac{8}{27}\right)^{-\frac{2}{3}}\).

Solve \(x^{\frac{3}{2}}=64\), for \(x\gt 0\).

Find the value of \(2\log_{6}3+\log_{6}4\).

Write \(3\ln2-\ln4+\ln5\) in the form \(\ln k\), where \(k\in\mathbb{Z}\).

Write down the value of \(\log_{5}1+\log_{5}5\).

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

Show that \(k=\dfrac{\ln2}{10}\).

Find the time taken for the temperature of the tea to fall to 37.5 °C.

Explain why, according to the model, the temperature of the tea never falls to 20 °C.

Write down the common ratio, \(r\).

Find the sum to infinity of the series.

A different infinite geometric series has sum to infinity 45 and common ratio \(\tfrac{1}{3}\). Find its first term.

Write down the row of Pascal’s triangle that gives the coefficients in the expansion of \((a+b)^{5}\).

Use the formula for \(\dbinom{n}{r}\) to show that \(\dbinom{7}{3}=35\).

Write down the value of \(\dbinom{7}{4}\).

Write down the term in \(a^{2}b^{5}\) in the expansion of \((a+b)^{7}\).

Find the coefficient of \(x^3\) in the expansion of \((3-2x)^5\).

Find the term independent of \(x\) in the expansion of \(\left(x^2+\frac{3}{x}\right)^6\).

Find the value of \(n\).

Hence find the coefficient of \(x^2\).

Prove that \((2n-1)(2n+1)+1\) is a perfect square for every positive integer \(n\), stating the side length of the square in terms of \(n\).

A panel is 19 tiles by 21 tiles. State the side length of the square formed when one extra tile is added.

Prove that \(\frac{1}{n}-\frac{1}{n+1}\equiv\frac{1}{n(n+1)}\).

Hence find the exact value of \(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+\dots+\frac{1}{24\times25}\).

Write the width of the hair in the form a×10^k, where 1 ≤ a < 10 and k ∈ ℤ.

Find how many transistor gate lengths, placed end to end, would span the width of the hair. Give your answer in the form a×10^k, where 1 ≤ a < 10 and k ∈ ℤ.

Find the total dose for this patient, in grams. Give your answer in the form a×10^k, where 1 ≤ a < 10 and k ∈ ℤ.

Write the total dose in milligrams.

Find the exact value of \(32^{-\frac{3}{5}}\).

Simplify \(\dfrac{(27x^{6})^{\frac{2}{3}}}{(4x^{-2})^{\frac{1}{2}}}\), where \(x>0\), giving your answer in the form \(ax^{n}\).

Find an expression for \(\log_b 40\).

Find an expression for \(\log_b\left(\dfrac{\sqrt{5}}{16}\right)\).

Find an expression for \(\log_b 0.4\).

Find the first day on which she walks more than 1 km.

Find the total distance she walks over the first 20 days of the programme.

Find the common ratio of the lengths of the swings.

Find the length of the fourth swing.

Find the common difference and the number of vines on terrace 1.

Find the total number of vines on all 30 terraces.

Find the common ratio and the number of new shares in hour 1.

Find the total number of new shares in the first 6 hours.

Write down the first term and the common ratio of the series.

Show that \(\displaystyle\sum_{k=1}^{n}3\times2^{k}=6(2^{n}-1)\).

Given that \(\displaystyle\sum_{k=1}^{n}3\times2^{k}=1530\), find the value of \(n\).

Show that the total storage of the data centre is \(2^{39}\) bytes.

A backup file has size \(16^{3}\) bytes. Find the number of such files that the data centre can store, giving your answer in the form \(2^{k}\).

The total storage can also be written as \(32^{n}\) bytes. Find the exact value of \(n\).

A vacuum cleaner produces a noise of intensity \(10^{6}I_0\). Find its sound level.

A drone produces a sound level of 85 dB. Express its intensity in the form \(10^{p}I_0\).

A stadium crowd's sound level rises by 20 dB when a goal is scored. Find the factor by which the sound intensity increases.

Solve \(9^{x+1}=27^{2x-1}\).

Solve \(2^{y}=7\), giving your answer as a logarithm.

Show that r = 25.

Find the value of the card 4 years after it was bought.

Assuming the same rate of growth applied before the purchase, find the value of the card one year before it was bought.

Find the time for a signal to travel from the ground station to the satellite. Give your answer in the form \(a \times 10^k\) where \(1 \le a < 10\) and \(k \in \mathbb{Z}\).

Write down the time for a signal to travel to the satellite and back to the ground station.

A message of \(4.8 \times 10^{6}\) bits is sent at \(1.2 \times 10^{6}\) bits per second. Find the time taken to send it.

Find the volume released in the 3rd hour.

Find the total volume that would be released if the process continued indefinitely.

Find the exact total volume released in the first 4 hours.

Find the common difference.

Find the number of books added in week 1.

Find the total number of books added in the first 20 weeks.

Find the total angle the rotor turns through, in degrees, if it keeps turning indefinitely under this model.

Hence show that the rotor never completes one full revolution.

Find the number of rotations in \(4\times10^{2}\) minutes.

The engine is overhauled after \(2.4\times10^{3}\) hours of running. Find the number of rotations between overhauls.

Find the number of trees in the 15th row.

Find the total number of trees in the first 15 rows.

Find the interval from a note of frequency 220 Hz to a note of frequency 1760 Hz.

A second note is 24 semitones below a note of frequency 220 Hz, that is, the interval from 220 Hz to this note is \(-24\). Find its frequency.

Write a proof.

Find the number of bees after 2 weeks. Give your answer in exact form.

Find the value of \(t\) for which the colony has 45 000 bees.

Find the volume delivered in the 5th hour.

The pump runs for a very long time. Find the limit that the total volume delivered approaches.

Express \(\log_5 (x^2 y)\) in terms of \(p\) and \(q\).

Express \(\log_5\left(25\sqrt{\dfrac{x}{y^3}}\right)\) in terms of \(p\) and \(q\).

Find the amount of the drug at the start of the fifth hour.

At the start of hour n there are 5 mg of the drug. Find the value of n.

Prove that the sum of any three consecutive even integers is divisible by 6.

Write down the common ratio of the series.

Give a reason why the series has a finite sum to infinity.

Find the sum to infinity.

Find the fourth term of the series.

Find the total distance the train travels before it comes to rest, according to this model.

Find the distance travelled in the 4th minute.

\(\log_3 10\)

\(\log_3\left(\dfrac{45}{8}\right)\)

Write down the common ratio.

Hence show that \(0.272727\ldots=\frac{3}{11}\).

Hence write \(3.272727\ldots\) as a fraction in the form \(\frac{a}{b}\), where \(a,b\) are integers.

Prove that the coach's claim is true for all consecutive integers.

Find the energy supplied to each household in one year. Give your answer in the form a × 10^k, where 1 ≤ a < 10 and k ∈ ℤ.

Each household in a different town uses 2.5 × 10³ kWh per year. Find how many such households the wind farm could supply. Give your answer in the form a × 10^k.

Evaluate 16^3/4.

Evaluate \(\left(\frac{81}{16}\right)^{-\frac{1}{4}}\).

Solve \(8^{x} = \frac{1}{4}\).

Find the value of \(16^{\frac{3}{4}}\).

Find the value of \(27^{-\frac{2}{3}}\).

Solve \(9^{x}=27^{x-1}\).

Find the value of \(k\).

Hence solve \(\ln x+3\ln 2+\ln 10-\ln 5=\ln 80\).

Find the value of \(a\) and the value of \(b\).

Hence solve \(\dfrac{(27x^{9})^{\frac23}}{(9x^{-2})^{\frac12}}=384\).

Prove that the sum of any three consecutive integers is divisible by 3.

Find the coefficient of \(x^3\).

Hence, or otherwise, find the coefficient of \(x^3\) in the expansion of \((3+2x)^5\).

Evaluate \(81^{\frac{3}{4}}\).

Solve \(4^{x+1}=32\).

Write down the number of terms in the expansion.

Find the coefficient of \(x^3\).

Solve the equation \(\log_3 x+\log_3 (x-8)=2\).

Prove that the difference between the squares of two consecutive odd integers is always a multiple of 8.

Find the common ratio and the number of coins in the level 1 chest.

Find the total number of coins in the chests on levels 1 to 8.

Prove that the sum of any three consecutive integers is a multiple of 3.

A student claims that the sum of any four consecutive integers is a multiple of 4. Show that the claim is false.

Find the value of \(r\).

Find the sum of the first five terms of the sequence.

Solve \(2^{x+1} = 32\).

Find the exact value of \(\log_2 40 - \log_2 5\).

Solve \(3^{2x} = 27^{x-1}\).

Find the distance travelled by the probe in \(5.0 \times 10^{7}\) seconds. Give your answer in the form \(a \times 10^k\) where \(1 \leq a < 10\) and \(k \in \mathbb{Z}\).

Find the distance remaining to the target at this time, giving your answer in the form \(a \times 10^k\).

Find the further time, in seconds, needed to reach the target at the same speed. Give your answer in the form \(a \times 10^k\).

Find the total power of the farm. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k ∈ ℤ.

Find the energy produced by the farm in one hour, in joules. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k ∈ ℤ.

Find the distance she swims in the 4th minute.

Find the total distance she would swim if the pattern continued indefinitely.

Find the total distance she swims in the first 4 minutes.

Write down the expansion in ascending powers of x, simplifying each coefficient.

By substituting a suitable value of x, use the first three terms of your expansion to estimate 1.03⁴.

Show your working.

Express \(\log_{10}\left(100x\sqrt{y}\right)\) in terms of \(p\) and \(q\).

Given that \(\log_{10}\left(\dfrac{x^3}{y}\right)=7\) and \(\log_{10}(xy)=5\), find the value of \(p\) and the value of \(q\).

Find the height the ball reaches after the third bounce. Give your answer as a fraction.

Find the total vertical distance the ball travels before it comes to rest.

Prove that the sum of the two bib numbers is always divisible by 4.

Two athletes have bib numbers 2023 and 2025. Use the result of part (a) to find the sum of their bib numbers divided by 4.

Find L when P = 1000P₀.

The power is multiplied by 10⁵. Find the change in L.

Find the common ratio, and the amplitude of the fifth echo as a fraction.

Find the sum of the amplitudes of all the echoes, if the sequence continued indefinitely.

Find the value of \(k\).

Find the power output when the wind speed is 8 m/s.

Find the wind speed at which the power output is 540 kW.

Show that the sum of the four hut numbers is \(4n+6\).

Hence prove that the sum of four consecutive integers is never a multiple of 4.

Simplify \(\left(27x^{-6}\right)^{-\frac{2}{3}}\), giving your answer in the form \(\dfrac{x^{p}}{q}\).

Solve \(4^{2x+1}=8^{x-2}\).

Show that \((2n+1)^{2}-(2n-1)^{2}=8n\).

Hence find the value of \(1001^{2}-999^{2}\).

Expand \((1+2x)^4\) in full.

Hence find the coefficient of \(x^2\) in the expansion of \((1+2x)^4(3-x)\).

Find the exact value of \(27^{-\frac{2}{3}}\times 4^{\frac{1}{2}}\).

Find the value of \(\log_5 75-\log_5 3\).

Find the common ratio.

Find the number of bees at the start of week 5.

Prove that the sum of the numbers of two adjacent bays is always a multiple of 4.

Prove that the product of the numbers of two adjacent bays is always odd.

Find the number of staff called at stage 8.

At one stage 3645 staff members are called. Find which stage this is. (You may use the fact that \(3^6 = 729\).)

Solve \(8^{x-1}=4^{x+2}\).

Write \(\log_2 40-\log_2 5\) as an integer.

Calculate the average power output of the wind farm, in W. Give your answer in the form a × 10^k where 1 ≤ a < 10 and k is an integer. (Power = energy ÷ time.)

Each home supplied uses 2.5 × 10³ W on average. Find the number of homes the wind farm can supply.

State the number of members when the club opened.

Find the number of members when \(t = 6\).

Find the value of \(t\) when the club has 80 members.

\(\ln 15\)

\(\ln 0.6\)

\(\ln \sqrt{45}\)

Find \(\sqrt{p}\). Give your answer in the form \(a\times10^{k}\), where \(1\le a\lt 10\) and \(k\in\mathbb{Z}\).

Find \(pq^{2}\). Give your answer in the form \(a\times10^{k}\), where \(1\le a\lt 10\) and \(k\in\mathbb{Z}\).

Find the integer \(n\) such that \(10^{n}\le\dfrac{p}{q^{3}}\lt 10^{n+1}\).

Write down the first three terms of the series.

Show that \(\displaystyle\sum_{k=1}^{n}(4k-1)=2n^{2}+n\).

Given that \(\displaystyle\sum_{k=1}^{n}(4k-1)=210\), find the value of \(n\).

Show that \(k^{2}-3k-4=0\).

The common difference of the sequence is positive. Find the value of \(k\).

Hence find the sum of the first 20 terms of the sequence.

Write down the number of terms in the series.

Find the sum of the series.

Write the series \(8+13+18+\cdots+203\) in sigma notation, in the form \(\displaystyle\sum_{r=1}^{n}(ar+b)\).

Find the common ratio, \(r\).

Find the first term, \(u_{1}\).

Find the sum of the first six terms of the sequence.

Explain why the sum to infinity of this sequence does not exist.

Show that \(k^{2}-11k+24=0\).

All the terms of the sequence are positive. Find the value of \(k\) and the common ratio.

Find the fifth term of the sequence.

Write down the first three terms of the series.

Find the sum of the series.

Hence find the value of \(\displaystyle\sum_{k=4}^{8}3\left(2^{k}\right)\).

Find the two consecutive integers between which \(\log_{10}4500\) lies. Justify your answer.

Write down the value of \(\mathrm{e}^{\ln 7}\).

Solve \(\mathrm{e}^{2x}=5\), giving your answer in the form \(a\ln b\), where \(a,b\in\mathbb{Q}\).

Given that \(2.7\lt\mathrm{e}\lt 2.8\), show that the solution to part (c) lies between \(\tfrac{1}{2}\) and \(1\).

Show that \(\dfrac{1}{n}-\dfrac{1}{n+2}\equiv\dfrac{2}{n(n+2)}\).

Hence write \(\dfrac{2}{63}\) in the form \(\dfrac{1}{n}-\dfrac{1}{n+2}\), where \(n\in\mathbb{Z}^{+}\).

Show that \((n-1)^{2}+n^{2}+(n+1)^{2}\equiv3n^{2}+2\).

Hence explain why the sum of the squares of any three consecutive integers is never a multiple of 3.

Find all sets of three consecutive integers whose squares have a sum of 110.

Show that \(2x^{2}-12x+23\equiv2(x-3)^{2}+5\).

Hence explain why \(2x^{2}-12x+23\gt 0\) for all \(x\in\mathbb{R}\).

Hence find the greatest value of \(\dfrac{10}{2x^{2}-12x+23}\), and the value of \(x\) at which it occurs.

Show that \((2n+1)^{2}\equiv4n(n+1)+1\).

Explain why \(n(n+1)\) is even.

Hence prove that the square of every odd integer is one more than a multiple of 8.

Write down the remainder when \(2025^{2}\) is divided by 8.

\(\log_{3}(a^{2}b)\);

\(\log_{3}\left(\dfrac{9a}{\sqrt{b}}\right)\);

\(\log_{9}b\).

Solve \(4^{x+1}=8^{2x-3}\).

By using the substitution \(y=5^{x}\), solve \(5^{2x}-6\left(5^{x}\right)+5=0\).

Solve \(5^{x}=3^{x+1}\), giving your answer in the form \(\dfrac{\ln a}{\ln b}\), where \(a,b\in\mathbb{Q}^{+}\).

Solve \(2^{x}\times3^{x+1}=12^{x}\), giving your answer in the form \(\log_{2}c\), where \(c\in\mathbb{Z}\).

Write \(\dfrac{x\sqrt{x}}{\sqrt[3]{x}}\) in the form \(x^{p}\), where \(x\gt 0\) and \(p\in\mathbb{Q}\).

Solve \(\sqrt[3]{x^{2}}=4\sqrt{x}\), for \(x\gt 0\).

Find the value of \(k\) for which \(\dfrac{\sqrt{8}\times4^{\frac{1}{3}}}{\sqrt[6]{2}}=2^{k}\).

State the set of values of \(x\) for which both logarithms in the equation are defined.

Solve the equation.

Write down the first three terms of this series and its common ratio.

Hence write \(0.\dot{3}\dot{6}\) as a fraction in its lowest terms.

Write \(0.1\dot{2}\dot{7}=0.1272727\ldots\) as a fraction in its lowest terms.