Courses

Let $f(x)$ be defined and positive for $x > 0$. Let $a$ and $b$ be real numbers with $0 < a < b$ and...

\begin{center} \begin{tikzpicture} % Setting up the same viewport/dimensions \clip (-0.45,1.84) rect...

\begin{questionparts} \item \noindent\vspace{-4cm} %%%%%%%The diagram requires scale of 1 unit = 15...

A circle $C$ is said to be {\em bisected} by a curve $X$ if $X$ meets $C$ in exactly two points and ...

The line passing through the point $(a,0)$ with gradient $b$ intersects the circle of unit radius ce...

The angle $A$ of triangle $ABC$ is a right angle and the sides $BC$, $CA$ and $AB$ are of lengths $...

\begin{questionparts} \item The point $A$ has coordinates $\l 5 \, , 16 \r$ and the point $B$ has c...

The three points $A$, $B$ and $C$ have coordinates $\l p_1 \, , \; q_1 \r$, $\l p_2 \, , \; q_2 \r$...

The line $y=d\,$, where $d>0\,$, intersects the circle $x^2+y^2=R^2$ at $G$ and $H$. Show that the a...

A pyramid stands on horizontal ground. Its base is an equilateral triangle with sides of length~$a...

Show that the equation of any circle passing through the points of intersection of the ellipse $(x+...

The points $A$, $B$ and $C$ lie on the sides of a square of side 1 cm and no two points lie on th...

Sketch on the same axes the two curves $C_1$ and $C_2$, given by \begin{center} \begin{align*} C_1...

A point moves in the $x$-$y$ plane so that the sum of the squares of its distances from the three fi...

Consider a fixed square $ABCD$ and a variable point $P$ in the plane of the square. We write the per...

The diagram shows a circle, of radius $r$ and centre $I$, touching the three sides of a triangle $AB...

The famous film star Birkhoff Maclane is sunning herself by the side of her enormous circular swimmi...

Show that the equation \[ ax^{2}+ay^{2}+2gx+2fy+c=0 \] where $a>0$ and $f^{2}+g^{2}>ac$ represents ...

My house has an attic consisting of a horizontal rectangular base of length $2q$ and breadth $2p$ (w...

\begin{questionparts} \item Prove that the intersection of the surface of a sphere with a plane is a...

Prove that the area of the zone of the surface of a sphere between two parallel planes cutting the s...

\begin{center} \begin{tikzpicture}[scale = 3] \draw[domain = 0:180, samples=50, variable...

Two points $P$ and $Q$ lie within, or on the boundary of, a square of side 1cm, one corner of which ...

\begin{questionparts} \item In the cubic equation $x^3-3pqx+pq(p+q)=0\,$, where $p$ and $q$ are di...

\begin{questionparts} \item The function f satisfies, for all $x$, the equation \[ \f(x) + (1- x)...

It is given that the two curves \[ y=4-x^2 \text{ and } m x = k-y^2\,, \] where $m > 0$, touch ex...

Given that \[ 5x^{2}+2y^{2}-6xy+4x-4y\equiv a\left(x-y+2\right)^{2} +b\left(cx+y\right)^{2}+d\,, \]...

Two curves have equations $\; x^4+y^4=u\;$ and $\; xy = v\;$, where $u$ and $v$ are positive constan...

A transformation $T$ of the real numbers is defined by \[ y=T(x)=\frac{ax-b}{cx-d}\,, \] where $a,b...

It is given that $x,y$ and $z$ are distinct and non-zero, and that they satisfy \[ x+\frac{1}{y}=...

Let $\mathrm{h}(x)=ax^{2}+bx+c,$ where $a,b$ and $c$ are constants, and $a\neq0$. Give a condition w...

The numbers $x,y$ and $z$ are non-zero, and satisfy \[ 2a-3y=\frac{\left(z-x\right)^{2}}{y}\quad\mbo...

If we split a set $S$ of integers into two subsets $A$ and $B$ whose intersection is empty and whose...

The sequence $u_0, u_1, \ldots$ is said to be a constant sequence if $u_n = u_{n+1}$ for $n = 0, 1, ...

Consider the following steps in a proof that $\sqrt{2} + \sqrt{3}$ is irrational. \begin{enumerate} ...

Find the set of positive integers $n$ for which $n$ does not divide $(n-1)!.$ Justify your answer. ...

\begin{questionparts} \item Find all pairs of positive integers $(n,p)$, where $p$ is a prime number...

The sequence of numbers $x_0$, $x_1$, $x_2$, $\ldots$ satisfies \[ x_{n+1} = \frac{ax_n-1}{x_n+b} \...

The set $S$ % = \{1, 5, 9, 13, \,\ldots \}$ consists of all the positive integers that leave a rem...

An operator $\rm D$ is defined, for any function $\f$, by \[ {\rm D}\f(x) = x\frac{\d\f(x)}{\d x} .\...

\begin{questionparts} \item In the following argument to show that $\sqrt2$ is irrational, give proo...

If $s_1$, $s_2$, $s_3$, $\ldots$ and $t_1$, $t_2$, $t_3$, $\ldots$ are sequences of positive numbers...

Show that: \begin{questionparts} \item $1+2+3+ \cdots + n = \frac12 n(n+1)$; \item if ...

\begin{questionparts} \item Let $\.f(x) = (x+2a)^3 -27 a^2 x$, where $a\ge 0$. By sketching $\.f(...

\textit{All numbers referred to in this question are non-negative integers.} \begin{questionparts} \...

In this question, you may assume that, if $a$, $b$ and $c$ are positive integers such that $a$ and $...

\begin{questionparts} \item Write down a solution of the equation \[ x^2-2y^2 =1\,, \tag{$*$} \] for...

\begin{questionparts} \item The point with coordinates $(a, b)$, where $a$ and $b$ are rational nu...

Write down the cubes of the integers $1, 2, \ldots , 10$. The positive integers $x$, $y$ and $z$, wh...

\begin{questionparts} \item The numbers $m$ and $n$ satisfy \[ m^3=n^3+n^2+1\,. \tag{$*$} \] \begin{...

The vertices $A$, $B$, $C$ and $D$ of a square have coordinates $(0,0)$, $(a,0)$, $(a,a)$ and $(0,a)...

\begin{questionparts} \item Suppose that $a$, $b$ and $c$ are integers that satisfy the equation \...

A {\em proper factor} of an integer $N$ is a positive integer, not $1$ or $N$, that divides $N$. \b...

A sequence of points $(x_1,y_1)$, $(x_2,y_2)$, $\ldots$ in the cartesian plane is generated by first...

The polynomial $\p(x)$ is given by \[ \ds \p(x)= x^n +\sum\limits_{r=0}^{n-1}a_rx^r\,, \] where $...

Prove that, if $c\ge a$ and $d\ge b$, then \[ ab+cd\ge bc+ad\,. \tag{$*$} \] \begin{questionparts} ...

What does it mean to say that a number $x$ is \textit{irrational}? Prove by contradiction statements...

$\triangle$ is an operation that takes polynomials in $x$ to polynomials in $x$; that is, given any...

\begin{questionparts} \item Show that $\displaystyle \big( 5 + \sqrt {24}\;\big)^4 + \frac{1 }{\bi...

\begin{questionparts} \item Show that, if $\l a \, , b\r$ is \textbf{any} point on the curve $x^2 -...

In this question $b$, $c$, $p$ and $q$ are real numbers. \begin{questionparts} \item By considering ...

Find the integer, $n$, that satisfies $n^2 < 33\,127< (n+1)^2$. Find also a small integer $m$ such t...

For any positive integer $N$, the function $\f(N)$ is defined by \[ \f(N) = N\Big(1-\frac1{p_1}\Big)...

The positive integers can be split into five distinct arithmetic progressions, as shown: \begin{ali...

Prove that the cube root of any irrational number is an irrational number. Let $\ds u_n = {5\vphanto...

Let $k$ be an integer satisfying $0\le k \le 9\,$. Show that $0\le 10k-k^2\le 25$ and that $k(k-1)...

The first question on an examination paper is: \hspace*{3cm} Solve for $x$ the equation \ \ \ $\ds \...

The numbers $x_n$, where $n=0$, $1$, $2$, $\ldots$ , satisfy \[ x_{n+1} = kx_n(1-x_n) \;. \] \begi...

A number of the form $1/N$, where $N$ is an integer greater than 1, is called a {\it unit fraction}...

A polyhedron is a solid bounded by $F$ plane faces, which meet in $E$ edges and $V$ vertices. You ma...

Show that, if $n$ is an integer such that $$(n-3)^3+n^3=(n+3)^3,\quad \quad {(*)}$$ then $n$ is even...

Find the integers $k$ satisfying the inequality $k\leqslant2(k-2).$ Given that $N$ is a strictly po...

A {\sl proper factor} of a positive integer $N$ is an integer $M$, with $M\ne 1$ and $M\ne N$, which...

Let $n$ be a positive integer. \begin{questionparts} \item Factorise $n^{5}-n^{3},$ and show that i...

The Tour de Clochemerle is not yet as big as the rival Tour de France. This year there were five rid...

In this question we consider only positive, non-zero integers written out in the usual (decimal) way...

For the real numbers $a_1$, $a_2$, $a_3$, $\ldots$, \begin{questionparts} \item prove that $a_1^2+a_...

In the game of ``Colonel Blotto'' there are two players, Adam and Betty. First Adam chooses three no...

\begin{questionparts} \item Find all the integer solutions with $1\leqslant p\leqslant q\leqslant r$...

Today's date is June 26th 1992 and the day of the week is Friday. Find which day of the week was Apr...

\begin{questionparts} \itemSuppose that the real number $x$ satisfies the $n$ inequalities \begin{al...

Six points $A,B,C,D,E$ and $F$ lie in three dimensional space and are in general positions, that is,...

Each of $m$ distinct points on the positive $y$-axis is joined by a line segment to each of $n$ dist...

Let $\lfloor x \rfloor$ denote the largest integer that satisfies $\lfloor x \rfloor \leq x$. For ex...

The \textit{Bernoulli polynomials} $P_{n}(x)$, where $n$ is a non-negative integer, are defined by $...

Let \[ S_n = \sum_{r=1}^n \frac 1 {\sqrt r \ } \,, \] where $n$ is a positive integer. \begin{questi...

Two sequences are defined by $a_1 = 1$ and $b_1 = 2$ and, for $n \ge 1$, \begin{equation*} \begin{sp...

Let \[ T _n = \left( \sqrt{a+1} + \sqrt a\right)^n\,, \] where $n$ is a positive integer and $a$...

The functions ${\rm T}_n(x)$, for $n=0$, 1, 2, $\ldots\,$, satisfy the recurrence relation \[ {\rm T...

A sequence of numbers, $F_1$, $F_2$, $\ldots$, is defined by $ F_1=1$, $F_2=1$, and \[ F_n=F_{n-1}+...

It is given that $\sum\limits_{r=-1}^ {n} r^2$ can be written in the form $pn^3 +qn^2+rn+s\,$, whe...

The $n$th Fermat number, $F_n$, is defined by \[ F_n = 2^{2^n} +1\, , \ \ \ \ \ \ \ n=0, \ 1, \ 2, ...

Let $$ {\rm S}_n(x)=\mathrm{e}^{x^3}{{\d^n}\over{\d x^n}}{(\mathrm{e}^{-x^3})}. $$ Show that ${\rm ...

Suppose that $$3=\frac{2}{ x_1}=x_1+\frac{2}{ x_2} =x_2+\frac{2}{ x_3}=x_3+\frac{2}{ x_4}=\cdots.$$ ...

The Fibonacci numbers $F_{n}$ are defined by the conditions $F_{0}=0$, $F_{1}=1$ and \[F_{n+1}=F_{n}...

I have $n$ fence posts placed in a line and, as part of my spouse's birthday celebrations, I wish to...

Suppose that $a_{i}>0$ for all $i>0$. Show that \[ a_{1}a_{2}\leqslant\left(\frac{a_{1}+a_{2}}{2}\r...

Let $\mathrm{g}(x)=ax+b.$ Show that, if $\mathrm{g}(0)$ and $\mathrm{g}(1)$ are integers, then $\mat...

A plane contains $n$ distinct given lines, no two of which are parallel, and no three of which inter...

In both parts of this question, $x$ is real and $0 < \theta < \pi$. \begin{questionparts} \item By c...

The triangle $ABC$ has side lengths $\left| BC \right| = a$, $\left| CA \right| = b$ and $\left| AB...

In the triangle $ABC$, angle $BAC = \alpha$ and angle $CBA= 2\alpha$, where $2\alpha$ is acute, and...

A prison consists of a square courtyard of side $b$ bounded by a perimeter wall and a square buildi...

In the triangle $ABC$, the base $AB$ is of length 1 unit and the angles at~$A$ and~$B$ are $\alpha...

A cyclic quadrilateral $ABCD$ has sides $AB$, $BC$, $CD$ and $DA$ of lengths $a$, $b$, $c$ and $d$, ...

Each edge of the tetrahedron $ABCD$ has unit length. The face $ABC$ is horizontal, and $P$ is the p...

Prove that \[ \tan \left ( \tfrac14 \pi -\tfrac12 x \right)\equiv \sec x -\tan x\,. \tag{$...

\begin{questionparts} \item The equation of the circle $C$ is \[ (x-2t)^2 +(y-t)^2 =t^2, \] where ...

The sides of a triangle have lengths $p-q$, $p$ and $p+q$, where $p>q> 0\,$. The largest and smalle...

{\it Note that the volume of a tetrahedron is equal to $\frac1 3$ $\times$ the area of the base ...

Given that $\alpha$ and $\beta$ are acute angles, show that $\alpha + \beta = \tfrac{1}{2}\pi$ if ...

Arthur and Bertha stand at a point $O$ on an inclined plane. The steepest line in the plane throug...

Let $f(x) = 7 - 2|x|$. A sequence $u_0, u_1, u_2, \ldots$ is defined by $u_0 = a$ and $u_n = f(u_{n-...

\begin{questionparts} \item The function $\f$ is defined by $\f(x)= |x-a| + |x-b| $, where $a < b$...

Sketch the following subsets of the $x$-$y$ plane: \begin{questionparts} \item $|x|+|y|\le 1$ ; \ite...

Find all the solutions of the equation \[|x+1|-|x|+3|x-1|-2|x-2|=x+2.\]...

The function $\mathrm{f}$ is defined for $x<2$ by \[ \mathrm{f}(x)=2| x^{2}-x|+|x^{2}-1|-2|x^{2}+x|...

The domain of the function f is the set of all $2 \times 2$ matrices and its range is the set of rea...

For any square matrix $\mathbf{A}$ such that $\mathbf{I-A}$ is non-singular (where $\mathbf{I}$ is...

For any two points $X$ and $Y$, with position vectors $\bf x$ and $\bf y$ respectively, $X*Y$ is def...

Verify that if \[ \mathbf{P}=\begin{pmatrix}1 & 2\\ 2 & -1 \end{pmatrix}\qquad\mbox{ and }\qquad\ma...

\textit{In this question, }\textbf{\textit{$\mathbf{A},\mathbf{B}$ }}\textit{and $\mathbf{X}$ are no...

The matrices $\mathbf{I}$ and $\mathbf{J}$ are \[ \mathbf{I}=\begin{pmatrix}1 & 0\\ 0 & 1 \end{pmat...

Two square matrices $\mathbf{A}$ and $\mathbf{B}$ satisfies $\mathbf{AB=0}.$ Show that either $\det\...

In a crude model of population dynamics of a community of aardvarks and buffaloes, it is assumed tha...

\begin{questionparts} \item $x_2$ and $y_2$ are defined in terms of $x_1$ and $y_1$ by the equation ...

The matrix A is given by $$\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$ \begin{quest...

Show that, if the lengths of the diagonals of a parallelogram are specified, then the parallogram ha...

Let $R_{\alpha}$ be the $2\times2$ matrix that represents a rotation through the angle $\alpha$ and ...

The transformation $T$ of the point $P$ in the $x$,$y$ plane to the point $P'$ is constructed as fol...

The transformation $T$ from $\begin{pmatrix} x \\ y \end{pmatrix}$ to $\begin{pmatrix} X \\ Y \end{p...

The linear transformation $\mathrm{T}$ is a shear which transforms a point $P$ to the point $P'$ def...

A pyramid has a horizontal rectangular base $ABCD$ and its vertex $V$ is vertically above the centre...

You are not required to consider issues of convergence in this question. For any sequence of numbers...

Show that if at least one of the four angles $A\pm B\pm C$ is a multiple of $\pi$, then \begin{alig...

In this question, you may use the following identity without proof: \[ \cos A + \cos B = 2\cos\tfra...

A function $\f(x)$ is said to be {\em concave} for $a< x < b$ if \[ \ t\,\f(x_1) +(1-t)\,\f(...

Use the identity \[ 2 \sin P\,\sin Q = \cos(Q-P)-\cos(Q+P)\, \] to show that \[ 2\sin\theta \,\big...

The points $R$ and $S$ have coordinates $(-a,\, 0)$ and $(2a,\, 0)$, respectively, where $a > 0\,$. ...

In this question, the $\mathrm{arctan}$ function satisfies $0\le \arctan x <\frac12 \pi$ for $x\ge0...

\begin{questionparts} \item The continuous function $\f$ is defined by \[ \tan \f(x) = x \ \ \ \...

\begin{questionparts} \item Show that $\cos 15^\circ = \dfrac{\sqrt3 +1}{2\sqrt2}$ and find a simila...

\begin{questionparts} \item The sequence of numbers $u_0, u_1, \ldots $ is given by $u_0=u$ an...

A thin circular path with diameter $AB$ is laid on horizontal ground. A vertical flagpole is erected...

\begin{questionparts} \item Find all the values of $\theta$, in the range $0^\circ <\theta<180^\cir...

Prove the identity \[ 4\sin\theta \sin(\tfrac13\pi-\theta) \sin (\tfrac13\pi+\theta)= \sin 3\theta\...

The points $P$, $Q$ and $R$ lie on a sphere of unit radius centred at the origin, $O$, which is fi...

Show that \[ \sin(x+y) -\sin(x-y) = 2 \cos x \, \sin y \] and deduce that \[ \sin A - \sin B = 2 ...

A curve has the equation $y=\f(x)$, where \[ \f(x) = \cos \Big( 2x+ \frac \pi 3\Big) + \sin \Big ( \...

The point $P$ has coordinates $(x,y)$ with respect to the origin $O$. By writing $x=r\cos\theta$ and...

In this question, $\f^2(x)$ denotes $\f(\f(x))$, $\f^3(x)$ denotes $\f( \f (\f(x)))\,$, and so on. ...

Given that $\cos A$, $\cos B$ and $\beta$ are non-zero, show that the equation \[ \alpha \sin(A-B) +...

\begin{questionparts} \item Given that $A = \arctan \frac12$ and that $B = \arctan\frac13\,$ (where ...

Show that $\sin A = \cos B$ if and only if $A = (4n+1)\frac{\pi}{2}\pm B$ for some integer $n$. Show...

The positive numbers $a$, $b$ and $c$ satisfy $bc=a^2+1$. Prove that $$ \arctan\left(\frac1 {a+b}\r...

The notation $\displaystyle \prod^n_{r=1} \f (r)$ denotes the product $\f (1) \times \f (2) \times ...

\begin{questionparts} \item Given that $\displaystyle \cos \theta = \frac35$ and that $\displaystyle...

Show that if $\, \cos(x - \alpha) = \cos \beta \,$ then either $\, \tan x = \tan ( \alpha + \beta)...

The triangle $OAB$ is isosceles, with $OA = OB$ and angle $AOB = 2 \alpha$ where $0< \alpha < {\pi ...

$\,$ \setlength{\unitlength}{1cm} \begin{center} \hspace{2cm} \begin{picture}(6,3.5) \put(-1.5,4.3){...

Write down a value of $\theta\,$ in the interval $\frac{1}{4}\pi< \theta <\frac{1}{2}\pi$ that satis...

Solve the inequality $$\frac{\sin\theta+1}{\cos\theta}\le1\;$$ where $0\le\theta<2\pi\,$ and $\cos\t...

\begin{questionparts} \item Show that $ 2\sin(\frac12\theta)=\sin \theta$ if and only if $\sin(\fr...

Prove that $\displaystyle \arctan a + \arctan b = \arctan \l {a + b \over 1-ab} \r\,$ when $0 < a ...

The lines $l_1$, $l_2$ and $l_3$ lie in an inclined plane $P$ and pass through a common point $A...

In this question, the function $\sin^{-1}$ is defined to have domain $ -1\le x \le 1$ and range \l...

Let $$ \f(x) = P \, {\sin x} + Q\, {\sin 2x} + R\, {\sin 3x} \;. $$ Show that if $Q^2 < 4R(P-R)$, ...

Show that $\displaystyle \tan 3\theta = \frac{3\tan\theta -\tan^3\theta}{1-3\tan^2\theta}$ . Given t...

Show that if $\alpha$ is a solution of the equation $$ 5{\cos x} + 12{\sin x} = 7, $$ then either ...

Let $a_{1}=\cos x$ with $0 < x < \pi/2$ and let $b_{1}=1$. Given that \begin{eqnarray*} a_{n+1}&=&{\...

Which of the following statements are true and which are false? Justify your answers. \begin{questio...

Show that, if $\,\tan^2\phi=2\tan\phi+1$, then $\tan2\phi=-1$. Find all solutions of the equation $$...

Let \[ \mathrm{C}_{n}(\theta)=\sum_{k=0}^{n}\cos k\theta \] and let \[ \mathrm{S}_{n}(\theta)=\sum...

Prove by induction, or otherwise, that, if $0<\theta<\pi$, \[ \frac{1}{2}\tan\frac{\theta}{2}+\frac...

The function $\mathrm{f}$ satisfies $\mathrm{f}(0)=1$ and \[ \mathrm{f}(x-y)=\mathrm{f}(x)\mathrm{f...

Let $y=\cos\phi+\cos2\phi$, where $\phi=\dfrac{2\pi}{5}.$ Verify by direct substitution that $y$ sat...

If $\theta+\phi+\psi=\tfrac{1}{2}\pi,$ show that \[ \sin^{2}\theta+\sin^{2}\phi+\sin^{2}\psi+2\sin\...

Prove that if $A+B+C+D=\pi,$ then \[ \sin\left(A+B\right)\sin\left(A+D\right)-\sin B\sin D=\sin A\s...

$\,$ \begin{center} \begin{tikzpicture}[scale=2] % Semicircle \def\r{2}; \coordinate (D)...

Prove that $\cos3\theta=4\cos^{3}\theta-3\cos\theta$. Show how the cubic equation \[ 24x^{3}-72x^{...

The curve $C$ has equation \[ y= a^{\sin (\pi \e^ x)}\,, \] where $a>1$. \begin{questionparts} \item...

The lengths of the sides $BC$, $CA$, $AB$ of the triangle $ABC$ are denoted by $a$, $b$, $c$, respe...

For this question, you may use the following approximations, valid if $\theta $ is small: \ $\sin...

Find the limit, as $n\rightarrow\infty,$ of each of the following. You should explain your reasoning...

Explain briefly, by means of a diagram, or otherwise, why \[ \mathrm{f}(\theta+\delta\theta)\approx...

By considering the graphs of $y=kx$ and $y=\sin x,$ show that the equation $kx=\sin x,$ where $k>0,$...

Show that \[ \cos\left(\frac{\alpha}{2}\right)\cos\left(\frac{\alpha}{4}\right)=\frac{\sin\alpha}{4\...

\begin{questionparts} \item The circle $x^2 + (y-a)^2 = r^2$ touches the parabola $2ky = x^2$, where...

\begin{questionparts} \item Sketch a graph of $y = \frac{\ln x}{x}$ for $x > 0$. \item Use your grap...

The function f satisfies $f(0) = 0$ and $f'(t) > 0$ for $t > 0$. Show by means of a sketch that, fo...

A straight line passes through the fixed point $(1 , k)$ and has gradient $- \tan \theta$, where $k ...

A woman stands in a field at a distance of $a\,\mathrm{m}$ from the straight bank of a river which f...

Find the stationary points of the function $\mathrm{f}$ given by \[ \mathrm{f}(x)=\mathrm{e}^{ax}\co...

The sequence of functions $y_0$, $y_1$, $y_2$, $\ldots\,$ is defined by $y_0=1$ and, for $n\ge1\,$,...

%In this question, %the definition of $a^b$ (for $a>0$) is %$ %a^b = \e^{b \ln a} \,. %$ %\\ The...

A circle of radius $a$ is centred at the origin $O$. A rectangle $PQRS$ lies in the minor sector $O...

For each non-negative integer $n$, the polynomial $\f_n$ is defined by \[ \f_n(x) = 1 + x + \frac{x^...

Differentiate, with respect to $x$, \[ (ax^2+bx+c)\,\ln \big( x+\sqrt{1+x^2}\big) +\big(dx+e\big)\s...

By simplifying $\sin(r+\frac12)x - \sin(r-\frac12)x$ or otherwise show that, for $\sin\frac12 x \n...

An accurate clock has an hour hand of length $a$ and a minute hand of length $b$ (where $b>a$), both...

\begin{questionparts} \item Find the value of $m$ for which the line $y = mx$ touches the curve $...

\begin{questionparts} \item Sketch the curve $y=\f(x)$, where \[ \f(x) = \frac 1 {(x-a)^2 -1} \hspa...

The line $L$ has equation $y=c-mx$, with $m>0$ and $c>0$. It passes through the point $R(a,b)$ and ...

In this question, you may assume without proof that any function $\f$ for which $\f'(x)\ge 0$ is ...

The number $E$ is defined by $\displaystyle E= \int_0^1 \frac{\e^x}{1+x} \, \d x\,.$ Show that \[ ...

Let $P$ be a given point on a given curve $C$. The {\em osculating circle} to $C$ at $P$ is defined...

The curve $\displaystyle y=\Bigl(\frac{x-a}{x-b}\Bigr)\e^{x}$, where $a$ and $b$ are constants, ...

\begin{questionparts} \item The gradient $y'$ of a curve at a point $(x,y)$ satisfies \[ (y')^2 -xy'...

A function $\f(x)$ is said to be \textit{convex} in the interval $a < x < b$ if $\f''(x)\ge0$ for al...

Using the series \[ \e^x = 1 + x +\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+\cdots\,, \] show...

By sketching on the same axes the graphs of $y=\sin x$ and $y=x$, show that, for $x>0$: \begin{que...

Find the three values of $x$ for which the derivative of $x^2 \e^{-x^2}$ is zero. Given that $a$ and...

Show that $(a+b)^2\le 2a^2+2b^2\,$. Find the stationary points on the curve $y=\big(a^2\cos^2\thet...

Let $f(x) = x^m(x-1)^n$, where $m$ and $n$ are both integers greater than $1$. Show that the curve $...

Let \[ {\f}(x)=a x-\frac{x^{3}}{1+x^{2}}, \] where $a$ is a constant. Show that, if $a\ge 9/8$, ...

\begin{questionparts} \item Let $\f(x)=(1+x^2)\e^x$. Show that $\f'(x)\ge 0$ and sketch the graph of...

Prove that $$ \sum_{k=0}^n \sin k\theta = \frac { \cos \half\theta - \cos (n+ \half) \theta} {2\sin...

Let $$ {\rm f}(x)=\sin^2x + 2 \cos x + 1 $$ for $0 \le x \le 2\pi$. Sketch the curve $y={\rm f}(x)$,...

\begin{eqnarray*} {\rm f}(x)&=& \tan x-x,\\ {\rm g}(x)&=& 2-2\cos x-x\sin x,\\ {\rm h}(x)&=& 2x+x\co...

By considering the maximum of $\ln x-x\ln a$, or otherwise, show that the equation $x=a^{x}$ has no ...

A cylindrical biscuit tin has volume $V$ and surface area $S$ (including the ends). Show that the mi...

Given that $a$ is constant, differentiate the following expressions with respect to $x$: \begin{que...

\begin{questionparts} \item Solve the differential equation \[ \frac{\mathrm{d}y}{\mathrm{d}x}-y-3y...

The diagram shows a coffee filter consisting of an inverted hollow right circular cone of height $H$...

According to the Institute of Economic Modelling Sciences, the Slakan economy has alternate years of...

$\ $ \begin{center} \begin{tikzpicture}[scale=1] % Shaded area with lightgray fill \fill[co...

Frosty the snowman is made from two uniform spherical snowballs, of initial radii $2R$ and $3R.$ The...

Prove that, for any integers $n$ and $r$, with $1\leqslant r\leqslant n,$ \[ \binom{n}{r}+\binom{n}{...

Let $\mathrm{f}(x)=\sin2x\cos x.$ Find the 1988th derivative of $\mathrm{f}(x).$ Show that the smal...

The function $\mathrm{f}$ and $\mathrm{g}$ are related (for all real $x$) by \[ \mathrm{g}(x)=\math...

\textit{You need not consider the convergence of the improper integrals in this question.} For $p, q...

You need not consider the convergence of the improper integrals in this question. \begin{questionpar...

\begin{questionparts} \item Let $$f(x) = \frac{x}{\sqrt{x^2 + p}},$$ where $p$ is a non-zero constan...

By first multiplying the numerator and the denominator of the integrand by $(1 - \sin x)$, evaluate ...

The function $f$ is defined, for $x > 1$, by $$f(x) = \int_1^x \sqrt{\frac{t-1}{t+1}} dt.$$ Do not a...

A definite integral can be evaluated approximately by means of the Trapezium rule: \[ \int_{x_{0}}...

Let \[ I=\int_{-\frac{1}{2}\pi}^{\frac{1}{2}\pi}\frac{\cos^{2}\theta}{1-\sin\theta\sin2\alpha}\,\m...

Explain why the use of the substitution $x=\dfrac{1}{t}$ does not demonstrate that the integrals \[...

Let $y=\mathrm{f}(x)$, $(0\leqslant x\leqslant a)$, be a continuous curve lying in the first quadran...

Using the substitution $x=\alpha\cos^{2}\theta+\beta\sin^{2}\theta,$ show that, if $\alpha<\beta$, ...

\begin{questionparts} \item Let \[ \f(x) = \frac 1 {1+\tan x} \] for $0\le x < \frac12\pi\,$. Sho...

The functions $\s$ and $\c$ satisfy $\s(0)= 0\,$, $\c(0)=1\,$ and \[ \s'(x) = \c(x)^2 ,\] \[ \c'(x)=...

The function $\f$ is defined by \[ \phantom{\ \ \ \ \ \ \ \ \ \ \ \ (x>0, \ \ x\ne1)} \f(x) = \...

\textit{In this question, you are not permitted to use any properties of trigonometric functions or ...

For any function $\f$ satisfying $\f(x) > 0$, we define the {\em geometric mean}, F, by \[...

The Schwarz inequality is \[ \left( \int_a^b \f(x)\, \g(x)\,\d x\right)^{\!\!2} \le \left( \int_a^b...

\textbf{Note:} In this question you may use without proof the result $ \dfrac{\d \ }{\d x}\big(\!\ar...

\begin{questionparts} \item The inequality $\dfrac 1 t \le 1$ holds for $t\ge1$. By integrating both...

\begin{questionparts} \item Use the substitution $u= x\sin x +\cos x$ to find \[ \int \frac{x }{x\...

\begin{questionparts} \item Given that \[ \int \frac {x^3-2}{(x+1)^2}\, \e ^x \d x = \frac{\P(x)}...

Show that \[ \int_0^a \f(x) \d x= \int _0^a \f(a-x) \d x\,, \tag{$*$} \] where f is any function fo...

\begin{questionparts} \item Show that \[ \mathrm{sec}^2\left(\tfrac14\pi-\tfrac12 x\right)=\frac{2...

\begin{questionparts} \item The function $\f$ is defined, for $x>0$, by \[ \f(x) =\int_{1}^3 (t-1)^{...

\begin{questionparts} \item Let \[ I = \int_0^1 \bigl((y')^2 -y^2\bigr)\d x \qquad\text{an...

\begin{questionparts} \item By using the substitution $u=1/x$, show that for $b>0$ \[ ...

This question concerns the inequality \begin{equation} \label{eq6:*} \int_0^\pi \bigl( \...

The numbers $a$ and $b$, where $b > a\ge0$, are such that \[ \int_a^b x^2 \d x = \left ( \int_...

\begin{questionparts} \item Show that $\int \ln (2-x) \d x = -(2-x)\ln (2-x) + (2-x) + c \,,\ $ whe...

Given that $t= \tan \frac12 x$, show that $\dfrac {\d t}{\d x} = \frac12(1+t^2)$ and $ \sin x = \df...

The function $\f$ satisfies $\f(x)>0$ for $x\ge0$ and is strictly decreasing (which means that $\...

\begin{questionparts} \item Show that, for $n> 0$, \[ \int_0^{\frac14\pi} \tan^n x \,\sec^2 x \, \...

Show that, for any function f (for which the integrals exist), \[ \int_0^\infty \f\big(x+\sqrt{1+x^2...

Show that \[ \int_0^{\frac14\pi} \sin (2x) \ln(\cos x)\, \d x = \frac14(\ln 2 -1)\,, \] and that \...

\begin{questionparts} \item Sketch the curve $y=\sin x$ for $0\le x \le \tfrac12 \pi$ and add to you...

For any given function $\f$, let \[ I = \int [\f'(x)]^2 \,[\f(x)]^n \d x\,, \tag{$*$} \] where $n$ i...

Given that $0 < k < 1$, show with the help of a sketch that the equation \[ \sin x = k x \tag{$*$}\]...

The curves $C_1$ and $C_2$ are defined by \[ y= \e^{-x} \ \ \ (x>0) \text{ \ \ \ and \ \ \ } y= \e...

\begin{questionparts} \item Let \[ I=\int_0^a \frac {\f(x)}{\f(x)+\f(a-x)} \, \d x\,. \] Use a su...

Prove that \[ \cos 3x = 4 \cos^3 x - 3 \cos x \,. \] Find and prove a similar result for $\sin 3x$...

Use the substitution $x=\dfrac{1}{t^{2}-1}\; $, where $t>1$, to show that, for $ x>0$, \[ \int ...

For any given (suitable) function $\f$, the \textit{Laplace transform} of $\f$ is the function $\F$ ...

Let $y= (x-a)^n \e^{bx} \sqrt{1+x^2}\,$, where $n$ and $a$ are constants and $b$ is a non-zero con...

Expand and simplify $(\sqrt{x-1}+1)^2\,$. \begin{questionparts} \item Evaluate \[ \int_{5}^{10} \fr...

Show that, for any integer $m$, \[ \int_0^{2\pi} \e^x \cos mx \, \d x = \frac {1}{m^2+1}\big(\e...

\begin{questionparts} \item Show that, for $m>0\,$, \[ \int_{1/m}^m \frac{x^2}{x+1} \, \d x = \fra...

Evaluate the integrals \[\int_0^{\frac{1}{2}\pi} \frac{\sin 2x}{1+\sin^2x} \d x \text{ and } \int_0...

The function $\f$ is defined by \[ \f(x) = \frac{\e^x-1}{\e-1}, \ \ \ \ \ x\ge0, \] and the function...

The functions $\s(x)$ ($0\le x<1$) and $t(x)$ ($x\ge0$), and the real number $p$, are defined by \...

By writing $x=a\tan\theta$, show that, for $a\ne0$, $\displaystyle \int \frac 1 {a^2+x^2}\, \d x =...

Prove the identities $\cos^4\theta -\sin^4\theta \equiv \cos 2\theta$ and $\cos^4 \theta + \sin^4 \...

Let \[ I = \int_{-\frac12 \pi}^{\frac12\pi} \frac {\cos^2\theta}{1-\sin\theta\sin2\alpha} \, \d\th...

By making the substitution $x=\pi-t\,$, show that \[ \! \int_0^\pi x\f(\sin x) \d x = \tfrac12 \pi ...

\begin{questionparts} \item Sketch on the same axes the functions ${\rm cosec}\, x$ and $2x/ \pi$, ...

\begin{questionparts} \item Use the substitution $u^2=2x+1$ to show that, for $x>4$, \[ \int \frac{...

Give a sketch, for $0 \le x \le \frac{1}{2}\pi$, of the curve $$ y = (\sin x - x\cos x)\;, $$ and ...

\begin{questionparts} \item Evaluate the integral \[ \int_0^1 \l x + 1 \r ^{k-1} \; \mathrm{d}x \] ...

Differentiate $\sec {t}$ with respect to $t$. \begin{questionparts} \item Use the substitution $x=\s...

The square bracket notation $\boldsymbol{[} x\boldsymbol{]}$ means the greatest integer less than ...

If $m$ is a positive integer, show that $\l 1+x \r^m + \l 1-x \r^m \ne 0$ for any real $x\,$. ...

The function $\f$ is defined by $$ \f(x)= \vert x-1 \vert\;, $$ where the domain is ${\bf R}\,$, ...

Evaluate the following integrals, in the different cases that arise according to the value of the po...

Find the area of the region between the curve $\displaystyle y = {\ln x \over x}\,$ and the $x$-axi...

Give a sketch to show that, if $\f(x) > 0$ for $p < x < q\,$, then $\displaystyle \int_p^{q} \f(x) ...

Show that \[ \int_{\frac{1}{6}\pi}^{\frac{1}{4}\pi} \frac 1{1-\cos2\theta} \;\d\theta = \frac{\sqrt3...

Let \[ I= \int_0^a \frac {\cos x}{\sin x + \cos x} \; \d x \, \quad \mbox{ and } \quad J= \int_0^a ...

Give a sketch of the curve $ \;\displaystyle y= \frac1 {1+x^2}\;$, for $x\ge0$. Find the equatio...

Show that \[ \int_0^1 \frac{x^4}{1+x^2} \, \d x = \frac \pi {4} - \frac 23 \;. \] Determine the valu...

Show that (for $t>0$) \begin{questionparts} \item \[ \int_0^1 \frac1{(1+tx)^2} \d x = \frac1{(1+t)} ...

Use the substitution $x = 2-\cos \theta $ to evaluate the integral $$ \int_{3/2}^2 \left(x...

Show that \[ \sin\theta = \frac {2t}{1+t^2}, \ \ \ \cos\theta = \frac{1-t^2}{1+t^2}, \ \ \ \frac{1+\...

It is required to approximate a given function $\f(x)$, over the interval $0 \le x \le 1$, by the li...

Show that \[ \int_{-1}^1 \vert \, x\e^x \,\vert \d x =- \int_{-1}^0 x\e^x \d x + \int_0^1 x\e^...

\begin{questionparts} \item Show that, for $0\le x\le 1$, the largest value of $\displaystyle \...

Find $\displaystyle \ \frac{\d y}{\d x} \ $ if $$ y = \frac{ax+b}{cx+d}. \eqno(*) $$ By using cha...

The function $\f$ satisfies $0\leqslant\f(t)\leqslant K$ when $0\leqslant t\leqslant x$. Explain by...

Let $$ {\rm I}(a,b) = \int_0^1 t^{a}(1-t)^{b} \, \d t \; \qquad (a\ge0,\ b\ge0) .$$ \begin{questio...

The integral $I_n$ is defined by $$I_n=\int_0^\pi(\pi/2-x)\sin(nx+x/2)\,{\rm cosec}\,(x/2)\,\d x,$$ ...

Show, by means of a suitable change of variable, or otherwise, that \[ \int_{0}^{\infty}\mathrm{f}((...

Find constants $a,\,b,\,c$ and $d$ such that $$\frac{ax+b}{ x^2+2x+2}+\frac{cx+d}{ x^2-2x+2}= \frac{...

Find constants $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{6}$ and $b$ such that \[x...

Find \[ \int_{0}^{\theta}\frac{1}{1-a\cos x}\,\mathrm{d}x\,, \] where $0 < \theta < \pi$ and $-1 < ...

Show that $\cos 4u=8\cos^{4}u-8\cos^{2}u+1$. If \[ I=\int_{-1}^{1} \frac{1}{\vphantom{{\big(}^2}\; ...

Let $\mathrm{f}(x)=\dfrac{\sin(n+\frac{1}{2})x}{\sin\frac{1}{2}x}$ for $0 < x\leqslant\pi.$ \begin{...

Show that \[ \int_{0}^{1}\frac{1}{x^{2}+2ax+1}\,\mathrm{d}x=\begin{cases} \dfrac{1}{\sqrt{1-a^{2}}}...

\begin{questionparts} \item Show that \[ \int_{0}^{1}\left(1+(\alpha-1)x\right)^{n}\,\mathrm{d}x=\f...

If \[ \mathrm{f}(x)=nx-\binom{n}{2}\frac{x^{2}}{2}+\binom{n}{3}\frac{x^{3}}{3}-\cdots+(-1)^{r+1}\bi...

\begin{questionparts} \item Suppose that \[ S=\int\frac{\cos x}{\cos x+\sin x}\,\mathrm{d}x\quad\m...

By considering the area of the region defined in terms of Cartesian coordinates $(x,y)$ by \[ \{(x,y...

If $\mathrm{Q}$ is a polynomial, $m$ is an integer, $m\geqslant1$ and $\mathrm{P}(x)=(x-a)^{m}\mathr...

By means of the change of variable $\theta=\frac{1}{4}\pi-\phi,$ or otherwise, show that \[ \int_{0...

Show that \begin{questionparts} \item $\dfrac{1-\cos\alpha}{\sin\alpha}=\tan\frac{1}{2}\alpha,$ \i...

\begin{questionparts} \item Evaluate \[ \int_{0}^{2\pi}\cos(mx)\cos(nx)\,\mathrm{d}x, \] where $m...

By making the change of variable $t=\pi-x$ in the integral \[ \int_{0}^{\pi}x\mathrm{f}(\sin x)\,\m...

In the figure, the large circle with centre $O$ has radius $4$ and the small circle with centre $P$ ...

Evaluate \begin{questionparts} \item ${\displaystyle \int_{-\pi}^{\pi}\left|\sin x\right|\,\mathrm...

\begin{questionparts} \item Prove that \[ \int_{0}^{\frac{1}{2}\pi}\ln(\sin x)\,\mathrm{d}x=\int_{0...

Show by means of a sketch, or otherwise, that if $0\leqslant\mathrm{f}(y)\leqslant\mathrm{g}(y)$ fo...

\begin{questionparts} \item By a substitution of the form $y=k-x$ for suitable $k$, prove that, for ...

The functions $\mathrm{x}$ and $\mathrm{y}$ are related by \[ \mathrm{x}(t)=\int_{0}^{t}\mathrm{y}(...

Let $A$ and $B$ be the points $(1,1)$ and $(b,1/b)$ respectively, where $b>1$. The tangents at $A$ a...

\begin{questionparts} \item Evaluate \[ \int_{1}^{3}\frac{1}{6x^{2}+19x+15}\,\mathrm{d}x\,. \] \i...

Sketch the graph of \[ y=\frac{x^{2}\mathrm{e}^{-x}}{1+x}, \] for $-\infty< x< \infty.$ Show that ...

The integral $I$ is defined by \[ I=\int_{1}^{2}\frac{(2-2x+x^{2})^{k}}{x^{k+1}}\,\mathrm{d}x \] wh...

Find the following integrals: \begin{questionparts} \item $\ {\displaystyle \int_{1}^{\mathrm{e}}\fr...

The curve $C$ is given parametrically by the equations $x = 3t^2$, $y = 2t^3$. Show that the equatio...

\begin{questionparts} \item Sketch, on $x$-$y$ axes, the set of all points satisfying $\sin y = \sin...

\begin{questionparts} \item Show, geometrically or otherwise, that the shortest distance between the...

\begin{questionparts} \item Show that the gradient of the curve $\; \dfrac a x + \dfrac by =1$, whe...

A right circular cone has base radius $r$, height $h$ and slant length $\ell$. Its volume $V$, and t...

A curve has the equation \[ y^3 = x^3 +a^3+b^3\,, \] where $a$ and $b$ ar...

A curve is given by \[x^2+y^2 +2axy = 1,\] where $a$ is a constant satisfying $0 < a < 1$. Show tha...

The variables $t$ and $x$ are related by $t=x+ \sqrt{x^2+2bx+c\;} \,$, where $b$ and $c$ are constan...

\begin{questionparts} \item If \[{\mathrm f}(x)=\tan^{-1}x+\tan^{-1}\left(\frac{1-x}{1+x}\right),\] ...

The function $\mathrm{g}$ satisfies, for all positive $x$ and $y$, \[ \mathrm{g}(x)+\mathrm{g}(y...

The equation of a hyperbola (with respect to axes which are displaced and rotated with respect to th...

Sketch the curve $y^{2}=1-\left|x\right|$. A rectangle, with sides parallel to the axes, is inscribe...

Note: You may assume that if the functions $y_1(x)$ and $y_2(x)$ both satisfy one of the differentia...

This question concerns solutions of the differential equation \[ (1-x^2) \left(\frac{\d y}{\d x}\r...

\begin{questionparts} \item Differentiate $\displaystyle \; \frac z {(1+z^2)^{\frac12}} \;$ with res...

\begin{questionparts} \item Use the substitution $y=ux$, where $u$ is a function of $x$, to show tha...

In this question, you may assume that $\ln (1+x) \approx x -\frac12 x^2$ when $\vert x \vert $ is sm...

Given that ${\rm P} (x) = {\rm Q} (x){\rm R}'(x) - {\rm Q}'(x){\rm R}(x)$, write down an expression...

\begin{questionparts} \item By writing $y=u{(1+x^2)\vphantom{\dot A}}^{\frac12}$, where $u$ is a ...

\begin{questionparts} \item Differentiate $\ln\big (x+\sqrt{3+x^2}\,\big)$ and $x\sqrt{3+x^2}$ and ...

Find the general solution of the differential equation $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}...

For $x \ge 0$ the curve $C$ is defined by $$ {\frac{\d y} {\d x}} = \frac{x^3y^2}{(1 + x^2)^{5/2}} ...

Show that, if $y^2 = x^k \f(x)$, then $\displaystyle 2xy \frac{\mathrm{d}y }{ \mathrm{d}x} = ky^2 +...

Let $x$ satisfy the differential equation $$ \frac {\d x}{\d t} = {\big( 1-x^n\big)\vphantom{\Big)}}...

Evaluate $\int_0^{{\pi}} x \sin x\,\d x$ and $\int_0^{{\pi}} x \cos x\,\d x\;$. The function $\f$ s...

It is given that $y$ satisfies $$ {{\d y} \over { \d t}} + k\left({{t^2-3t+2} \over {t+1}}\right)y...

A liquid of fixed volume $V$ is made up of two chemicals $A$ and $B\,$. A reaction takes place in w...

Find all the solution curves of the differential equation \[ y^4 \l {\mathrm{d}y \over \mathrm{d}x }...

Find $y$ in terms of $x$, given that: \begin{eqnarray*} \mbox{for $x < 0\,$}, && \frac{\d y}{\d ...

Sketch the graph of the function $\ln x - {1 \over 2} x^2$. Show that the differential equation \[ ...

The function $\f$ satisfies $\f(x+1)= \f(x)$ and $\f(x)>0$ for all $x$. \begin{questionparts} \item...

The curve $C_1$ passes through the origin in the $x$--$y$ plane and its gradient is given by $$ \fra...

In a cosmological model, the radius $\rm R$ of the universe is a function of the age $t$ of the univ...

Fluid flows steadily under a constant pressure gradient along a straight tube of circular cross-sec...

\begin{questionparts} \item At time $t=0$ a tank contains one unit of water. Water flows out of the ...

\begin{questionparts} \item In the differential equation \[ \frac{1}{y^{2}}\frac{\mathrm{d}y}{\math...

Find the two solutions of the differential equation \[ \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)...

In the manufacture of Grandma's Home Made Ice-cream, chemicals $A$ and $B$ pour at constant rates $a...

A set of curves $S_{1}$ is defined by the equation \[ y=\frac{x}{x-a}, \] where $a$ is a constant w...

Suppose that $y$ satisfies the differential equation \[ y=x\frac{\mathrm{d}y}{\mathrm{d}x}-\cosh\le...

A damped system with feedback is modelled by the equation \[ \mathrm{f}'(t)+\mathrm{f}(t)-k\mathr...

Given that \[ \frac{\mathrm{d}x}{\mathrm{d}t}=4(x-y)\qquad\mbox{ and }\qquad\frac{\mathrm{d}y}{\mat...

The normal to the curve $y=\mathrm{f}(x)$ at the point $P$ with coordinates $(x,\mathrm{f}(x))$ cuts...

Four greyhounds $A,B,C$ and $D$ are held at positions such that $ABCD$ is a large square. At a given...

The vectors ${\bf a}$ and ${\bf b}$ lie in the plane $\Pi\,$. Given that $\vert {\bf a} \vert= 1$ ...

The position vectors of the points $A\,$, $B\,$ and $P$ with respect to an origin $O$ are $a{\bf i...

Consider the equations \begin{alignat*}{2} ax-&y- \ z && =3 \;,\\ 2ax -&y -3z && = 7 \;,\\ 3ax-&y-5...

For all values of $a$ and $b,$ either solve the simultaneous equations \begin{alignat*}{1} x+y+az & ...

Consider the system of equations \begin{alignat*}{1} 2yz+zx-5xy & =2\\ yz-zx+2xy & =1\\ yz-2zx+6xy &...

Find the simultaneous solutions of the three linear equations \begin{alignat*}{1} a^{2}x+ay+z & =a^{...

The matrices $\mathbf{A},\mathbf{B}$ and $\mathbf{M}$ are given by \[ \mathbf{A}=\begin{pmatrix}a &...

The point $P$ moves on a straight line in three-dimensional space. The position of $P$ is observed f...

The distinct points $P_{1},P_{2},P_{3},Q_{1},Q_{2}$ and $Q_{3}$ in the Argand diagram are represente...

State carefully the conditions which the fixed vectors $\mathbf{a,b,u}$ and $\mathbf{v}$ must satisf...

If $x=\log_bc\,$, express $c$ in terms of $b$ and $x$ and prove that $ \dfrac{\log_a c}{\log_a b} = ...

A function $\f(x)$ is said to be concave on some interval if $\f''(x)<0$ in that interval. Show that...

To nine decimal places, $\log_{10}2=0.301029996$ and $\log_{10}3=0.477121255$. \begin{questionparts...

Let $x=10^{100}$, $y=10^{x}$, $z=10^{y}$, and let $$ a_1=x!, \quad a_2=x^y,\quad a_3=y^x,\quad a...

Let \[\mathrm{f}(t)=\frac{\ln t}t\quad\text{ for }t>0.\] Sketch the graph of $\mathrm{f}(t)$ and fin...

Let $a_{1}=3$, $a_{n+1}=a_{n}^{3}$ for $n\geqslant 1$. (Thus $a_{2}=3^{3}$, $a_{3}=(3^{3})^{3}$ and ...

Sketch the graph of the function $\mathrm{h}$, where \[ \mathrm{h}(x)=\frac{\ln x}{x},\qquad(x>0). ...

\begin{questionparts} \item Let $r$ be a real number with $\vert r \vert<1$ and let \[ S = \sum_{n=...

The two sequences $a_0$, $a_1$, $a_2$, $\ldots$ and $b_0$, $b_1$, $b_2$, $\ldots$ have general t...

Let $x_{\low1}$, $x_{\low2}$, \ldots, $x_n$ and $x_{\vphantom {\dot A} n+1}$ be any fixed real numb...

The first four terms of a sequence are given by $F_0=0$, $F_1=1$, $F_2=1$ and $F_3=2$. The general...

The Fibonacci sequence $F_1$, $F_2$, $F_3$, $\ldots$ is defined by $F_1=1$, $F_2= 1$ and \[ F_{n+1...

I borrow $C$ pounds at interest rate $100\alpha \,\%$ per year. The interest is added at the end of ...

How many integers greater than or equal to zero and less than a million are not divisible by 2 or 5?...

My bank pays $\rho$\% interest at the end of each year. I start with nothing in my account. Then for...

Find the sum of those numbers between 1000 and 6000 every one of whose digits is one of the numbers ...

\begin{questionparts} \item By using the formula for the sum of a geometric series, or otherwise, ex...

\begin{questionparts} \item If $\mathrm{f}(r)$ is a function defined for $r=0,1,2,3,\ldots,$ show t...

From the facts \begin{alignat*}{2} 1 & \quad=\quad & & 0\\ 2+3+4 & \quad=\quad & & 1+8\\ 5+6+7+8+...

If $\left|r\right|\neq1,$ show that \[ 1+r^{2}+r^{4}+\cdots+r^{2n}=\frac{1-r^{2n+2}}{1-r^{2}}\,. \]...

The sequence $a_{1},a_{2},\ldots,a_{n},\ldots$ forms an arithmetic progression. Establish a formula,...

A firm of engineers obtains the right to dig and exploit an undersea tunnel. Each day the firm borro...

The definition of the derivative $f'$ of a (differentiable) function f is $$f'(x) = \lim_{h\to 0} \f...

Let $f(x) = (x-p)g(x)$, where g is a polynomial. Show that the tangent to the curve $y = f(x)$ at th...

The real numbers $a_1$, $a_2$, $a_3$, $\ldots$ are all positive. For each positive integer $n$, $A_n...

The line $y=a^2 x$ and the curve $y=x(b-x)^2$, where $0 < a < b\,$, intersect at the origin $...

The points $P(ap^2, 2ap)$ and $Q(aq^2, 2aq)$, where $p>0$ and $q<0$, lie on the curve $C$ with equa...

Let \[ \f(x) = 3ax^2 - 6x^3\, \] and, for each real number $a$, let ${\rm M}(a)$ be the greatest v...

Let $L_a$ denote the line joining the points $(a,0)$ and $(0, 1-a)$, where $0< a < 1$. The li...

\begin{questionparts} \item A function $\f(x)$ satisfies $\f(x) = \f(1-x)$ for all $x$. Show, by dif...

\begin{questionparts} \item Sketch the curve $y= x^4-6x^2+9$ giving the coordinates of the stationar...

\begin{questionparts} \item Find the coordinates of the turning points of the curve $y=27x^3-27x^2...

A small goat is tethered by a rope to a point at ground level on a side of a square barn which stan...

The point $P$ has coordinates $\l p^2 , 2p \r$ and the point $Q$ has coordinates $\l q^2 , 2q \...

Prove that the rectangle of greatest perimeter which can be inscribed in a given circle is a square....

\begin{questionparts} \item By considering $(1+x+x^{2}+\cdots+x^{n})(1-x)$ show that, if $x\neq1$, ...

$\lozenge$ is an operation which take polynomials in $x$ to polynomials in $x$; that is, given a pol...

Given a curve described by $y=\mathrm{f}(x)$, and such that $y\geqslant0$, a \textit{push-off }of th...

The function $\mathrm{f}$ is defined by \[ \mathrm{f}(x)=ax^{2}+bx+c. \] Show that \[ \mathrm{f}'(...

In this question, you may assume that, if a continuous function takes both positive and negative v...

A spherical loaf of bread is cut into parallel slices of equal thickness. Show that, after any numb...

For any number $x$, the largest integer less than or equal to $x$ is denoted by $[x]$. For example, ...

If ${\rm f}(t)\ge {\rm g}(t)$ for $a\le t\le b$, explain very briefly why $\displaystyle \int_a^b ...

Find constants $a_{1}$, $a_{2}$, $u_{1}$ and $u_{2}$ such that, whenever ${\mathrm P}$ is a cubic po...

Explain diagrammatically, or otherwise, why \[ \frac{\mathrm{d}}{\mathrm{d}x}\int_{a}^{x}\mathrm{f}...

Criticise each step of the following arguments. You should correct the arguments where necessary and...

\begin{questionparts} \item Prove that \[ \sum_{r=1}^{n}r(r+1)(r+2)(r+3)(r+4)=\tfrac{1}{6}n(n+1)(...

The function $\mathrm{f}$ satisfies the condition $\mathrm{f}'(x)>0$ for $a\leqslant x\leqslant b$, ...

Three points, $A$, $B$ and $C$, lie in a horizontal plane, but are not collinear. The point $O$ lies...

\begin{questionparts} \item The points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf...

Let $\mathbf{r}$ be the position vector of a point in three-dimensional space. Describe fully the lo...

$ABC$ is a triangle whose vertices have position vectors $\mathbf{a,b,c}$brespectively, relative to ...

The vertices of a plane quadrilateral are labelled $A$, $B$, $A'$ and $B'$, in clockwise order. A po...

Three non-collinear points $A$, $B$ and $C$ lie in a horizontal ceiling. A particle $P$ of weight $W...

\textit{ Note: a regular octahedron is a polyhedron with eight faces each of which is an equilateral...

By considering a suitable scalar product, prove that \[ (ax+by+cz)^2 \le (a^2+b^2+c^2)(x^2+y^2+z^2)...

In 3-dimensional space, the lines $m_1$ and $m_2$ pass through the origin and have directions $\bf ...

Three ships $A$, $B$ and $C$ move with velocities ${\bf v}_1$, ${\bf v}_2$ and $\bf u$ respectively...

The plane \[ {x \over a} + {y \over b} +{z \over c} = 1 \] meets the co-ordinate axes at the points ...

The cuboid $ABCDEFGH$ is such $AE$, $BF$, $CG$, $DH$ are perpendicular to the opposite faces $ABCD...

A ship sails at $20$ kilometres/hour in a straight line which is, at its closest, 1 kilometre from a...

\begin{questionparts} \item Show that four vertices of a cube, no two of which are adjacent, form th...

Points $\mathbf{A},\mathbf{B},\mathbf{C}$ in three dimensions have coordinate vectors $\mathbf{a},\m...

A single stream of cars, each of width $a$ and exactly in line, is passing along a straight road of ...

Four rigid rods $AB$, $BC$, $CD$ and $DA$ are freely jointed together to form a quadrilateral in the...

A ship is sailing due west at $V$ knots while a plane, with an airspeed of $kV$ knots, where $k>\sqr...

Let $A,B,C$ be three non-collinear points in the plane. Explain briefly why it is possible to choose...

The island of Gammaland is totally flat and subject to a constant wind of $w$ kh$^{-1},$ blowing fro...

Describe geometrically the possible intersections of a plane with a sphere. Let $P_{1}$ and $P_{2}...

A square pyramid has its base vertices at the points $A$ $(a,0,0)$, $B$ $(0,a,0)$, $C$ $(-a,0,0)$ an...

The points $P$ and $R$ lie on the sides $AB$ and $AD,$ respectively, of the parallelogram $ABCD.$ Th...

The tetrahedron $ABCD$ has $A$ at the point $(0,4,-2)$. It is symmetrical about the plane $y+z=2,$ w...

A straight stick of length $h$ stands vertically. On a sunny day, the stick casts a shadow on flat h...

A set of $n$ distinct vectors $\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{n},$ where $n\geqsla...

The distinct points $O\,(0,0,0),$ $A\,(a^{3},a^{2},a),$ $B\,(b^{3},b^{2},b)$ and $C\,(c^{3},c^{2},c)...

Let $ABCD$ be a parallelogram. By using vectors, or otherwise, prove that: \begin{questionparts} \i...

In the triangle $OAB,$ $\overrightarrow{OA}=\mathbf{a},$ $\overrightarrow{OB}=\mathbf{b}$ and $OA=OB...

$ABCD$ is a skew (non-planar) quadrilateral, and its pairs of opposite sides are equal, i.e. $AB=CD$...

A bus has the shape of a cuboid of length $a$ and height $h$. It is travelling northwards on a journ...

I stand at the top of a vertical well. The depth of the well, from the top to the surface of the wat...

On the (flat) planet Zog, the acceleration due to gravity is $g$ up to height $h$ above the surface...

The Norman army is advancing with constant speed $u$ towards the Saxon army, which is at rest. When ...

A particle is travelling in a straight line. It accelerates from its initial velocity $u$ to velo...

Point $B$ is a distance $d$ due south of point $A$ on a horizontal plane. Particle $P$ is at rest a...

A competitor in a Marathon of $42 \frac38$ km runs the first $t$ hours of the race at a constant s...

A tortoise and a hare have a race to the vegetable patch, a distance $X$ kilometres from the startin...

A particle moves so that ${\bf r}$, its displacement from a fixed origin at time $t$, is given by ...

A train is made up of two engines, each of mass $M$, and $n$ carriages, each of mass $m$. One of th...

A particle of mass $m$ is pulled along the floor of a room in a straight line by a light string whic...

A light smoothly jointed planar framework in the form of a regular hexagon $ABCDEF$ is suspended smo...

A projectile of mass $m$ is fired horizontally from a toy cannon of mass $M$ which slides freely on ...

$\ $\vspace{-1cm} \noindent \begin{center} \psset{xunit=1.1cm,yunit=0.7cm,algebraic=true,dotstyle=o...

A librarian wishes to pick up a row of identical books from a shelf, by pressing her hands on the ou...

One end of a thin uniform inextensible, but perfectly flexible, string of length $l$ and uniform mas...

The diagrams below show two separate systems of particles, strings and pulleys. In both systems,...

The diagram shows two particles, $A$ of mass $5m$ and $B$ of mass $3m$, connected by a light inexte...

A plane is inclined at an angle $\arctan \frac34$ to the horizontal and a small, smooth, light ...

A long light inextensible string passes over a fixed smooth light pulley. A particle of mass 4~kg ...

Hank's Gold Mine has a very long vertical shaft of height $l$. A light chain of length $l$ passes ov...

\noindent \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dotstyle=o,dotsi...

A small smooth ring $R$ of mass $m$ is free to slide on a fixed smooth horizontal rail. A light in...

A triangular wedge is fixed to a horizontal surface. The base angles of the wedge are $\alpha$ and...

A block of mass $4\,$kg is at rest on a smooth, horizontal table. A smooth pulley $P$ is fixed to on...

A wedge of mass $M$ rests on a smooth horizontal surface. The face of the wedge is a smooth plane in...

$\,$ \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth...

A horizontal rail is fixed parallel to a vertical wall and at a distance $d$ from the wall. A~unifor...

$\,$ \begin{center} \begin{tikzpicture} % Set the clipping area to match pspicture* \clip (-...

A hollow circular cylinder of internal radius $r$ is held fixed with its axis horizontal. A uniform ...

A wedge of mass $km$ has the shape (in cross-section) of a right-angled triangle. It stands on a smo...

A straight uniform rod has mass $m$. Its ends $P_1$ and $P_2$ are attached to small light rings th...

A particle of weight $W$ is placed on a rough plane inclined at an angle of $\theta$ to the horizon...

Two particles, $A$ and $B$, of masses $m$ and $2m$, respectively, are placed on a line of greatest ...

A bead $B$ of mass $m$ can slide along a rough horizontal wire. A light inextensible string of lengt...

A rod $AB$ of length 0.81 m and mass 5 kg is in equilibrium with the end $A$ on a rough floor and th...

A small bullet of mass $m$ is fired into a block of wood of mass $M$ which is at rest. The speed of...

A small light ring is attached to the end $A$ of a uniform rod $AB$ of weight $W$ and length $2a$. T...

Two identical rough cylinders of radius $r$ and weight $W$ rest, not touching each other but a ...

Two long circular cylinders of equal radius lie in equilibrium on an inclined plane, in \mbox{conta...

The diagram shows three identical discs in equilibrium in a vertical plane. Two discs rest, not in ...

A thin non-uniform bar $AB$ of length $7d$ has centre of mass at a point $G$, where $AG=3d$. A lig...

A uniform rod $AB$ of length $4L $ and weight $W$ is inclined at an angle $\theta$ to the horizon...

A painter of weight $kW$ uses a ladder to reach the guttering on the outside wall of a house. The ...

$AB$ is a uniform rod of weight $W\,$. The point $C$ on $AB$ is such that $AC>CB\,$. The rod is in...

A rigid straight beam $AB$ has length $l$ and weight $W$. Its weight per unit length at a distance ...

$\,$ \vspace*{-0.5in} \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dots...

\noindent \begin{center} \psset{xunit=0.8cm,yunit=0.8cm,algebraic=true,dotstyle=o,dotsize=3pt 0,line...

Two identical uniform cylinders, each of mass $m,$ lie in contact with one another on a horizontal p...

A light rod of length $2a$ is hung from a point $O$ by two light inextensible strings $OA$ and $OB$ ...

Show that the point $T$ with coordinates \[ \left( \frac{a(1-t^2)}{1+t^2} \; , \; \frac{2bt}{1+t^2}\...

A curve $C$ is determined by the parametric equations \[ x=at^2 \, , \; y = 2at\,, \] where $a>0$\,....

The curve $C_1$ has parametric equations $x=t^2$, $y= t^3$, where $-\infty < t < \infty\,$. Let $O...

The midpoint of a rod of length $2b$ slides on the curve $y =\frac14 x^2$, $x\ge0$, in such a way t...

The curve $C$ has equation $xy = \frac12$. The tangents to $C$ at the distinct points $P\big(p, \fra...

The distinct points $P$ and $Q$, with coordinates $(ap^2,2ap)$ and $(aq^2,2aq)$ respectively, lie on...

A curve is given parametrically by \begin{align*} x&= a\big( \cos t +\ln \tan \tfrac12 t\big)\,,\\ y...

An ellipse has equation $\dfrac{x^2}{a^2} +\dfrac {y^2}{b^2} = 1$. Show that the equation of the t...

A curve is defined parametrically by \[ x=t^2 \;, \ \ \ y=t (1 + t^2 ) \;. \] The...

Show that the equation $x^3 + px + q=0$ has exactly one real solution if $p \ge 0\,$. A parabola $C...

A closed curve is given by the equation $$ x^{2/n} + y^{2/n} = a^{2/n} \eqno(*) $$ where $n$ is an o...

Two curves are given parametrically by \[ x_{1}=(\theta+\sin\theta),\qquad y_{1}=(1+\cos\theta),\tag...

Sketch the curve $C_{1}$ whose parametric equations are $x=t^{2},$ $y=t^{3}.$ The circle $C_{2}$ pas...

A particle $P$ is projected from a point $O$ on horizontal ground with speed $u$ and angle of projec...

In this question, the $x$-axis is horizontal and the positive $y$-axis is vertically upwards. A part...

Two thin vertical parallel walls, each of height $2a$, stand a distance $a$ apart on horizontal grou...

A particle is projected at speed $u$ from a point $O$ on a horizontal plane. It passes through a fix...

\begin{questionparts} \item Two particles move on a smooth horizontal surface. The positions, in Car...

The point $O$ is at the top of a vertical tower of height $h$ which stands in the middle of a large...

A short-barrelled machine gun stands on horizontal ground. The gun fires bullets, from ground level...

A particle is projected from a point $O$ on horizontal ground with initial speed $u$ and at an angl...

A particle of mass $m$ is projected due east at speed $U$ from a point on horizontal ground at a...

A particle is projected at an angle of elevation $\alpha$ (where $\alpha>0$) from a point $A$ on ho...

Two particles, $A$ and $B$, are projected simultaneously towards each other from two points which ar...

A tennis ball is projected from a height of $2h$ above horizontal ground with speed $u$ and at an a...

A tall shot-putter projects a small shot from a point $2.5\,$m above the ground, which is horizontal...

A particle is projected from a point on a horizontal plane, at speed $u$ and at an angle~$\theta$ a...

A particle is projected at an angle $\theta$ above the horizontal from a point on a horizontal plane...

Two points $A$ and $B$ lie on horizontal ground. A particle $P_1$ is projected from $A$ towards $...

A particle is projected under gravity from a point $P$ and passes through a point $Q$. The angles of...

Two particles $P$ and $Q$ are projected simultaneously from points $O$ and $D$, respectively, where~...

In this question, use $g=10\,$m\,s$^{-2}$. In cricket, a fast bowler projects a ball at $40\,$m\,s$^...

A particle is projected from a point on a plane that is inclined at an angle~$\phi$ to the horizo...

{\sl In this question take the acceleration due to gravity to be $10\,{\rm m \,s}^{-2}$ and negle...

A solid right circular cone, of mass $M$, has semi-vertical angle $\alpha$ and smooth surfaces. I...

A smooth, straight, narrow tube of length $L$ is fixed at an angle of $30^\circ$ to the horizontal....

A projectile of unit mass is fired in a northerly direction from a point on a horizontal plain at sp...

A particle $P$ is projected in the $x$-$y$ plane, where the $y$-axis is vertical and the $x$-axis i...

The points $A$ and $B$ are $180$ metres apart and lie on horizontal ground. A missile is launche...

A particle is projected over level ground with a speed $u$ at an angle $\theta$ above the horizont...

A particle $P_1$ is projected with speed $V$ at an angle of elevation ${\alpha}\,\,\,( > 45^{\circ})...

A particle is projected with speed $V$ at an angle $\theta$ above the horizontal. The particle p...

A particle is projected from a point $O$ on a horizontal plane with speed $V$ and at an angle of ele...

A two-stage missile is projected from a point $A$ on the ground with horizontal and vertical velocit...

A gun is sited on a horizontal plain and can fire shells in any direction and at any elevation at sp...

A child is playing with a toy cannon on the floor of a long railway carriage. The carriage is moving...

A fielder, who is perfectly placed to catch a ball struck by the batsman in a game of cricket, watch...

A shell explodes on the surface of horizontal ground. Earth is scattered in all directions with vary...

A tennis player serves from height $H$ above horizontal ground, hitting the ball downwards with spe...

A particle is projected under the influence of gravity from a point $O$ on a level plane in such a w...

A cannon is situated at the bottom of a plane inclined at angle $\beta$ to the horizontal. A (small)...

A particle is projected from a point $O$ with speed $\sqrt{2gh},$ where $g$ is the acceleration due ...

As part of a firework display a shell is fired vertically upwards with velocity $v$ from a point on ...

A cannon-ball is fired from a cannon at an initial speed $u$. After time $t$ it has reached height $...

In a clay pigeon shoot the target is launched vertically from ground level with speed $v$. At a time...

I am standing next to an ice-cream van at a distance $d$ from the top of a vertical cliff of height ...

The Ruritanian army is supplied with shells which may explode at any time in flight but not before ...

A shell of mass $m$ is fired at elevation $\pi/3$ and speed $v$. Superman, of mass $2m$, catches the...

A shot-putter projects a shot at an angle $\theta$ above the horizontal, releasing it at height $h$ ...

A sniper at the top of a tree of height $h$ is hit by a bullet fired from the undergrowth covering t...

Prove that: \begin{questionparts} \item if $a+2b+3c=7x$, then \[ a^{2}+b^{2}+c^{2}=\left(x-a\right...

Let \[ y=\dfrac{x^2+x\sin\theta+1}{x^2+x\cos\theta+1} \,.\] \begin{questionparts} \item Given that ...

\begin{questionparts} \item Use the substitution $\sqrt x = y$ (where $y\ge0$) to find the real root...

\begin{questionparts} \item By considering the equation $x^2+x-a=0\,$, show that the equation $x={...

Let $\f(x)=x^2+px+q$ and $\g(x)=x^2+rx+s\,$. Find an expression for $\f ( \g (x))$ and hence f...

In this question $a$ and $b$ are distinct, non-zero real numbers, and $c$ is a real number. \begin{...

Prove that, if $\vert \alpha\vert < 2\sqrt{2},$ then there is no value of $x$ for which \begin{equat...

Find all real values of $x$ that satisfy: \begin{questionparts} \item $ \ds \sqrt{3x^2+1} + \sqrt{x}...

\begin{questionparts} \item Express $\left(3+2\sqrt{5} \, \right)^3$ in the form $a+b\sqrt{5}$ wher...

Give a condition that must be satisfied by $p$, $q$ and $r$ for it to be possible to write the qua...

Show that if $x$ and $y$ are positive and $x^3 + x^2 = y^3 - y^2$ then $x < y\,$. Show further that...

Let \[\f(x) = a \sqrt{x} - \sqrt{x - b}\;, \] where $x\ge b >0$ and $a>1\,$. Sketch the graph of $...

Show that setting $z - z^{-1}=w$ in the quartic equation \[ z^4 +5z^3 +4z^2 -5z +1=0 \] results in t...

Consider the equation \[ x^2 - b x + c = 0 \;, \] where $b$ and $c$ are real numbers. \begin{questi...

Solve the inequalities \begin{questionparts} \item $1+2x-x^2 >2/x \quad (x\ne0)$ , \item $\sqrt{3x+...

Show that \[ x^2-y^2 +x+3y-2 = (x-y+2)(x+y-1) \] and hence, or otherwise, indicate by means of a s...

Consider the quadratic equation $$ nx^2+2x \sqrt{pn^2+q} + rn + s = 0, \tag{*} $$ where $p>0$, $...

\begin{questionparts} \item Find the greatest and least values of $bx+a$ for $-10\leqslant x \leqsla...

The $n$ positive numbers $x_{1},x_{2},\dots,x_{n}$, where $n\ge3$, satisfy $$ x_{1}=1+\frac{1}{x_...

\begin{questionparts} \item Find the real values of $x$ for which \[ x^{3}-4x^{2}-x+4\geqslant0. \...

\begin{questionparts} \item By setting $y=x+x^{-1},$ find the solutions of \[ x^{4}+10x^{3}+26x^{2...

Let $a,b,c,d,p$ and $q$ be positive integers. Prove that: \begin{questionparts} \item if $b > a$ and...

Prove that both $x^{4}-2x^{3}+x^{2}$ and $x^{2}-8x+17$ are non-negative for all real $x$. By conside...

The function $\mathrm{Min}$ is defined as \[ \mathrm{Min}(a, b) = \begin{cases} a & \text{if } a \le...

The \textit{Devil's Curve} is given by $$y^2(y^2 - b^2) = x^2(x^2 - a^2),$$ where $a$ and $b$ are po...

In this question, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x...

\begin{questionparts} \item Sketch the curve $y = \e^x (2x^2 -5x+ 2)\,.$ Hence determine how many...

In this question, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x...

\begin{questionparts} \item Sketch the curve $y=\sqrt{1-x} + \sqrt{3+x}\;$. Use your sketch to show...

A curve has equation $y=2x^3-bx^2+cx$. It has a maximum point at $(p,m)$ and a minimum point at $(q,...

Sketch the curve with cartesian equation \[ y = \frac{2x(x^2-5)}{x^2-4} \] and give the equations o...

The notation ${\lfloor } x \rfloor$ denotes the greatest integer less than or equal to the real numb...

The equation of a curve is $y=\f ( x )$ where \[ \f ( x ) = x-4-\frac{16 \l 2x+1 \r^2}{x^2 \l x - 4 ...

The curve $C$ has equation $$ y = x(x+1)(x-2)^4. $$ Determine the coordinates of all the stationary ...

Find the coordinates of the turning point on the curve $y = x^2 - 2bx + c\,$. Sketch the curve in th...

Sketch the graph of the function $[x/N]$, for $0 < x < 2N$, where the notation $[y]$ means the...

The function $\f(x)$ is defined by $$ \f(x) = \frac{x( x - 2 )(x-a)}{ x^2 - 1}. $$ Prove algeb...

The curve $C$ has equation $$ y = \frac x {\sqrt{x^2-2x+a}}\; , $$ where the square root is positiv...

Let $$y^2=x^2(a^2-x^2),$$ where $a$ is a real constant. Find, in terms of $a$, the maximum and minim...

`24 Hour Spares' stocks a small, widely used and cheap component. Every $T$ hours $X$ units arrive b...

\begin{questionparts} \item Show that the equation \[ (x-1)^{4}+(x+1)^{4}=c \] has exactly two real...

Sketch the curve \[ \mathrm{f}(x)=x^{3}+Ax^{2}+B \] first in the case $A>0$ and $B>0$, and then in ...

Let $N=10^{100}.$ The graph of \[ \mathrm{f}(x)=\frac{x^{N}}{1+x^{N}}+2 \] for $-3\leqslant x\leqsl...

Sketch the curves given by \[ y=x^{3}-2bx^{2}+c^{2}x, \] where $b$ and $c$ are non-negative, in the...

Sketch the curve whose cartesian equation is \[ y=\frac{2x(x^{2}-5)}{x^{2}-4}, \] and give the equa...

The function $\mathrm{f}$ is defined by \[ \mathrm{f}(x)=\frac{\left(x-a\right)\left(x-b\right)}{\l...

Sketch the graph of $8y=x^{3}-12x$ for $-4\leqslant x\leqslant4$, marking the coordinates of the tur...

For any two real numbers $x_1$ and $x_2$, show that $$|x_1 + x_2| \leq |x_1| + |x_2|.$$ Show further...

\begin{questionparts} \item Find integers $m$ and $n$ such that $$\sqrt{3+2\sqrt{2}} = m + n\sqrt{2...

Show that, if $k$ is a root of the quartic equation \[ x^4 + ax^3 + bx^2 + ax + 1 = 0\,, \tag{$*$} \...

\begin{questionparts} \item Write down the most general polynomial of degree 4 that leaves a remain...

Use the factor theorem to show that $a+b-c$ is a factor of \[ (a+b+c)^3 -6(a+b+c)(a^2+b^2+c^2) +8(a...

\begin{questionparts} \item For $n=1$, $2$, $3$ and $4$, the functions $\p_n$ and $\q_n$ are defined...

\begin{questionparts} \item Given that the cubic equation $x^3+3ax^2 + 3bx +c=0$ has three distinct...

If $\p(x)$ and $\q(x)$ are polynomials of degree $m$ and $n$, respectively, what is the degree of $\...

The polynomial $\f(x)$ is defined by \[ \f(x) = x^n + a_{\low{n-1}}x^{n-1} + \cdots + a_{\lo...

\begin{questionparts} \item The number $\alpha$ is a common root of the equations $x^2 +ax +b=0$ an...

\begin{questionparts} \item By considering the positions of its turning points, show that the curv...

The points $S$, $T$, $U$ and $V$ have coordinates $(s,ms)$, $(t,mt)$, $(u,nu)$ and $(v,nv)$, respect...

The polynomial $\p(x)$ is of degree 9 and $\p(x)-1$ is exactly divisible by $(x-1)^5$. \begin{que...

A curve is given by the equation \[ y = ax^3 - 6ax^2+ \left( 12a + 12 \right)x - \left( 8a + 16 \rig...

\begin{questionparts} \item Given that $x^2 - y^2 = \left( x - y \right)^3$ and that $x-y = d$ (wh...

Show that $x^3-3xbc + b^3 + c^3$ can be written in the form $\left( x+ b+ c \right) {\rm Q}( x)$, w...

\begin{questionparts} \item Show that $x-3$ is a factor of \begin{equation} x^3-5x^2+2x^2y+xy^2-8xy...

Let \[ \f(x) = x^n + a_1 x^{n-1} + \cdots + a_n\;, \] where $a_1$, $a_2$, $\ldots$, $a_n$ are given...

Sketch, without calculating the stationary points, the graph of the function $\f(x)$ given by \[ \f(...

Given that \[ x^4 + p x^2 + q x + r = ( x^2 - a x + b ) ( x^2 + a x + c ) , \] express $p$, $q...

Prove that if ${(x-a)^{2}}$ is a factor of the polynomial $\p(x)$, then $\p'(a)=0$. Prove a correspo...

Consider the cubic equation \[ x^3-px^2+qx-r=0\;, \] where $p\ne0$ and $r\ne 0$. \begin{questionpart...

Show that, when the polynomial ${\rm p} (x)$ is divided by $(x-a)$, where $a$ is a real number, the...

Let $\mathrm{p}_{0}(x)=(1-x)(1-x^{2})(1-x^{4}).$ Show that $(1-x)^{3}$ is a factor of $\mathrm{p}_{0...

\begin{questionparts} \item By considering the binomial expansion of $(1+x)^{2m+1}$, prove that \[ ...

For positive integers $n$, $a$ and $b$, the integer $c_r$ ($0\le r\le n$) is defined to be the coe...

By considering the coefficient of $x^r$ in the series for $(1+x)(1+x)^n$, or otherwise, obtain the f...

By considering the expansion of $\left(1+x\right)^{n}$ where $n$ is a positive integer, or otherwi...

Show that the coefficient of $x^{-12}$ in the expansion of \[ \left(x^{4}-\frac{1}{x^{2}}\right)^{5...

By considering the expansions in powers of $x$ of both sides of the identity $$ {(1+x)^n}{(1+x)^n}\...

By considering the coefficient of $x^{n}$ in the identity $(1-x)^{n}(1+x)^{n}=(1-x^{2})^{n},$ or oth...

Write down the binomial expansion of $(1+x)^{n}$, where $n$ is a positive integer. \begin{questionp...

\begin{questionparts} \item The functions $\mathrm{a, b, c}$ and $\mathrm{d}$ are defined by \begin...

The point $P$ has coordinates $(x,y)$ which satisfy \[ x^2+y^2 + kxy +3x +y =0\,. \] \begin{question...

Let $a_n$ be the coefficient of $x^n$ in the series expansion, in ascending powers of $x$, of \[\di...

The function $\mathrm{f}$ is defined, for $x\neq1$ and $x\neq2$ by \[ \mathrm{f}(x)=\frac{1}{\left(x...

In this question, the definition of $\displaystyle\binom pq$ is taken to be \[ \binom pq = \begin{c...

Given an infinite sequence of numbers $u_0$, $u_1$, $u_2$, $\ldots\,$, we define the {\em generatin...

Write down the general term in the expansion in powers of $x$ of $(1-x^6)^{-2}\,$. \begin{question...

Use the binomial expansion to show that the coefficient of $x^r$ in the expansion of $(1-x)^{-3}...

\begin{questionparts} \item Show that $1.3.5.7. \;\ldots \;.(2n-1)=\dfrac {(2n)!}{2^n n!}\;$ and th...

\textit{In this question, you are not required to justify the accuracy of the approximations.} \beg...

\begin{questionparts} \item Write down the general term in the expansion in powers of $x$ of $(1...

\begin{questionparts} \item The function $\f(x)$ is defined for $\vert x \vert < \frac15$ by \[ \f(...

Show that $\ds ^{2r} \! {\rm C}_r =\frac{1\times3\times\dots\times (2r-1)}{r!} \, \times 2^r ...

\begin{questionparts} \item In the binomial expansion of $(2x+1/x^{2})^{6}\;$ for $x\ne0$, show that...

Use the binomial expansion to obtain a polynomial of degree $2$ which is a good approximation to ...

Use the first four terms of the binomial expansion of $(1-1/50)^{1/2}$, writing $1/50 = 2/100$ to s...

\begin{questionparts} \item Find the coefficient of $x^{6}$ in \[(1-2x+3x^{2}-4x^{3}+5x^{4})^{3}.\]...

Given that $b>a>0$, find, by using the binomial theorem, coefficients $c_{m}$ ($m=0,1,2,\ldots$) suc...

\begin{questionparts} \item \begin{enumerate} \item Show that if the complex number $z$ satisfies t...

The point $P$ in the Argand diagram is represented by the the complex number $z$, which satisfies $$...

\begin{questionparts} \item The distinct points $A$, $Q$ and $C$ lie on a straight line in the Argan...

A quadrilateral drawn in the complex plane has vertices $A$, $B$, $C$ and~$D$, labelled anticlockw...

Let $z$ and $w$ be complex numbers. Use a diagram to show that $\vert z-w \vert \le \vert z\vert + ...

Let $x+{\rm i} y$ be a root of the quadratic equation $z^2 + pz +1=0$, where $p$ is a real number...

The complex numbers $z$ and $w$ are related by \[ w= \frac{1+\mathrm{i}z}{\mathrm{i}+z}\,. \] Let $z...

The points $A$, $B$ and $C$ in the Argand diagram are the vertices of an equilateral triangle d...

The distinct points $P$, $Q$, $R$ and $S$ in the Argand diagram lie on a circle of radius $a$ centre...

Show that the distinct complex numbers $\alpha$, $\beta$ and $\gamma$ represent the vertices of an ...

In this question, $a$ and $c$ are distinct non-zero complex numbers. The complex conjugate of any ...

Four complex numbers $u_1$, $u_2$, $u_3$ and $u_4$ have unit modulus, and arguments $\theta_1$, $\...

\begin{questionparts} \item Prove that the equations $$ \left|z - (1 + \mathrm{i}) \right|^2 = 2 \eq...

In an Argand diagram, $O$ is the origin and $P$ is the point $2+0\mathrm{i}$. The points $Q$, $R$ ...

Define the modulus of a complex number $z$ and give the geometric interpretation of $\vert\,z_1-z_2\...

\begin{questionparts} \item In the Argand diagram, the points $Q$ and $A$ represent the complex numb...

The complex numbers $w=u+\mathrm{i}v$ and $z=x+\mathrm{i}y$ are related by the equation $$z= (\cos v...

If $$ z^{4}+z^{3}+z^{2}+z+1=0\tag{*} $$ and $u=z+z^{-1}$, find the ...

\begin{questionparts} \item Find all rational numbers $r$ and $s$ which satisfy \[ (r+s\sqrt{3})^{2...

The variable non-zero complex number $z$ is such that \[ \left|z-\mathrm{i}\right|=1. \] Find the m...

Let $a,b,c,d,p,q,r$ and $s$ be real numbers. By considering the determinant of the matrix product \...

The four points $A,B,C,D$ in the Argand diagram (complex plane) correspond to the complex numbers $a...

The function $\mathrm{f}$ is defined, for any complex number $z$, by \[ \mathrm{f}(z)=\frac{\mathrm...

The point in the Argand diagram representing the complex number $z$ lies on the circle with centre $...

\textit{In this question, the argument of a complex number is chosen to satisfy $0\leqslant\arg z<2\...

If $z=x+\mathrm{i}y$ where $x$ and $y$ are real, define $\left|z\right|$ in terms of $x$ and $y$. Sh...

Let $\alpha$ be a fixed angle, $0 < \alpha \leqslant\frac{1}{2}\pi.$ In each of the following cases,...

Sketch the following subsets of the complex plane using Argand diagrams. Give reasons for your answe...

The distinct points $L,M,P$ and $Q$ of the Argand diagram lie on a circle $S$ centred on the origin ...

Give a parametric form for the curve in the Argand diagram determined by $\left|z-\mathrm{i}\right|=...

The complex number $w$ is such that $w^{2}-2w$ is real. \begin{questionparts} \item Sketch the locu...

The quadratic equation $x^{2}+bx+c=0$, where $b$ and $c$ are real, has the properly that if $k$ is a...

The complex numbers $z_{1},z_{2},\ldots,z_{6}$ are represented by six distinct points $P_{1},P_{2},\...

Three rods have lengths $a$, $b$ and $c$, where $a< b< c$. The three rods can be made into a tri...

A positive integer with $2n$ digits (the first of which must not be $0$) is called a \textit{balance...

$47231$ is a five-digit number whose digits sum to $4+7+2+3+1 = 17\,$. \begin{questionparts} \item S...

How many integers between $10\,000$ and $100\,000$ (inclusive) contain exactly two different digits?...

Show that you can make up 10 pence in eleven ways using 10p, 5p, 2p and 1p coins. In how many ways c...

I have two dice whose faces are all painted different colours. I number the faces of one of them $1,...

A $3\times3$ magic square is a $3\times3$ array \[ \begin{array}{ccc} a & b & c\\ d & e & f\\ g & h...

Let $X$ be a random variable which takes only the finite number of different possible real values $x...

\begin{questionparts} \item Alice tosses a fair coin twice and Bob tosses a fair coin three times. C...

From the integers $1, 2, \ldots , 52$, I choose seven (distinct) integers at random, all choices be...

Each day, I have to take $k$ different types of medicine, one tablet of each. The tablets are identi...

I choose at random an integer in the range 10000 to 99999, all choices being equally likely. Given ...

I am selling raffle tickets for $\pounds1$ per ticket. In the queue for tickets, there are $m$ peop...

Three married couples sit down at a round table at which there are six chairs. All of the possible s...

\begin{questionparts} \item A bag contains $N$ sweets (where $N \ge 2$), of which $a$ are red. Two s...

Three pirates are sharing out the contents of a treasure chest containing $n$ gold coins and $2$ lea...

On the basis of an interview, the $N$ candidates for admission to a college are ranked in order acco...

A school has $n$ pupils, of whom $r$ play hocket, where $n\geqslant r\geqslant2.$ All $n$ pupils are...

A set of $2N+1$ rods consists of one of each length $1,2,\ldots,2N,2N+1$, where $N$ is an integer gr...

Four children, $A$, $B$, $C$ and $D$, are playing a version of the game `pass the parcel'. They st...

A bag contains three coins. The probabilities of their showing heads when tossed are $p_1$, $p_2...

I have a sliced loaf which initially contains $n$ slices of bread. Each time I finish setting a STE...

Four players $A$, $B$, $C$ and $D$ play a coin-tossing game with a fair coin. Each player chooses a ...

A modern villa has complicated lighting controls. In order for the light in the swimming pool to be ...

Xavier and Younis are playing a match. The match consists of a series of games and each game consist...

Rosalind wants to join the Stepney Chess Club. In order to be accepted, she must play a challenge m...

Satellites are launched using two different types of rocket: the Andover and the Basingstoke. The An...

Prove that, for any real numbers $x$ and $y$, $x^2+y^2\ge2xy\,$. \begin{questionparts} \item Carol h...

Bag $P$ and bag $Q$ each contain $n$ counters, where $n\ge2$. The counters are identical in shape a...

In the High Court of Farnia, the outcome of each case is determined by three judges: the ass, the b...

Four students, Arthur, Bertha, Chandra and Delilah, exchange gossip. When Arthur hears a rumour, he ...

\begin{questionparts} \item Prove that if $x>0$ then $x+x^{-1}\ge2.\;$ I have a pair of six-faced di...

A pack of $2n$ (where $n\geqslant4$) cards consists of two each of $n$ different sorts. If four card...

Starting with the result $\P(A\cup B) = \P(A)+P(B) - \P(A\cap B)$, prove that \[ \P(A\cup B\cup C) ...

\begin{questionparts} \item The probability that a hobbit smokes a pipe is 0.7 and the probability t...

Explain why, if $\mathrm{A, B}$ and $\mathrm{C}$ are three events, \[ \mathrm{P(A \cup B \cup C) =...

The set $S$ is the set of all integers from 1 to $n$. The set $T$ is the set of all distinct subsets...

Given that $0 < r < n$ and $r$ is much smaller than $n$, show that $\dfrac {n-r}n \approx \e^{-r/n}...

The mountain villages $A,B,C$ and $D$ lie at the vertices of a tetrahedron, and each pair of village...

Suppose that a solution $(X,Y,Z)$ of the equation \[X+Y+Z=20,\] with $X$, $Y$ and $Z$ non-negative i...

In certain forms of Tennis two players $A$ and $B$ serve alternate games. Player $A$ has probability...

A point moves in unit steps on the $x$-axis starting from the origin. At each step the point is equa...

The four towns $A,B,C$ and $D$ are linked by roads $AB,AC,CB,BD$ and $CD.$ The probability that any ...

A patient arrives with blue thumbs at the doctor's surgery. With probability $p$ the patient is suff...

My two friends, who shall remain nameless, but whom I shall refer to as $P$ and $Q$, both told me th...

A multiple-choice test consists of five questions. For each question, $n$ answers are given ($n\ge2...

A bag contains eleven small discs, which are identical except that six of the discs are blank and fi...

I know that ice-creams come in $n$ different sizes, but I don't know what the sizes are. I am offer...

Oxtown and Camville are connected by three roads, which are at risk of being blocked by flooding. ...

The twins Anna and Bella share a computer and never sign their e-mails. When I e-mail them, only th...

In a rabbit warren, underground chambers $A, B, C$ and $D$ are at the vertices of a square, and bu...

In a game, a player tosses a biased coin repeatedly until two successive tails occur, when the game...

The random variable $U$ takes the values $+1$, $0$ and $-1\,$, each with probability $\frac13\,$. Th...

In a game for two players, a fair coin is tossed repeatedly. Each player is assigned a sequence o...

Every person carries two genes which can each be either of type $A$ or of type $B$. It is known ...

It is known that there are three manufacturers $A, B, C,$ who can produce micro chip MB666. The pr...

Write down the probability of obtaining $k$ heads in $n$ tosses of a fair coin. Now suppose that $k$...

The diagnostic test AL has a probability 0.9 of giving a positive result when applied to a person su...

I have a bag initially containing $r$ red fruit pastilles (my favourites) and $b$ fruit pastilles of...

Mr Blond returns to his flat to find it in complete darkness. He knows that this means that one of f...

It has been observed that Professor Ecks proves three types of theorems: 1, those that are correct a...

A biased coin, with a probability $p$ of coming up heads and a probability $q=1-p$ of coming up tail...

A message of $10^{k}$ binary digits is sent along a fibre optic cable with high probabilities $p_{0}...

Bread roll throwing duels at the Drones' Club are governed by a strict etiquette. The two duellists ...

Calamity Jane sits down to play the game of craps with Buffalo Bill. In this game she rolls two fair...

There are 28 colleges in Cambridge, of which two (New Hall and Newnham) are for women only; the othe...

Captain Spalding is on a visit to the idyllic island of Gambriced. The population of the island cons...

A taxi driver keeps a packet of toffees and a packet of mints in her taxi. From time to time she tak...

At any instant the probability that it is safe to cross a busy road is $0.1$. A toad is waiting to c...

Each day, I choose at random between my brown trousers, my grey trousers and my expensive but fashio...

A bus is supposed to stop outside my house every hour on the hour. From long observation I know that...

A and B play a guessing game. Each simultaneously names one of the numbers $1,2,3.$ If the numbers d...

\begin{questionparts} \item Show that, for any functions $f$ and $g$, and for any $m \geq 0$, $$\sum...

A game in a casino is played with a fair coin and an unbiased cubical die whose faces are labelled ...

A discrete random variable $X$ takes only positive integer values. Define $\E(X)$ for this case, an...

I seat $n$ boys and $3$ girls in a line at random, so that each order of the $n+3$ children is a...

A box contains $n$ pieces of string, each of which has two ends. I select two string ends at random...

A frog jumps towards a large pond. Each jump takes the frog either $1\,$m or $2\,$m nearer to the...

A bag contains $b$ balls, $r$ of them red and the rest white. In a game the player must remove ball...

In a bag are $n$ balls numbered 1, 2, $\ldots\,$, $n\,$. When a ball is taken out of the bag, each b...

You play the following game. You throw a six-sided fair die repeatedly. You may choose to stop after...

A bag contains 5 white balls, 3 red balls and 2 black balls. In the game of Blackball, a player draw...

\begin{questionparts} \item The three integers $n_1$, $n_2$ and $n_3$ satisfy $0 < n_1 < n_2 < n_3$ ...

\begin{questionparts} \item $X_{1},X_{2},\ldots,X_{n}$ are independent identically distributed rando...

In a lottery, each of the $N$ participants pays $\pounds c$ to the organiser and picks a number fr...

A random number generator prints out a sequence of integers $I_1, I_2, I_3, \dots$. Each integer is ...

Integers $n_{1},n_{2},\ldots,n_{r}$ (possibly the same) are chosen independently at random from the ...

An examination consists of several papers, which are marked independently. The mark given for each p...

A cricket team has only three bowlers, Arthur, Betty and Cuba, each of whom bowls 30 balls in any ma...

Bar magnets are placed randomly end-to-end in a straight line. If adjacent magnets have ends of oppo...

By considering the coefficients of $t^{n}$ in the equation \[(1+t)^{n}(1+t)^{n}=(1+t)^{2n},\] or oth...

Widgets are manufactured in batches of size $(n+N)$. Any widget has a probability $p$ of being fault...

\begin{questionparts} \item By considering the sum of a geometric series, or otherwise, show that \[...

\begin{questionparts} \item Two people adopt the following procedure for deciding where to go for a ...

$A,B$ and $C$ play a table tennis tournament. The winner of the tournament will be the first person ...

In a game, I toss a coin repeatedly. The probability, $p$, that the coin shows Heads on any given ...

A biased coin has probability $p$ of showing a head and probability $q$ of showing a tail, where $p\...

The infinite series $S$ is given by \[ S = 1 + (1 + d)r + (1 + 2d)r^2 + \cdots + (1+nd)r^n +\...

I have two identical dice. When I throw either one of them, the probability of it showing a 6 is $p$...

A men's endurance competition has an unlimited number of rounds. In each round, a competitor has, in...

The life of a certain species of elementary particles can be described as follows. Each particle has...

If a football match ends in a draw, there may be a "penalty shoot-out". Initially the teams each t...

I have $k$ different keys on my key ring. When I come home at night I try one key after another unt...

The game of Cambridge Whispers starts with the first participant Albert flipping an un-biased coin a...

The makers of Cruncho (`The Cereal Which Cares') are giving away a series of cards depicting $n$ gre...

An unbiased twelve-sided die has its faces marked $A,A,A,B,B,B,B,B,B,B,B,B.$ In a series of throws o...

\begin{questionparts} \item A bag of sweets contains one red sweet and $n$ blue sweets. I take a sw...

A pack of cards consists of $n+1$ cards, which are printed with the integers from $0$ to $n$. A...

A bag contains $b$ black balls and $w$ white balls. Balls are drawn at random from the bag and when ...

A group of biologists attempts to estimate the magnitude, $N$, of an island population of voles ({\...

An examiner has to assign a mark between 1 and $m$ inclusive to each of $n$ examination scripts ($n\...

I have a bag containing $M$ tokens, $m$ of which are red. I remove $n$ tokens from the bag at random...

Balls are chosen at random without replacement from an urn originally containing $m$ red balls and $...

In a television game show, a contestant has to open a door using a key. The contestant is given a b...

A fair die with faces numbered $1$, $\ldots\,$, $6$ is thrown repeatedly. The events $A$, $B$, $C$,...

In this question, the notation $\lfloor x \rfloor$ denotes the greatest integer less than or equal t...

The probability of throwing a head with a certain coin is $p$ and the probability of throwing a tail...

A coin has probability $p$ ($0 < p < 1$) of showing a head when tossed. Give a careful argument to s...

In a certain factory, microchips are made by two machines. Machine A makes a proportion~$\lambda$ o...

The probability of throwing a 6 with a biased die is $p\,$. It is known that $p$ is equal to one or...

\begin{questionparts} \item Find the maximum value of $\sqrt{p(1-p)}$ as $p$ varies between $0$ and ...

In Fridge football, each team scores two points for a goal and one point for a foul committed by the...

The random variable $X$ has the probability density function on the interval $[0, 1]$: $$f(x) = \beg...

A continuous random variable $X$ has a \textit{triangular} distribution, which means that it has a ...

Fire extinguishers may become faulty at any time after manufacture and are tested annually on the a...

In this question, you may use without proof the following result: \[ \int \sqrt{4-x^2}\, \d x = 2 \...

Sketch the graph of \[ y= \dfrac1 { x \ln x} \text{ for $x>0$, $x\ne1$}.\] You may assume that $x\...

The random variable $X$ can take the value \mbox{$X=-1$}, and also any value in the range \mbox{$0\l...

The life times of a large batch of electric light bulbs are independently and identically distribute...

A stick is broken at a point, chosen at random, along its length. Find the probability that the rati...

Each of my $n$ students has to hand in an essay to me. Let $T_{i}$ be the time at which the $i$th es...

When he sets out on a drive Mr Toad selects a speed $V$ kilometres per minute where $V$ is a random ...

The prevailing winds blow in a constant southerly direction from an enchanted castle. Each year, acc...

Wondergoo is applied to all new cars. It protects them completely against rust for three years, but ...

The random variable $X$ has mean $\mu$ and variance $\sigma^2$, and the function ${\rm V}$ is defi...

In this question, you may use without proof the results: \[ \sum_{r=1}^n r = \tfrac12 n(n+1) \qquad...

\begin{questionparts} \item Three real numbers are drawn independently from the continuous rectangu...

The random variable $X$ is uniformly distributed on the interval $[-1,1]$. Find $\E(X^2)$ and $\var ...

Two computers, LEP and VOZ are programmed to add numbers after first approximating each number by an...

Trains leave Barchester Station for London at 12 minutes past the hour, taking 60 minutes to complet...

A train of length $l_{1}$ and a lorry of length $l_{2}$ are heading for a level crossing at speeds $...

A point $P$ is chosen at random (with uniform distribution) on the circle $x^{2}+y^{2}=1$. The rando...

\begin{questionparts} \item A point $P$ lies in an equilateral triangle $ABC$ of height 1. The perp...

\begin{questionparts} \item My favourite dartboard is a disc of unit radius and centre $O$. I never...

Two points are chosen independently at random on the perimeter (including the diameter) of a semicir...

A densely populated circular island is divided into $N$ concentric regions $R_1$, $R_2$, $\ldots\,$...

Harry the Calculating Horse will do any mathematical problem I set him, providing the answer is 1, ...

The cakes in our canteen each contain exactly four currants, each currant being randomly placed in t...

When I throw a dart at a target, the probability that it lands a distance $X$ from the centre is a ...

To celebrate the opening of the financial year the finance minister of Genland flings a Slihing, a c...

\item A needle of length two cm is dropped at random onto a large piece of paper ruled with parallel...

Three points, $P,Q$ and $R$, are independently randomly chosen on the perimeter of a circle. Prove t...

When Septimus Moneybags throws darts at a dart board they are certain to end on the board (a disc o...

By making the substitution $y=\cos^{-1}t,$ or otherwise, show that \[ \int_{0}^{1}\cos^{-1}t\,\mathr...

Two points are chosen independently at random on the perimeter (including the diameter) of a semicir...

\begin{questionparts} \item The random variable $Z$ has a Normal distribution with mean $0$ and var...

A continuous random variable $X$ has probability density function given by \[ \f(x) = \begin{cases}...

The probability density function $\f(x)$ of the random variable $X$ is given by $$ \f(x) = k\left[{\...

The random variable $X$ has mean $\mu$ and standard deviation $\sigma$. The distribution of $X$ is s...

{\it Tabulated values of ${\Phi}(\cdot)$, the cumulative distribution function of a standard normal ...

Suppose $X$ is a random variable with probability density \[ \mathrm{f}(x)=Ax^{2}\exp(-x^{2}/2) \] ...

The time taken for me to set an acceptable examination question it $T$ hours. The distribution of $T...

Find the probability that the quadratic equation \[ X^{2}+2BX+1=0 \] has real roots when $B$ is nor...

The parliament of Laputa consists of 60 Preservatives and 40 Progressives. Preservatives never chan...

Fifty times a year, 1024 tourists disembark from a cruise liner at a port. From there they must trav...

On $K$ consecutive days each of $L$ identical coins is thrown $M$ times. For each coin, the probabi...

Each time it rains over the Cabbibo dam, a volume $V$ of water is deposited, almost instanetaneously...

It is believed that the population of Ruritania can be described as follows: \begin{questionparts} ...

Let $X$ be a standard normal random variable. If $M$ is any real number, the random variable $X_{M}$...

The function f is defined by $$\f(x) = 2\sin x - x\,.$$ Show graphically that the equation $\f(x)=0$...

Sketch the graphs of $y=\sec x$ and $y=\ln(2\sec x)$ for $0\leqslant x\leqslant\frac{1}{2}\pi$. Show...

Let $a$ and $b$ be positive integers such that $b<2a-1$. For any given positive integer $n$, the int...

\begin{questionparts} \item The coefficients in the series \[ S= \tfrac13 x + \tfrac 16 x^2 + \tfrac...

The sequence $a_n$ is defined by $a_0 = 1$ , $a_1 = 1$ , and $$ a_n = {1 + a_{n - 1}^2 \over a_{n ...

The value $V_N$ of a bond after $N$ days is determined by the equation $$ V_{N+1} = (1+c) V_{N} -d \...

Each day, books returned to a library are placed on a shelf in order of arrival, and left there. Whe...

The real numbers $u_{0},u_{1},u_{2},\ldots$ satisfy the difference equation \[ au_{n+2}+bu_{n+1}+cu...

A secret message consists of the numbers $1,3,7,23,24,37,39,43,43,43,45,47$ arranged in some order a...

The integers $a,b$ and $c$ satisfy \[ 2a^{2}+b^{2}=5c^{2}. \] By considering the possible values of...

Find the ratio, over one revolution, of the distance moved by a wheel rolling on a flat surface to t...

The curve $P$ has the parametric equations $$ x= \sin\theta, \quad y=\cos2\theta \qquad\hbox{ for ...

A kingdom consists of a vast plane with a central parabolic hill. In a vertical cross-section throu...

Let $(G,*)$ and $(H,\circ)$ be two groups and $G\times H$ be the set of ordered pairs $(g,h)$ with $...

The set $S$ consists of $N(>2)$ elements $a_{1},a_{2},\ldots,a_{N}.$ $S$ is acted upon by a binary o...

Let $a$ be a non-zero real number and define a binary operation on the set of real numbers by $$ x*y...

\begin{questionparts} \item Let $S$ be the set of matrices of the form \[ \begin{pmatrix}a & a\\ a ...

Consider the following sets with the usual definition of multiplication appropriate to each. In each...

Let $S_{3}$ be the group of permutations of three objects and $Z_{6}$ be the group of integers under...

The set $S$ consists of ordered pairs of complex numbers $(z_1,z_2)$ and a binary operation $\circ$...

Explain what is meant by the order of an element $g$ of a group $G$. The set $S$ consists of all ...

Let $G$ be the set of all matrices of the form \[ \begin{pmatrix}a & b\\ 0 & c \end{pma...

The elements $a,b,c,d$ belong to the group $G$ with binary operation $*.$ Show that \begin{questionp...

The matrix $\mathbf{M}$ is given by \[ \mathbf{M}=\begin{pmatrix}\cos(2\pi/m) & -\sin(2\pi/m)\\ \si...

Let $\Omega=\exp(\mathrm{i}\pi/3).$ Prove that $\Omega^{2}-\Omega+1=0.$ Two transformations, $R$ ...

Let $G$ be a finite group with identity $e.$ For each element $g\in G,$ the order of $g$, $o(g),$ is...

Give a careful argument to show that, if $G_{1}$ and $G_{2}$ are subgroups of a finite group $G$ suc...

Let ${\displaystyle I_{m,n}=\int\cos^{m}x\sin nx\,\mathrm{d}x,}$ where $m$ and $n$ are non-negative ...

Let \[ \displaystyle I_n= \int_{-\infty}^\infty \frac 1 {(x^2+2ax+b)^n} \, \d x \] where $a$ and $b...

\begin{questionparts} \item Let \[ I_n= \int_0^\infty \frac 1 {(1+u^2)^n}\, \d u \,, \] where $n$ i...

For $n\ge 0$, let \[ I_n = \int_0^1 x^n(1-x)^n\d x\,. \] \begin{questionparts} \item For $n\ge 1$, ...

Let $m$ be a positive integer and let $n$ be a non-negative integer. \begin{qu...

For $n=1$, $2$, $3$, $\ldots\,$, let \[ I_n = \int_0^1 {t^{n-1} \over \l t+1 \r^n} \, \mathrm{d} t \...

If \[ \mathrm{I}_{n}=\int_{0}^{a}x^{n+\frac{1}{2}}(a-x)^{\frac{1}{2}}\,\mathrm{d}x, \] show that $\...

Let \[ u_{n}=\int_{0}^{\frac{1}{2}\pi}\sin^{n}t\,\mathrm{d}t \] for each integer $n\geqslant0$. By ...

Calculate \[ \int_{0}^{x}\mathrm{sech}\, t\,\mathrm{d}t. \] Find the reduction formula involving $I...

Given that ${\displaystyle I_{n}=\int_{0}^{\pi}\frac{x\sin^{2}(nx)}{\sin^{2}x}\,\mathrm{d}x,}$ where...

\begin{questionparts} \item The integral $I_{k}$ is defined by \[ I_{k}=\int_{0}^{\theta}\cos^{k}x\...

Give a rough sketch of the function $\tan^{k}\theta$ for $0\leqslant\theta\leqslant\frac{1}{4}\pi$ ...

A plank $AB$ of length $L$ initially lies horizontally at rest along the $x$-axis on a flat surface,...

A box has the shape of a uniform solid cuboid of height $h$ and with a square base of side $b$, wher...

A long, inextensible string passes through a small fixed ring. One end of the string is attached to ...

A rough ring of radius $a$ is fixed so that it lies in a plane inclined at an angle $\alpha$ to the ...

A plane makes an acute angle $\alpha$ with the horizontal. A box in the shape of a cube is fixed on...

Two uniform ladders $AB$ and $BC$ of equal length are hinged smoothly at $B$. The weight of $AB$ is ...

A sphere of radius $a$ and weight $W$ rests on horizontal ground. A thin uniform beam of weight $3...

A uniform solid sphere of diameter $d$ and mass $m$ is drawn very slowly and without slipping from ...

Two rough solid circular cylinders, of equal radius and length and of uniform density, lie side by s...

\begin{center} \begin{tikzpicture}[scale=0.5] % Circle \def\a{5} \def\b{10} \def\k{\...

The diagram shows a crude step-ladder constructed by smoothly hinging-together two light ladders $AB...

\begin{center} \begin{tikzpicture} % Coordinate axes lines \coordinate (O) at (0,0); \co...

$\ $\vspace{-1.5cm} \noindent \begin{center} \psset{xunit=0.8cm,yunit=0.8cm,algebraic=true,dotstyle...

The edges $OA,OB,OC$ of a rigid cube are taken as coordinate axes and $O',A',B',C'$ are the vertices...

A rough circular cylinder of mass $M$ and radius $a$ rests on a rough horizontal plane. The curved s...

A uniform ladder of mass $M$ rests with its upper end against a smooth vertical wall, and with its l...

A uniform ladder of length $l$ and mass $m$ rests with one end in contact with a smooth ramp incline...

A straight road leading to my house consists of two sections. The first section is inclined downward...

In this question take $g = 10 ms^{-2}.$ The point $A$ lies on a fixed rough plane inclined at $30^{\...

In a certain race, runners run 5$\,$km in a straight line to a fixed point and then turn and run bac...

A particle $P$ of mass $m$ is projected with speed $u_0$ along a smooth horizontal floor directly to...

Two small beads, $A$ and $B$, of the same mass, are threaded onto a vertical wire on which they slid...

A railway truck, initially at rest, can move forwards without friction on a long straight \mbox{hor...

Particles $P_1$, $P_2$, $\ldots$ are at rest on the $x$-axis, and the $x$-coordinate of $P_n$ is $n...

Four particles $A$, $B$, $C$ and $D$ are initially at rest on a smooth horizontal table. They lie ...

\begin{questionparts} \item A uniform spherical ball of mass $M$ and radius $R$ is released from r...

Three identical particles lie, not touching one another, in a straight line on a smooth horizontal s...

Two parallel vertical barriers are fixed a distance $d$ apart on horizontal ice. A small ice hockey ...

A small block of mass $km$ is initially at rest on a smooth horizontal surface. Particles $P_1$, $P_...

Two particles, $A$ of mass $2m$ and $B$ of mass $m$, are moving towards each other in a straight lin...

A particle, $A$, is dropped from a point $P$ which is at a height $h$ above a horizontal plane. A~...

A bullet of mass $m$ is fired horizontally with speed $u$ into a wooden block of mass $M$ at rest o...

\begin{questionparts} \item In an experiment, a particle $A$ of mass $m$ is at rest on a smooth hori...

Two particles move on a smooth horizontal table and collide. The masses of the particles are $m$ and...

A lift of mass $M$ and its counterweight of mass $M$ are connected by a light inextensible cable w...

Three particles, $A$, $B$ and $C$, of masses $m$, $km$ and $3m$ respectively, are initially at rest ...

Particles $A_1$, $A_2$, $A_3$, $\ldots$, $A_n$ (where $n\ge 2$) lie at rest in that order in a smo...

Two particles, A and B, move without friction along a horizontal line which is perpendicular to a ...

Three collinear, non-touching particles $A$, $B$ and $C$ have masses $a$, $b$ and $c$, respectively,...

A smooth plane is inclined at an angle $\alpha$ to the horizontal. $A$ and $B$ are two points a dist...

A bicycle pump consists of a cylinder and a piston. The piston is pushed in with steady speed~$u$. A...

Two particles $A$ and $B$ of masses $m$ and $km$, respectively, are at rest on a smooth horizontal s...

Three particles $P_1$, $P_2$ and $P_3$ of masses $m_{1}$, $m_{2}$ and $m_{3}$ respectively lie at ...

A chain of mass $m$ and length $l$ is composed of $n$ small smooth links. It is suspended verticall...

$N$ particles $P_1$, $P_2$, $P_3$, $\ldots$, $P_N$ with masses $m$, $qm$, $q^2m$, $\ldots$ , ${q^{N-...

\noindent{\it In this question the effect of gravity is to be neglected.} A small body of mass $M$ i...

A spaceship of mass $M$ is at rest. It separates into two parts in an explosion in which the total k...

Three small spheres of masses $m_{1},m_{2}$ and $m_{3},$ move in a straight line on a smooth horizon...

A ball of mass $m$ is thrown vertically upwards from the floor of a room of height $h$ with speed $\...

$\,$ \begin{center} \begin{tikzpicture} % Draw vertical lines \draw (0,0) -- (0,4) node[abov...

Two particles $P_{1}$ and $P_{2}$, each of mass $m$, are joined by a light smooth inextensible strin...

A piledriver consists of a weight of mass $M$ connected to a lighter counterweight of mass $m$ by a ...

A uniform smooth wedge of mass $m$ has congruent triangular end faces $A_{1}B_{1}C_{1}$ and $A_{2}B_...

The identical uniform smooth spherical marbles $A_{1},A_{2},\ldots,A_{n},$ where $n\geqslant3,$ each...

The axles of the wheels of a motorbike of mass $m$ are a distance $b$ apart. Its centre of mass is ...

A thin uniform wire is bent into the shape of an isosceles triangle $ABC$, where $AB$ and $AC$ are o...

A uniform rectangular lamina $ABCD$ rests in equilibrium in a vertical plane with the $A$ in contact...

\begin{center} \begin{tikzpicture} \begin{scope}[rotate=60] \coordinate (O) at (...

A non-uniform rod $AB$ has weight $W$ and length $3l$. When the rod is suspended horizontally in equ...

$ABCD$ is a uniform rectangular lamina and $X$ is a point on $BC\,$. The lengths of $AD$, $AB$ and $...

A piece of uniform wire is bent into three sides of a square $ABCD$ so that the side $AD$ is missing...

A thin non-uniform rod $PQ$ of length $2a$ has its centre of gravity a distance $a+d$ from $P$. It h...

\begin{questionparts} \item A horizontal disc of radius $r$ rotates about a vertical axis through it...

Three particles, $A$, $B$ and $C$, each of mass $m$, lie on a smooth horizontal table. Particles $A$...

A particle $P$ of mass $m$ is connected by two light inextensible strings to two fixed points $A$ an...

A particle $P$ moves so that, at time $t$, its displacement $ \bf r $ from a fixed origin is given ...

An automated mobile dummy target for gunnery practice is moving anti-clockwise around the circumfere...

The force of attraction between two stars of masses $m_{1}$ and $m_{2}$ a distance $r$ apart is $\ga...

$\ $\vspace{-1.5cm} \noindent \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=...

A small heavy bead can slide smoothly in a vertical plane on a fixed wire with equation \[ y=x-\fra...

A skater of mass $M$ is skating inattentively on a smooth frozen canal. She suddenly realises that s...

Two identical smooth spheres $P$ and $Q$ can move on a smooth horizontal table. Initially, $P$ moves...

Ice snooker is played on a rectangular horizontal table, of length $L$ and width $B$, on which a sma...

$\,$ \vspace{-1cm} \begin{center} \begin{tikzpicture}[scale=1.3] % Setting up the same viewport/dim...

Two particles of masses $m$ and $M$, with $M>m$, lie in a smooth circular groove on a horizontal pl...

$\,$ \begin{center} \psset{xunit=1.5cm,yunit=1.5cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth...

The lengths of the sides of a rectangular billiards table $ABCD$ are given by $AB = DC = a$ and $AD...

A particle moves on a smooth triangular horizontal surface $AOB$ with angle $AOB = 30^\circ$. Th...

$\,$ \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=...

Two small discs of masses $m$ and $\mu m$ lie on a smooth horizontal surface. The disc of mass $...

Two identical spherical balls, moving on a horizontal, smooth table, collide in such a way that both...

A particle rests at a point $A$ on a horizontal table and is joined to a point $O$ on the table by a...

A smooth particle $P_{1}$ is projected from a point $O$ on the horizontal floor of a room with has a...

A straight staircase consists of $N$ smooth horizontal stairs each of height $h$. A particle slides ...

A smooth uniform sphere, with centre $A$, radius $2a$ and mass $3m,$ is suspended from a fixed point...

A smooth billiard ball moving on a smooth horizontal table strikes another identical ball which is a...

The lower end of a rigid uniform rod of mass $m$ and length $a$ rests at point $M$ on rough horizont...

A rubber band band of length $2\pi$ and modulus of elasticity $\lambda$ encircles a smooth cylinder ...

Two particles $A$ and $B$ of masses $m$ and $2 m$, respectively, are connected by a light spr...

Three pegs $P$, $Q$ and $R$ are fixed on a smooth horizontal table in such a way that they form the...

An equilateral triangle, comprising three light rods each of length $\sqrt3a$, has a particle of mas...

One end of a thin heavy uniform inextensible perfectly flexible rope of length $2L$ and mass $2M$ is...

Particles $P$ and $Q$, each of mass $m$, lie initially at rest a distance $a$ apart on a smooth hor...

A train consists of an engine and $n$ trucks. It is travelling along a straight horizontal section ...

A long string consists of $n$ short light strings joined together, each of natural length $\ell$ and...

Two small beads, $A$ and $B$, each of mass $m$, are threaded on a smooth horizontal circular hoop...

A long, light, inextensible string passes through a small, smooth ring fixed at the point $O$. One...

Two thin discs, each of radius $r$ and mass $m$, are held on a rough horizontal surface with their c...

A smooth cylinder with circular cross-section of radius $a$ is held with its axis horizontal. A~ligh...

A uniform rigid rod $BC$ is suspended from a fixed point $A$ by light stretched springs $AB,AC$. The...

A bungee-jumper of mass $m$ is attached by means of a light rope of natural length $l$ and modulus o...

A small ball of mass $m$ is suspended in equilibrium by a light elastic string of natural length $l$...

A step-ladder has two sections $AB$ and $AC,$ each of length $4a,$ smoothly hinged at $A$ and connec...

A truck is towing a trailer of mass $m$ across level ground by means of an elastic rope of natural l...

One end $A$ of a light elastic string of natural length $l$ and modulus of elasticity $\lambda$ is f...

$ABCD$ is a horizontal line with $AB=CD=a$ and $BC=6a$. There are fixed smooth pegs at $B$ and $C$...

A small lamp of mass $m$ is at the end $A$ of a light rod $AB$ of length $2a$ attached at $B$ to a v...

$\,$ \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=3p...

In the figure, $W_{1}$ and $W_{2}$ are wheels, both of radius $r$. Their centres $C_{1}$ and $C_{2}$...

Three light elastic strings $AB,BC$ and $CD$, each of natural length $a$ and modulus of elasticity $...

$\,$ \begin{center} \begin{tikzpicture}[scale=1.0] % Define coordinates \coordinate (A) at (...

A lift of mass $M$ and its counterweight of mass $M$ are connected by a light inextensible cable whi...

A regular tetrahedron $ABCD$ of mass $M$ is made of 6 identical uniform rigid rods, each of length $...

A piece of circus apparatus consists of a rigid uniform plank of mass 1000$\,$kg, suspended in a hor...

In this question, $n \geq 2$. \begin{questionparts} \item A solid, of uniform density, is formed by ...

It is given that the gravitational force between a disc, of radius $a,$ thickness $\delta x$ and uni...

The end $A$ of an inextensible string $AB$ of length $\pi$ is attached to a point on the circumfere...

\begin{questionparts} \item A uniform lamina $OXYZ$ is in the shape of the trapezium shown in the di...

A solid figure is composed of a uniform solid cylinder of density $\rho$ and a uniform solid hemis...

The base of a non-uniform solid hemisphere, of mass $M,$ has radius $r.$ The distance of the centre ...

A tall container made of light material of negligible thickness has the form of a prism, with a squa...

A child's toy consists of a solid cone of height $\lambda a$ and a solid hemisphere of radius $a$, m...

A uniform rectangular lamina of sides $2a$ and $2b$ rests in a vertical plane. It is supported in eq...

Derive a formula for the position of the centre of mass of a uniform circular arc of radius $r$ whic...

Points $A$ and $B$ are at the same height and a distance $\sqrt{2}r$ apart. Two small, spherical pa...

In this question, $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors and $\mathbf{j}$ is v...

A firework consists of a uniform rod of mass $M$ and length $2a$, pivoted smoothly at one end so tha...

A smooth sphere of radius $r$ stands fixed on a horizontal floor. A particle of mass $m$ is displace...

A particle is attached to one end of a light inextensible string of length $b$. The other end of the...

A smooth plane is inclined at an angle $\alpha$ to the horizontal. A particle $P$ of mass $m$ is at...

An equilateral triangle $ABC$ is made of three light rods each of length $a$. It is free to rotate i...

A small ring of mass $m$ is free to slide without friction on a hoop of radius $a$. The hoop is f...

A thin uniform circular disc of radius $a$ and mass $m$ is held in equilibrium in a horizontal plane...

Particles $P$ and $Q$ have masses $3m$ and $4m$, respectively. They lie on the outer curved surface ...

$\,$ \begin{center} \newrgbcolor{wwwwww}{0.4 0.4 0.4} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,...

Two identical particles $P$ and $Q$, each of mass $m$, are attached to the ends of a diameter of a ...

\begin{questionparts} \item A wheel consists of a thin light circular rim attached by light spokes ...

A horizontal spindle rotates freely in a fixed bearing. Three light rods are each attached by one e...

A circular hoop of radius $a$ is free to rotate about a fixed horizontal axis passing through a poi...

A light hollow cylinder of radius $a$ can rotate freely about its axis of symmetry, which is fixed...

The point $A$ is vertically above the point $B$. A light inextensible string, with a smooth ring $P...

A particle of mass $m$ is at rest on top of a smooth fixed sphere of radius $a$. Show that, if the p...

The plot of `Rhode Island Red and the Henhouse of Doom' calls for the heroine to cling on to the cir...

$\,$ \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=3p...

$\,$ \begin{center} \begin{tikzpicture} % Draw horizontal line \coordinate (O) at (0,0); ...

\begin{center} \begin{tikzpicture} % Main lines \draw (1,1) -- (5,7); \draw (5,7) -- (5,...

A smooth tube whose axis is horizontal has an elliptic cross-section in the form of the curve with p...

A particle $P$ is projected, from the lowest point, along the smooth inside surface of a fixed spher...

A smooth horizontal plane rotates with constant angular velocity $\Omega$ about a fixed vertical axi...

One end of a light inextrnsible string of length $l$ is fixed to a point on the upper surface of a t...

Two particles of mass $M$ and $m$ $(M>m)$ are attached to the ends of a light rod of length $2l.$ Th...

A heavy particle lies on a smooth horizontal table, and is attached to one end of a light inextensib...

A particle of mass $m$ moves along the $x$-axis. At time $t=0$ it passes through $x=0$ with velocity...

A uniform elastic string lies on a smooth horizontal table. One end of the string is attached to a ...

A car of mass $m$ makes a journey of distance $2d$ in a straight line. It experiences air resis...

A car of mass $m$ travels along a straight horizontal road with its engine working at a constant r...

A light rod of length $2a$ has a particle of mass $m$ attached to each end and it moves in a vertic...

A particle $P$ of mass $m$ moves on a smooth fixed straight horizontal rail and is attached to a f...

A particle of mass $m$ is projected with velocity $\+ u$. It is acted upon by the force $m\+g$ due...

A comet in deep space picks up mass as it travels through a large stationary dust cloud. It is subj...

A particle of mass $m$ is initially at rest on a rough horizontal surface. The particle experience...

Particles $P$, of mass $2$, and $Q$, of mass $1$, move along a line. Their distances from a fixed...

The maximum power that can be developed by the engine of train $A$, of mass $m$, when travelling a...

A particle moves along the $x$-axis in such a way that its acceleration is $kx \dot{x}\,$ where $k$...

A particle of unit mass is projected vertically upwards with speed $u$. At height $x$, while the par...

In an aerobatics display, Jane and Karen jump from a great height and go through a period of free f...

In the $Z$--universe, a star of mass $M$ suddenly blows up, and the fragments, with various initial ...

A particle of unit mass is projected vertically upwards in a medium whose resistance is $k$ times th...

Two identical particles of unit mass move under gravity in a medium for which the magnitude of the r...

\textit{In this question, take the value of $g$ to be $10\ \mathrm{ms}^{-2}.$} A body of mass $m$ kg...

A train starts from a station. The tractive force exerted by the engine is at first constant and equ...

A comet, which may be regarded as a particle of mass $m$, moving in the sun's gravitational field, a...

The current in a straight river of constant width $h$ flows at uniform speed $\alpha v$ parallel to...

A spaceship of mass $M$ is travelling at constant speed $V$ in a straight line when it enters a forc...

A goalkeeper stands on the goal-line and kicks the football directly into the wind, at an angle $\al...

A thin uniform elastic band of mass $m,$ length $l$ and modulus of elasticity $\lambda$ is pushed on...

A particle $P$ of mass $m$ is attached to points $A$ and $B$, where $A$ is a distance $9a$ vertical...

A particle $P$ of mass $m$ is constrained to move on a vertical circle of smooth wire with centre~$...

$B_1$ and $B_2$ are parallel, thin, horizontal fixed beams. $B_1$ is a vertical distance $d \sin \a...

The string $AP$ has a natural length of $1\!\cdot5\!$ metres and modulus of elasticity equal to $5g$...

A particle is attached to a point $P$ of an unstretched light uniform spring $AB$ of modulus of ela...

Consider a simple pendulum of length $l$ and angular displacement $\theta$, which is {\bf not} assum...

Two small spheres $A$ and $B$ of equal mass $m$ are suspended in contact by two light inextensible s...

A smooth circular wire of radius $a$ is held fixed in a vertical plane with light elastic strings of...

A particle hangs in equilibrium from the ceiling of a stationary lift, to which it is attached by an...

A smooth, axially symmetric bowl has its vertical cross-sections determined by $s=2\sqrt{ky},$ where...

The force $F$ of repulsion between two particles with positive charges $Q$ and $Q'$ is given by $F=k...

A particle is attached to one end $B$ of a light elastic string of unstretched length $a$. Initially...

The points $A,B,C,D$ and $E$ lie on a thin smooth horizontal table and are equally spaced on a circl...

Prove that \[ \tan^{-1}t=t-\frac{t^{3}}{3}+\frac{t^{5}}{5}-\cdots+\frac{(-1)^{n}t^{2n+1}}{2n+1}+(-1...

If $y=\mathrm{f}(x)$, then the inverse of $\mathrm{f}$ (when it exists) can be obtained from \textit...

Show that the sum of the infinite series \[ \log_{2}\mathrm{e}-\log_{4}\mathrm{e}+\log_{16}\mathrm{...

Prove that, for any numbers $a_1$, $a_2$, $\ldots$\,, and $b_1$, $b_2$, $\ldots$\,, and for $n\ge1$,...

\begin{questionparts} \item Prove that, for any positive integers $n$ and $r$, \[ \frac{1}{^{n+r}\C_...

\begin{questionparts} \item By considering \ $\displaystyle \frac1 {1+ x^r} - \frac1 {1+ x^{r +1}} ...

Evaluate the integral \[ \hphantom{ \ \ \ \ \ \ \ \ \ (m> \tfrac12)\,.} \int_{m-\frac12} ^\infty \...

The numbers $\.f(r)$ satisfy $\.f(r)>\.f(r+1)$ for $r=1$, $2$, \dots. Show that, for any non-nega...

In this question, the following theorem may be used.\newline {\sl Let $u_1$, $u_2$, $\ldots$ be a s...

The sequence $F_0$, $F_1$, $F_2$, $\ldots\,$ is defined by $F_0=0$, $F_1=1$ and, for $n\ge0$, \[ F_{...

In this question, $\vert x \vert <1$ and you may ignore issues of convergence. \begin{questionparts}...

The positive numbers $\alpha$, $\beta$ and $q$ satisfy $\beta-\alpha >q$. Show that \[ \frac{\alpha^...

A sequence of numbers $t_0$, $t_1$, $t_2$, $\ldots\,$ satisfies \[ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \...

Given that $y = \cos(m \arcsin x)$, for $\vert x \vert <1$, prove that \[ (1-x^2) \frac {\d^2 y}{\d ...

Let $S_k(n) \equiv \sum\limits_{r=0}^n r^k\,$, where $k$ is a positive integer, so that \[ S_1(n) \e...

The sequence of real numbers $u_1$, $u_2$, $u_3$, $\ldots$ is defined by \begin{equation*} u_1=2 \,,...

The sequence $u_n$ ($n= 1, 2, \ldots$) satisfies the recurrence relation \[ u_{n+2}= \frac{u_{n+1}...

Given a sequence $w_0$, $w_1$, $w_2$, $\ldots\,$, the sequence $F_1$, $F_2$, $\ldots$ is defined by ...

Given that $\f''(x) > 0$ when $a \le x \le b\,$, explain with the aid of a sketch why \[ (b-a) \, \...

A sequence $t_0$, $t_1$, $t_2$, $...$ is said to be {\sl strictly increasing} if $t_{n+1} > t_n$ f...

Show that \[ 2\sin \frac12 \theta \, \cos r\theta = \sin\big(r+\frac12\big)\theta - \sin\big(r-\...

Show that, if $n>0\,$, then $$ \int_{e^{1/n}}^\infty\,{{\ln x} \over {x^{n+1}}}\,\d x = {2 \over ...

Given that $$\e = 1 + {1 \over 1 !} + {1 \over 2 !} + {1 \over 3 !} + \cdots + {1 \over r !} + \cdo...

The sequence $u_0$, $u_1$, $u_2$, ... is defined by $$ u_0=1,\hspace{0.2in} u_1=1, \quad u_{n+1}=u...

Justify, by means of a sketch, the formula $$ \lim_{n\rightarrow\infty}\left\{{1\over n}\sum_{m=1}^n...

Show that the sum $S_N$ of the first $N$ terms of the series $$\frac{1}{1\cdot2\cdot3}+\frac{3}{\cdo...

For each positive integer $n$, let \begin{align*} a_n&=\frac1{n+1}+\frac1{(n+1)(n+2)}+\frac1{(n+1)(n...

Sum the following infinite series. \begin{questionparts} \item \[ 1 + \frac13 \bigg({\frac12}\bigg...

Obtain the sum to infinity of each of the following series. \begin{questionparts} \item $1{\display...

For $x>0$ find $\int x\ln x\,\mathrm{d}x$. By approximating the area corresponding to $\int_{0}^{1}...

\begin{questionparts} \item Let $a$, $b$ and $c$ be three non-zero complex numbers with the properti...

The $n$th degree polynomial P$(x)$ is said to be \textit{reflexive} if: \begin{enumerate} \item[(a)]...

\begin{questionparts} \item The function $\f$ is given by \[ \f(\beta)=\beta - \frac 1 \beta - \fr...

Let $\alpha$, $\beta$, $\gamma$ and $\delta$ be the roots of the quartic equation \[ x^4 +px^3 +qx^...

\begin{questionparts} \item Let $w$ and $z$ be complex numbers, and let $u= w+z$ and $v=w^2+z^2$. Pr...

Let $a$, $b$ and $c$ be real numbers such that $a+b+c=0$ and let \[(1+ax)(1+bx)(1+cx) = 1+qx^2 +rx^3...

The numbers $x$, $y$ and $z$ satisfy \begin{align*} x+y+z&= 1\\ x^2+y^2+z^2&=2\\ x^3+y^3+z^3&=3\,. \...

Find all values of $a$, $b$, $x$ and $y$ that satisfy the simultaneous equations \begin{alignat*}{3}...

\textit{In this question, do not consider the special cases in which the denominators of any of your...

In this question, you may assume that if $k_1,\dots,k_n$ are distinct positive real numbers, then \[...

\begin{questionparts} \item If $x+y+z=\alpha,$ $xy+yz+zx=\beta$ and $xyz=\gamma,$ find numbers $A,...

The cubic equation \[ x^{3}-px^{2}+qx-r=0 \] has roots $a,b$ and $c$. Express $p,q$ and $r$ in term...

The equation \[ x^{n}-qx^{n-1}+r=0, \] where $n\geqslant5$ and $q$ and $r$ are real constants, has...

The point $P(a\sec \theta, b\tan \theta )$ lies on the hyperbola \[ \dfrac{x^{2}}{a^{2}}-\dfrac{y^...

The point with cartesian coordinates $(x,y)$ lies on a curve with polar equation $r=\f(\theta)\,$. ...

\begin{questionparts} \item Show that under the changes of variable $x= r\cos\theta$ and $y = r\sin\...

In this question, $r$ and $\theta$ are polar coordinates with $r \ge0$ and $- \pi < \theta\le \pi$...

A movable point $P$ has cartesian coordinates $(x,y)$, where $x$ and $y$ are functions of $t$. T...

Show that in polar coordinates the gradient of any curve at the point $(r,\theta)$ is \[ \frac{ \ \...

Show that the equation (in plane polar coordinates) $r=\cos\theta$, for $-\frac{1}{2}\pi \le \theta ...

The curve $C$ has the equation $x^3+y^3 = 3xy$. \begin{questionparts} \item Show that there is no po...

\noindent \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,line...

Sketch the curve $C$ whose polar equation is \[ r=4a\cos2\theta\qquad\mbox{ for }-\tfrac{1}{4}\pi<\...

The parametric equations $E_{1}$ and $E_{2}$ define the same ellipse, in terms of the parameters $\t...

The curve $C$ has the differential equation in polar coordinates \[ \frac{\mathrm{d}^{2}r}{\mathrm{d...

Show by means of a sketch that the parabola $r(1+\cos\theta)=1$ cuts the interior of the cardioid $r...

Show that, for a given constant $\gamma$ $(\sin\gamma\neq0)$ and with suitable choice of the constan...

\begin{questionparts} \item Show that in polar coordinates, the gradient of any curve at the poin...

The distinct points $P(ap^2 , 2ap)$, $Q(aq^2 , 2aq)$ and $R(ar^2,2ar)$ lie on the parabola $y^2 = ...

\begin{questionparts} \item The line $L$ has equation $y=mx+c$, where $m>0$ and $c>0$. Show th...

The point $P(a\cos\theta\,,\, b\sin\theta)$, where $a>b>0$, lies on the ellipse \[\dfrac {x^2}{a^2}...

Let $P$ be the point on the curve $y=ax^2+bx+c$ (where $a$ is non-zero) at which the gradient is $m...

In the $x$--$y$ plane, the point $A$ has coordinates $(a\,,0)$ and the point $B$ has coordinates...

A parabola has the equation $y=x^{2}.$ The points $P$ and $Q$ with coordinates $(p,p^{2})$ and $(q,q...

The straight line $OSA,$ where $O$ is the origin, bisects the angle between the positive $x$ and $y$...

Find the equations of the tangent and normal to the parabola $y^{2}=4ax$ at the point $(at^{2},2at)....

In this question, you should ignore issues of convergence. \begin{questionparts} \item Let \[ I = \...

In this question, you should ignore issues of convergence. \begin{questionparts} \item Write down t...

\begin{questionparts} \item By use of calculus, show that $x- \ln(1+x)$ is positive for all positive...

In this question, you may ignore questions of convergence. Let $y= \dfrac {\arcsin x}{\sqrt{1-x^2}}\...

\begin{questionparts} \item Show that \[ \sum_{n=1} ^\infty \frac{n+1}{n!} = 2\e - 1 \] and \...

In this question, you may assume that the infinite series \[ \ln(1+x) = x-\frac{x^2}2 + \frac{x^3}{...

The function $\f(t)$ is defined, for $t\ne0$, by \[ \f(t) = \frac t {\e^t-1}\,. \] \begin{quest...

The function $f$ satisfies the identity \begin{equation} f(x) +f(y) \equiv f(x+y) \tag{$*$} \end{eq...

\begin{questionparts} \item Let \[ \tan x = \sum\limits_{n=0}^\infty a_n x^n \text{ and } \cot x = ...

Given that $y = \ln ( x + \sqrt{x^2 + 1})$, show that $ \displaystyle \frac{\d y}{\d x} = \frac1 {\...

Sketch the graph of ${\rm f}(s)={ \e}^s(s-3)+3$ for $0\le s < \infty$. Taking ${\e\approx 2.7}$, fin...

The exponential of a square matrix ${\bf A}$ is defined to be $$ \exp ({\bf A}) = \sum_{r=0}^\infty ...

\begin{questionparts} \item By considering the series expansion of $(x^2+5x+4){\rm \; e}^x$ show tha...

The function $\mathrm{f}$ is given by $\mathrm{f}(x)=\sin^{-1}x$ for $-1 < x < 1.$ Prove that \[ (1...

\begin{questionparts} \item Evaluate \[ \sum_{r=1}^{n}\frac{6}{r(r+1)(r+3)}. \] \item Expand $\ln(1...

The points $P\,(0,a),$ $Q\,(a,0)$ and $R\,(a,-a)$ lie on the curve $C$ with cartesian equation \[ x...

Let \begin{alignat*}{2} \tan x & =\ \ \, \quad{\displaystyle \sum_{n=0}^{\infty}a_{n}x^{n}} & & \t...

\begin{center} \begin{tikzpicture}[scale=0.5] \draw[domain = -2.5:2.5, samples=180, vari...

Show, by finding $R$ and $\gamma$, that $A \sinh x + B\cosh x $ can be written in the form $R\cosh...

Starting from the result that \[ \.h(t) >0\ \mathrm{for}\ 0< t < x \Longrightarrow \int_0^x \.h(...

Let $y = \ln (x^2-1)\,$, where $x >1$, and let $r$ and $\theta$ be functions of $x$ determined by ...

\begin{questionparts} \item Solve the equation $u^2+2u\sinh x -1=0$ giving $u$ in terms of $x$. Find...

Define $\cosh x$ and $\sinh x$ in terms of exponentials and prove, from your definitions, that \[ \...

The real numbers $x$ and $y$ satisfy the simultaneous equations $$ \sinh (2x) = \cosh y \qquad\hbox{...

\begin{questionparts} \item Given that \[ \mathrm{f}(x)=\ln(1+\mathrm{e}^{x}), \] prove that $\ln[\...

The transformation $T$ from $\binom{x}{y}$ to $\binom{x'}{y'}$ in two-dimensional space is given by ...

Solve the quadratic equation $u^{2}+2u\sinh x-1=0$, giving $u$ in terms of $x$. Find the solution...

The real variables $\theta$ and $u$ are related by the equation $\tan\theta=\sinh u$ and $0\leqslant...

Given that $y=\cosh(n\cosh^{-1}x),$ for $x\geqslant1,$ prove that \[ y=\frac{(x+\sqrt{x^{2}-1})^{n}+...

The real numbers $x$ and $y$ are related to the real numbers $u$ and $v$ by \[ 2(u+\mathrm{i}v)=\...

Show that the following functions are positive when $x$ is positive: \begin{questionparts} \item[ $...

Let $f(x) = \sqrt{x^2 + 1} - x$. \begin{questionparts} \item Using a binomial series, or otherwise, ...

\begin{questionparts} \item Show, by means of the substitution $u=\cosh x\,$, that \[ \in...

The definite integrals $T$, $U$, $V$ and $X$ are defined by \begin{align*} T&= \int_{\frac13}^{\frac...

The following result applies to any function $\f$ which is continuous, has positive gradient and...

In this question, $a$ is a positive constant. \begin{questionparts} \item Express $\cosh a$ in term...

\begin{questionparts} \item Show, with the aid of a sketch, that $y> \tanh (y/2)$ for $y>0$ and d...

Show that if $\displaystyle \int\frac1{u \, \f(u)}\; \d u = \F(u) + c\;$, then $\displaystyle ...

Show that \[ \int_0^a \frac{\sinh x}{2\cosh^2 x -1} \, \mathrm{d} x = \frac{1}{2 \sqrt{2}} \ln \l \f...

Given that $x+a>0$ and $x+b>0\,$, and that $b>a\,$, show that \[ \frac{\mathrm{d} \ }{\mathrm{d...

Show that $ \cosh^{-1} x = \ln ( x + \sqrt{x^2-1})$. Show that the area of the region defined by th...

Calculate the following integrals \begin{questionparts} \item ${\displaystyle \int\frac{x}{(x-1)(x^...

\begin{questionparts} \item The four points $P$, $Q$, $R$ and $S$ are the vertices of a plane quadri...

The points $O$, $A$ and $B$ are the vertices of an acute-angled triangle. The points $M$ and $N$ li...

All vectors in this question lie in the same plane. The vertices of the non-right-angled triangle...

The sides $OA$ and $CB$ of the quadrilateral $OABC$ are parallel. The point $X$ lies on $OA$, betwee...

\noindent \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dotstyle=o,dotsi...

The four distinct points $P_i$ ($i=1$, $2$, $3$, $4$) are the vertices, labelled anticlockwise, ...

In the triangle $OAB$, the point $D$ divides the side $BO$ in the ratio $r:1$ (so that $BD = rDO$)...

The four vertices $P_i$ ($i= 1, 2, 3, 4$) of a regular tetrahedron lie on the surface of a sphere ...

Three distinct points, $X_1$, $X_2$ and $X_3$, with position vectors ${\bf x}_1$, ${\bf x}_2$ and ${...

The points $A$ and $B$ have position vectors $\bf a $ and $\bf b$ with respect to an origin $O$, a...

The points $A$ and $B$ have position vectors $\bf i +j+k$ and $5{\bf i} - {\bf j} -{\bf k}$, respect...

Relative to a fixed origin $O$, the points $A$ and $B$ have position vectors $\bf{a}$ and $\bf{b}...

The non-collinear points $A$, $B$ and $C$ have position vectors $\bf a$, $\bf b$ and $\bf c$, respe...

The points $A$ and $B$ have position vectors $\bf a$ and $\bf b$, respectively, relative to the or...

The points $B$ and $C$ have position vectors $\mathbf{b}$ and $\mathbf{c}$, respectively, relative t...

\begin{questionparts} \item The line $L_1$ has vector equation $\displaystyle {\bf r} = \begin{pmatr...

Show that the line through the points with position vectors $\bf x$ and $\bf y$ has equation \[{\bf...

The position vectors, relative to an origin $O$, at time $t$ of the particles $P$ and $Q$ are $$\co...

Given two non-zero vectors $\mathbf{a}=\begin{pmatrix}a_{1}\\ a_{2} \end{pmatrix}$ and $\mathbf{b}...

The line $l$ has vector equation ${\bf r} = \lambda {\bf s}$, where \[ {\bf s} = (\cos\theta+\sqrt...

\begin{questionparts} %\item[(i)] Consider the sphere of radius $a$ and centre the origin. %Show tha...

A plane $\pi$ in 3-dimensional space is given by the vector equation $\mathbf{r}\cdot\mathbf{n}=p,$ ...

Two non-parallel lines in 3-dimensional space are given by $\mathbf{r}=\mathbf{p}_{1}+t_{1}\mathbf{m...

Let $\mathbf{a},\mathbf{b}$ and $\mathbf{c}$ be the position vectors of points $A,B$ and $C$ in thre...

The points $A,B$ and $C$ lie on the surface of the ground, which is an inclined plane. The point $B$...

The surface $S$ in 3-dimensional space is described by the equation \[ \mathbf{a}\cdot\mathbf{r}+ar=...

By substituting $y(x)=xv(x)$ in the differential equation \[ x^{3}\frac{\mathrm{d}v}{\mathrm{d}x}+x...

\begin{questionparts} \item Use the substitution $v= \sqrt y$ to solve the differential equation \[...

Given that $y=xu$, where $u$ is a function of $x$, write down an expression for $\dfrac {\d y}{\d x}...

\begin{questionparts} \item Show that substituting $y=xv$, where $v$ is a function of $x$, in the ...

Show that if \[ {\mathrm{d}y \over \mathrm{d} x}=\f(x)y + {\g(x) \over y} \] then the substitution $...

\begin{questionparts} \item Show that the gradient at a point $\l x\,, \, y \r$ on the curve \[ \l ...

\begin{questionparts} \item Let $y$ be the solution of the differential equation \[ \frac{\d y}{\d x...

If there are $x$ micrograms of bacteria in a nutrient medium, the population of bacteria will grow a...

Let $P,Q$ and $R$ be functions of $x$. Prove that, for any function $y$ of $x$, the function \[ Py'...

Let $y,u,v,P$ and $Q$ all be functions of $x$. Show that the substitution $y=uv$ in the differential...

The matrix $\mathbf{F}$ is defined by \[ \mathbf{F}=\mathbf{I}+\sum_{n=1}^{\infty}\frac{1}{n!}t^{n}...

\begin{questionparts} \item Show that $$z^{m+1} - \frac{1}{z^{m+1}} = \left(z - \frac{1}{z}\right)\...

\begin{questionparts} \item If $z=x+\mathrm{i}y,$ with $x,y$ real, show that \[ \left|x\right|\...

Explain the geometrical relationship between the points in the Argand diagram represented by the com...

Sum each of the series \[ \sin\left(\frac{2\pi}{23}\right)+\sin\left(\frac{6\pi}{23}\right)+\sin\lef...

\begin{questionparts} \item Use De Moivre's theorem to show that, if $\sin\theta\ne0$\,, then \[ \fr...

The transformation $R$ in the complex plane is a rotation (anticlockwise) by an angle $\theta$ about...

Let $\omega = \e^{2\pi {\rm i}/n}$, where $n$ is a positive integer. Show that, for any complex num...

\begin{questionparts} \item If $a$, $b$ and $c$ are all real, show that the equation \[ z^3+az^2+bz+...

Evaluate $\displaystyle \sum_{r=0}^{n-1} \e^{2i(\alpha + r\pi/n)}$ where $\alpha$ is a fixed angle a...

Show that $(z-\e^{i\theta})(z-\e^{-i\theta})=z^2 -2z\cos\theta +1\,$. Write down the $(2n)$th roots ...

Show that, provided $q^2\ne 4p^3$, the polynomial \[ \hphantom{(p\ne0, \ q\ne0)\hspace{2cm}} x^3-3...

For any given positive integer $n$, a number $a$ (which may be complex) is said to be a \textit{prim...

Show that $\big\vert \e^{\i\beta} -\e^{\i\alpha}\big\vert = 2\sin\frac12 (\beta-\alpha)\,$ for $0<\a...

In this question, you may use without proof the results \[ 4 \cosh^3 y - 3 \cosh y = \cosh (3y) \ \ ...

Given that $\alpha = \e^{\mathrm{i} \pi/3}$ , prove that $1 + \alpha^2 = \alpha$. A triangle i...

Prove that \[ (\cos\theta +\mathrm{i}\sin\theta) (\cos\phi +\mathrm{i}\sin\phi) = \cos(\theta+\phi) ...

By considering the solutions of the equation $z^n-1=0$, or otherwise, show that \[(z-\omega)(z-\ome...

Show, using de Moivre's theorem, or otherwise, that \[ \tan7\theta=\frac{t(t^{6}-21t^{4}+35t^{2}-7)...

If $u$ and $v$ are the two roots of $z^{2}+az+b=0,$ show that $a=-u-v$ and $b=uv.$ Let $\alpha=\cos...

By applying de Moivre's theorem to $\cos5\theta+\mathrm{i}\sin5\theta,$ expanding the result using t...

Show that \[ \sin(2n+1)\theta=\sin^{2n+1}\theta\sum_{r=0}^{n}(-1)^{n-r}\binom{2n+1}{2r}\cot^{2r}\th...

A path is made up in the Argand diagram of a series of straight line segments $P_{1}P_{2},$ $P_{2}P_...

Given that $\sin\beta\neq0,$ sum the series \[ \cos\alpha+\cos(\alpha+2\beta)+\cdots+\cos(\alpha+2r...

Show, using de Moivre's theorem, or otherwise, that \[ \tan9\theta=\frac{t(t^{2}-3)(t^{6}-33t^{4}+2...

Let $\omega=\mathrm{e}^{2\pi\mathrm{i}/3}.$ Show that $1+\omega+\omega^{2}=0$ and calculate the modu...

By using de Moivre's theorem, or otherwise, show that \begin{questionparts} \item $\cos4\theta=8\co...

By considering the imaginary part of the equation $z^{7}=1,$ or otherwise, find all the roots of the...

The differential equation \[\frac{d^2x}{dt^2} = 2x\frac{dx}{dt}\] describes the motion of a particl...

The coordinates of a particle at time $t$ are $x$ and $y$. For $t \geq 0$, they satisfy the pair of ...

The functions $x(t)$ and $y(t)$ satisfy the simultaneous differential equations \begin{alignat*}{1}...

Show that the second-order differential equation \[ x^2y''+(1-2p) x\, y' + (p^2-q^2) \, y= \f(x) \,...

Two particles $X$ and $Y$, of equal mass $m$, lie on a smooth horizontal table and are connected by ...

A sphere of radius $R$ and uniform density $\rho_{\text{s}}$ is floating in a large tank of liquid ...

\begin{questionparts} \item Let $y(x)$ be a solution of the differential equation $ \dfrac {\d^2 y...

A pain-killing drug is injected into the bloodstream. It then diffuses into the brain, where it is ...

Given that $\displaystyle z = y^n \left( \frac{\d y}{\d x}\right)^{\!2}$, show that \[ \frac{\d z}{\...

\begin{questionparts} \item Find the general solution of the differential equation \[ \frac{\d u}{\d...

A small bead $B$, of mass $m$, slides without friction on a fixed horizontal ring of radius $a$. Th...

Show that, if $y=\e^x$, then \[ (x-1) \frac{\d^2 y}{\d x^2} -x \frac{\d y}{\d x} +y=0\,. \tag{$*...

A light spring is fixed at its lower end and its axis is vertical. When a certain particle $P$ rest...

\begin{questionparts} \item The functions $\f_n(x)$ are defined for $n=0$, $1$, $2$, $\ldots$\, ,...

\begin{questionparts} \item Let $\displaystyle y= \sum_{n=0}^\infty a_n x^n\,$, where the coefficien...

In this question, $p$ denotes $\dfrac{\d y}{\d x}\,$. \begin{questionparts} \item Given that \[ y=p...

\begin{questionparts} \item Find functions ${\rm a}(x)$ and ${\rm b}(x)$ such that $u=x$ and $u=\e...

Given that $y=x$ and $y=1-x^2$ satisfy the differential equation $$ \frac{\d^2 {y}}{\d x^2} + \p(x)...

The function $y(x)$ is defined for $x\ge0$ and satisfies the conditions \[ y=0 \mbox{ \ \ and \ \...

Show that $\sin(k\sin^{-1} x)$, where $k$ is a constant, satisfies the differential equation $$ (1...

Suppose that $y_n$ satisfies the equations \[(1-x^2)\frac{{\rm d}^2y_n}{{\rm d}x^2}-x\frac{{\rm d}y_...

Suppose that \[{\rm f}''(x)+{\rm f}(-x)=x+3\cos 2x\] and ${\rm f}(0)=1$, ${\rm f}'(0)=-1$. If ${\rm ...

Show that $y=\sin^{2}(m\sin^{-1}x)$ satisfies the differential equation \[ (1-x^{2})y^{(2)}=xy^{(1)}...

What is the general solution of the differential equation \[ \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}...

Find functions $\mathrm{f,g}$ and $\mathrm{h}$ such that \[ \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}...

By means of the substitution $x^{\alpha},$ where $\alpha$ is a suitably chosen constant, find the ge...

For $n=0,1,2,\ldots,$ the functions $y_{n}$ satisfy the differential equation \[ \frac{\mathrm{d}^{...

I have a Penny Black stamp which I want to sell to my friend Jim, but we cannot agree a price. So I ...

Let $X$ be a Poisson random variable with mean $\lambda$ and let $p_r = P(X = r)$, for $r = 0, 1, 2,...

The number of customers arriving at a builders' merchants each day follows a Poisson distribution wi...

Adam and Eve are catching fish. The number of fish, $X$, that Adam catches in any time interval is...

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. \textit{Chebyshev's i...

The number $X$ of casualties arriving at a hospital each day follows a Poisson distribution with me...

The random variable $U$ has a Poisson distribution with parameter $\lambda$. The random variables $X...

In this question, you may assume that $\displaystyle \int_0^\infty \!\!\! \e^{-x^2/2} \d x = \sqrt{\...

The number of texts that George receives on his mobile phone can be modelled by a Poisson random va...

The discrete random variable $X$ has a Poisson distribution with mean $\lambda$. \begin{questionpart...

A very generous shop-owner is hiding small diamonds in chocolate bars. Each diamond is hidden indep...

The number of printing errors on any page of a large book of $N$ pages is modelled by a Poisson va...

Brief interruptions to my work occur on average every ten minutes and the number of interruptions i...

Jane goes out with any of her friends who call, except that she never goes out with more than two ...

Four students, one of whom is a mathematician, take turns at washing up over a long period of time. ...

In a lottery, any one of $N$ numbers, where $N$ is large, is chosen at random and independently fo...

In the basic version of Horizons (H1) the player has a maximum of $n$ turns, where $n \ge 1$. At ea...

The staff of Catastrophe College are paid a salary of $A$ pounds per year. With a Teaching Assessmen...

Traffic enters a tunnel which is 9600 metres long, and in which overtaking is impossible. The number...

Whenever I go cycling I start with my bike in good working order. However if all is well at time $t$...

Fly By Night Airlines run jumbo jets which seat $N$ passengers. From long experience they know that ...

A scientist is checking a sequence of microscope slides for cancerous cells, marking each cancerous ...

At the terminus of a bus route, passengers arrive at an average rate of 4 per minute according to a ...

The probability that there are exactly $n$ misprints in an issue of a newspaper is $\mathrm{e}^{-\la...

I can choose one of three routes to cycle to school. Via Angle Avenue the distance is 5$\,$km, and I...

The random variable $X$ takes the values $k=1$, $2$, $3$, $\dotsc$, and has probability distribution...

\begin{questionparts} \item The random variable $X$ has a binomial distribution with parameters $n...

The national lottery of Ruritania is based on the positive integers from $1$ to $N$, where $N$ is ve...

The random variable $X$ takes only non-negative integer values and has probability generating functi...

A 6-sided fair die has the numbers 1, 2, 3, 4, 5, 6 on its faces. The die is thrown $n$ times, th...

I play a game which has repeated rounds. Before the first round, my score is $0$. Each round can h...

The random variable $N$ takes positive integer values and has pgf (probability generating function) ...

\begin{questionparts} \item Albert tosses a fair coin $k$ times, where $k$ is a given positive int...

Write down the probability generating function for the score on a standard, fair six-faced die whos...

A set of $n$ dice is rolled repeatedly. For each die the probability of showing a six is $p$. Sh...

\begin{questionparts} \item I toss a biased coin which has a probability $p$ of landing heads and a ...

\begin{questionparts} \item Let $\lambda > 0$. The independent random variables $X_1, X_2, \ldots, X...

The random variable $X$ has probability density function $\f(x)$ (which you may assume is different...

The lifetime of a fly (measured in hours) is given by the continuous random variable~$T$ with prob...

\begin{questionparts} \item The continuous random variable $X$ satisfies $0\le X\le 1$, and has prob...

What property of a distribution is measured by its {\em skewness}? \begin{questionparts} \item One...

The continuous random variable $X$ has probability density function $\f(x)$, where \[ \f(x) = \beg...

The random variable $X$ has a continuous probability density function $\f(x)$ given by \begin{equat...

In this question, you may use the result \[ \displaystyle \int_0^\infty \frac{t^m}{(t+k)^{n+2}} \; ...

Sketch the graph, for $x \ge 0\,$, of $$ y = kx\e^{-ax^2} \;, $$ where $a$ and $k$ are positive c...

Let $\F(x)$ be the cumulative distribution function of a random variable $X$, which satisfies $\F(a)...

\begin{questionparts} \item Let $X_{1}$, $X_{2}$, \dots, $X_{n}$ be independent random variables ea...

A goat $G$ lies in a square field $OABC$ of side $a$. It wanders randomly round its field, so that a...

A target consists of a disc of unit radius and centre $O$. A certain marksman never misses the targe...

The continuous random variable $X$ is uniformly distributed over the interval $[-c,c].$ Write down e...

An internet tester sends $n$ e-mails simultaneously at time $t=0$. Their arrival times at their de...

The maximum height $X$ of flood water each year on a certain river is a random variable with probabi...

A continuous random variable is said to have an exponential distribution with parameter $\lambda$ i...

In order to get money from a cash dispenser I have to punch in an identification number. I have f...

A random variable $X$ has the probability density function \[ \mathrm{f}(x)=\begin{cases} \lambda\ma...

The maximum height $X$ of flood water each year on a certain river is a random variable with densit...

During his performance a trapeze artist is supported by two identical ropes, either of which can bea...

A random process generates, independently, $n$ numbers each of which is drawn from a uniform (rectan...

The discrete random variables $X$ and $Y$ can each take the values $1$, $\ldots\,$, $n$ (where $n\g...

Given a random variable $X$ with mean $\mu$ and standard deviation $\sigma$, we define the \textit{k...

Each of the two independent random variables $X$ and $Y$ is uniformly distributed on the interval~$...

A list consists only of letters $A$ and $B$ arranged in a row. In the list, there are $a$ letter $A$...

In this question, ${\rm Corr}(U,V)$ denotes the product moment correlation coefficient between the r...

I choose a number from the integers $1, 2, \ldots$, $(2n-1)$ and the outcome is the random variable~...

For any random variables $X_1$ and $X_2$, state the relationship between $\E(aX_1+bX_2)$ and $\E(X_1...

Five independent timers time a runner as she runs four laps of a track. Four of the timers measure t...

A team of $m$ players, numbered from $1$ to $m$, puts on a set of a $m$ shirts, similarly numbered ...

Prove that, for any two discrete random variables $X$ and $Y$, \[ \mathrm{Var} \left(X + Y \right) ...

The random variable $X$ takes only the values $x_1$ and $x_2$ (where $ x_1 \not= x_2 $), and the ...

An industrial process produces rectangular plates of mean length $\mu_{1}$ and mean breadth $\mu_{2}...

The random variables $X$ and $Y$ are independently normally distributed with means 0 and variances 1...

The random variables $X$ and $Y$ take integer values $x$ and $y$ respectively which are restricted b...

\begin{questionparts} \item A rod of unit length is cut into pieces of length $X$ and $1-X$; the lat...

Let $X$ be a random variable with a Laplace distribution, so that its probability density functi...

In the game of endless cricket the scores $X$ and $Y$ of the two sides are such that \[ \P (X=j,\ Y...

Let $X$ and $Y$ be independent standard normal random variables: the probability density function, $...

A candidate finishes examination questions in time $T$, where $T$ has probability density function ...

\begin{questionparts} \item The point $P$ lies on the circumference of a circle of unit radius and ...

A random variable $X$ is distributed uniformly on $[\, 0\, , \, a\,]$. Show that the variance of $X...

Two coins $A$ and $B$ are tossed together. $A$ has probability $p$ of showing a head, and $B$ has ...

The random variables $X_1$, $X_2$, $\ldots$ , $X_{2n+1}$ are independently and uniformly distribute...

A hostile naval power possesses a large, unknown number $N$ of submarines. Interception of radio si...

An experiment produces a random number $T$ uniformly distributed on $[0,1]$. Let $X$ be the larger ...

The random variable $X$ is uniformly distributed on $[0,1]$. A new random variable $Y$ is defined by...

The average number of pedestrians killed annually in road accidents in Poldavia during the period 19...

A fair coin is thrown $n$ times. On each throw, 1 point is scored for a head and 1 point is lost for...

\textit{In this question, $\Phi(z)$ is the cumulative distribution function of a standard normal ra...

A uniform rod, of mass $3m$ and length $2a,$ is freely hinged at one end and held by the other end i...

A uniform disc with centre $O$ and radius $a$ is suspended from a point $A$ on its circumference, so...

A uniform rod $PQ$ of mass $m$ and length $3a$ is freely hinged at $P$. The rod is held horizontall...

A uniform rod $AB$ has mass $M$ and length $2a$. The point $P$ lies on the rod a distance $a-x$ from...

A pulley consists of a disc of radius $r$ with centre $O$ and a light thin axle through $O$ perpendi...

A circular wheel of radius $r$ has moment of inertia $I$ about its axle, which is fixed in a horizon...

A disc rotates freely in a horizontal plane about a vertical axis through its centre. The moment of...

A uniform cylinder of radius $a$ rotates freely about its axis, which is fixed and horizontal. The m...

A thin beam is fixed at a height $2a$ above a horizontal plane. A uniform straight rod $ACB$ of len...

Calculate the moment of inertia of a uniform thin circular hoop of mass $m$ and radius $a$ about a...

The gravitational force between two point particles of masses $m$ and $m'$ is mutually attractive a...

A uniform right circular cone of mass $m$ has base of radius $a$ and perpendicular height $h$ from ...

$\,$ \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth...

By pressing a finger down on it, a uniform spherical marble of radius $a$ is made to slide along a ...

A thin circular disc of mass $m$, radius $r$ and with its centre of mass at its centre $C$ can rotat...

$\ $\vspace{-1cm} \noindent \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotst...

\textit{In this question, all gravitational forces are to be neglected. } A rigid frame is construc...

A non-uniform rod $AB$ of mass $m$ is pivoted at one end $A$ so that it can swing freely in a vertic...

The points $O,A,B$ and $C$ are the vertices of a uniform square lamina of mass $M.$ The lamina can t...

A disc is free to rotate in a horizontal plane about a vertical axis through its centre. The moment ...

\begin{questionparts} \item A solid circular disc has radius $a$ and mass $m.$ The density is propor...

A body of mass $m$ and centre of mass $O$ is said to be dynamically equivalent to a system of partic...