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 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 \(k\).

Show that the sum is equal to \(n(n+2k)\).

Explain why, if the sum is 120, both \(n\) and \(n+2k\) must be even.

Hence find all the ways in which 120 can be written as a sum of at least two consecutive odd positive integers.

Show that \((n-2)^{2}+(n-1)^{2}+n^{2}+(n+1)^{2}+(n+2)^{2}\equiv5\left(n^{2}+2\right)\).

By considering the possible final digits of \(n^{2}\), show that \(n^{2}+2\) is never a multiple of 5.

Hence prove that the sum of the squares of five consecutive integers is never a perfect square.

Show that \(\log_{2}5=pq\).

Hence express \(\log_{12}45\) in terms of \(p\) and \(q\).

For stage \(n\):

write down an expression for the number of sides;

show that the perimeter is \(P_{n}=27\left(\dfrac{4}{3}\right)^{n}\) cm.

Find the first stage at which the perimeter is greater than 100 cm.

Show that the areas added at stages \(1, 2, 3, \ldots\) form a geometric sequence with first term \(\dfrac{27\sqrt3}{4}\) cm² and common ratio \(\dfrac{4}{9}\).

Find the exact area that the shapes approach as \(n\to\infty\).

Comment on the perimeter and the area of the shape as \(n\to\infty\).

Show that \(\dfrac{{}^{n}C_{r}}{{}^{n}C_{r-1}}=\dfrac{n-r+1}{r}\).

In the expansion of \((1+x)^{n}\), the coefficients of \(x^{r-1}\), \(x^{r}\) and \(x^{r+1}\) are in the ratio \(1:3:6\). Find the value of \(n\) and the value of \(r\).

Express \(f(x)\) in partial fractions.

Hence find the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^{3}\).

Show that the coefficient of \(x^{n}\) in the expansion of \(f(x)\) is \(2^{n+1}+(-1)^{n}\).

State the set of values of \(x\) for which the expansion of \(f(x)\) is valid, justifying your answer.

Find the value of \(n\) for which the coefficient of \(x^{n}\) is 2049.

By taking the modulus of both sides, show that \(|z|=0\) or \(|z|=1\).

Find all the solutions of the equation, giving your answers in the form \(a+bi\), where \(a,b\in\mathbb{R}\).

Write down the complex number represented by \(\overrightarrow{\text{PQ}}\).

Explain why the complex number represented by \(\overrightarrow{\text{PR}}\) is \((4-2i)\,\mathrm{e}^{\frac{\pi}{3}i}\).

Find the complex number represented by R, giving your answer in the form \(a+bi\), where \(a,b\) are exact.

Find the exact area of triangle PQR.

Write \(w\) in the form \(r\mathrm{e}^{i\theta}\).

Show that \(w^{n+1}-w^{n}=i\,w^{n}\).

Hence explain why, for every \(n\), triangle \(\text{OP}_{n}\text{P}_{n+1}\) is right-angled and isosceles.

Show that the area of triangle \(\text{OP}_{n}\text{P}_{n+1}\) is \(2^{n-1}\).

Find the least value of \(N\) for which the total area of the triangles \(\text{OP}_{0}\text{P}_{1},\ \text{OP}_{1}\text{P}_{2},\ \ldots,\ \text{OP}_{N-1}\text{P}_{N}\) exceeds 1000.

Find the least positive integer \(n\) for which \(\text{P}_{n}\) lies on the positive real axis, and write down the distance \(\text{OP}_{n}\) for this \(n\).

Show that \(1+\omega+\omega^{2}+\omega^{3}+\omega^{4}=0\).

Hence show that \(\cos\dfrac{2\pi}{5}+\cos\dfrac{4\pi}{5}=-\dfrac12\).

Hence find the exact value of \(\cos\dfrac{2\pi}{5}\).

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

Find the least positive integer \(n\) for which \(w^{n}\) is real, and find the value of \(w^{n}\) for this \(n\).

Use De Moivre’s theorem to show that \(\cos4\theta=8c^{4}-8c^{2}+1\).

Hence show that the solutions of the equation \(8x^{4}-8x^{2}+1=0\) are \(x=\cos\dfrac{k\pi}{8}\) for \(k=1,3,5,7\).

By solving \(8x^{4}-8x^{2}+1=0\) as a quadratic equation in \(x^{2}\), find the exact value of \(\cos^{2}\dfrac{\pi}{8}\).

By considering the product of the roots of \(8x^{4}-8x^{2}+1=0\), find the exact value of \(\cos\dfrac{\pi}{8}\cos\dfrac{3\pi}{8}\).

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

Hence solve the equation \(\sin x+\sin3x+\sin5x+\sin7x=0\), for \(0\lt x\lt\pi\).

Prove by contradiction that \(\sqrt{6}\) is irrational.

Hence prove by contradiction that \(\sqrt{2}+\sqrt{3}\) is irrational.

By considering \(\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)\), deduce that \(\sqrt{3}-\sqrt{2}\) is irrational.

Show that 7 cannot be written in this form, and hence explain why the conjecture is false.

Show that, for every integer \(k\), \(k^{2}\) leaves a remainder of 0, 1 or 4 when divided by 8.

Hence prove that no integer of the form \(8n+7\), \(n\in\mathbb{Z}\), can be written as the sum of three integer squares.

Show that 101 and 102 can each be written as the sum of three integer squares, and hence find the least integer greater than 100 that cannot.

Show that the product of two integers of the form \(4k+1\) is also of the form \(4k+1\).

Hence explain why every positive integer of the form \(4k+3\) has at least one prime factor of the form \(4k+3\).

Assume that there are only finitely many primes of the form \(4k+3\), and that they are \(3,\,p_{2},\,p_{3},\,\ldots,\,p_{m}\). Let \(N=4\left(3p_{2}p_{3}\cdots p_{m}\right)-1\).

Show that \(N\) is of the form \(4k+3\).

Explain why \(N\) is not divisible by any of \(3,\,p_{2},\,\ldots,\,p_{m}\).

Hence complete a proof by contradiction that there are infinitely many primes of the form \(4k+3\).

A student suggests that the number \(N\) constructed from any finite list of such primes is itself prime.

Find \(N\) for the list \(3,\,7\), and show that it is prime.

Find \(N\) for the list \(3,\,7,\,11\), and hence show that the student’s suggestion is false.

Show that the system reduces to \(x+y+az=1\), \((a-1)(y-z)=0\), \((1-a)(a+2)z=b-a\).

For \(a\neq1\) and \(a\neq-2\), find the unique solution, giving \(x\), \(y\) and \(z\) in terms of \(a\) and \(b\).

Consider the case \(a=1\).

State the condition on \(b\) for the system to be consistent, and describe the solution set in this case.

Describe the geometric arrangement of the three planes when \(b\neq1\).

Consider the case \(a=-2\).

Find the value of \(b\) for which the system is consistent.

For this value of \(b\), find the general solution of the system.

Hence find the solution when \(a=3\) and \(b=5\).

Find \(Q_{10}\).

Show that \(P_{n}\gt Q_{n}\) when \(\left(\dfrac{4}{3}\right)^{n-1}\gt \dfrac{200}{3}\).

Hence find the first hour at the start of which the area of colony P is greater than the area of colony Q.

Find \(p(1)\).

Show that \(p(1)+p(2)+\cdots+p(9)=1\).

Find the exact proportion of numbers whose leading digit is at least 5, giving your answer as a single logarithm.

Find the least value of \(k\) for which the proportion of numbers whose leading digit is at most \(k\) is greater than 0.75.

Find the number of different tickets possible.

Find the number of tickets that match exactly 4 of the 6 winning numbers.

The organisers want to choose 6 numbers from the integers 1 to \(n\), so that there are at least 20 000 000 possible tickets. Find the least value of \(n\).

\(n^{2}+n+17\) is prime for all \(n\in\mathbb{Z}^{+}\).

\(2^{p}-1\) is prime for every prime number \(p\).

For all real numbers \(a\) and \(b\), if \(a^{2}\gt b^{2}\) then \(a\gt b\).

Write down a system of three linear equations in \(x\), \(y\) and \(z\).

Find the value of \(x\), of \(y\) and of \(z\).

Mia wonders whether she could split €20 000 so that the amount in C is still €2000 less than the total in A and B, but the first-year interest is €1100. Show that this is not possible.

Find the price of a coffee, of a sandwich and of a pastry.

A fourth office orders 45 coffees, 20 sandwiches and 18 pastries. Find the total cost of this order.

Find the number of different passwords of length 16. Give your answer in the form \(a\times10^{k}\), where \(1\le a\lt 10\), \(k\in\mathbb{Z}\) and \(a\) is given correct to three significant figures.

Find the time, in years, that the computer needs to test every password of length 16. Give your answer in the form \(a\times10^{k}\), where \(1\le a\lt 10\) and \(k\in\mathbb{Z}\).

Find the least password length for which the computer needs more than \(10^{9}\) years to test every password of that length.

Find the percentage of passwords of length 16 in which no symbol is used more than once.

Find the value of Ana’s investment after 5 years. Give your answer to the nearest cent.

Find the least number of complete years after which the value of Ana’s investment is greater than the value of Ben’s investment.

Carla invests €11 000 in an account that pays a nominal annual interest rate of \(r\%\), compounded quarterly. After 8 years the value of Carla’s investment is equal to the value of Ana’s investment. Find the value of \(r\), correct to two decimal places.

Find \(z_{1}+z_{2}\) in the form \(a+bi\).

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

By considering the parallelogram formed by \(z_{1}\), \(z_{2}\) and \(z_{1}+z_{2}\) on an Argand diagram, explain why \(|z_{1}+z_{2}|^{2}=34+30\cos1.5\).

Find the apparent power.

Find the power factor.

A capacitor bank is installed. It reduces \(Q\) but leaves \(P\) unchanged, and the power factor becomes 0.95. Find the new value of \(Q\).

Find \(\dfrac{w}{z}\) in the form \(a+bi\).

Hence find the size of angle ZOW.

Find the area of triangle OZW.

Find the modulus and argument of \(3+4i\).

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

The roots are represented on an Argand diagram by the vertices of a regular pentagon. Find the perimeter of the pentagon.

Write down a system of three linear equations in \(a\), \(b\) and \(c\).

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

Find the maximum height of the ball according to the model.

Find the time at which the ball reaches the ground.

Interpret the value of \(c\) in this context.

Show that the annual rate of depreciation is 16.7%, correct to three significant figures.

Find the value of the car after 6 years.

Marta intends to replace the car with a new model that costs €31 200 today. The price of the new model increases by 2.3% at the end of every year. Find the price of the new model in 6 years’ time.

Find the least number of complete years after which the value of Marta’s car is less than half of the price of the new model at that time.

Marta will replace the car after 6 years. She will sell her car for its value after 6 years and pay the rest of the price of the new model from savings. Find the amount she must invest today, at a nominal annual interest rate of 4.5% compounded monthly, so that she has exactly the amount she needs after 6 years.

Marta instead has €20 000 to invest today, compounded monthly. Find the least nominal annual interest rate, correct to two decimal places, that would give her at least the amount she needs after 6 years.

Find \(A_{1}\) and \(A_{2}\).

Show that \(A_{n}=60\,000-10\,000(1.05)^{n}\).

Find the number of full withdrawals of €3000 that Lina can make.

Lina’s brother also invests €50 000 in this account, and withdraws €\(W\) at the end of each year, immediately after the interest has been added. Find the greatest value of \(W\) for which the amount in his account never decreases.

Find \(|Z_{1}|\) and \(\arg Z_{1}\).

Show that \(\dfrac{1}{Z_{1}}=0.012-0.016i\).

Show that \(\dfrac{1}{Z_{2}}=\dfrac{20+Xi}{400+X^{2}}\).

The circuit is tuned so that the total impedance \(Z\) is a real number. Find the two possible values of \(X\).

For the larger value of \(X\), find \(Z\).

For the smaller value of \(X\), a real voltage of 230 V is applied across the circuit. The current \(I\), in amperes, is given by \(I=\dfrac{230}{Z}\). Find \(I\).

Write down a system of three linear equations in \(x\), \(y\) and \(z\).

Show that the system does not have a unique solution.

Find the general solution of the system, using \(z=\lambda\).

Find the range of possible values of \(\lambda\) in this context.

Fertilisers A, B and C cost €1.20, €1.50 and €2.00 per kg respectively. Find the minimum cost of a suitable mixture, and the masses of A, B and C used.

Show that a mixture containing exactly 4 kg of nitrogen, 3 kg of phosphate and 3.5 kg of potassium cannot be made.

Consider \(m=1\).

Express \(\dfrac{1}{r(r+1)}\) in partial fractions.