Number and algebra

Show that \(r^{3}=27\).

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

Find the least value of \(n\) for which the sum of the first \(n\) terms is greater than \(10^{6}\).

Show that \(\dfrac{3^{n+2}-3^{n}}{8\times3^{n-1}}=3\) for all \(n\in\mathbb{Z}\).

Find the integers \(a\) and \(b\) for which \(12^{a}\times18^{b}=2^{8}\times3^{7}\).

Solve \(5^{2k}\times25^{-1}=125^{k-2}\).

Find the number of different teams that can be chosen.

Find the number of different teams that contain exactly 2 seniors.

Find the number of different teams that contain at least one senior.

Factorise \(x^{2}-x-6\).

Hence express \(\dfrac{7x-1}{x^{2}-x-6}\) in partial fractions.

Find \(z+2w\).

Find \(zw\).

Find \(\dfrac{z}{w}\).

Write down the value of \(i^{4}\).

Find \(i^{2027}\).

Find \(\displaystyle\sum_{k=1}^{50}i^{k}\), giving your answer in the form \(a+bi\).

Write down the least positive integer \(n\) for which \(\displaystyle\sum_{k=1}^{n}i^{k}=0\).

Find \(z\) in the form \(a+bi\), where \(a,b\in\mathbb{R}\).

Hence find \(\dfrac{z}{z^{*}}\) in the form \(c+di\), where \(c,d\in\mathbb{R}\).

Write down \(z_{1}\) and \(z_{2}\) in the form \(a+bi\).

Find \(|z_{1}|\), \(|z_{2}|\) and \(|z_{2}-z_{1}|\).

Show that triangle OAB is right-angled and isosceles.

Write down the size of angle AOB.

Write down the complex number represented by the reflection of B in the real axis.

Given that \(w\) is purely imaginary, find the value of \(k\).

For this value of \(k\), find \(|w|\).

Express \(-1-\sqrt3\,i\) in the form \(r\mathrm{e}^{i\theta}\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\).

Express \(4\operatorname{cis}\dfrac{5\pi}{6}\) in the form \(a+bi\).

Express \(-3\operatorname{cis}\dfrac{\pi}{5}\) in the form \(r\operatorname{cis}\theta\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\).

Write down a second root, giving a reason for your answer.

Find a quadratic factor of \(z^{3}-7z^{2}+17z-15\) with real coefficients.

Find the third root.

Express \(z\) in the form \(r\mathrm{e}^{i\theta}\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\).

Use De Moivre’s theorem to find \(z^{9}\) in the form \(a+bi\).

Find the least positive integer \(n\) for which \(z^{n}\) is real and positive.

Find the three roots of the equation, giving your answers in the form \(a+bi\).

State why the two non-real roots are complex conjugates of each other.

Write down the sum of the three roots.

Find \(u_{2}\), \(u_{3}\) and \(u_{4}\).

Prove by mathematical induction that \(u_{n}=3^{n}-2^{n}\), for \(n\in\mathbb{Z}^{+}\).

Prove by mathematical induction that \(\displaystyle\sum_{r=1}^{n}r\times r!=(n+1)!-1\), for \(n\in\mathbb{Z}^{+}\).

Hence find the value of \(n\) for which \(\displaystyle\sum_{r=1}^{n}r\times r!=719\).

Prove by contradiction that \(a+b\) is irrational.

Given also that \(a\neq0\), prove by contradiction that \(ab\) is irrational.

Use a counterexample to show that the sum of two irrational numbers is not always irrational.

If \(f'(a)=0\), then the graph of \(y=f(x)\) has a local maximum or a local minimum point at \(x=a\).

For all functions \(f\) and \(g\) with domain \(\mathbb{R}\), \((f\circ g)(x)=(g\circ f)(x)\) for every \(x\in\mathbb{R}\).

For all \(a,b\gt0\), \(\ln(a+b)\neq\ln a+\ln b\).

Use row reduction to solve the system.

Give a geometric interpretation of your answer.

Find the value of \(a\).

Hence find the coefficient of \(x^3\).

Find the first term and the common difference.

Find an expression for \(u_n\) in terms of \(n\).

Show that 500 is a term of the sequence and state which term it is.

Find the exact value of \(x\).

Find \(\log_5 125\).

Find \(\log_4 2\).

Solve \(\log_x 81=4\).

Write down the total impedance when the components are connected in series.

Find \(Z_1Z_2\) in the form \(a+bi\).

Find the total impedance when the components are connected in parallel. Give your answer in the form \(a+bi\), where \(a, b \in \mathbb{Q}\).

Find \(z\).

Show that \(\frac{1}{k}-\frac{1}{k+1} = \frac{1}{k(k+1)}\).

Hence find the exact value of \(\sum_{k=1}^{99} \frac{1}{k(k+1)}\).

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

Show that the statement “\(n^2+n+41\) is prime for every positive integer \(n\)” is false.

Statement 1: “n² − n + 11 is prime for every positive integer n.” Use a counterexample to show that Statement 1 is false.

Statement 2: “For all real numbers a and b, if a² > b² then a > b.” Use a counterexample to show that Statement 2 is false.

Prove that there are no integers a and b such that 6a + 9b = 1.

Prove that if x is rational and y is irrational, then x + y is irrational.

Express \(G(s)\) in the form \(\frac{A}{s-3}+\frac{B}{s+2}\), where \(A,B\in\mathbb{Z}\).

Write \(z\) in the form \(re^{i\theta}\), where \(r>0\) and \(-\pi<\theta\le\pi\).

The arm rotates anticlockwise about its base through an angle of \(\frac{\pi}{3}\). Find the new position of the tip, giving your answer in the form \(re^{i\theta}\) with \(-\pi<\theta\le\pi\), and in the form \(a+bi\).

Write \(w\) in the form \(re^{i\theta}\).

Hence find \(w^8\).

Find the smallest positive integer \(n\) for which \(w^n\) is a negative real number, and state the value of \(w^n\) for this \(n\).

Find the daily resting energy requirement of a dog of mass 81 kg.

A second animal has a daily resting energy requirement of 560 kcal. Find its mass.

Show that \(\log_9 x = \frac{1}{2}\log_3 x\).

Hence solve the equation.

Write down a second complex root.

Find the quadratic factor of \(z^3 + az^2 + bz - 20\) with these two roots.

Find the real root, and the values of \(a\) and \(b\).

Write \(z\) in the form \(r\,\text{cis}\,\theta\), where \(r > 0\) and \(-\pi < \theta \le \pi\).

Use De Moivre's theorem to find \(z^6\).

Find the least positive integer \(n\) for which \(z^n\) is a positive real number.

Find the number of possible release orders in which X and Y are released consecutively, in either order.

Find the number of possible release orders in which X and Y are not released consecutively.

Find the value of r.

Find the arc length of the fourth swing.

Find the number of hard drives in the data centre. Give your answer in the form \(a\times10^k\), where \(1\le a<10\) and \(k\in\mathbb{Z}\).

Each drive uses 6.5 W of power. Find the total power used by all the drives, giving your answer in the form \(a\times10^k\), where \(1\le a<10\) and \(k\in\mathbb{Z}\).

Find the common ratio.

Find the first term.

Find the sum of the first eight terms.

Write w in the form r(cos θ + i sin θ), where r > 0 and 0 ≤ θ < 2π.

Use De Moivre's theorem to find w⁹ in the form a + bi, where a and b are integers.

Find the number of different teams that can be chosen.

Two of the surgeons, Dr A and Dr B, cannot work in the same team. Find the number of possible teams.

Express \(\dfrac{5x+1}{(x-1)(x+2)}\) in the form \(\dfrac{A}{x-1}+\dfrac{B}{x+2}\), where \(A\) and \(B\) are constants.

Hence find the exact value of \(\dfrac{5x+1}{(x-1)(x+2)}\) when \(x=2\).

Find \(zw\) in the form \(a+b\mathrm{i}\).

Find \(\dfrac zw\) in the form \(a+b\mathrm{i}\), where \(a,b\in\mathbb{Q}\).

Find the modulus of \(z\) and the argument of \(z\), in radians, with \(-\pi<\arg z\le\pi\).

Write \(z\) in the form \(re^{i\theta}\).

Write \(w\) in the form \(r(\cos\theta+i\sin\theta)\) with \(r>0\).

Use de Moivre's theorem to find \(w^6\) in the form \(a+bi\).

Find the common ratio of the sequence.

Find the total volume delivered from day 3 onwards, if the pump ran for ever.

Show that the claim is true for \(n=1\), \(n=2\) and \(n=3\).

Show that the claim is false by finding a suitable value of \(n\).

Write \(p\) as the sum of an infinite geometric series and state its first term and common ratio.

Hence show that \(p=\frac{4}{11}\).

Find the number of different orders in which the six robots can queue.

Find the number of orders in which \(R_1\) and \(R_2\) are next to each other.

Find the number of orders in which \(R_1\) and \(R_2\) are not next to each other.

Find the common ratio.

Find the length of the 1st movement.

Find the total length of the first 8 movements.

Write down another root.

Find the third root.

State, with a reason, whether the controller is stable.

Find the value of k.

Find the common difference.

Find the distance run in week 10.

Find the total distance run in the first 10 weeks.

Write down three linear equations for a, c and s.

Solve the system to find a, c and s.

Find the total price for 2 adults, 3 children and 1 senior.

Find the modulus and the argument of \(z\), where \(-\pi<\arg z\le\pi\).

Write \(z\) in the form \(re^{i\theta}\).

Find \(z^3\) in its simplest form.

Find \(z_1z_2\), giving your answer in modulus-argument form.

Find \(\dfrac{z_1}{z_2}\) in modulus-argument form.

Write \(z_1z_2\) in the form \(a+bi\).

Evaluate \(27^{\frac23}\times16^{-\frac34}\), giving your answer as a fraction.

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

Prove that the number of extra tiles needed is always a multiple of 8.

Find the number of extra tiles needed when \(n=31\).

Write \(z\) in the form \(r(\cos\theta+i\sin\theta)\), where \(r>0\) and \(-\pi<\theta\le\pi\).

Use de Moivre's theorem to find the position of the fifth anchor, \(z^{5}\), in the form \(a+bi\) where \(a\) and \(b\) are exact.

Find the total mass deposited by the colony each second. Give your answer in the form a×10^k, where 1 ≤ a < 10 and k is an integer.

Find the time, in seconds, for the colony to deposit 3 kg of calcium carbonate. Give your answer in standard form.

Write a complete proof.

Claim 1: "For every positive integer \(n\), the number \(n^2 + n + 41\) is prime." Show that this claim is false.

Claim 2: "Exactly 100 tablets can be packed into boxes, each box holding either 6 or 9 tablets, with every box full." Use proof by contradiction to show that this is impossible.

Find the sum of all the multiples of 3 in this range.

Find the sum of all the multiples of 15 in this range.

Hence find the sum of all the integers from 1 to 500 inclusive that are divisible by neither 3 nor 5.

Show that \(n^{2}-17n+72=0\).

Find the value of \(n\), justifying your answer.

Show that there is no convex polygon whose interior angles form an arithmetic sequence with first term \(100^{\circ}\) and common difference \(20^{\circ}\).

Show that \(a=6\).

Find the two possible values of \(r\).

For the value of \(r\) with \(r\lt 1\), the three terms are the first three terms of an infinite geometric series. Find the sum to infinity.

The squares of the terms of this infinite series form another infinite geometric series. Find the sum to infinity of this new series.

Show that \(n(n+3)=n^{2}+3n\) and \((n+1)(n+2)=n^{2}+3n+2\).

Hence show that \(n(n+1)(n+2)(n+3)+1\equiv\left(n^{2}+3n+1\right)^{2}\).

Hence write \(2024\times2025\times2026\times2027+1\) in the form \(m^{2}\), where \(m\in\mathbb{Z}^{+}\).

Solve \(x^{\log_{10}x}=100x\), for \(x\gt0\).

Show that \(4^{\log_{2}x}=x^{2}\), for \(x\gt0\).

Hence solve \(4^{\log_{2}x}+2x=15\), for \(x\gt0\).

Write \(Q(t)\) in the form \(50\mathrm{e}^{kt}\), giving the exact value of \(k\).

Show that the two populations are equal when \(t=\dfrac{4\ln2}{\ln3-0.7}\).

Explain why \(Q(t)\gt P(t)\) for all values of \(t\) greater than the value in part (b).

Show that \(u_{1}+u_{3}+u_{5}+\cdots=\dfrac{u_{1}}{1-r^{2}}\).

Find the value of \(r\) and the value of \(u_{1}\).

Write down the sum of the terms in the even positions, \(u_{2}+u_{4}+u_{6}+\cdots\).

Show that, for every infinite geometric series with \(u_{1}\gt0\) and \(0\lt|r|\lt1\), the sum of the terms in the odd positions is more than half of the sum to infinity.

Write down the first four terms, in ascending powers of \(x\), of the expansion of \((1+x)^{n}\).

Hence show that \(1.01^{100}\gt2.6\).

Find the coefficient of \(x^{2}\) and the coefficient of \(x^{3}\) in the expansion of \((1+x)^{100}(1-x)^{100}\).

Find the number of possible arrangements.

Find the number of arrangements in which Ana and Ben sit next to each other.

Find the number of arrangements in which no two of Chloe, Dev and Eva sit next to each other.

Use the formula for \({}^{n}P_{r}\) to find the number of possible codes.

Find the number of codes that are even.

Find the number of codes that are even and greater than 6000.

Find the first four terms, in ascending powers of \(x\), of the expansion of \((1-4x)^{-\frac{1}{2}}\).

Write down the set of values of \(x\) for which the expansion is valid.

Show that, when \(x=0.01\), \((1-4x)^{-\frac12}=\dfrac{5\sqrt6}{12}\).

Hence find an approximation for \(\sqrt6\), giving your answer to six decimal places.

Show that the first three terms of the expansion of \((4-x)^{-\frac12}\), in ascending powers of \(x\), are \(\dfrac{1}{2}+\dfrac{x}{16}+\dfrac{3x^{2}}{256}\).

State the set of values of \(x\) for which this expansion is valid.

Hence find the first three terms of the expansion of \(\dfrac{2+x}{\sqrt{4-x}}\).

Find the value of \(k\) for which the coefficient of \(x^{2}\) in the expansion of \(\dfrac{k+x}{\sqrt{4-x}}\) is zero.

Express \(\dfrac{4}{(2k-1)(2k+3)}\) in partial fractions.

Hence show that \(\displaystyle\sum_{k=1}^{n}\dfrac{4}{(2k-1)(2k+3)}=\dfrac{4}{3}-\dfrac{1}{2n+1}-\dfrac{1}{2n+3}\).

Write down the value of \(\displaystyle\sum_{k=1}^{\infty}\dfrac{4}{(2k-1)(2k+3)}\).

Find the two possible pairs of values of \(a\) and \(b\).

Hence solve the equation \(z^{2}+2z+6-12i=0\), giving your answers in the form \(p+qi\), where \(p,q\in\mathbb{Z}\).

Show that \(zw^{*}+z^{*}w=2(ac+bd)\).

Hence explain why OZ is perpendicular to OW if and only if \(zw^{*}\) is purely imaginary, for non-zero \(z\) and \(w\).

Given that \(z=3+2i\) and \(w=k-6i\), where \(k\in\mathbb{R}\), and that OZ is perpendicular to OW, find \(k\).

Find the modulus and argument of \(z_{1}\) and of \(z_{2}\), giving each argument in the interval \(-\pi\lt\theta\le\pi\).

Find \(z_{1}z_{2}\) in the form \(r\mathrm{e}^{i\theta}\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\).

Find \(\dfrac{z_{1}}{z_{2}}\) in the form \(r\mathrm{e}^{i\theta}\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\). Hence write down \(\dfrac{z_{1}}{z_{2}}\) in the form \(a+bi\).

By considering a rotation about O, find the complex number represented by C.

Write down the complex number represented by B.

The square is rotated anticlockwise about O through an angle of \(\dfrac{\pi}{3}\) and then enlarged with centre O and scale factor 2. Find, in the form \(a+bi\), the complex number represented by the image of B.

Show that \(z=2\cos\left(\dfrac{\theta}{2}\right)\mathrm{e}^{\frac{\theta}{2}i}\).

For \(0\lt\theta\lt\pi\), write down \(|z|\) and \(\arg z\) in terms of \(\theta\).

For \(\pi\lt\theta\lt 2\pi\), find \(|z|\) and \(\arg z\) in terms of \(\theta\), where \(-\pi\lt\arg z\le\pi\).

Find \(uv\) in the form \(a+bi\).

By writing \(u\) and \(v\) in Euler form, find \(uv\) in the form \(r\mathrm{e}^{i\theta}\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\).

Hence find the exact values of \(\cos\dfrac{11\pi}{12}\) and \(\sin\dfrac{11\pi}{12}\).

Find the values of \(a\), \(b\), \(c\) and \(d\).

Explain why \(p(x)\gt 0\) for all \(x\in\mathbb{R}\).

Express \(-8+8\sqrt3\,i\) in the form \(r\mathrm{e}^{i\theta}\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\).

Find the four roots of the equation, giving your answers in the form \(r\mathrm{e}^{i\theta}\), where \(r\gt 0\) and \(-\pi\lt\theta\le\pi\).

Write down the root that lies in the first quadrant in the form \(a+bi\), and hence write down the other three roots in the form \(a+bi\).

Prove by mathematical induction that \((\operatorname{cis}\theta)^{n}=\operatorname{cis}(n\theta)\) for all \(n\in\mathbb{Z}^{+}\).

Show that \((\operatorname{cis}\theta)^{-1}=\operatorname{cis}(-\theta)\), and hence show that \((\operatorname{cis}\theta)^{-m}=\operatorname{cis}(-m\theta)\) for all \(m\in\mathbb{Z}^{+}\).

Use De Moivre’s theorem to show that \(z^{n}+z^{-n}=2\cos n\theta\), for \(n\in\mathbb{Z}^{+}\).

Expand \(\left(z+\dfrac1z\right)^{4}\).

Hence show that \(\cos^{4}\theta=\dfrac18\left(\cos4\theta+4\cos2\theta+3\right)\).

Hence find the exact value of \(\displaystyle\int_{0}^{\frac{\pi}{4}}\cos^{4}\theta\,\mathrm{d}\theta\).

Prove by mathematical induction that \(\displaystyle\sum_{r=1}^{n}r\,2^{r}=(n-1)2^{n+1}+2\), for \(n\in\mathbb{Z}^{+}\).

Hence find \(\displaystyle\sum_{r=11}^{20}r\,2^{r}\), giving your answer in the form \(p\times2^{q}\), where \(p\) is an odd integer and \(q\in\mathbb{Z}^{+}\).

Prove by mathematical induction that \(5^{2n}+3n-1\) is divisible by 9 for all \(n\in\mathbb{Z}^{+}\).

Find \(f'(x)\) and \(f''(x)\).

Prove by mathematical induction that \(f^{(n)}(x)=2^{n-1}(2x+n)\,\mathrm{e}^{2x}\), for \(n\in\mathbb{Z}^{+}\).

Hence write down, in terms of \(n\), the solution of the equation \(f^{(n)}(x)=0\).

Show that \(P(4)\) is false.

Prove by mathematical induction that \(P(n)\) is true for all integers \(n\ge5\).

The argument used in the inductive step of part (b) is valid for every \(k\ge2\). Explain why this does not mean that \(P(n)\) is true for all \(n\in\mathbb{Z}^{+}\).

Prove by mathematical induction that \(\displaystyle\prod_{r=2}^{n}\left(1-\frac{1}{r^{2}}\right)=\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\cdots\left(1-\frac{1}{n^{2}}\right)=\frac{n+1}{2n}\), for all integers \(n\ge2\).

Find the least value of \(n\) for which the product is less than \(0.501\).

Prove by mathematical induction that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x^{n}\right)=nx^{n-1}\), for \(n\in\mathbb{Z}^{+}\).

Hence, by writing \(x^{-m}=\dfrac{1}{x^{m}}\), show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x^{-m}\right)=-mx^{-m-1}\), for \(m\in\mathbb{Z}^{+}\) and \(x\neq0\).

Show that, if the claim is true for \(n=k\), then it is true for \(n=k+1\).

Show that the claim is false for \(n=1\).

Explain why part (a) does not prove the claim.

Write down the correct expression for \(\displaystyle\sum_{r=1}^{n}r(r+1)\).

Prove by contradiction that \(\log_{2}3\) is irrational.

Hence prove that \(\log_{2}12\) is irrational.

Prove by contradiction that there are no integers \(a\) and \(b\) such that \(a^{2}-4b=2\).

Hence prove that there is no quadratic equation \(x^{2}+px+q=0\), with \(p,q\in\mathbb{Z}\), whose two roots differ by \(\sqrt{2}\).

Show that the equation \(f(x)=0\) has exactly one real root, and that this root lies between \(-1\) and \(0\).

Prove by contradiction that this root is irrational.

Show that the system can be reduced to \(x+y+z=2\), \(y+2z=3\), \((a-4)z=b-7\).

Write down the value of \(a\) for which the system does not have a unique solution.

For this value of \(a\),

find the value of \(b\) for which the system has infinitely many solutions;

find the general solution of the system in this case.

For the value of \(a\) found in part (b) and \(b\neq7\), describe the geometric arrangement of the three planes represented by the equations.

Show that \(3E+F=-9\), and find two further linear equations in \(D\), \(E\) and \(F\).

Solve the system of equations to find \(D\), \(E\) and \(F\).

Hence find the coordinates of the centre and the radius of the circle.

By writing \(x=y-d\) and \(z=y+d\), use the first two equations to find the value of \(y\) and the value of \(d\).

Hence find the value of \(k\).

Show that the system has a unique solution for every value of \(k\).

Find the term independent of \(x\).

Hence find the term independent of \(x\) in the expansion of \((1+x^3)\left(x^2-\dfrac{2}{x}\right)^6\).

Find the first four terms of the binomial expansion of \(f(x)\) in ascending powers of \(x\).

State the set of values of \(x\) for which the expansion is valid.

By substituting \(x=\frac{1}{10}\), find an approximation for \(\sqrt{10}\), giving your answer as a single fraction.

Use the binomial theorem to express \(z^5\) in the form \(a+b\mathrm{i}\), where \(a,b\in\mathbb{Z}\).

Hence write down \((2-\mathrm{i})^5\) and find \((2+\mathrm{i})^5+(2-\mathrm{i})^5\).

By considering \((2+\mathrm{i})^5(2-\mathrm{i})^5\), show that \(38^2+41^2=5^5\).

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

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

State the values of \(x\) for which the expansion is valid.

Show that, for \(k\neq7\), the system has a unique solution, and find this solution. Show that it does not depend on \(k\).

For \(k=7\), explain why the system has infinitely many solutions, and find the general solution.

Show that, using the substitution \(u=3^{x}\), the equation becomes \(u^{2}-12u+27=0\).

Hence solve the equation for \(x\).