Part IA, 2012

# Part IA, 2012

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Paper 1, Section I, $4 \mathbf{F}$

commentLet $f, g:[0,1] \rightarrow \mathbb{R}$ be continuous functions with $g(x) \geqslant 0$ for $x \in[0,1]$. Show that

$\int_{0}^{1} f(x) g(x) d x \leqslant M \int_{0}^{1} g(x) d x$

where $M=\sup \{|f(x)|: x \in[0,1]\}$.

Prove there exists $\alpha \in[0,1]$ such that

$\int_{0}^{1} f(x) g(x) d x=f(\alpha) \int_{0}^{1} g(x) d x$

[Standard results about continuous functions and their integrals may be used without proof, if clearly stated.]

Paper 1, Section I, E

commentWhat does it mean to say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $x_{0} \in \mathbb{R}$ ?

Give an example of a continuous function $f:(0,1] \rightarrow \mathbb{R}$ which is bounded but attains neither its upper bound nor its lower bound.

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and non-negative, and satisfies $f(x) \rightarrow 0$ as $x \rightarrow \infty$ and $f(x) \rightarrow 0$ as $x \rightarrow-\infty$. Show that $f$ is bounded above and attains its upper bound.

[Standard results about continuous functions on closed bounded intervals may be used without proof if clearly stated.]

Paper 1, Section II, D

commentLet $f$ be a continuous function from $(0,1)$ to $(0,1)$ such that $f(x)<x$ for every $0<x<1$. We write $f^{n}$ for the $n$-fold composition of $f$ with itself (so for example $\left.f^{2}(x)=f(f(x))\right)$.

(i) Prove that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

(ii) Must it be the case that for every $\epsilon>0$ there exists $n$ with the property that $f^{n}(x)<\epsilon$ for all $0<x<1$ ? Justify your answer.

Now suppose that we remove the condition that $f$ be continuous.

(iii) Give an example to show that it need not be the case that for every $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$.

(iv) Must it be the case that for some $0<x<1$ we have $f^{n}(x) \rightarrow 0$ as $n \rightarrow \infty$ ? Justify your answer.

Paper 1, Section II, D

commentLet $\left(a_{n}\right)$ be a sequence of reals.

(i) Show that if the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent then so is the sequence $\left(\frac{a_{n}}{n}\right)$.

(ii) Give an example to show the sequence $\left(\frac{a_{n}}{n}\right)$ being convergent does not imply that the sequence $\left(a_{n+1}-a_{n}\right)$ is convergent.

(iii) If $a_{n+k}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for each positive integer $k$, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

(iv) If $a_{n+f(n)}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$ for every function $f$ from the positive integers to the positive integers, does it follow that $\left(a_{n}\right)$ is convergent? Justify your answer.

Paper 1, Section II, E

comment(a) What does it mean to say that the sequence $\left(x_{n}\right)$ of real numbers converges to $\ell \in \mathbb{R} ?$

Suppose that $\left(y_{n}^{(1)}\right),\left(y_{n}^{(2)}\right), \ldots,\left(y_{n}^{(k)}\right)$ are sequences of real numbers converging to the same limit $\ell$. Let $\left(x_{n}\right)$ be a sequence such that for every $n$,

$x_{n} \in\left\{y_{n}^{(1)}, y_{n}^{(2)}, \ldots, y_{n}^{(k)}\right\}$

Show that $\left(x_{n}\right)$ also converges to $\ell$.

Find a collection of sequences $\left(y_{n}^{(j)}\right), j=1,2, \ldots$ such that for every $j,\left(y_{n}^{(j)}\right) \rightarrow \ell$ but the sequence $\left(x_{n}\right)$ defined by $x_{n}=y_{n}^{(n)}$ diverges.

(b) Let $a, b$ be real numbers with $0<a<b$. Sequences $\left(a_{n}\right),\left(b_{n}\right)$ are defined by $a_{1}=a, b_{1}=b$ and

$\text { for all } n \geqslant 1, \quad a_{n+1}=\sqrt{a_{n} b_{n}}, \quad b_{n+1}=\frac{a_{n}+b_{n}}{2} \text {. }$

Show that $\left(a_{n}\right)$ and $\left(b_{n}\right)$ converge to the same limit.

Paper 1, Section II, F

comment(a) (i) State the ratio test for the convergence of a real series with positive terms.

(ii) Define the radius of convergence of a real power series $\sum_{n=0}^{\infty} a_{n} x^{n}$.

(iii) Prove that the real power series $f(x)=\sum_{n} a_{n} x^{n}$ and $g(x)=\sum_{n}(n+1) a_{n+1} x^{n}$ have equal radii of convergence.

(iv) State the relationship between $f(x)$ and $g(x)$ within their interval of convergence.

(b) (i) Prove that the real series

$f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}, \quad g(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}$

have radius of convergence $\infty$.

(ii) Show that they are differentiable on the real line $\mathbb{R}$, with $f^{\prime}=-g$ and $g^{\prime}=f$, and deduce that $f(x)^{2}+g(x)^{2}=1$.

[You may use, without proof, general theorems about differentiating within the interval of convergence, provided that you give a clear statement of any such theorem.]

Paper 2, Section I, A

commentFind the constant solutions (those with $u_{n+1}=u_{n}$ ) of the discrete equation

$u_{n+1}=\frac{1}{2} u_{n}\left(1+u_{n}\right)$

and determine their stability.

Paper 2, Section I, A

commentFind two linearly independent solutions of

$y^{\prime \prime}+4 y^{\prime}+4 y=0 .$

Find the solution in $x \geqslant 0$ of

$y^{\prime \prime}+4 y^{\prime}+4 y=e^{-2 x},$

subject to $y=y^{\prime}=0$ at $x=0$.

Paper 2, Section II, A

commentConsider the function

$V(x, y)=x^{4}-x^{2}+2 x y+y^{2}$

Find the critical (stationary) points of $V(x, y)$. Determine the type of each critical point. Sketch the contours of $V(x, y)=$ constant.

Now consider the coupled differential equations

$\frac{d x}{d t}=-\frac{\partial V}{\partial x}, \quad \frac{d y}{d t}=-\frac{\partial V}{\partial y}$

Show that $V(x(t), y(t))$ is a non-increasing function of $t$. If $x=1$ and $y=-\frac{1}{2}$ at $t=0$, where does the solution tend to as $t \rightarrow \infty$ ?

Paper 2, Section II, A

commentFind the first three non-zero terms in the series solutions $y_{1}(x)$ and $y_{2}(x)$ for the differential equation

$x^{2} y^{\prime \prime}-2 x y^{\prime}+\left(2-x^{2}\right) y=0$

that satisfy

$\begin{array}{llc} y_{1}^{\prime}(0)=a & \text { and } & y_{1}^{\prime \prime}(0)=0 \\ y_{2}^{\prime}(0)=0 & \text { and } & y_{2}^{\prime \prime}(0)=2 b \end{array}$

Identify these solutions in closed form.

Paper 2, Section II, A

commentConsider the second-order differential equation for $y(t)$ in $t \geqslant 0$

$\ddot{y}+2 k \dot{y}+\left(k^{2}+\omega^{2}\right) y=f(t) .$

(i) For $f(t)=0$, find the general solution $y_{1}(t)$ of $(*)$.

(ii) For $f(t)=\delta(t-a)$ with $a>0$, find the solution $y_{2}(t, a)$ of $(*)$ that satisfies $y=0$ and $\dot{y}=0$ at $t=0$.

(iii) For $f(t)=H(t-b)$ with $b>0$, find the solution $y_{3}(t, b)$ of $(*)$ that satisfies $y=0$ and $\dot{y}=0$ at $t=0 .$

(iv) Show that

$y_{2}(t, b)=-\frac{\partial y_{3}}{\partial b}$

Paper 2, Section II, A

commentFind the solution to the system of equations

$\begin{gathered} \frac{d x}{d t}+\frac{-4 x+2 y}{t}=-9, \\ \frac{d y}{d t}+\frac{x-5 y}{t}=3 \end{gathered}$

in $t \geqslant 1$ subject to

$x=0 \quad \text { and } \quad y=0 \quad \text { at } \quad t=1$

[Hint: powers of t.]

Paper 4, Section I, B

commentLet $S$ and $S^{\prime}$ be inertial frames in 2-dimensional spacetime with coordinate systems $(t, x)$ and $\left(t^{\prime}, x^{\prime}\right)$ respectively. Suppose that $S^{\prime}$ moves with positive velocity $v$ relative to $S$ and the spacetime origins of $S$ and $S^{\prime}$ coincide. Write down the Lorentz transformation relating the coordinates of any event relative to the two frames.

Show that events which occur simultaneously in $S$ are not generally seen to be simultaneous when viewed in $S^{\prime}$.

In $S$ two light sources $A$ and $B$ are at rest and placed a distance $d$ apart. They simultaneously each emit a photon in the positive $x$ direction. Show that in $S^{\prime}$ the photons are separated by a constant distance $d \sqrt{\frac{c+v}{c-v}}$.

Paper 4, Section I, B

commentTwo particles of masses $m_{1}$ and $m_{2}$ have position vectors $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ respectively. Particle 2 exerts a force $\mathbf{F}_{12}\left(\mathbf{r}\right.$ ) on particle 1 (where $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$ ) and there are no external forces.

Prove that the centre of mass of the two-particle system will move at constant speed along a straight line.

Explain how the two-body problem of determining the motion of the system may be reduced to that of a single particle moving under the force $\mathbf{F}_{12}$.

Suppose now that $m_{1}=m_{2}=m$ and that

$\mathbf{F}_{12}=-\frac{G m^{2}}{r^{3}} \mathbf{r}$

is gravitational attraction. Let $C$ be a circle fixed in space. Is it possible (by suitable choice of initial conditions) for the two particles to be traversing $C$ at the same constant angular speed? Give a brief reason for your answer.

Paper 4, Section II, B

comment(a) Define the 4-momentum $\mathbf{P}$ of a particle of rest mass $m$ and 3 -velocity $\mathbf{v}$, and the 4-momentum of a photon of frequency $\nu$ (having zero rest mass) moving in the direction of the unit vector $e$.

Show that if $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ are timelike future-pointing 4-vectors then $\mathbf{P}_{1} \cdot \mathbf{P}_{2} \geqslant 0$ (where the dot denotes the Lorentz-invariant scalar product). Hence or otherwise show that the law of conservation of 4 -momentum forbids a photon to spontaneously decay into an electron-positron pair. [Electrons and positrons have equal rest masses $m>0$.]

(b) In the laboratory frame an electron travelling with velocity u collides with a positron at rest. They annihilate, producing two photons of frequencies $\nu_{1}$ and $\nu_{2}$ that move off at angles $\theta_{1}$ and $\theta_{2}$ to $\mathbf{u}$, in the directions of the unit vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively. By considering 4-momenta in the laboratory frame, or otherwise, show that

$\frac{1+\cos \left(\theta_{1}+\theta_{2}\right)}{\cos \theta_{1}+\cos \theta_{2}}=\sqrt{\frac{\gamma-1}{\gamma+1}}$

where $\gamma=\left(1-\frac{u^{2}}{c^{2}}\right)^{-1 / 2}$

Paper 4, Section II, B

comment(a) State the parallel axis theorem for moments of inertia.

(b) A uniform circular disc $D$ of radius $a$ and total mass $m$ can turn frictionlessly about a fixed horizontal axis that passes through a point $A$ on its circumference and is perpendicular to its plane. Initially the disc hangs at rest (in constant gravity $g$ ) with its centre $O$ being vertically below $A$. Suppose the disc is disturbed and executes free oscillations. Show that the period of small oscillations is $2 \pi \sqrt{\frac{3 a}{2 g}}$.

(c) Suppose now that the disc is released from rest when the radius $O A$ is vertical with $O$ directly above $A$. Find the angular velocity and angular acceleration of $O$ about $A$ when the disc has turned through angle $\theta$. Let $\mathbf{R}$ denote the reaction force at $A$ on the disc. Find the acceleration of the centre of mass of the disc. Hence, or otherwise, show that the component of $\mathbf{R}$ parallel to $O A$ is $m g(7 \cos \theta-4) / 3$.

Paper 4, Section II, B

commentFor any frame $S$ and vector $\mathbf{A}$, let $\left[\frac{d \mathbf{A}}{d t}\right]_{S}$ denote the derivative of $\mathbf{A}$ relative to $S$. A frame of reference $S^{\prime}$ rotates with constant angular velocity $\omega$ with respect to an inertial frame $S$ and the two frames have a common origin $O$. [You may assume that for any vector $\left.\mathbf{A},\left[\frac{d \mathbf{A}}{d t}\right]_{S}=\left[\frac{d \mathbf{A}}{d t}\right]_{S^{\prime}}+\boldsymbol{\omega} \times \mathbf{A} .\right]$

(a) If $\mathbf{r}(t)$ is the position vector of a point $P$ from $O$, show that

$\left[\frac{d^{2} \mathbf{r}}{d t^{2}}\right]_{S}=\left[\frac{d^{2} \mathbf{r}}{d t^{2}}\right]_{S^{\prime}}+2 \boldsymbol{\omega} \times \mathbf{v}^{\prime}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$

where $\mathbf{v}^{\prime}=\left[\frac{d \mathbf{r}}{d t}\right]_{S^{\prime}}$ is the velocity in $S^{\prime}$.

Suppose now that $\mathbf{r}(t)$ is the position vector of a particle of mass $m$ moving under a conservative force $\mathbf{F}=-\nabla \phi$ and a force $\mathbf{G}$ that is always orthogonal to the velocity $\mathbf{v}^{\prime}$ in $S^{\prime}$. Show that the quantity

$E=\frac{1}{2} m \mathbf{v}^{\prime} \cdot \mathbf{v}^{\prime}+\phi-\frac{m}{2}(\boldsymbol{\omega} \times \mathbf{r}) \cdot(\boldsymbol{\omega} \times \mathbf{r})$

is a constant of the motion. [You may assume that $\nabla[(\boldsymbol{\omega} \times \mathbf{r}) \cdot(\boldsymbol{\omega} \times \mathbf{r})]=-2 \boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$.]

(b) A bead slides on a frictionless circular hoop of radius $a$ which is forced to rotate with constant angular speed $\omega$ about a vertical diameter. Let $\theta(t)$ denote the angle between the line from the centre of the hoop to the bead and the downward vertical. Using the results of (a), or otherwise, show that

$\ddot{\theta}+\left(\frac{g}{a}-\omega^{2} \cos \theta\right) \sin \theta=0 .$

Deduce that if $\omega^{2}>g / a$ there are two equilibrium positions $\theta=\theta_{0}$ off the axis of rotation, and show that these are stable equilibria.

Paper 4, Section II, B

commentLet $(r, \theta)$ be polar coordinates in the plane. A particle of mass $m$ moves in the plane under an attractive force of $m f(r)$ towards the origin $O$. You may assume that the acceleration a is given by

$\mathbf{a}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+\frac{1}{r} \frac{d}{d t}\left(r^{2} \dot{\theta}\right) \hat{\theta}$

where $\hat{\mathbf{r}}$ and $\hat{\theta}$ are the unit vectors in the directions of increasing $r$ and $\theta$ respectively, and the dot denotes $d / d t$.

(a) Show that $l=r^{2} \dot{\theta}$ is a constant of the motion. Introducing $u=1 / r$ show that $\dot{r}=-l \frac{d u}{d \theta}$ and derive the geometric orbit equation

$l^{2} u^{2}\left(\frac{d^{2} u}{d \theta^{2}}+u\right)=f\left(\frac{1}{u}\right)$

(b) Suppose now that

$f(r)=\frac{3 r+9}{r^{3}}$

and that initially the particle is at distance $r_{0}=1$ from $O$, moving with speed $v_{0}=4$ in a direction making angle $\pi / 3$ with the radial vector pointing towards $O$.

Show that $l=2 \sqrt{3}$ and find $u$ as a function of $\theta$. Hence or otherwise show that the particle returns to its original position after one revolution about $O$ and then flies off to infinity.

Paper 3, Section I, E

commentWhat is a cycle in the symmetric group $S_{n}$ ? Show that a cycle of length $p$ and a cycle of length $q$ in $S_{n}$ are conjugate if and only if $p=q$.

Suppose that $p$ is odd. Show that any two $p$-cycles in $A_{p+2}$ are conjugate. Are any two 3 -cycles in $A_{4}$ conjugate? Justify your answer.

Paper 3, Section I, E

commentState Lagrange's Theorem. Deduce that if $G$ is a finite group of order $n$, then the order of every element of $G$ is a divisor of $n$.

Let $G$ be a group such that, for every $g \in G, g^{2}=e$. Show that $G$ is abelian. Give an example of a non-abelian group in which every element $g$ satisfies $g^{4}=e$.

Paper 3, Section II, $7 \mathrm{E}$

commentLet $\mathbb{F}_{p}$ be the set of (residue classes of) integers $\bmod p$, and let

$G=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right): a, b, c, d \in \mathbb{F}_{p}, a d-b c \neq 0\right\}$

Show that $G$ is a group under multiplication. [You may assume throughout this question that multiplication of matrices is associative.]

Let $X$ be the set of 2-dimensional column vectors with entries in $\mathbb{F}_{p}$. Show that the mapping $G \times X \rightarrow X$ given by

$\left(\left(\begin{array}{ll} a & b \\ c & d \end{array}\right),\left(\begin{array}{l} x \\ y \end{array}\right)\right) \mapsto\left(\begin{array}{l} a x+b y \\ c x+d y \end{array}\right)$

is a group action.

Let $g \in G$ be an element of order $p$. Use the orbit-stabilizer theorem to show that there exist $x, y \in \mathbb{F}_{p}$, not both zero, with

$g\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} x \\ y \end{array}\right)$

Deduce that $g$ is conjugate in $G$ to the matrix

$\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)$

Paper 3, Section II, E

commentLet $p$ be a prime number, and $a$ an integer with $1 \leqslant a \leqslant p-1$. Let $G$ be the Cartesian product

$G=\{(x, u) \mid x \in\{0,1, \ldots, p-2\}, u \in\{0,1, \ldots, p-1\}\}$

Show that the binary operation

$(x, u) *(y, v)=(z, w)$

where

$\begin{aligned} z & \equiv x+y(\bmod p-1) \\ w & \equiv a^{y} u+v(\bmod p) \end{aligned}$

makes $G$ into a group. Show that $G$ is abelian if and only if $a=1$.

Let $H$ and $K$ be the subsets

$H=\{(x, 0) \mid x \in\{0,1, \ldots, p-2\}\}, \quad K=\{(0, u) \mid u \in\{0,1, \ldots, p-1\}\}$

of $G$. Show that $K$ is a normal subgroup of $G$, and that $H$ is a subgroup which is normal if and only if $a=1$.

Find a homomorphism from $G$ to another group whose kernel is $K$.

Paper 3, Section II, E

commentLet $G$ be $S L_{2}(\mathbb{R})$, the groups of real $2 \times 2$ matrices of determinant 1 , acting on $\mathbb{C} \cup\{\infty\}$ by MÃ¶bius transformations.

For each of the points $0, i,-i$, compute its stabilizer and its orbit under the action of $G$. Show that $G$ has exactly 3 orbits in all.

Compute the orbit of $i$ under the subgroup

$H=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d=1\right\} \subset G .$

Deduce that every element $g$ of $G$ may be expressed in the form $g=h k$ where $h \in H$ and for some $\theta \in \mathbb{R}$,

$k=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$

How many ways are there of writing $g$ in this form?

Paper 3, Section II, E

comment(i) State and prove the Orbit-Stabilizer Theorem.

Show that if $G$ is a finite group of order $n$, then $G$ is isomorphic to a subgroup of the symmetric group $S_{n}$.

(ii) Let $G$ be a group acting on a set $X$ with a single orbit, and let $H$ be the stabilizer of some element of $X$. Show that the homomorphism $G \rightarrow \operatorname{Sym}(X)$ given by the action is injective if and only if the intersection of all the conjugates of $H$ equals $\{e\}$.

(iii) Let $Q_{8}$ denote the quaternion group of order 8 . Show that for every $n<8, Q_{8}$ is not isomorphic to a subgroup of $S_{n}$.

Paper 4, Section I, $2 \mathrm{D}$

commentWhat is an equivalence relation on a set $X$ ? If $R$ is an equivalence relation on $X$, what is an equivalence class of $R$ ? Prove that the equivalence classes of $R$ form a partition of $X$.

Let $R$ and $S$ be equivalence relations on a set $X$. Which of the following are always equivalence relations? Give proofs or counterexamples as appropriate.

(i) The relation $V$ on $X$ given by $x V y$ if both $x R y$ and $x S y$.

(ii) The relation $W$ on $X$ given by $x W y$ if $x R y$ or $x S y$.

Paper 4, Section I, D

comment(i) Find integers $x$ and $y$ such that $18 x+23 y=101$.

(ii) Find an integer $x$ such that $x \equiv 3(\bmod 18)$ and $x \equiv 2(\bmod 23)$.

Paper 4, Section II, D

commentShow that there is no injection from the power-set of $\mathbb{R}$ to $\mathbb{R}$. Show also that there is an injection from $\mathbb{R}^{2}$ to $\mathbb{R}$.

Let $X$ be the set of all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x)=x$ for all but finitely many $x$. Determine whether or not there exists an injection from $X$ to $\mathbb{R}$.

Paper 4, Section II, D

commentProve that each of the following numbers is irrational: (i) $\sqrt{2}+\sqrt{3}$ (ii) $e$ (iii) The real root of the equation $x^{3}+4 x-7=0$ (iv) $\log _{2} 3$.

Paper 4, Section II, D

commentState Fermat's Theorem and Wilson's Theorem.

For which prime numbers $p$ does the equation $x^{2} \equiv-1(\bmod p)$ have a solution? Justify your answer.

For a prime number $p$, and an integer $x$ that is not a multiple of $p$, the order of $x$ $(\bmod p)$ is the least positive integer $d$ such that $x^{d} \equiv 1(\bmod p)$. Show that if $x$ has order $d$ and also $x^{k} \equiv 1(\bmod p)$ then $d$ must divide $k$.

For a positive integer $n$, let $F_{n}=2^{2^{n}}+1$. If $p$ is a prime factor of $F_{n}$, determine the order of $2(\bmod p)$. Hence show that the $F_{n}$ are pairwise coprime.

Show that if $p$ is a prime of the form $4 k+3$ then $p$ cannot be a factor of any $F_{n}$. Give, with justification, a prime $p$ of the form $4 k+1$ such that $p$ is not a factor of any $F_{n}$.

Paper 4, Section II, D

commentLet $X$ be a set, and let $f$ and $g$ be functions from $X$ to $X$. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.

(i) If $f g$ is the identity map then $g f$ is the identity map.

(ii) If $f g=g$ then $f$ is the identity map.

(iii) If $f g=f$ then $g$ is the identity map.

How (if at all) do your answers change if we are given that $X$ is finite?

Determine which sets $X$ have the following property: if $f$ is a function from $X$ to $X$ such that for every $x \in X$ there exists a positive integer $n$ with $f^{n}(x)=x$, then there exists a positive integer $n$ such that $f^{n}$ is the identity map. [Here $f^{n}$ denotes the $n$-fold composition of $f$ with itself.]

Paper 2, Section I, F

commentDefine the probability generating function $G(s)$ of a random variable $X$ taking values in the non-negative integers.

A coin shows heads with probability $p \in(0,1)$ on each toss. Let $N$ be the number of tosses up to and including the first appearance of heads, and let $k \geqslant 1$. Find the probability generating function of $X=\min \{N, k\}$.

Show that $E(X)=p^{-1}\left(1-q^{k}\right)$ where $q=1-p$.

Paper 2, Section I, F

commentGiven two events $A$ and $B$ with $P(A)>0$ and $P(B)>0$, define the conditional probability $P(A \mid B)$.

Show that

$P(B \mid A)=P(A \mid B) \frac{P(B)}{P(A)}$

A random number $N$ of fair coins are tossed, and the total number of heads is denoted by $H$. If $P(N=n)=2^{-n}$ for $n=1,2, \ldots$, find $P(N=n \mid H=1)$.

Paper 2, Section II, F

commentLet $X, Y$ be independent random variables with distribution functions $F_{X}, F_{Y}$. Show that $U=\min \{X, Y\}, V=\max \{X, Y\}$ have distribution functions

$F_{U}(u)=1-\left(1-F_{X}(u)\right)\left(1-F_{Y}(u)\right), \quad F_{V}(v)=F_{X}(v) F_{Y}(v)$

Now let $X, Y$ be independent random variables, each having the exponential distribution with parameter 1. Show that $U$ has the exponential distribution with parameter 2 , and that $V-U$ is independent of $U$.

Hence or otherwise show that $V$ has the same distribution as $X+\frac{1}{2} Y$, and deduce the mean and variance of $V$.

[You may use without proof that $X$ has mean 1 and variance 1.]

Paper 2, Section II, F

comment(i) Let $X_{n}$ be the size of the $n^{\text {th }}$generation of a branching process with familysize probability generating function $G(s)$, and let $X_{0}=1$. Show that the probability generating function $G_{n}(s)$ of $X_{n}$ satisfies $G_{n+1}(s)=G\left(G_{n}(s)\right)$ for $n \geqslant 0$.

(ii) Suppose the family-size mass function is $P\left(X_{1}=k\right)=2^{-k-1}, k=0,1,2, \ldots$ Find $G(s)$, and show that

$G_{n}(s)=\frac{n-(n-1) s}{n+1-n s} \quad \text { for }|s|<1+\frac{1}{n} .$

Deduce the value of $P\left(X_{n}=0\right)$.

(iii) Write down the moment generating function of $X_{n} / n$. Hence or otherwise show that, for $x \geqslant 0$,

$P\left(X_{n} / n>x \mid X_{n}>0\right) \rightarrow e^{-x} \quad \text { as } n \rightarrow \infty$

[You may use the continuity theorem but, if so, should give a clear statement of it.]

Paper 2, Section II, F

comment(i) Define the distribution function $F$ of a random variable $X$, and also its density function $f$ assuming $F$ is differentiable. Show that

$f(x)=-\frac{d}{d x} P(X>x)$

(ii) Let $U, V$ be independent random variables each with the uniform distribution on $[0,1]$. Show that

$P\left(V^{2}>U>x\right)=\frac{1}{3}-x+\frac{2}{3} x^{3 / 2}, \quad x \in(0,1)$

What is the probability that the random quadratic equation $x^{2}+2 V x+U=0$ has real roots?

Given that the two roots $R_{1}, R_{2}$ of the above quadratic are real, what is the probability that both $\left|R_{1}\right| \leqslant 1$ and $\left|R_{2}\right| \leqslant 1 ?$

Paper 2, Section II, F

comment(i) Define the moment generating function $M_{X}(t)$ of a random variable $X$. If $X, Y$ are independent and $a, b \in \mathbb{R}$, show that the moment generating function of $Z=a X+b Y$ is $M_{X}(a t) M_{Y}(b t)$.

(ii) Assume $T>0$, and $M_{X}(t)<\infty$ for $|t|<T$. Explain the expansion

$M_{X}(t)=1+\mu t+\frac{1}{2} s^{2} t^{2}+\mathrm{o}\left(t^{2}\right)$

where $\mu=E(X)$ and $s^{2}=E\left(X^{2}\right) . \quad$ [You may assume the validity of interchanging expectation and differentiation.]

(iii) Let $X, Y$ be independent, identically distributed random variables with mean 0 and variance 1 , and assume their moment generating function $M$ satisfies the condition of part (ii) with $T=\infty$.

Suppose that $X+Y$ and $X-Y$ are independent. Show that $M(2 t)=M(t)^{3} M(-t)$, and deduce that $\psi(t)=M(t) / M(-t)$ satisfies $\psi(t)=\psi(t / 2)^{2}$.

Show that $\psi(h)=1+\mathrm{o}\left(h^{2}\right)$ as $h \rightarrow 0$, and deduce that $\psi(t)=1$ for all $t$.

Show that $X$ and $Y$ are normally distributed.

Paper 3, Section I, C

commentWhat does it mean for a second-rank tensor $T_{i j}$ to be isotropic? Show that $\delta_{i j}$ is isotropic. By considering rotations through $\pi / 2$ about the coordinate axes, or otherwise, show that the most general isotropic second-rank tensor in $\mathbb{R}^{3}$ has the form $T_{i j}=\lambda \delta_{i j}$, for some scalar $\lambda$.

Paper 3, Section I, C

commentDefine what it means for a differential $P d x+Q d y$ to be exact, and derive a necessary condition on $P(x, y)$ and $Q(x, y)$ for this to hold. Show that one of the following two differentials is exact and the other is not:

$\begin{aligned} &y^{2} d x+2 x y d y \\ &y^{2} d x+x y^{2} d y \end{aligned}$

Show that the differential which is not exact can be written in the form $g d f$ for functions $f(x, y)$ and $g(y)$, to be determined.

Paper 3, Section II, C

comment(i) Let $V$ be a bounded region in $\mathbb{R}^{3}$ with smooth boundary $S=\partial V$. Show that Poisson's equation in $V$

$\nabla^{2} u=\rho$

has at most one solution satisfying $u=f$ on $S$, where $\rho$ and $f$ are given functions.

Consider the alternative boundary condition $\partial u / \partial n=g$ on $S$, for some given function $g$, where $n$ is the outward pointing normal on $S$. Derive a necessary condition in terms of $\rho$ and $g$ for a solution $u$ of Poisson's equation to exist. Is such a solution unique?

(ii) Find the most general spherically symmetric function $u(r)$ satisfying

$\nabla^{2} u=1$

in the region $r=|\mathbf{r}| \leqslant a$ for $a>0$. Hence in each of the following cases find all possible solutions satisfying the given boundary condition at $r=a$ : (a) $u=0$, (b) $\frac{\partial u}{\partial n}=0$.

Compare these with your results in part (i).

Paper 3, Section II, C

comment(a) Prove the identity

$\nabla(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \nabla) \mathbf{G}+(\mathbf{G} \cdot \nabla) \mathbf{F}+\mathbf{F} \times(\nabla \times \mathbf{G})+\mathbf{G} \times(\nabla \times \mathbf{F})$

(b) If $\mathbf{E}$ is an irrotational vector field (i.e. $\nabla \times \mathbf{E}=\mathbf{0}$ everywhere), prove that there exists a scalar potential $\phi(\mathbf{x})$ such that $\mathbf{E}=-\nabla \phi$.

Show that the vector field

$\left(x y^{2} z e^{-x^{2} z},-y e^{-x^{2} z}, \frac{1}{2} x^{2} y^{2} e^{-x^{2} z}\right)$

is irrotational, and determine the corresponding potential $\phi$.

Paper 3, Section II, C

commentConsider the transformation of variables

$x=1-u, \quad y=\frac{1-v}{1-u v} .$

Show that the interior of the unit square in the $u v$ plane

$\{(u, v): 0<u<1,0<v<1\}$

is mapped to the interior of the unit square in the $x y$ plane,

$R=\{(x, y): 0<x<1,0<y<1\} .$

[Hint: Consider the relation between $v$ and $y$ when $u=\alpha$, for $0<\alpha<1$ constant.]

Show that

$\frac{\partial(x, y)}{\partial(u, v)}=\frac{(1-(1-x) y)^{2}}{x}$

Now let

$u=\frac{1-t}{1-w t}, \quad v=1-w$

By calculating

$\frac{\partial(x, y)}{\partial(t, w)}=\frac{\partial(x, y)}{\partial(u, v)} \frac{\partial(u, v)}{\partial(t, w)}$

as a function of $x$ and $y$, or otherwise, show that

$\int_{R} \frac{x(1-y)}{(1-(1-x) y)\left(1-\left(1-x^{2}\right) y\right)^{2}} d x d y=1$

Paper 3, Section II, C

commentState Stokes' Theorem for a vector field $\mathbf{B}(\mathbf{x})$ on $\mathbb{R}^{3}$.

Consider the surface $S$ defined by

$z=x^{2}+y^{2}, \quad \frac{1}{9} \leqslant z \leqslant 1 .$

Sketch the surface and calculate the area element $d \mathbf{S}$ in terms of suitable coordinates or parameters. For the vector field

$\mathbf{B}=\left(-y^{3}, x^{3}, z^{3}\right)$

compute $\nabla \times \mathbf{B}$ and calculate $I=\int_{S}(\nabla \times \mathbf{B}) \cdot d \mathbf{S}$.

Use Stokes' Theorem to express $I$ as an integral over $\partial S$ and verify that this gives the same result.

Paper 1, Section I, $1 \mathrm{C}$

comment(a) Let $R$ be the set of all $z \in \mathbb{C}$ with real part 1 . Draw a picture of $R$ and the image of $R$ under the map $z \mapsto e^{z}$ in the complex plane.

(b) For each of the following equations, find all complex numbers $z$ which satisfy it:

(i) $e^{z}=e$,

(ii) $(\log z)^{2}=-\frac{\pi^{2}}{4}$.

(c) Prove that there is no complex number $z$ satisfying $|z|-z=i$.

Paper 1, Section I, A

commentDefine what is meant by the terms rotation, reflection, dilation and shear. Give examples of real $2 \times 2$ matrices representing each of these.

Consider the three $2 \times 2$ matrices

$A=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array}\right), \quad B=\frac{1}{\sqrt{2}}\left(\begin{array}{ll} 1 & 1 \\ 1 & 3 \end{array}\right) \quad \text { and } \quad C=A B$

Identify the three matrices in terms of your definitions above.

Paper 1, Section II, $5 \mathrm{C}$

commentThe equation of a plane $\Pi$ in $\mathbb{R}^{3}$ is

$\mathbf{x} \cdot \mathbf{n}=d$

where $d$ is a constant scalar and $\mathbf{n}$ is a unit vector normal to $\Pi$. What is the distance of the plane from the origin $O$ ?

A sphere $S$ with centre $\mathbf{p}$ and radius $r$ satisfies the equation

$|\mathbf{x}-\mathbf{p}|^{2}=r^{2}$

Show that the intersection of $\Pi$ and $S$ contains exactly one point if $|\mathbf{p} \cdot \mathbf{n}-d|=r$.

The tetrahedron $O A B C$ is defined by the vectors $\mathbf{a}=\overrightarrow{O A}, \mathbf{b}=\overrightarrow{O B}$, and $\mathbf{c}=\overrightarrow{O C}$with $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0$. What does the condition $\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})>0$ imply about the set of vectors $\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$ ? A sphere $T_{r}$ with radius $r>0$ lies inside the tetrahedron and intersects each of the three faces $O A B, O B C$, and $O C A$ in exactly one point. Show that the centre $P$ of $T_{r}$ satisfies

$\overrightarrow{O P}=r \frac{|\mathbf{b} \times \mathbf{c}| \mathbf{a}+|\mathbf{c} \times \mathbf{a}| \mathbf{b}+|\mathbf{a} \times \mathbf{b}| \mathbf{c}}{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})}$

Given that the vector $\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}$ is orthogonal to the plane $\Psi$ of the face $A B C$, obtain an equation for $\Psi$. What is the distance of $\Psi$ from the origin?

Paper 1, Section II, 7B

comment(a) Consider the matrix

$M=\left(\begin{array}{rrr} 2 & 1 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 4 \end{array}\right)$

Determine whether or not $M$ is diagonalisable.

(b) Prove that if $A$ and $B$ are similar matrices then $A$ and $B$ have the same eigenvalues with the same corresponding algebraic multiplicities.

Is the converse true? Give either a proof (if true) or a counterexample with a brief reason (if false).

(c) State the Cayley-Hamilton theorem for a complex matrix $A$ and prove it in the case when $A$ is a $2 \times 2$ diagonalisable matrix.

Suppose that an $n \times n$ matrix $B$ has $B^{k}=\mathbf{0}$ for some $k>n$ (where $\mathbf{0}$ denotes the zero matrix). Show that $B^{n}=\mathbf{0}$.

Paper 1, Section II, A

commentExplain why the number of solutions $\mathbf{x}$ of the simultaneous linear equations $A \mathbf{x}=\mathbf{b}$ is 0,1 or infinity, where $A$ is a real $3 \times 3$ matrix and $\mathbf{x}$ and $\mathbf{b}$ are vectors in $\mathbb{R}^{3}$. State necessary and sufficient conditions on $A$ and $\mathrm{b}$ for each of these possibilities to hold.

Let $A$ and $B$ be real $3 \times 3$ matrices. Give necessary and sufficient conditions on $A$ for there to exist a unique real $3 \times 3$ matrix $X$ satisfying $A X=B$.

Find $X$ when

$A=\left(\begin{array}{ccc} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 1 & 2 & 0 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{ccc} 4 & 0 & 1 \\ 2 & 1 & 0 \\ 3 & -1 & -1 \end{array}\right)$

Paper 1, Section II, B

comment(a) (i) Find the eigenvalues and eigenvectors of the matrix

$A=\left(\begin{array}{lll} 3 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 2 \end{array}\right)$

(ii) Show that the quadric $\mathcal{Q}$ in $\mathbb{R}^{3}$ defined by

$3 x^{2}+2 x y+2 y^{2}+2 x z+2 z^{2}=1$

is an ellipsoid. Find the matrix $B$ of a linear transformation of $\mathbb{R}^{3}$ that will map $\mathcal{Q}$ onto the unit sphere $x^{2}+y^{2}+z^{2}=1$.

(b) Let $P$ be a real orthogonal matrix. Prove that:

(i) as a mapping of vectors, $P$ preserves inner products;

(ii) if $\lambda$ is an eigenvalue of $P$ then $|\lambda|=1$ and $\lambda^{*}$ is also an eigenvalue of $P$.

Now let $Q$ be a real orthogonal $3 \times 3$ matrix having $\lambda=1$ as an eigenvalue of algebraic multiplicity 2. Give a geometrical description of the action of $Q$ on $\mathbb{R}^{3}$, giving a reason for your answer. [You may assume that orthogonal matrices are always diagonalisable.]