Part IA, 2007

# Part IA, 2007

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Paper 1, Section I, $1 A$

comment(i) The spherical polar unit basis vectors $\mathbf{e}_{r}, \mathbf{e}_{\phi}$ and $\mathbf{e}_{\theta}$ in $\mathbb{R}^{3}$ are given in terms of the Cartesian unit basis vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ by

$\begin{aligned} &\mathbf{e}_{r}=\mathbf{i} \cos \phi \sin \theta+\mathbf{j} \sin \phi \sin \theta+\mathbf{k} \cos \theta \\ &\mathbf{e}_{\theta}=\mathbf{i} \cos \phi \cos \theta+\mathbf{j} \sin \phi \cos \theta-\mathbf{k} \sin \theta \\ &\mathbf{e}_{\phi}=-\mathbf{i} \sin \phi+\mathbf{j} \cos \phi \end{aligned}$

Express $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ in terms of $\mathbf{e}_{r}, \mathbf{e}_{\phi}$ and $\mathbf{e}_{\theta}$.

(ii) Use suffix notation to prove the following identity for the vectors $\mathbf{A}, \mathbf{B}$, and $\mathbf{C}$ in $\mathbb{R}^{3}$ :

$(\mathbf{A} \times \mathbf{B}) \times(\mathbf{A} \times \mathbf{C})=(\mathbf{A} \cdot \mathbf{B} \times \mathbf{C}) \mathbf{A}$

Paper 1, Section I, B

commentFor the equations

$\begin{gathered} p x+y+z=1 \\ x+2 y+4 z=t \\ x+4 y+10 z=t^{2} \end{gathered}$

find the values of $p$ and $t$ for which

(i) there is a unique solution;

(ii) there are infinitely many solutions;

(iii) there is no solution.

Paper 1, Section II, A

comment(i) Show that any line in the complex plane $\mathbb{C}$ can be represented in the form

$\bar{c} z+c \bar{z}+r=0,$

where $c \in \mathbb{C}$ and $r \in \mathbb{R}$.

(ii) If $z$ and $u$ are two complex numbers for which

$\left|\frac{z+u}{z+\bar{u}}\right|=1$

show that either $z$ or $u$ is real.

(iii) Show that any Möbius transformation

$w=\frac{a z+b}{c z+d} \quad(b c-a d \neq 0)$

that maps the real axis $z=\bar{z}$ into the unit circle $|w|=1$ can be expressed in the form

$w=\lambda \frac{z+k}{z+\bar{k}}$

where $\lambda, k \in \mathbb{C}$ and $|\lambda|=1$.

Paper 1, Section II, C

commentLet $\mathbf{x}$ and $\mathbf{y}$ be non-zero vectors in a real vector space with scalar product denoted by $\mathbf{x} \cdot \mathbf{y}$. Prove that $(\mathbf{x} \cdot \mathbf{y})^{2} \leqslant(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y})$, and prove also that $(\mathbf{x} \cdot \mathbf{y})^{\mathbf{2}}=(\mathbf{x} \cdot \mathbf{x})(\mathbf{y} \cdot \mathbf{y})$ if and only if $\mathbf{x}=\lambda \mathbf{y}$ for some scalar $\lambda$.

(i) By considering suitable vectors in $\mathbb{R}^{3}$, or otherwise, prove that the inequality $x^{2}+y^{2}+z^{2} \geqslant y z+z x+x y$ holds for any real numbers $x, y$ and $z$.

(ii) By considering suitable vectors in $\mathbb{R}^{4}$, or otherwise, show that only one choice of real numbers $x, y, z$ satisfies $3\left(x^{2}+y^{2}+z^{2}+4\right)-2(y z+z x+x y)-4(x+y+z)=0$, and find these numbers.

Paper 1, Section II, C

commentLet $\mathcal{M}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be the linear map defined by

$\mathbf{x} \mapsto \mathbf{x}^{\prime}=a \mathbf{x}+b(\mathbf{n} \times \mathbf{x})$

where $a$ and $b$ are positive scalar constants, and $\mathbf{n}$ is a unit vector.

(i) By considering the effect of $\mathcal{M}$ on $\mathbf{n}$ and on a vector orthogonal to $\mathbf{n}$, describe geometrically the action of $\mathcal{M}$.

(ii) Express the map $\mathcal{M}$ as a matrix $M$ using suffix notation. Find $a, b$ and $\mathbf{n}$ in the case

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

(iii) Find, in the general case, the inverse map (i.e. express $\mathbf{x}$ in terms of $\mathbf{x}^{\prime}$ in vector form).

Paper 1, Section II, C

comment(i) Describe geometrically the following surfaces in three-dimensional space:

(a) $\mathbf{r} \cdot \mathbf{u}=\alpha|\mathbf{r}|$, where $0<|\alpha|<1$

(b) $|\mathbf{r}-(\mathbf{r} \cdot \mathbf{u}) \mathbf{u}|=\beta$, where $\beta>0$.

Here $\alpha$ and $\beta$ are fixed scalars and $\mathbf{u}$ is a fixed unit vector. You should identify the meaning of $\alpha, \beta$ and $\mathbf{u}$ for these surfaces.

(ii) The plane $\mathbf{n} \cdot \mathbf{r}=p$, where $\mathbf{n}$ is a fixed unit vector, and the sphere with centre $\mathbf{c}$ and radius $a$ intersect in a circle with centre $\mathbf{b}$ and radius $\rho$.

(a) Show that $\mathbf{b}-\mathbf{c}=\lambda \mathbf{n}$, where you should give $\lambda$ in terms of $a$ and $\rho$.

(b) Find $\rho$ in terms of $\mathbf{c}, \mathbf{n}, a$ and $p$.

Paper 3, Section I, D

commentWhat does it mean to say that groups $G$ and $H$ are isomorphic?

Prove that no two of $C_{8}, C_{4} \times C_{2}$ and $C_{2} \times C_{2} \times C_{2}$ are isomorphic. [Here $C_{n}$ denotes the cyclic group of order $n$.]

Give, with justification, a group of order 8 that is not isomorphic to any of those three groups.

Paper 3, Section I, D

commentProve that every permutation of $\{1, \ldots, n\}$ may be expressed as a product of disjoint cycles.

Let $\sigma=(1234)$ and let $\tau=(345)(678)$. Write $\sigma \tau$ as a product of disjoint cycles. What is the order of $\sigma \tau ?$

Paper 3, Section II, $6 \mathrm{D}$

commentWhat does it mean to say that a subgroup $H$ of a group $G$ is normal? Give, with justification, an example of a subgroup of a group that is normal, and also an example of a subgroup of a group that is not normal.

If $H$ is a normal subgroup of $G$, explain carefully how to make the set of (left) cosets of $H$ into a group.

Let $H$ be a normal subgroup of a finite group $G$. Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.

(i) If $G$ is cyclic then $H$ and $G / H$ are cyclic.

(ii) If $H$ and $G / H$ are cyclic then $G$ is cyclic.

(iii) If $G$ is abelian then $H$ and $G / H$ are abelian.

(iv) If $H$ and $G / H$ are abelian then $G$ is abelian.

Paper 3, Section II, D

commentLet $x$ be an element of a finite group $G$. What is meant by the order of $x$ ? Prove that the order of $x$ must divide the order of $G$. [No version of Lagrange's theorem or the Orbit-Stabilizer theorem may be used without proof.]

If $G$ is a group of order $n$, and $d$ is a divisor of $n$ with $d<n$, is it always true that $G$ must contain an element of order $d$ ? Justify your answer.

Prove that if $m$ and $n$ are coprime then the group $C_{m} \times C_{n}$ is cyclic.

If $m$ and $n$ are not coprime, can it happen that $C_{m} \times C_{n}$ is cyclic?

[Here $C_{n}$ denotes the cyclic group of order $n$.]

Paper 3, Section II, D

commentIn the group of Möbius maps, what is the order of the Möbius map $z \mapsto \frac{1}{z}$ ? What is the order of the Möbius map $z \mapsto \frac{1}{1-z}$ ?

Prove that every Möbius map is conjugate either to a map of the form $z \mapsto \mu z$ (some $\mu \in \mathbb{C}$ ) or to the $\operatorname{map} z \mapsto z+1$. Is $z \mapsto z+1$ conjugate to a map of the form $z \mapsto \mu z ?$

Let $f$ be a Möbius map of order $n$, for some positive integer $n$. Under the action on $\mathbb{C} \cup\{\infty\}$ of the group generated by $f$, what are the various sizes of the orbits? Justify your answer.

Paper 3, Section II, D

commentLet $A$ be a real symmetric $n \times n$ matrix. Prove that every eigenvalue of $A$ is real, and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Indicate clearly where in your argument you have used the fact that $A$ is real.

What does it mean to say that a real $n \times n$ matrix $P$ is orthogonal ? Show that if $P$ is orthogonal and $A$ is as above then $P^{-1} A P$ is symmetric. If $P$ is any real invertible matrix, must $P^{-1} A P$ be symmetric? Justify your answer.

Give, with justification, real $2 \times 2$ matrices $B, C, D, E$ with the following properties:

(i) $B$ has no real eigenvalues;

(ii) $C$ is not diagonalisable over $\mathbb{C}$;

(iii) $D$ is diagonalisable over $\mathbb{C}$, but not over $\mathbb{R}$;

(iv) $E$ is diagonalisable over $\mathbb{R}$, but does not have an orthonormal basis of eigenvectors.

Paper 1, Section I, $3 F$

commentProve that, for positive real numbers $a$ and $b$,

$2 \sqrt{a b} \leqslant a+b$

For positive real numbers $a_{1}, a_{2}, \ldots$, prove that the convergence of

$\sum_{n=1}^{\infty} a_{n}$

implies the convergence of

$\sum_{n=1}^{\infty} \frac{\sqrt{a_{n}}}{n}$

Paper 1, Section I, D

commentLet $\sum_{n=0}^{\infty} a_{n} z^{n}$ be a complex power series. Show that there exists $R \in[0, \infty]$ such that $\sum_{n=0}^{\infty} a_{n} z^{n}$ converges whenever $|z|<R$ and diverges whenever $|z|>R$.

Find the value of $R$ for the power series

$\sum_{n=1}^{\infty} \frac{z^{n}}{n}$

Paper 1, Section II, D

commentExplain carefully what it means to say that a bounded function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

Prove that every continuous function $f:[0,1] \rightarrow \mathbb{R}$ is Riemann integrable.

For each of the following functions from $[0,1]$ to $\mathbb{R}$, determine with proof whether or not it is Riemann integrable:

(i) the function $f(x)=x \sin \frac{1}{x}$ for $x \neq 0$, with $f(0)=0$;

(ii) the function $g(x)=\sin \frac{1}{x}$ for $x \neq 0$, with $g(0)=0$.

Paper 1, Section II, E

commentLet $a<b$ be real numbers, and let $f:[a, b] \rightarrow \mathbb{R}$ be continuous. Show that $f$ is bounded on $[a, b]$, and that there exist $c, d \in[a, b]$ such that for all $x \in[a, b]$, $f(c) \leqslant f(x) \leqslant f(d)$.

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that

$\lim _{x \rightarrow+\infty} g(x)=\lim _{x \rightarrow-\infty} g(x)=0$

Show that $g$ is bounded. Show also that, if $a$ and $c$ are real numbers with $0<c \leqslant g(a)$, then there exists $x \in \mathbb{R}$ with $g(x)=c$.

Paper 1, Section II, E

commentState and prove the Mean Value Theorem.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that, for every $x \in \mathbb{R}, f^{\prime \prime}(x)$ exists and is non-negative.

(i) Show that if $x \leqslant y$ then $f^{\prime}(x) \leqslant f^{\prime}(y)$.

(ii) Let $\lambda \in(0,1)$ and $a<b$. Show that there exist $x$ and $y$ such that

$f(\lambda a+(1-\lambda) b)=f(a)+(1-\lambda)(b-a) f^{\prime}(x)=f(b)-\lambda(b-a) f^{\prime}(y)$

and that

$f(\lambda a+(1-\lambda) b) \leqslant \lambda f(a)+(1-\lambda) f(b) .$

Paper 1, Section II, F

commentLet $a_{1}=\sqrt{2}$, and consider the sequence of positive real numbers defined by

$a_{n+1}=\sqrt{2+\sqrt{a}_{n}}, \quad n=1,2,3, \ldots$

Show that $a_{n} \leqslant 2$ for all $n$. Prove that the sequence $a_{1}, a_{2}, \ldots$ converges to a limit.

Suppose instead that $a_{1}=4$. Prove that again the sequence $a_{1}, a_{2}, \ldots$ converges to a limit.

Prove that the limits obtained in the two cases are equal.

Paper 2, Section I, B

commentInvestigate the stability of:

(i) the equilibrium points of the equation

$\frac{d y}{d t}=\left(y^{2}-4\right) \tan ^{-1}(y)$

(ii) the constant solutions $\left(u_{n+1}=u_{n}\right)$ of the discrete equation

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

Paper 2, Section I, B

commentFind the solution $y(x)$ of the equation

$y^{\prime \prime}-6 y^{\prime}+9 y=\cos (2 x) \mathrm{e}^{3 x}$

that satisfies $y(0)=0$ and $y^{\prime}(0)=1$.

Paper 2, Section II, B

comment(i) Find the general solution of the difference equation

$u_{k+1}+5 u_{k}+6 u_{k-1}=12 k+1$

(ii) Find the solution of the equation

$y_{k+1}+5 y_{k}+6 y_{k-1}=2^{k}$

that satisfies $y_{0}=y_{1}=1$. Hence show that, for any positive integer $n$, the quantity $2^{n}-26(-3)^{n}$ is divisible by $10 .$

Paper 2, Section II, B

comment(i) Find, in the form of an integral, the solution of the equation

$\alpha \frac{d y}{d t}+y=f(t)$

that satisfies $y \rightarrow 0$ as $t \rightarrow-\infty$. Here $f(t)$ is a general function and $\alpha$ is a positive constant.

Hence find the solution in each of the cases:

(a) $f(t)=\delta(t)$;

(b) $f(t)=H(t)$, where $H(t)$ is the Heaviside step function.

(ii) Find and sketch the solution of the equation

$\frac{d y}{d t}+y=H(t)-H(t-1)$

given that $y(0)=0$ and $y(t)$ is continuous.

Paper 2, Section II, B

commentFind the most general solution of the equation

$6 \frac{\partial^{2} u}{\partial x^{2}}-5 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=1$

by making the change of variables

$\xi=x+2 y, \quad \eta=x+3 y .$

Find the solution that satisfies $u=0$ and $\partial u / \partial y=x$ when $y=0$.

Paper 2, Section II, B

comment(i) The function $y(z)$ satisfies the equation

$y^{\prime \prime}+p(z) y^{\prime}+q(z) y=0$

Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.

(ii) For the equation

$4 z y^{\prime \prime}+2 y^{\prime}+y=0,$

classify the point $z=0$ according to the definitions you gave in (i), and find the series solutions about $z=0$. Identify these solutions in closed form.

Paper 4, Section I, C

commentSketch the graph of $y=3 x^{2}-2 x^{3}$.

A particle of unit mass moves along the $x$ axis in the potential $V(x)=3 x^{2}-2 x^{3}$. Sketch the phase plane, and describe briefly the motion of the particle on the different trajectories.

Paper 4, Section I, C

commentA rocket, moving vertically upwards, ejects gas vertically downwards at speed $u$ relative to the rocket. Derive, giving careful explanations, the equation of motion

$m \frac{d v}{d t}=-u \frac{d m}{d t}-g m$

where $v$ and $m$ are the speed and total mass of the rocket (including fuel) at time $t$.

If $u$ is constant and the rocket starts from rest with total mass $m_{0}$, show that

$m=m_{0} e^{-(g t+v) / u}$

Paper 4, Section II, $12 \mathrm{C}$

commentThe $i$ th particle of a system of $N$ particles has mass $m_{i}$ and, at time $t$, position vector $\mathbf{r}_{i}$ with respect to an origin $O$. It experiences an external force $\mathbf{F}_{i}^{e}$, and also an internal force $\mathbf{F}_{i j}$ due to the $j$ th particle (for each $j=1, \ldots, N, j \neq i$ ), where $\mathbf{F}_{i j}$ is parallel to $\mathbf{r}_{i}-\mathbf{r}_{j}$ and Newton's third law applies.

(i) Show that the position of the centre of mass, $\mathbf{X}$, satisfies

$M \frac{d^{2} \mathbf{X}}{d t^{2}}=\mathbf{F}^{e}$

where $M$ is the total mass of the system and $\mathbf{F}^{e}$ is the sum of the external forces.

(ii) Show that the total angular momentum of the system about the origin, $\mathbf{L}$, satisfies

$\frac{d \mathbf{L}}{d t}=\mathbf{N}$

where $\mathbf{N}$ is the total moment about the origin of the external forces.

(iii) Show that $\mathbf{L}$ can be expressed in the form

$\mathbf{L}=M \mathbf{X} \times \mathbf{V}+\sum_{i} m_{i} \mathbf{r}_{i}^{\prime} \times \mathbf{v}_{i}^{\prime}$

where $\mathbf{V}$ is the velocity of the centre of mass, $\mathbf{r}_{i}^{\prime}$ is the position vector of the $i$ th particle relative to the centre of mass, and $\mathbf{v}_{i}^{\prime}$ is the velocity of the $i$ th particle relative to the centre of mass.

(iv) In the case $N=2$ when the internal forces are derived from a potential $U(|\mathbf{r}|)$, where $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$, and there are no external forces, show that

$\frac{d T}{d t}+\frac{d U}{d t}=0$

where $T$ is the total kinetic energy of the system.

Paper 4, Section II, C

commentA particle moves in the gravitational field of the Sun. The angular momentum per unit mass of the particle is $h$ and the mass of the Sun is $M$. Assuming that the particle moves in a plane, write down the equations of motion in polar coordinates, and derive the equation

$\frac{d^{2} u}{d \theta^{2}}+u=k$

where $u=1 / r$ and $k=G M / h^{2}$.

Write down the equation of the orbit ( $u$ as a function of $\theta$ ), given that the particle moves with the escape velocity and is at the perihelion of its orbit, a distance $r_{0}$ from the Sun, when $\theta=0$. Show that

$\sec ^{4}(\theta / 2) \frac{d \theta}{d t}=\frac{h}{r_{0}^{2}}$

and hence that the particle reaches a distance $2 r_{0}$ from the Sun at time $8 r_{0}^{2} /(3 h)$.

Paper 4, Section II, C

commentA particle of mass $m$ experiences, at the point with position vector $\mathbf{r}$, a force $\mathbf{F}$ given by

$\mathbf{F}=-k \mathbf{r}-e \dot{\mathbf{r}} \times \mathbf{B},$

where $k$ and $e$ are positive constants and $\mathbf{B}$ is a constant, uniform, vector field.

(i) Show that $m \dot{\mathbf{r}} \cdot \dot{\mathbf{r}}+k \mathbf{r} \cdot \mathbf{r}$ is constant. Give a physical interpretation of each term and a physical explanation of the fact that $\mathbf{B}$ does not arise in this expression.

(ii) Show that $m(\dot{\mathbf{r}} \times \mathbf{r}) \cdot \mathbf{B}+\frac{1}{2} e(\mathbf{r} \times \mathbf{B}) \cdot(\mathbf{r} \times \mathbf{B})$ is constant.

(iii) Given that the particle was initially at rest at $\mathbf{r}_{0}$, derive an expression for $\mathbf{r} \cdot \mathbf{B}$ at time $t$.

Paper 4, Section II, C

commentA small ring of mass $m$ is threaded on a smooth rigid wire in the shape of a parabola given by $x^{2}=4 a z$, where $x$ measures horizontal distance and $z$ measures distance vertically upwards. The ring is held at height $z=h$, then released.

(i) Show by dimensional analysis that the period of oscillations, $T$, can be written in the form

$T=(a / g)^{1 / 2} G(h / a)$

for some function $G$.

(ii) Show that $G$ is given by

$G(\beta)=2 \sqrt{2} \int_{-1}^{1}\left(\frac{1+\beta u^{2}}{1-u^{2}}\right)^{\frac{1}{2}} d u$

and find, to first order in $h / a$, the period of small oscillations.

Paper 4 , Section II, E

commentState and prove the Inclusion-Exclusion principle.

The keypad on a cash dispenser is broken. To withdraw money, a customer is required to key in a 4-digit number. However, the key numbered 0 will only function if either the immediately preceding two keypresses were both 1 , or the very first key pressed was 2. Explaining your reasoning clearly, use the Inclusion-Exclusion Principle to find the number of 4-digit codes which can be entered.

Paper 4, Section I, $1 E$

comment(i) Use Euclid's algorithm to find all pairs of integers $x$ and $y$ such that

$7 x+18 y=1$

(ii) Show that, if $n$ is odd, then $n^{3}-n$ is divisible by 24 .

Paper 4, Section I, $2 E$

commentFor integers $k$ and $n$ with $0 \leqslant k \leqslant n$, define $\left(\begin{array}{l}n \\ k\end{array}\right)$. Arguing from your definition, show that

$\left(\begin{array}{c} n-1 \\ k \end{array}\right)+\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right)=\left(\begin{array}{l} n \\ k \end{array}\right)$

for all integers $k$ and $n$ with $1 \leqslant k \leqslant n-1$.

Use induction on $k$ to prove that

$\sum_{j=0}^{k}\left(\begin{array}{c} n+j \\ j \end{array}\right)=\left(\begin{array}{c} n+k+1 \\ k \end{array}\right)$

for all non-negative integers $k$ and $n$.

Paper 4, Section II, $6 \mathrm{E}$

commentStating carefully any results about countability you use, show that for any $d \geqslant 1$ the set $\mathbb{Z}\left[X_{1}, \ldots, X_{d}\right]$ of polynomials with integer coefficients in $d$ variables is countable. By taking $d=1$, deduce that there exist uncountably many transcendental numbers.

Show that there exists a sequence $x_{1}, x_{2}, \ldots$ of real numbers with the property that $f\left(x_{1}, \ldots, x_{d}\right) \neq 0$ for every $d \geqslant 1$ and for every non-zero polynomial $f \in \mathbb{Z}\left[X_{1}, \ldots, X_{d}\right]$.

[You may assume without proof that $\mathbb{R}$ is uncountable.]

Paper 4, Section II, $7 \mathbf{E}$

commentLet $x_{n}(n=1,2, \ldots)$ be real numbers.

What does it mean to say that the sequence $\left(x_{n}\right)_{n=1}^{\infty}$ converges?

What does it mean to say that the series $\sum_{n=1}^{\infty} x_{n}$ converges?

Show that if $\sum_{n=1}^{\infty} x_{n}$ is convergent, then $x_{n} \rightarrow 0$. Show that the converse can be false.

Sequences of positive real numbers $x_{n}, y_{n}(n \geqslant 1)$ are given, such that the inequality

$y_{n+1} \leqslant y_{n}-\frac{1}{2} \min \left(x_{n}, y_{n}\right)$

holds for all $n \geqslant 1$. Show that, if $\sum_{n=1}^{\infty} x_{n}$ diverges, then $y_{n} \rightarrow 0$.

Paper 4, Section II, E

comment(i) Let $p$ be a prime number, and let $x$ and $y$ be integers such that $p$ divides $x y$. Show that at least one of $x$ and $y$ is divisible by $p$. Explain how this enables one to prove the Fundamental Theorem of Arithmetic.

[Standard properties of highest common factors may be assumed without proof.]

(ii) State and prove the Fermat-Euler Theorem.

Let $1 / 359$ have decimal expansion $0 \cdot a_{1} a_{2} \ldots$ with $a_{n} \in\{0,1, \ldots, 9\}$. Use the fact that $60^{2} \equiv 10(\bmod 359)$ to show that, for every $n, a_{n}=a_{n+179}$.

Paper 2, Section I, F

commentLet $X$ be a normally distributed random variable with mean 0 and variance 1 . Define, and determine, the moment generating function of $X$. Compute $\mathbb{E} X^{r}$ for $r=0,1,2,3,4$.

Let $Y$ be a normally distributed random variable with mean $\mu$ and variance $\sigma^{2}$. Determine the moment generating function of $Y$.

Paper 2, Section I, F

commentLet $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Let $U=\min (X, Y)$ and $V=\max (X, Y)$. Show that $\mathbb{E} U=\frac{1}{3}$, and hence find the covariance of $U$ and $V$.

Paper 2, Section II, F

commentLet $A, B$ and $C$ be three random points on a sphere with centre $O$. The positions of $A, B$ and $C$ are independent, and each is uniformly distributed over the surface of the sphere. Calculate the probability density function of the angle $\angle A O B$ formed by the lines $O A$ and $O B$.

Calculate the probability that all three of the angles $\angle A O B, \angle A O C$ and $\angle B O C$ are acute. [Hint: Condition on the value of the angle $\angle A O B$.]

Paper 2, Section II, F

commentLet $A_{1}, A_{2}, \ldots, A_{n}(n \geqslant 2)$ be events in a sample space. For each of the following statements, either prove the statement or provide a counterexample.

(i)

$P\left(\bigcap_{k=2}^{n} A_{k} \mid A_{1}\right)=\prod_{k=2}^{n} P\left(A_{k} \mid \bigcap_{r=1}^{k-1} A_{r}\right), \quad \text { provided } P\left(\bigcap_{k=1}^{n-1} A_{k}\right)>0$

(ii)

$\text { If } \sum_{k=1}^{n} P\left(A_{k}\right)>n-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0$

(iii)

$\text { If } \sum_{i<j} P\left(A_{i} \cap A_{j}\right)>\left(\begin{array}{c} n \\ 2 \end{array}\right)-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0 \text {. }$

(iv) If $B$ is an event and if, for each $k,\left\{B, A_{k}\right\}$ is a pair of independent events, then $\left\{B, \cup_{k=1}^{n} A_{k}\right\}$ is also a pair of independent events.

Paper 2, Section II, F

commentLet $X$ and $Y$ be independent non-negative random variables, with densities $f$ and $g$ respectively. Find the joint density of $U=X$ and $V=X+a Y$, where $a$ is a positive constant.

Let $X$ and $Y$ be independent and exponentially distributed random variables, each with density

$f(x)=\lambda e^{-\lambda x}, \quad x \geqslant 0$

Find the density of $X+\frac{1}{2} Y$. Is it the same as the density of the random variable $\max (X, Y) ?$

Paper 2, Section II, F

commentLet $N$ be a non-negative integer-valued random variable with

$P\{N=r\}=p_{r}, \quad r=0,1,2, \ldots$

Define $\mathbb{E} N$, and show that

$\mathbb{E} N=\sum_{n=1}^{\infty} P\{N \geqslant n\} .$

Let $X_{1}, X_{2}, \ldots$ be a sequence of independent and identically distributed continuous random variables. Let the random variable $N$ mark the point at which the sequence stops decreasing: that is, $N \geqslant 2$ is such that

$X_{1} \geqslant X_{2} \geqslant \ldots \geqslant X_{N-1}<X_{N},$

where, if there is no such finite value of $N$, we set $N=\infty$. Compute $P\{N=r\}$, and show that $P\{N=\infty\}=0$. Determine $\mathbb{E} N$.

Paper 3 , Section I, A

comment(i) Give definitions for the unit tangent vector $\hat{\mathbf{T}}$ and the curvature $\kappa$ of a parametrised curve $\mathbf{x}(t)$ in $\mathbb{R}^{3}$. Calculate $\hat{\mathbf{T}}$ and $\kappa$ for the circular helix

$\mathbf{x}(t)=(a \cos t, a \sin t, b t),$

where $a$ and $b$ are constants.

(ii) Find the normal vector and the equation of the tangent plane to the surface $S$ in $\mathbb{R}^{3}$ given by

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

at the point $x=1, y=1, z=1$.

Paper 3, Section I, A

commentBy using suffix notation, prove the following identities for the vector fields $\mathbf{A}$ and B in $\mathbb{R}^{3}$ :

$\begin{gathered} \nabla \cdot(\mathbf{A} \times \mathbf{B})=\mathbf{B} \cdot(\nabla \times \mathbf{A})-\mathbf{A} \cdot(\nabla \times \mathbf{B}) \\ \nabla \times(\mathbf{A} \times \mathbf{B})=(\mathbf{B} \cdot \nabla) \mathbf{A}-\mathbf{B}(\nabla \cdot \mathbf{A})-(\mathbf{A} \cdot \nabla) \mathbf{B}+\mathbf{A}(\nabla \cdot \mathbf{B}) \end{gathered}$

Paper 3, Section II, A

commentShow that any second rank Cartesian tensor $P_{i j}$ in $\mathbb{R}^{3}$ can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that $P_{i j}$ can be decomposed into the following terms

$\tag{†} P_{i j}=P \delta_{i j}+S_{i j}+\epsilon_{i j k} A_{k},$

where $S_{i j}$ is symmetric and traceless. Give expressions for $P, S_{i j}$ and $A_{k}$ explicitly in terms of $P_{i j}$.

For an isotropic material, the stress $P_{i j}$ can be related to the strain $T_{i j}$ through the stress-strain relation, $P_{i j}=c_{i j k l} T_{k l}$, where the elasticity tensor is given by

$c_{i j k l}=\alpha \delta_{i j} \delta_{k l}+\beta \delta_{i k} \delta_{j l}+\gamma \delta_{i l} \delta_{j k}$

and $\alpha, \beta$ and $\gamma$ are scalars. As in $(†)$, the strain $T_{i j}$ can be decomposed into its trace $T$, a symmetric traceless tensor $W_{i j}$ and a vector $V_{k}$. Use the stress-strain relation to express each of $T, W_{i j}$ and $V_{k}$ in terms of $P, S_{i j}$ and $A_{k}$.

Hence, or otherwise, show that if $T_{i j}$ is symmetric then so is $P_{i j}$. Show also that the stress-strain relation can be written in the form

$P_{i j}=\lambda \delta_{i j} T_{k k}+\mu T_{i j}$

where $\mu$ and $\lambda$ are scalars.

Paper 3, Section II, A

commentThe function $\phi(x, y, z)$ satisfies $\nabla^{2} \phi=0$ in $V$ and $\phi=0$ on $S$, where $V$ is a region of $\mathbb{R}^{3}$ which is bounded by the surface $S$. Prove that $\phi=0$ everywhere in $V$.

Deduce that there is at most one function $\psi(x, y, z)$ satisfying $\nabla^{2} \psi=\rho$ in $V$ and $\psi=f$ on $S$, where $\rho(x, y, z)$ and $f(x, y, z)$ are given functions.

Given that the function $\psi=\psi(r)$ depends only on the radial coordinate $r=|\mathbf{x}|$, use Cartesian coordinates to show that

$\nabla \psi=\frac{1}{r} \frac{d \psi}{d r} \mathbf{x}, \quad \nabla^{2} \psi=\frac{1}{r} \frac{d^{2}(r \psi)}{d r^{2}}$

Find the general solution in this radial case for $\nabla^{2} \psi=c$ where $c$ is a constant.

Find solutions $\psi(r)$ for a solid sphere of radius $r=2$ with a central cavity of radius $r=1$ in the following three regions:

(i) $0 \leqslant r \leqslant 1$ where $\nabla^{2} \psi=0$ and $\psi(1)=1$ and $\psi$ bounded as $r \rightarrow 0$;

(ii) $1 \leqslant r \leqslant 2$ where $\nabla^{2} \psi=1$ and $\psi(1)=\psi(2)=1$;

(iii) $r \geqslant 2$ where $\nabla^{2} \psi=0$ and $\psi(2)=1$ and $\psi \rightarrow 0$ as $r \rightarrow \infty$.

Paper 3, Section II, A

commentFor a given charge distribution $\rho(x, y, z)$ and divergence-free current distribution $\mathbf{J}(x, y, z)$ (i.e. $\nabla \cdot \mathbf{J}=0)$ in $\mathbb{R}^{3}$, the electric and magnetic fields $\mathbf{E}(x, y, z)$ and $\mathbf{B}(x, y, z)$ satisfy the equations

$\nabla \times \mathbf{E}=0, \quad \nabla \cdot \mathbf{B}=0, \quad \nabla \cdot \mathbf{E}=\rho, \quad \nabla \times \mathbf{B}=\mathbf{J}$

The radiation flux vector $\mathbf{P}$ is defined by $\mathbf{P}=\mathbf{E} \times \mathbf{B}$. For a closed surface $S$ around a region $V$, show using Gauss' theorem that the flux of the vector $\mathbf{P}$ through $S$ can be expressed as

$\iint_{S} \mathbf{P} \cdot \mathbf{d} \mathbf{S}=-\iiint_{V} \mathbf{E} \cdot \mathbf{J} d V$

For electric and magnetic fields given by

$\mathbf{E}(x, y, z)=(z, 0, x), \quad \mathbf{B}(x, y, z)=(0,-x y, x z)$

find the radiation flux through the quadrant of the unit spherical shell given by

$x^{2}+y^{2}+z^{2}=1, \quad \text { with } \quad 0 \leqslant x \leqslant 1, \quad 0 \leqslant y \leqslant 1, \quad-1 \leqslant z \leqslant 1$

[If you use (*), note that an open surface has been specified.]

Paper 3, Section II, A

comment(i) Define what is meant by a conservative vector field. Given a vector field $\mathbf{A}=\left(A_{1}(x, y), A_{2}(x, y)\right)$ and a function $\psi(x, y)$ defined in $\mathbb{R}^{2}$, show that, if $\psi \mathbf{A}$ is a conservative vector field, then

$\psi\left(\frac{\partial A_{1}}{\partial y}-\frac{\partial A_{2}}{\partial x}\right)=A_{2} \frac{\partial \psi}{\partial x}-A_{1} \frac{\partial \psi}{\partial y}$

(ii) Given two functions $P(x, y)$ and $Q(x, y)$ defined in $\mathbb{R}^{2}$, prove Green's theorem,

$\oint_{C}(P d x+Q d y)=\iint_{R}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) d x d y$

where $C$ is a simple closed curve bounding a region $R$ in $\mathbb{R}^{2}$.

Through an appropriate choice for $P$ and $Q$, find an expression for the area of the region $R$, and apply this to evaluate the area of the ellipse bounded by the curve

$x=a \cos \theta, \quad y=b \sin \theta, \quad 0 \leqslant \theta \leqslant 2 \pi$