Part IA, 2015

# Part IA, 2015

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Paper 1, Section I, $3 \mathbf{F}$

commentFind the following limits: (a) $\lim _{x \rightarrow 0} \frac{\sin x}{x}$ (b) $\lim _{x \rightarrow 0}(1+x)^{1 / x}$ (c) $\lim _{x \rightarrow \infty} \frac{(1+x)^{\frac{x}{1+x}} \cos ^{4} x}{e^{x}}$

Carefully justify your answers.

[You may use standard results provided that they are clearly stated.]

Paper 1, Section I, E

commentLet $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $0 \leqslant R \leqslant \infty$.

Find the radius of convergence of $\sum_{n \geqslant 0} p(n) z^{n}$, where $p(n)$ is a fixed polynomial in $n$ with coefficients in $\mathbb{C}$.

Paper 1, Section II, $11 D$

comment(i) State and prove the intermediate value theorem.

(ii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. The chord joining the points $(\alpha, f(\alpha))$ and $(\beta, f(\beta))$ of the curve $y=f(x)$ is said to be horizontal if $f(\alpha)=f(\beta)$. Suppose that the chord joining the points $(0, f(0))$ and $(1, f(1))$ is horizontal. By considering the function $g$ defined on $\left[0, \frac{1}{2}\right]$ by

$g(x)=f\left(x+\frac{1}{2}\right)-f(x)$

or otherwise, show that the curve $y=f(x)$ has a horizontal chord of length $\frac{1}{2}$ in $[0,1]$. Show, more generally, that it has a horizontal chord of length $\frac{1}{n}$ for each positive integer $n$.

Paper 1, Section II, 10D

comment(a) For real numbers $a, b$ such that $a<b$, let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function. Prove that $f$ is bounded on $[a, b]$, and that $f$ attains its supremum and infimum on $[a, b]$.

(b) For $x \in \mathbb{R}$, define

$g(x)=\left\{\begin{array}{ll} |x|^{\frac{1}{2}} \sin (1 / \sin x), & x \neq n \pi \\ 0, & x=n \pi \end{array} \quad(n \in \mathbb{Z})\right.$

Find the set of points $x \in \mathbb{R}$ at which $g(x)$ is continuous.

Does $g$ attain its supremum on $[0, \pi] ?$

Does $g$ attain its supremum on $[\pi, 3 \pi / 2]$ ?

Justify your answers.

Paper 1, Section II, E

commentLet $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function, and let $\mathcal{D}_{n}$ denote the dissection $0<\frac{1}{n}<\frac{2}{n}<\cdots<\frac{n-1}{n}<1$ of $[0,1]$. Prove that $f$ is Riemann integrable if and only if the difference between the upper and lower sums of $f$ with respect to the dissection $\mathcal{D}_{n}$ tends to zero as $n$ tends to infinity.

Suppose that $f$ is Riemann integrable and $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable. Prove that $g \circ f$ is Riemann integrable.

[You may use the mean value theorem provided that it is clearly stated.]

Paper 1, Section II, F

commentLet $\left(a_{n}\right),\left(b_{n}\right)$ be sequences of real numbers. Let $S_{n}=\sum_{j=1}^{n} a_{j}$ and set $S_{0}=0$. Show that for any $1 \leqslant m \leqslant n$ we have

$\sum_{j=m}^{n} a_{j} b_{j}=S_{n} b_{n}-S_{m-1} b_{m}+\sum_{j=m}^{n-1} S_{j}\left(b_{j}-b_{j+1}\right)$

Suppose that the series $\sum_{n \geqslant 1} a_{n}$ converges and that $\left(b_{n}\right)$ is bounded and monotonic. Does $\sum_{n \geqslant 1} a_{n} b_{n}$ converge?

Assume again that $\sum_{n \geqslant 1} a_{n}$ converges. Does $\sum_{n \geqslant 1} n^{1 / n} a_{n}$ converge?

Justify your answers.

[You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]

Paper 2, Section $I$, B

commentFind the general solution of the equation

$2 \frac{d y}{d t}=y-y^{3} .$

Compute all possible limiting values of $y$ as $t \rightarrow \infty$.

Find a non-zero value of $y(0)$ such that $y(t)=y(0)$ for all $t$.

Paper 2, Section I, B

commentFind the general solution of the equation

$\frac{d y}{d x}-2 y=e^{\lambda x}$

where $\lambda$ is a constant not equal to 2 .

By subtracting from the particular integral an appropriate multiple of the complementary function, obtain the limit as $\lambda \rightarrow 2$ of the general solution of $(*)$ and confirm that it yields the general solution for $\lambda=2$.

Solve equation $(*)$ with $\lambda=2$ and $y(1)=2$.

Paper 2, Section II, B

commentConsider the equation

$2 \frac{\partial^{2} u}{\partial x^{2}}+3 \frac{\partial^{2} u}{\partial y^{2}}-7 \frac{\partial^{2} u}{\partial x \partial y}=0$

for the function $u(x, y)$, where $x$ and $y$ are real variables. By using the change of variables

$\xi=x+\alpha y, \quad \eta=\beta x+y$

where $\alpha$ and $\beta$ are appropriately chosen integers, transform $(*)$ into the equation

$\frac{\partial^{2} u}{\partial \xi \partial \eta}=0$

Hence, solve equation $(*)$ supplemented with the boundary conditions

$u(0, y)=4 y^{2}, \quad u(-2 y, y)=0, \quad \text { for all } y$

Paper 2, Section II, B

commentWrite as a system of two first-order equations the second-order equation

$\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}\left|\frac{d \theta}{d t}\right|+\sin \theta=0$

where $c$ is a small, positive constant, and find its equilibrium points. What is the nature of these points?

Draw the trajectories in the $(\theta, \omega)$ plane, where $\omega=d \theta / d t$, in the neighbourhood of two typical equilibrium points.

By considering the cases of $\omega>0$ and $\omega<0$ separately, find explicit expressions for $\omega^{2}$ as a function of $\theta$. Discuss how the second term in $(*)$ affects the nature of the equilibrium points.

Paper 2, Section II, B

commentSuppose that $\mathbf{x}(t) \in \mathbb{R}^{3}$ obeys the differential equation

$\frac{d \mathbf{x}}{d t}=M \mathbf{x}$

where $M$ is a constant $3 \times 3$ real matrix.

(i) Suppose that $M$ has distinct eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ with corresponding eigenvectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. Explain why $\mathbf{x}$ may be expressed in the form $\sum_{i=1}^{3} a_{i}(t) \mathbf{e}_{i}$ and deduce by substitution that the general solution of $(*)$ is

$\mathbf{x}=\sum_{i=1}^{3} A_{i} e^{\lambda_{i} t} \mathbf{e}_{i}$

where $A_{1}, A_{2}, A_{3}$ are constants.

(ii) What is the general solution of $(*)$ if $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but there are still three linearly independent eigenvectors?

(iii) Suppose again that $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but now there are only two linearly independent eigenvectors: $\mathbf{e}_{1}$ corresponding to $\lambda_{1}$ and $\mathbf{e}_{2}$ corresponding to $\lambda_{2}$. Suppose that a vector $\mathbf{v}$ satisfying the equation $\left(M-\lambda_{2} I\right) \mathbf{v}=\mathbf{e}_{2}$ exists, where $I$ denotes the identity matrix. Show that $\mathbf{v}$ is linearly independent of $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$, and hence or otherwise find the general solution of $(*)$.

Paper 2, Section II, B

commentSuppose that $u(x)$ satisfies the equation

$\frac{d^{2} u}{d x^{2}}-f(x) u=0$

where $f(x)$ is a given non-zero function. Show that under the change of coordinates $x=x(t)$,

$\frac{d^{2} u}{d t^{2}}-\frac{\ddot{x}}{\dot{x}} \frac{d u}{d t}-\dot{x}^{2} f(x) u=0$

where a dot denotes differentiation with respect to $t$. Furthermore, show that the function

$U(t)=\dot{x}^{-\frac{1}{2}} u(x)$

satisfies

$\frac{d^{2} U}{d t^{2}}-\left[\dot{x}^{2} f(x)+\dot{x}^{-\frac{1}{2}}\left(\frac{\ddot{x}}{\dot{x}} \frac{d}{d t}\left(\dot{x}^{\frac{1}{2}}\right)-\frac{d^{2}}{d t^{2}}\left(\dot{x}^{\frac{1}{2}}\right)\right)\right] U=0$

Choosing $\dot{x}=(f(x))^{-\frac{1}{2}}$, deduce that

$\frac{d^{2} U}{d t^{2}}-(1+F(t)) U=0$

for some appropriate function $F(t)$. Assuming that $F$ may be neglected, deduce that $u(x)$ can be approximated by

$u(x) \approx A(x)\left(c_{+} e^{G(x)}+c_{-} e^{-G(x)}\right),$

where $c_{+}, c_{-}$are constants and $A, G$ are functions that you should determine in terms of $f(x)$.

Paper 4, Section I, C

commentWrite down the 4-momentum of a particle with energy $E$ and 3-momentum p. State the relationship between the energy $E$ and wavelength $\lambda$ of a photon.

An electron of mass $m$ is at rest at the origin of the laboratory frame: write down its 4 -momentum. The electron is scattered by a photon of wavelength $\lambda_{1}$ travelling along the $x$-axis: write down the initial 4-momentum of the photon. Afterwards, the photon has wavelength $\lambda_{2}$ and has been deflected through an angle $\theta$. Show that

$\lambda_{2}-\lambda_{1}=\frac{2 h}{m c} \sin ^{2}\left(\frac{1}{2} \theta\right)$

where $c$ is the speed of light and $h$ is Planck's constant.

Paper 4, Section I, C

commentFind the moment of inertia of a uniform sphere of mass $M$ and radius $a$ about an axis through its centre.

The kinetic energy $T$ of any rigid body with total mass $M$, centre of mass $\mathbf{R}$, moment of inertia $I$ about an axis of rotation through $\mathbf{R}$, and angular velocity $\omega$ about that same axis, is given by $T=\frac{1}{2} M \dot{\mathbf{R}}^{2}+\frac{1}{2} I \omega^{2}$. What physical interpretation can be given to the two parts of this expression?

A spherical marble of uniform density and mass $M$ rolls without slipping at speed $V$ along a flat surface. Explaining any relationship that you use between its speed and angular velocity, show that the kinetic energy of the marble is $\frac{7}{10} M V^{2}$.

Paper 4, Section II, $\mathbf{9 C}$

commentA particle is projected vertically upwards at speed $V$ from the surface of the Earth, which may be treated as a perfect sphere. The variation of gravity with height should not be ignored, but the rotation of the Earth should be. Show that the height $z(t)$ of the particle obeys

$\ddot{z}=-\frac{g R^{2}}{(R+z)^{2}},$

where $R$ is the radius of the Earth and $g$ is the acceleration due to gravity measured at the Earth's surface.

Using dimensional analysis, show that the maximum height $H$ of the particle and the time $T$ taken to reach that height are given by

$H=R F(\lambda) \quad \text { and } \quad T=\frac{V}{g} G(\lambda)$

where $F$ and $G$ are functions of $\lambda=V^{2} / g R$.

Write down the equation of conservation of energy and deduce that

$T=\int_{0}^{H} \sqrt{\frac{R+z}{V^{2} R-\left(2 g R-V^{2}\right) z}} d z$

Hence or otherwise show that

$F(\lambda)=\frac{\lambda}{2-\lambda} \quad \text { and } \quad G(\lambda)=\int_{0}^{1} \sqrt{\frac{2-\lambda+\lambda x}{(2-\lambda)^{3}(1-x)}} d x$

Paper 4, Section II, C

commentWrite down the Lorentz transform relating the components of a 4-vector between two inertial frames.

A particle moves along the $x$-axis of an inertial frame. Its position at time $t$ is $x(t)$, its velocity is $u=d x / d t$, and its 4 -position is $X=(c t, x)$, where $c$ is the speed of light. The particle's 4-velocity is given by $U=d X / d \tau$ and its 4 -acceleration is $A=d U / d \tau$, where proper time $\tau$ is defined by $c^{2} d \tau^{2}=c^{2} d t^{2}-d x^{2}$. Show that

$U=\gamma(c, u) \quad \text { and } \quad A=\gamma^{4} \dot{u}(u / c, 1)$

where $\gamma=\left(1-u^{2} / c^{2}\right)^{-\frac{1}{2}}$ and $\dot{u}=d u / d t$.

The proper 3-acceleration a of the particle is defined to be the spatial component of its 4-acceleration measured in the particle's instantaneous rest frame. By transforming $A$ to the rest frame, or otherwise, show that

$a=\gamma^{3} \dot{u}=\frac{d}{d t}(\gamma u)$

Given that the particle moves with constant proper 3 -acceleration starting from rest at the origin, show that

$x(t)=\frac{c^{2}}{a}\left(\sqrt{1+\frac{a^{2} t^{2}}{c^{2}}}-1\right)$

and that, if $a t \ll c$, then $x \approx \frac{1}{2} a t^{2}$.

Paper 4, Section II, C

commentConsider a particle with position vector $r(t)$ moving in a plane described by polar coordinates $(r, \theta)$. Obtain expressions for the radial $(r)$ and transverse $(\theta)$ components of the velocity $\dot{\mathbf{r}}$ and acceleration $\ddot{\mathbf{r}}$.

A charged particle of unit mass moves in the electric field of another charge that is fixed at the origin. The electrostatic force on the particle is $-p / r^{2}$ in the radial direction, where $p$ is a positive constant. The motion takes place in an unusual medium that resists radial motion but not tangential motion, so there is an additional radial force $-k \dot{r} / r^{2}$ where $k$ is a positive constant. Show that the particle's motion lies in a plane. Using polar coordinates in that plane, show also that its angular momentum $h=r^{2} \dot{\theta}$ is constant.

Obtain the equation of motion

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

where $u=r^{-1}$, and find its general solution assuming that $k /|h|<2$. Show that so long as the motion remains bounded it eventually becomes circular with radius $h^{2} / p$.

Obtain the expression

$E=\frac{1}{2} h^{2}\left(u^{2}+\left(\frac{d u}{d \theta}\right)^{2}\right)-p u$

for the particle's total energy, that is, its kinetic energy plus its electrostatic potential energy. Hence, or otherwise, show that the energy is a decreasing function of time.

Paper 4, Section II, C

commentA particle of mass $m$ and charge $q$ has position vector $\mathbf{r}(t)$ and moves in a constant, uniform magnetic field $\mathbf{B}$ so that its equation of motion is

$m \ddot{\mathbf{r}}=q \dot{\mathbf{r}} \times \mathbf{B}$

Let $\mathbf{L}=m \mathbf{r} \times \dot{\mathbf{r}}$ be the particle's angular momentum. Show that

$\mathbf{L} \cdot \mathbf{B}+\frac{1}{2} q|\mathbf{r} \times \mathbf{B}|^{2}$

is a constant of the motion. Explain why the kinetic energy $T$ is also constant, and show that it may be written in the form

$T=\frac{1}{2} m \mathbf{u} \cdot\left((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}-r^{2} \ddot{\mathbf{u}}\right)$

where $\mathbf{v}=\dot{\mathbf{r}}, r=|\mathbf{r}|$ and $\mathbf{u}=\mathbf{r} / r$.

[Hint: Consider u $\cdot \dot{\mathbf{u}} .]$

Paper 3, Section I, D

commentHow many cyclic subgroups (including the trivial subgroup) does $S_{5}$ contain? Exhibit two isomorphic subgroups of $S_{5}$ which are not conjugate.

Paper 3, Section I, D

commentSay that a group is dihedral if it has two generators $x$ and $y$, such that $x$ has order $n$ (greater than or equal to 2 and possibly infinite), $y$ has order 2 , and $y x y^{-1}=x^{-1}$. In particular the groups $C_{2}$ and $C_{2} \times C_{2}$ are regarded as dihedral groups. Prove that:

(i) any dihedral group can be generated by two elements of order 2 ;

(ii) any group generated by two elements of order 2 is dihedral; and

(iii) any non-trivial quotient group of a dihedral group is dihedral.

Paper 3, Section II, D

comment(a) Let $G$ be a non-trivial group and let $Z(G)=\{h \in G: g h=h g$ for all $g \in G\}$. Show that $Z(G)$ is a normal subgroup of $G$. If the order of $G$ is a power of a prime, show that $Z(G)$ is non-trivial.

(b) The Heisenberg group $H$ is the set of all $3 \times 3$ matrices of the form

$\left(\begin{array}{lll} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right)$

with $x, y, z \in \mathbb{R}$. Show that $H$ is a subgroup of the group of non-singular real matrices under matrix multiplication.

Find $Z(H)$ and show that $H / Z(H)$ is isomorphic to $\mathbb{R}^{2}$ under vector addition.

(c) For $p$ prime, the modular Heisenberg group $H_{p}$ is defined as in (b), except that $x, y$ and $z$ now lie in the field of $p$ elements. Write down $\left|H_{p}\right|$. Find both $Z\left(H_{p}\right)$ and $H_{p} / Z\left(H_{p}\right)$ in terms of generators and relations.

Paper 3, Section II, D

comment(a) State and prove Lagrange's theorem.

(b) Let $G$ be a group and let $H, K$ be fixed subgroups of $G$. For each $g \in G$, any set of the form $H g K=\{h g k: h \in H, k \in K\}$ is called an $(H, K)$ double coset, or simply a double coset if $H$ and $K$ are understood. Prove that every element of $G$ lies in some $(H, K)$ double coset, and that any two $(H, K)$ double cosets either coincide or are disjoint.

Let $G$ be a finite group. Which of the following three statements are true, and which are false? Justify your answers.

(i) The size of a double coset divides the order of $G$.

(ii) Different double cosets for the same pair of subgroups have the same size.

(iii) The number of double cosets divides the order of $G$.

Paper 3, Section II, D

commentLet $G, H$ be groups and let $\varphi: G \rightarrow H$ be a function. What does it mean to say that $\varphi$ is a homomorphism with kernel $K$ ? Show that if $K=\{e, \xi\}$ has order 2 then $x^{-1} \xi x=\xi$ for each $x \in G$. [If you use any general results about kernels of homomorphisms, then you should prove them.]

Which of the following four statements are true, and which are false? Justify your answers.

(a) There is a homomorphism from the orthogonal group $\mathrm{O}(3)$ to a group of order 2 with kernel the special orthogonal group $\mathrm{SO}(3)$.

(b) There is a homomorphism from the symmetry group $S_{3}$ of an equilateral triangle to a group of order 2 with kernel of order 3 .

(c) There is a homomorphism from $\mathrm{O}(3)$ to $\mathrm{SO}(3)$ with kernel of order 2 .

(d) There is a homomorphism from $S_{3}$ to a group of order 3 with kernel of order 2 .

Paper 3, Section II, D

commentWhat does it mean for a group $G$ to act on a set $X$ ? For $x \in X$, what is meant by the orbit $\operatorname{Orb}(x)$ to which $x$ belongs, and by the stabiliser $G_{x}$ of $x$ ? Show that $G_{x}$ is a subgroup of $G$. Prove that, if $G$ is finite, then $|G|=\left|G_{x}\right| \cdot|\operatorname{Orb}(x)|$.

(a) Prove that the symmetric group $S_{n}$ acts on the set $P^{(n)}$ of all polynomials in $n$ variables $x_{1}, \ldots, x_{n}$, if we define $\sigma \cdot f$ to be the polynomial given by

$(\sigma \cdot f)\left(x_{1}, \ldots, x_{n}\right)=f\left(x_{\sigma(1)}, \ldots, x_{\sigma(n)}\right)$

for $f \in P^{(n)}$ and $\sigma \in S_{n}$. Find the orbit of $f=x_{1} x_{2}+x_{3} x_{4} \in P^{(4)}$ under $S_{4}$. Find also the order of the stabiliser of $f$.

(b) Let $r, n$ be fixed positive integers such that $r \leqslant n$. Let $B_{r}$ be the set of all subsets of size $r$ of the set $\{1,2, \ldots, n\}$. Show that $S_{n}$ acts on $B_{r}$ by defining $\sigma \cdot U$ to be the set $\{\sigma(u): u \in U\}$, for any $U \in B_{r}$ and $\sigma \in S_{n}$. Prove that $S_{n}$ is transitive in its action on $B_{r}$. Find also the size of the stabiliser of $U \in B_{r}$.

Paper 4 , Section I, E

commentState the Chinese remainder theorem and Fermat's theorem. Prove that

$p^{4} \equiv 1 \quad(\bmod 240)$

for any prime $p>5$.

Paper 4, Section I, E

comment(a) Find all integers $x$ and $y$ such that

$6 x+2 y \equiv 3 \quad(\bmod 53) \quad \text { and } \quad 17 x+4 y \equiv 7 \quad(\bmod 53)$

(b) Show that if an integer $n>4$ is composite then $(n-1) ! \equiv 0(\bmod n)$.

Paper 4, Section II, E

commentWhat does it mean for a set to be countable? Prove that

(a) if $B$ is countable and $f: A \rightarrow B$ is injective, then $A$ is countable;

(b) if $A$ is countable and $f: A \rightarrow B$ is surjective, then $B$ is countable.

Prove that $\mathbb{N} \times \mathbb{N}$ is countable, and deduce that

(i) if $X$ and $Y$ are countable, then so is $X \times Y$;

(ii) $\mathbb{Q}$ is countable.

Let $\mathcal{C}$ be a collection of circles in the plane such that for each point $a$ on the $x$-axis, there is a circle in $\mathcal{C}$ passing through the point $a$ which has the $x$-axis tangent to the circle at $a$. Show that $\mathcal{C}$ contains a pair of circles that intersect.

Paper 4, Section II, E

commentState the inclusion-exclusion principle.

Let $n \in \mathbb{N}$. A permutation $\sigma$ of the set $\{1,2,3, \ldots, n\}$ is said to contain a transposition if there exist $i, j$ with $1 \leqslant i<j \leqslant n$ such that $\sigma(i)=j$ and $\sigma(j)=i$. Derive a formula for the number, $f(n)$, of permutations which do not contain a transposition, and show that

$\lim _{n \rightarrow \infty} \frac{f(n)}{n !}=e^{-\frac{1}{2}}$

Paper 4, Section II, E

commentLet $p$ be a prime. A base $p$ expansion of an integer $k$ is an expression

$k=k_{0}+p \cdot k_{1}+p^{2} \cdot k_{2}+\cdots+p^{\ell} \cdot k_{\ell}$

for some natural number $\ell$, with $0 \leqslant k_{i}<p$ for $i=0,1, \ldots, \ell$.

(i) Show that the sequence of coefficients $k_{0}, k_{1}, k_{2}, \ldots, k_{\ell}$ appearing in a base $p$ expansion of $k$ is unique, up to extending the sequence by zeroes.

(ii) Show that

$\left(\begin{array}{l} p \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0<j<p$

and hence, by considering the polynomial $(1+x)^{p}$ or otherwise, deduce that

$\left(\begin{array}{c} p^{i} \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0<j<p^{i}$

(iii) If $n_{0}+p \cdot n_{1}+p^{2} \cdot n_{2}+\cdots+p^{\ell} \cdot n_{\ell}$ is a base $p$ expansion of $n$, then, by considering the polynomial $(1+x)^{n}$ or otherwise, show that

$\left(\begin{array}{l} n \\ k \end{array}\right) \equiv\left(\begin{array}{l} n_{0} \\ k_{0} \end{array}\right)\left(\begin{array}{l} n_{1} \\ k_{1} \end{array}\right) \cdots\left(\begin{array}{l} n_{\ell} \\ k_{\ell} \end{array}\right) \quad(\bmod p)$

Paper 4, Section II, E

comment(i) Let $\sim$ be an equivalence relation on a set $X$. What is an equivalence class of $\sim$ ? What is a partition of $X ?$ Prove that the equivalence classes of $\sim$ form a partition of $X$.

(ii) Let $\sim$ be the relation on the natural numbers $\mathbb{N}=\{1,2,3, \ldots\}$ defined by

$m \sim n \Longleftrightarrow \exists a, b \in \mathbb{N} \text { such that } m \text { divides } n^{a} \text { and } n \text { divides } m^{b} .$

Show that $\sim$ is an equivalence relation, and show that it has infinitely many equivalence classes, all but one of which are infinite.

Paper 2, Section I, F

commentLet $A, B$ be events in the sample space $\Omega$ such that $0<P(A)<1$ and $0<P(B)<1$. The event $B$ is said to attract $A$ if the conditional probability $P(A \mid B)$ is greater than $P(A)$, otherwise it is said that $A$ repels $B$. Show that if $B$ attracts $A$, then $A$ attracts $B$. Does $B^{c}=\Omega \backslash B$ repel $A ?$

Paper 2, Section I, F

commentLet $U$ be a uniform random variable on $(0,1)$, and let $\lambda>0$.

(a) Find the distribution of the random variable $-(\log U) / \lambda$.

(b) Define a new random variable $X$ as follows: suppose a fair coin is tossed, and if it lands heads we set $X=U^{2}$ whereas if it lands tails we set $X=1-U^{2}$. Find the probability density function of $X$.

Paper 2, Section II, F

commentWhen coin $A$ is tossed it comes up heads with probability $\frac{1}{4}$, whereas coin $B$ comes up heads with probability $\frac{3}{4}$. Suppose one of these coins is randomly chosen and is tossed twice. If both tosses come up heads, what is the probability that coin $B$ was tossed? Justify your answer.

In each draw of a lottery, an integer is picked independently at random from the first $n$ integers $1,2, \ldots, n$, with replacement. What is the probability that in a sample of $r$ successive draws the numbers are drawn in a non-decreasing sequence? Justify your answer.

Paper 2, Section II, F

commentState and prove Markov's inequality and Chebyshev's inequality, and deduce the weak law of large numbers.

If $X$ is a random variable with mean zero and finite variance $\sigma^{2}$, prove that for any $a>0$,

$P(X \geqslant a) \leqslant \frac{\sigma^{2}}{\sigma^{2}+a^{2}}$

[Hint: Show first that $P(X \geqslant a) \leqslant P\left((X+b)^{2} \geqslant(a+b)^{2}\right)$ for every $b>0$.]

Paper 2, Section II, F

commentConsider the function

$\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad x \in \mathbb{R}$

Show that $\phi$ defines a probability density function. If a random variable $X$ has probability density function $\phi$, find the moment generating function of $X$, and find all moments $E\left[X^{k}\right]$, $k \in \mathbb{N}$.

Now define

$r(x)=\frac{P(X>x)}{\phi(x)}$

Show that for every $x>0$,

$\frac{1}{x}-\frac{1}{x^{3}}<r(x)<\frac{1}{x}$

Paper 2, Section II, F

commentLionel and Cristiana have $a$ and $b$ million pounds, respectively, where $a, b \in \mathbb{N}$. They play a series of independent football games in each of which the winner receives one million pounds from the loser (a draw cannot occur). They stop when one player has lost his or her entire fortune. Lionel wins each game with probability $0<p<1$ and Cristiana wins with probability $q=1-p$, where $p \neq q$. Find the expected number of games before they stop playing.

Paper 3, Section I, A

commentThe smooth curve $\mathcal{C}$ in $\mathbb{R}^{3}$ is given in parametrised form by the function $\mathbf{x}(u)$. Let $s$ denote arc length measured along the curve.

(a) Express the tangent $\mathbf{t}$ in terms of the derivative $\mathbf{x}^{\prime}=d \mathbf{x} / d u$, and show that $d u / d s=\left|\mathbf{x}^{\prime}\right|^{-1}$.

(b) Find an expression for $d \mathbf{t} / d s$ in terms of derivatives of $\mathbf{x}$ with respect to $u$, and show that the curvature $\kappa$ is given by

$\kappa=\frac{\left|\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right|}{\left|\mathbf{x}^{\prime}\right|^{3}}$

[Hint: You may find the identity $\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime \prime}\right) \mathbf{x}^{\prime}-\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime}\right) \mathbf{x}^{\prime \prime}=\mathbf{x}^{\prime} \times\left(\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right)$ helpful.]

(c) For the curve

$\mathbf{x}(u)=\left(\begin{array}{c} u \cos u \\ u \sin u \\ 0 \end{array}\right)$

with $u \geqslant 0$, find the curvature as a function of $u$.

Paper 3, Section I, A

comment(i) For $r=|\mathbf{x}|$ with $\mathbf{x} \in \mathbb{R}^{3} \backslash\{\mathbf{0}\}$, show that

$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r} \quad(i=1,2,3) .$

(ii) Consider the vector fields $\mathbf{F}(\mathbf{x})=r^{2} \mathbf{x}, \mathbf{G}(\mathbf{x})=(\mathbf{a} \cdot \mathbf{x}) \mathbf{x}$ and $\mathbf{H}(\mathbf{x})=\mathbf{a} \times \hat{\mathbf{x}}$, where $\mathbf{a}$ is a constant vector in $\mathbb{R}^{3}$ and $\hat{\mathbf{x}}$ is the unit vector in the direction of $\mathbf{x}$. Using suffix notation, or otherwise, find the divergence and the curl of each of $\mathbf{F}, \mathbf{G}$ and $\mathbf{H}$.

Paper 3, Section II, A

comment(a) Let $t_{i j}$ be a rank 2 tensor whose components are invariant under rotations through an angle $\pi$ about each of the three coordinate axes. Show that $t_{i j}$ is diagonal.

(b) An array of numbers $a_{i j}$ is given in one orthonormal basis as $\delta_{i j}+\epsilon_{1 i j}$ and in another rotated basis as $\delta_{i j}$. By using the invariance of the determinant of any rank 2 tensor, or otherwise, prove that $a_{i j}$ is not a tensor.

(c) Let $a_{i j}$ be an array of numbers and $b_{i j}$ a tensor. Determine whether the following statements are true or false. Justify your answers.

(i) If $a_{i j} b_{i j}$ is a scalar for any rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

(ii) If $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

(iii) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

(iv) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any antisymmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

Paper 3, Section II, A

comment(i) Starting with the divergence theorem, derive Green's first theorem

$\int_{V}\left(\psi \nabla^{2} \phi+\nabla \psi \cdot \nabla \phi\right) d V=\int_{\partial V} \psi \frac{\partial \phi}{\partial n} d S$

(ii) The function $\phi(\mathbf{x})$ satisfies Laplace's equation $\nabla^{2} \phi=0$ in the volume $V$ with given boundary conditions $\phi(\mathbf{x})=g(\mathbf{x})$ for all $\mathbf{x} \in \partial V$. Show that $\phi(\mathbf{x})$ is the only such function. Deduce that if $\phi(\mathbf{x})$ is constant on $\partial V$ then it is constant in the whole volume $V$.

(iii) Suppose that $\phi(\mathbf{x})$ satisfies Laplace's equation in the volume $V$. Let $V_{r}$ be the sphere of radius $r$ centred at the origin and contained in $V$. The function $f(r)$ is defined by

$f(r)=\frac{1}{4 \pi r^{2}} \int_{\partial V_{r}} \phi(\mathbf{x}) d S$

By considering the derivative $d f / d r$, and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that $f(r)$ is constant and that $f(r)=\phi(\mathbf{0})$.

(iv) Let $M$ denote the maximum of $\phi$ on $\partial V_{r}$ and $m$ the minimum of $\phi$ on $\partial V_{r}$. By using the result from (iii), or otherwise, show that $m \leqslant \phi(\mathbf{0}) \leqslant M$.

Paper 3, Section II, A

commentState Stokes' theorem.

Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=x^{2}+y^{2}+1-\lambda$, where $0 \leqslant z \leqslant 1$ and $\lambda$ is a positive constant. Sketch the surface $S$ for representative values of $\lambda$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

Compute $\nabla \times \mathbf{F}$ for the vector field

$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ z \end{array}\right)$

and verify Stokes' theorem for $\mathbf{F}$ on the surface $S$ for every value of $\lambda$.

Now compute $\nabla \times \mathbf{G}$ for the vector field

$\mathbf{G}(\mathbf{x})=\frac{1}{x^{2}+y^{2}}\left(\begin{array}{c} -y \\ x \\ 0 \end{array}\right)$

and find the line integral $\int_{\partial S} \mathbf{G} \cdot \mathbf{d x}$ for the boundary $\partial S$ of the surface $S$. Is it possible to obtain this result using Stokes' theorem? Justify your answer.

Paper 3, Section II, A

commentThe vector field $\mathbf{F}(\mathbf{x})$ is given in terms of cylindrical polar coordinates $(\rho, \phi, z)$ by

$\mathbf{F}(\mathbf{x})=f(\rho) \mathbf{e}_{\rho}$

where $f$ is a differentiable function of $\rho$, and $\mathbf{e}_{\rho}=\cos \phi \mathbf{e}_{x}+\sin \phi \mathbf{e}_{y}$ is the unit basis vector with respect to the coordinate $\rho$. Compute the partial derivatives $\partial F_{1} / \partial x, \partial F_{2} / \partial y$, $\partial F_{3} / \partial z$ and hence find the divergence $\nabla \cdot \mathbf{F}$ in terms of $\rho$ and $\phi$.

The domain $V$ is bounded by the surface $z=\left(x^{2}+y^{2}\right)^{-1}$, by the cylinder $x^{2}+y^{2}=1$, and by the planes $z=\frac{1}{4}$ and $z=1$. Sketch $V$ and compute its volume.

Find the most general function $f(\rho)$ such that $\nabla \cdot \mathbf{F}=0$, and verify the divergence theorem for the corresponding vector field $\mathbf{F}(\mathbf{x})$ in $V$.

Paper 1, Section I, $2 \mathrm{C}$

commentPrecisely one of the four matrices specified below is not orthogonal. Which is it?

Give a brief justification.

$\frac{1}{\sqrt{6}}\left(\begin{array}{rcc} 1 & -\sqrt{3} & \sqrt{2} \\ 1 & \sqrt{3} & \sqrt{2} \\ -2 & 0 & \sqrt{2} \end{array}\right) \quad \frac{1}{3}\left(\begin{array}{ccc} 1 & 2 & -2 \\ 2 & -2 & -1 \\ 2 & 1 & 2 \end{array}\right) \quad \frac{1}{\sqrt{6}}\left(\begin{array}{rrr} 1 & -2 & 1 \\ -\sqrt{6} & 0 & \sqrt{6} \\ 1 & 1 & 1 \end{array}\right) \quad \frac{1}{9}\left(\begin{array}{rrr} 7 & -4 & -4 \\ -4 & 1 & -8 \\ -4 & -8 & 1 \end{array}\right)$

Given that the four matrices represent transformations of $\mathbb{R}^{3}$ corresponding (in no particular order) to a rotation, a reflection, a combination of a rotation and a reflection, and none of these, identify each matrix. Explain your reasoning.

[Hint: For two of the matrices, $A$ and $B$ say, you may find it helpful to calculate $\operatorname{det}(A-I)$ and $\operatorname{det}(B-I)$, where $I$ is the identity matrix.]

Paper 1, Section I, B

comment(a) Describe geometrically the curve

$|\alpha z+\beta \bar{z}|=\sqrt{\alpha \beta}(z+\bar{z})+(\alpha-\beta)^{2},$

where $z \in \mathbb{C}$ and $\alpha, \beta$ are positive, distinct, real constants.

(b) Let $\theta$ be a real number not equal to an integer multiple of $2 \pi$. Show that

$\sum_{m=1}^{N} \sin (m \theta)=\frac{\sin \theta+\sin (N \theta)-\sin (N \theta+\theta)}{2(1-\cos \theta)}$

and derive a similar expression for $\sum_{m=1}^{N} \cos (m \theta)$.

Paper 1, Section II, $6 \mathrm{C}$

comment(i) Consider the map from $\mathbb{R}^{4}$ to $\mathbb{R}^{3}$ represented by the matrix

$\left(\begin{array}{rrrr} \alpha & 1 & 1 & -1 \\ 2 & -\alpha & 0 & -2 \\ -\alpha & 2 & 1 & 1 \end{array}\right)$

where $\alpha \in \mathbb{R}$. Find the image and kernel of the map for each value of $\alpha$.

(ii) Show that any linear map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ may be written in the form $f(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}$ for some fixed vector $\mathbf{a} \in \mathbb{R}^{n}$. Show further that $\mathbf{a}$ is uniquely determined by $f$.

It is given that $n=4$ and that the vectors

$\left(\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right),\left(\begin{array}{r} 2 \\ -1 \\ 0 \\ -2 \end{array}\right),\left(\begin{array}{r} -1 \\ 2 \\ 1 \\ 1 \end{array}\right)$

lie in the kernel of $f$. Determine the set of possible values of a.

Paper 1, Section II, 5B

comment(i) State and prove the Cauchy-Schwarz inequality for vectors in $\mathbb{R}^{n}$. Deduce the inequalities

$|\mathbf{a}+\mathbf{b}| \leqslant|\mathbf{a}|+|\mathbf{b}| \text { and }|\mathbf{a}+\mathbf{b}+\mathbf{c}| \leqslant|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|$

for $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^{n}$.

(ii) Show that every point on the intersection of the planes

$\mathbf{x} \cdot \mathbf{a}=A, \quad \mathbf{x} \cdot \mathbf{b}=B$

where $\mathbf{a} \neq \mathbf{b}$, satisfies

$|\mathbf{x}|^{2} \geqslant \frac{(A-B)^{2}}{|\mathbf{a}-\mathbf{b}|^{2}}$

What happens if $\mathbf{a}=\mathbf{b} ?$

(iii) Using your results from part (i), or otherwise, show that for any $\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{y}_{1}, \mathbf{y}_{2} \in \mathbb{R}^{n}$,

$\left|\mathbf{x}_{1}-\mathbf{y}_{1}\right|-\left|\mathbf{x}_{1}-\mathbf{y}_{2}\right| \leqslant\left|\mathbf{x}_{2}-\mathbf{y}_{1}\right|+\left|\mathbf{x}_{2}-\mathbf{y}_{2}\right|$

Paper 1, Section II, A

comment(a) A matrix is called normal if $A^{\dagger} A=A A^{\dagger}$. Let $A$ be a normal $n \times n$ complex matrix.

(i) Show that for any vector $\mathbf{x} \in \mathbb{C}^{n}$,

$|A \mathbf{x}|=\left|A^{\dagger} \mathbf{x}\right|$

(ii) Show that $A-\lambda I$ is also normal for any $\lambda \in \mathbb{C}$, where $I$ denotes the identity matrix.

(iii) Show that if $\mathbf{x}$ is an eigenvector of $A$ with respect to the eigenvalue $\lambda \in \mathbb{C}$, then $\mathbf{x}$ is also an eigenvector of $A^{\dagger}$, and determine the corresponding eigenvalue.

(iv) Show that if $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are eigenvectors of $A$ with respect to distinct eigenvalues $\lambda$ and $\mu$ respectively, then $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are orthogonal.

(v) Show that if $A$ has a basis of eigenvectors, then $A$ can be diagonalised using an orthonormal basis. Justify your answer.

[You may use standard results provided that they are clearly stated.]

(b) Show that any matrix $A$ satisfying $A^{\dagger}=A$ is normal, and deduce using results from (a) that its eigenvalues are real.

(c) Show that any matrix $A$ satisfying $A^{\dagger}=-A$ is normal, and deduce using results from (a) that its eigenvalues are purely imaginary.

(d) Show that any matrix $A$ satisfying $A^{\dagger}=A^{-1}$ is normal, and deduce using results from (a) that its eigenvalues have unit modulus.

Paper 1, Section II, A

comment(i) Find the eigenvalues and eigenvectors of the following matrices and show that both are diagonalisable:

$A=\left(\begin{array}{rrr} 1 & 1 & -1 \\ -1 & 3 & -1 \\ -1 & 1 & 1 \end{array}\right), \quad B=\left(\begin{array}{rcr} 1 & 4 & -3 \\ -4 & 10 & -4 \\ -3 & 4 & 1 \end{array}\right)$

(ii) Show that, if two real $n \times n$ matrices can both be diagonalised using the same basis transformation, then they commute.

(iii) Suppose now that two real $n \times n$ matrices $C$ and $D$ commute and that $D$ has $n$ distinct eigenvalues. Show that for any eigenvector $\mathbf{x}$ of $D$ the vector $C \mathbf{x}$ is a scalar multiple of $\mathbf{x}$. Deduce that there exists a common basis transformation that diagonalises both matrices.

(iv) Show that $A$ and $B$ satisfy the conditions in (iii) and find a matrix $S$ such that both of the matrices $S^{-1} A S$ and $S^{-1} B S$ are diagonal.