Part IA, 2004

# Part IA, 2004

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1.I.1B

commentThe linear map $H: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ represents reflection in the plane through the origin with normal $\mathbf{n}$, where $|\mathbf{n}|=1$, and $\mathbf{n}=\left(n_{1}, n_{2}, n_{3}\right)$ referred to the standard basis. The map is given by $\mathbf{x} \mapsto \mathbf{x}^{\prime}=\mathbf{M} \mathbf{x}$, where $\mathbf{M}$ is a $(3 \times 3)$ matrix.

Show that

$M_{i j}=\delta_{i j}-2 n_{i} n_{j}$

Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors such that $(\mathbf{u}, \mathbf{v}, \mathbf{n})$ is an orthonormal set. Show that

$\mathbf{M n}=-\mathbf{n}, \quad \mathbf{M u}=\mathbf{u}, \quad \mathbf{M v}=\mathbf{v}$

and find the matrix $\mathbf{N}$ which gives the mapping relative to the basis $(\mathbf{u}, \mathbf{v}, \mathbf{n})$.

1.I.2C

commentShow that

$\sum_{i=1}^{n} a_{i} b_{i} \leqslant\left(\sum_{i=1}^{n} a_{i}^{2}\right)^{1 / 2}\left(\sum_{i=1}^{n} b_{i}^{2}\right)^{1 / 2}$

for any real numbers $a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{n}$. Using this inequality, show that if $\mathbf{a}$ and $\mathbf{b}$ are vectors of unit length in $\mathbb{R}^{n}$ then $|\mathbf{a} \cdot \mathbf{b}| \leqslant 1$.

1.II.5B

commentThe vector $\mathbf{x}=\left(\begin{array}{l}x \\ y \\ z\end{array}\right)$ satisfies the equation

$\mathbf{A} \mathbf{x}=\mathbf{b}$

where $\mathbf{A}$ is a $(3 \times 3)$ matrix and $\mathbf{b}$ is a $(3 \times 1)$ column vector. State the conditions under which this equation has (a) a unique solution, (b) an infinity of solutions, (c) no solution for $\mathbf{x}$.

Find all possible solutions for the unknowns $x, y$ and $z$ which satisfy the following equations:

$\begin{array}{r} x+y+z=1 \\ x+y+\lambda z=2 \\ x+2 y+\lambda z=4 \end{array}$

in the cases (a) $\lambda=0$, and (b) $\lambda=1$.

1.II.6A

commentExpress the product $\mathbf{a} \cdot(\mathbf{a} \times \mathbf{b})$ in suffix notation and thence prove that the result is zero.

Silver Beard the space pirate believed people relied so much on space-age navigation techniques that he could safely write down the location of his treasure using the ancient art of vector algebra. Spikey the space jockey thought he could follow the instructions, by moving by the sequence of vectors $\mathbf{a}, \mathbf{b}, \ldots, \mathbf{f}$ one stage at a time. The vectors (expressed in 1000 parsec units) were defined as follows:

Start at the centre of the galaxy, which has coordinates $(0,0,0)$.

Vector a has length $\sqrt{3}$, is normal to the plane $x+y+z=1$ and is directed into the positive quadrant.

Vector $\mathbf{b}$ is given by $\mathbf{b}=(\mathbf{a} \cdot \mathbf{m}) \mathbf{a} \times \mathbf{m}$, where $\mathbf{m}=(2,0,1)$.

Vector $\mathbf{c}$ has length $2 \sqrt{2}$, is normal to $\mathbf{a}$ and $\mathbf{b}$, and moves you closer to the $x$ axis.

Vector $\mathbf{d}=(1,-2,2)$.

Vector $\mathbf{e}$ has length $\mathbf{a} \cdot \mathbf{b}$. Spikey was initially a little confused with this one, but then realised the orientation of the vector did not matter.

Vector $\mathbf{f}$ has unknown length but is parallel to $\mathbf{m}$ and takes you to the treasure located somewhere on the plane $2 x-y+4 z=10$.

Determine the location of the way-points Spikey will use and thence the location of the treasure.

1.II.7A

commentSimplify the fraction

$\zeta=\frac{1}{\bar{z}+\frac{1}{z+\frac{1}{\bar{z}}}},$

where $\bar{z}$ is the complex conjugate of $z$. Determine the geometric form that satisfies

$\operatorname{Re}(\zeta)=\operatorname{Re}\left(\frac{z+\frac{1}{4}}{|z|^{2}}\right)$

Find solutions to

$\operatorname{Im}(\log z)=\frac{\pi}{3}$

and

$z^{2}=x^{2}-y^{2}+2 i x,$

where $z=x+i y$ is a complex variable. Sketch these solutions in the complex plane and describe the region they enclose. Derive complex equations for the circumscribed and inscribed circles for the region. [The circumscribed circle is the circle that passes through the vertices of the region and the inscribed circle is the largest circle that fits within the region.]

1.II.8C

comment(i) The vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$ in $\mathbb{R}^{3}$ satisfy $\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3} \neq 0$. Are $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$ necessarily linearly independent? Justify your answer by a proof or a counterexample.

(ii) The vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}$ in $\mathbb{R}^{n}$ have the property that every subset comprising $(n-1)$ of the vectors is linearly independent. Are $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}$ necessarily linearly independent? Justify your answer by a proof or a counterexample.

(iii) For each value of $t$ in the range $0 \leqslant t<1$, give a construction of a linearly independent set of vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$ in $\mathbb{R}^{3}$ satisfying

$\mathbf{a}_{i} \cdot \mathbf{a}_{j}=\delta_{i j}+t\left(1-\delta_{i j}\right),$

where $\delta_{i j}$ is the Kronecker delta.

3.I.1D

commentState Lagrange's Theorem.

Show that there are precisely two non-isomorphic groups of order 10 . [You may assume that a group whose elements are all of order 1 or 2 has order $2^{k}$.]

3.I.2D

commentDefine the Möbius group, and describe how it acts on $\mathbb{C} \cup\{\infty\}$.

Show that the subgroup of the Möbius group consisting of transformations which fix 0 and $\infty$ is isomorphic to $\mathbb{C}^{*}=\mathbb{C} \backslash\{0\}$.

Now show that the subgroup of the Möbius group consisting of transformations which fix 0 and 1 is also isomorphic to $\mathbb{C}^{*}$.

3.II.5D

commentLet $G=\left\langle g, h \mid h^{2}=1, g^{6}=1, h g h^{-1}=g^{-1}\right\rangle$ be the dihedral group of order 12 .

i) List all the subgroups of $G$ of order 2 . Which of them are normal?

ii) Now list all the remaining proper subgroups of $G$. [There are $6+3$ of them.]

iii) For each proper normal subgroup $N$ of $G$, describe the quotient group $G / N$.

iv) Show that $G$ is not isomorphic to the alternating group $A_{4}$.

3.II.6D

commentState the conditions on a matrix $A$ that ensure it represents a rotation of $\mathbb{R}^{3}$ with respect to the standard basis.

Check that the matrix

$A=\frac{1}{3}\left(\begin{array}{ccc} -1 & 2 & -2 \\ 2 & 2 & 1 \\ 2 & -1 & -2 \end{array}\right)$

represents a rotation. Find its axis of rotation $\mathbf{n}$.

Let $\Pi$ be the plane perpendicular to the axis $\mathbf{n}$. The matrix $A$ induces a rotation of $\Pi$ by an angle $\theta$. Find $\cos \theta$.

3.II.7D

commentLet $A$ be a real symmetric matrix. Show that all the eigenvalues of $A$ are real, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal to each other.

Find the eigenvalues and eigenvectors of

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

Give an example of a non-zero complex $(2 \times 2)$ symmetric matrix whose only eigenvalues are zero. Is it diagonalisable?

3.II.8D

commentCompute the characteristic polynomial of

$A=\left(\begin{array}{ccc} 3 & -1 & 2 \\ 0 & 4-s & 2 s-2 \\ 0 & -2 s+2 & 4 s-1 \end{array}\right)$

Find the eigenvalues and eigenvectors of $A$ for all values of $s$.

For which values of $s$ is $A$ diagonalisable?

1.I.3D

commentDefine the supremum or least upper bound of a non-empty set of real numbers.

State the Least Upper Bound Axiom for the real numbers.

Starting from the Least Upper Bound Axiom, show that if $\left(a_{n}\right)$ is a bounded monotonic sequence of real numbers, then it converges.

1.I.4E

commentLet $f(x)=(1+x)^{1 / 2}$ for $x \in(-1,1)$. Show by induction or otherwise that for every integer $r \geq 1$,

$f^{(r)}(x)=(-1)^{r-1} \frac{(2 r-2) !}{2^{2 r-1}(r-1) !}(1+x)^{\frac{1}{2}-r}$

Evaluate the series

$\sum_{r=1}^{\infty}(-1)^{r-1} \frac{(2 r-2) !}{8^{r} r !(r-1) !}$

[You may use Taylor's Theorem in the form

$f(x)=f(0)+\sum_{r=1}^{n} \frac{f^{(r)}(0)}{r !} x^{r}+\int_{0}^{x} \frac{(x-t)^{n} f^{(n+1)}(t)}{n !} d t$

without proof.]

1.II.10E

commentDefine, for an integer $n \geq 0$,

$I_{n}=\int_{0}^{\pi / 2} \sin ^{n} x d x$

Show that for every $n \geq 2, n I_{n}=(n-1) I_{n-2}$, and deduce that

$I_{2 n}=\frac{(2 n) !}{\left(2^{n} n !\right)^{2}} \frac{\pi}{2} \quad \text { and } \quad I_{2 n+1}=\frac{\left(2^{n} n !\right)^{2}}{(2 n+1) !}$

Show that $0<I_{n}<I_{n-1}$, and that

$\frac{2 n}{2 n+1}<\frac{I_{2 n+1}}{I_{2 n}}<1$

Hence prove that

$\lim _{n \rightarrow \infty} \frac{2^{4 n+1}(n !)^{4}}{(2 n+1)(2 n) !^{2}}=\pi .$

1.II.11F

commentLet $f$ be defined on $\mathbb{R}$, and assume that there exists at least one point $x_{0} \in \mathbb{R}$ at which $f$ is continuous. Suppose also that, for every $x, y \in \mathbb{R}, f$ satisfies the equation

$f(x+y)=f(x)+f(y)$

Show that $f$ is continuous on $\mathbb{R}$.

Show that there exists a constant $c$ such that $f(x)=c x$ for all $x \in \mathbb{R}$.

Suppose that $g$ is a continuous function defined on $\mathbb{R}$ and that, for every $x, y \in \mathbb{R}$, $g$ satisfies the equation

$g(x+y)=g(x) g(y) .$

Show that if $g$ is not identically zero, then $g$ is everywhere positive. Find the general form of $g$.

1.II.12F

comment(i) Show that if $a_{n}>0, b_{n}>0$ and

$\frac{a_{n+1}}{a_{n}} \leqslant \frac{b_{n+1}}{b_{n}}$

for all $n \geqslant 1$, and if $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ converges.

(ii) Let

$c_{n}=\left(\begin{array}{c} 2 n \\ n \end{array}\right) 4^{-n} .$

By considering $\log c_{n}$, or otherwise, show that $c_{n} \rightarrow 0$ as $n \rightarrow \infty$.

[Hint: $\log (1-x) \leqslant-x$ for $x \in(0,1)$.]

(iii) Determine the convergence or otherwise of

$\sum_{n=1}^{\infty}\left(\begin{array}{c} 2 n \\ n \end{array}\right) x^{n}$

for (a) $x=\frac{1}{4}$, (b) $x=-\frac{1}{4}$.

1.II.9D

commenti) State Rolle's theorem.

Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous functions which are differentiable on $(a, b)$.

ii) Prove that for some $c \in(a, b)$,

$(f(b)-f(a)) g^{\prime}(c)=(g(b)-g(a)) f^{\prime}(c) .$

iii) Suppose that $f(a)=g(a)=0$, and that $\lim _{x \rightarrow a+} \frac{f^{\prime}(x)}{g^{\prime}(x)}$ exists and is equal to $L$.

Prove that $\lim _{x \rightarrow a+} \frac{f(x)}{g(x)}$ exists and is also equal to $L$.

[You may assume there exists a $\delta>0$ such that, for all $x \in(a, a+\delta), g^{\prime}(x) \neq 0$ and $g(x) \neq 0 .]$

iv) Evaluate $\lim _{x \rightarrow 0} \frac{\log \cos x}{x^{2}}$.

2.I.1B

commentBy writing $y(x)=m x$ where $m$ is a constant, solve the differential equation

$\frac{d y}{d x}=\frac{x-2 y}{2 x+y}$

and find the possible values of $m$.

Describe the isoclines of this differential equation and sketch the flow vectors. Use these to sketch at least two characteristically different solution curves.

Now, by making the substitution $y(x)=x u(x)$ or otherwise, find the solution of the differential equation which satisfies $y(0)=1$.

2.I.2B

commentFind two linearly independent solutions of the differential equation

$\frac{d^{2} y}{d x^{2}}+2 p \frac{d y}{d x}+p^{2} y=0$

Find also the solution of

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

that satisfies

$y=0, \frac{d y}{d x}=0 \quad \text { at } x=0$

2.II.5B

commentConstruct a series solution $y=y_{1}(x)$ valid in the neighbourhood of $x=0$, for the differential equation

$\frac{d^{2} y}{d x^{2}}+4 x^{3} \frac{d y}{d x}+x^{2} y=0$

satisfying

$y_{1}=1, \frac{d y_{1}}{d x}=0 \quad \text { at } x=0 .$

Find also a second solution $y=y_{2}(x)$ which satisfies

$y_{2}=0, \frac{d y_{2}}{d x}=1 \quad \text { at } x=0 .$

Obtain an expression for the Wronskian of these two solutions and show that

$y_{2}(x)=y_{1}(x) \int_{0}^{x} \frac{e^{-\xi^{4}}}{y_{1}^{2}(\xi)} d \xi$

2.II.6B

commentTwo solutions of the recurrence relation

$x_{n+2}+b(n) x_{n+1}+c(n) x_{n}=0$

are given as $p_{n}$ and $q_{n}$, and their Wronskian is defined to be

$W_{n}=p_{n} q_{n+1}-p_{n+1} q_{n}$

Show that

$W_{n+1}=W_{1} \prod_{m=1}^{n} c(m)$

Suppose that $b(n)=\alpha$, where $\alpha$ is a real constant lying in the range $[-2,2]$, and that $c(n)=1$. Show that two solutions are $x_{n}=e^{i n \theta}$ and $x_{n}=e^{-i n \theta}$, where $\cos \theta=-\alpha / 2$. Evaluate the Wronskian of these two solutions and verify $(*)$.

2.II.7B

commentShow how a second-order differential equation $\ddot{x}=f(x, \dot{x})$ may be transformed into a pair of coupled first-order equations. Explain what is meant by a critical point on the phase diagram for a pair of first-order equations. Hence find the critical points of the following equations. Describe their stability type, sketching their behaviour near the critical points on a phase diagram.

$\begin{aligned} &\text { (i) } \ddot{x}+\cos x=0 \\ &\text { (ii) } \ddot{x}+x\left(x^{2}+\lambda x+1\right)=0, \quad \text { for } \lambda=1,5 / 2 . \end{aligned}$

Sketch the phase portraits of these equations marking clearly the direction of flow.

2.II.8B

commentConstruct the general solution of the system of equations

$\begin{aligned} &\dot{x}+4 x+3 y=0 \\ &\dot{y}+4 y-3 x=0 \end{aligned}$

in the form

$\left(\begin{array}{l} x(t) \\ y(t) \end{array}\right)=\mathbf{x}=\sum_{j=1}^{2} a_{j} \mathbf{x}^{(j)} e^{\lambda_{j} t}$

and find the eigenvectors $\mathbf{x}^{(j)}$ and eigenvalues $\lambda_{j}$.

Explain what is meant by resonance in a forced system of linear differential equations.

Consider the forced system

$\begin{aligned} &\dot{x}+4 x+3 y=\sum_{j=1}^{2} p_{j} e^{\lambda_{j} t} \\ &\dot{y}+4 y-3 x=\sum_{j=1}^{2} q_{j} e^{\lambda_{j} t} \end{aligned}$

Find conditions on $p_{j}$ and $q_{j}(j=1,2)$ such that there is no resonant response to the forcing.

4.I.3A

commentA lecturer driving his car of mass $m_{1}$ along the flat at speed $U_{1}$ accidentally collides with a stationary vehicle of mass $m_{2}$. As both vehicles are old and very solidly built, neither suffers damage in the collision: they simply bounce elastically off each other in a straight line. Determine how both vehicles are moving after the collision if neither driver applied their brakes. State any assumptions made and consider all possible values of the mass ratio $R=m_{1} / m_{2}$. You may neglect friction and other such losses.

An undergraduate drives into a rigid rock wall at speed $V$. The undergraduate's car of mass $M$ is modern and has a crumple zone of length $L$ at its front. As this zone crumples upon impact, it exerts a net force $F=(L-y)^{-1 / 2}$ on the car, where $y$ is the amount the zone has crumpled. Determine the value of $y$ at the point the car stops moving forwards as a function of $V$, where $V<2 L^{\frac{1}{4}} / M^{\frac{1}{2}}$.

4.I.4A

commentA small spherical bubble of radius a containing carbon dioxide rises in water due to a buoyancy force $\rho g V$, where $\rho$ is the density of water, $g$ is gravitational attraction and $V$ is the volume of the bubble. The drag on a bubble moving at speed $u$ is $6 \pi \mu a u$, where $\mu$ is the dynamic viscosity of water, and an accelerating bubble acts like a particle of mass $\alpha \rho V$, for some constant $\alpha$. Find the location at time $t$ of a bubble released from rest at $t=0$ and show the bubble approaches a steady rise speed

$U=\frac{2}{9} \frac{\rho g}{\mu} a^{2}$

Under some circumstances the carbon dioxide gradually dissolves in the water, which leads to the bubble radius varying as $a^{2}=a_{0}^{2}-\beta t$, where $a_{0}$ is the bubble radius at $t=0$ and $\beta$ is a constant. Under the assumption that the bubble rises at speed given by $(*)$, determine the height to which it rises before it disappears.

4.II.10A

commentA small probe of mass $m$ is in low orbit about a planet of mass $M$. If there is no drag on the probe then its orbit is governed by

$\ddot{\mathbf{r}}=-\frac{G M}{|\mathbf{r}|^{3}} \mathbf{r}$

where $\mathbf{r}$ is the location of the probe relative to the centre of the planet and $G$ is the gravitational constant. Show that the basic orbital trajectory is elliptical. Determine the orbital period for the probe if it is in a circular orbit at a distance $r_{0}$ from the centre of the planet.

Data returned by the probe shows that the planet has a very extensive but diffuse atmosphere. This atmosphere induces a drag on the probe that may be approximated by the linear law $\mathbf{D}=-A \dot{\mathbf{r}}$, where $\mathbf{D}$ is the drag force and $A$ is a constant. Show that the angular momentum of the probe about the planet decays exponentially.

4.II.11A

commentA particle of mass $m$ and charge $q$ moves through a magnetic field $\mathbf{B}$. There is no electric field or external force so that the particle obeys

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

where $\mathbf{r}$ is the location of the particle. Prove that the kinetic energy of the particle is preserved.

Consider an axisymmetric magnetic field described by $\mathbf{B}=(0,0, B(r))$ in cylindrical polar coordinates $\mathbf{r}=(r, \theta, z)$. Determine the angular velocity of a circular orbit centred on $\mathbf{r}=\mathbf{0}$.

For a general orbit when $B(r)=B_{0} / r$, show that the angular momentum about the $z$-axis varies as $L=L_{0}-q B_{0}\left(r-r_{0}\right)$, where $L_{0}$ is the angular momentum at radius $r_{0}$. Determine and sketch the relationship between $\dot{r}^{2}$ and $r$. [Hint: Use conservation of energy.] What is the escape velocity for the particle?

4.II.12A

commentA circular cylinder of radius $a$, length $L$ and mass $m$ is rolling along a surface. Show that its moment of inertia is given by $\frac{1}{2} m a^{2}$.

At $t=0$ the cylinder is at the bottom of a slope making an angle $\alpha$ to the horizontal, and is rolling with velocity $V$ and angular velocity $V / a$. Assuming slippage does not occur, determine the position of the cylinder as a function of time. What is the maximum height that the cylinder reaches?

The frictional force between the cylinder and surface is given by $\mu m g \cos \alpha$, where $\mu$ is the friction coefficient. Show that the cylinder begins to slip rather than roll if $\tan \alpha>3 \mu$. Determine as a function of time the location, speed and angular velocity of the cylinder on the slope if this condition is satisfied. Show that slipping continues as the cylinder ascends and descends the slope. Find also the maximum height the cylinder reaches, and its speed and angular velocity when it returns to the bottom of the slope.

4.II.9A

commentA horizontal table oscillates with a displacement $\mathbf{A} \sin \omega t$, where $\mathbf{A}=\left(A_{x}, 0, A_{z}\right)$ is the amplitude vector and $\omega$ the angular frequency in an inertial frame of reference with the $z$ axis vertically upwards, normal to the table. A block sitting on the table has mass $m$ and linear friction that results in a force $\mathbf{F}=-\lambda \mathbf{u}$, where $\lambda$ is a constant and $\mathbf{u}$ is the velocity difference between the block and the table. Derive the equations of motion for this block in the frame of reference of the table using axes $(\xi, \eta, \zeta)$ on the table parallel to the axes $(x, y, z)$ in the inertial frame.

For the case where $A_{z}=0$, show that at late time the block will approach the steady orbit

$\xi=\xi_{0}-A_{x} \sin \theta \cos (\omega t-\theta),$

where

$\sin ^{2} \theta=\frac{m^{2} \omega^{2}}{\lambda^{2}+m^{2} \omega^{2}}$

and $\xi_{0}$ is a constant.

Given that there are no attractive forces between block and table, show that the block will only remain in contact with the table if $\omega^{2} A_{z}<g$.

4.I.1E

comment(a) Use Euclid's algorithm to find positive integers $m, n$ such that $79 m-100 n=1$.

(b) Determine all integer solutions of the congruence

$237 x \equiv 21(\bmod 300)$

(c) Find the set of all integers $x$ satisfying the simultaneous congruences

$\begin{aligned} &x \equiv 8(\bmod 79) \\ &x \equiv 11(\bmod 100) \end{aligned}$

4.I.2E

commentProve by induction the following statements:

i) For every integer $n \geq 1$,

$1^{2}+3^{2}+\cdots+(2 n-1)^{2}=\frac{1}{3}\left(4 n^{3}-n\right)$

ii) For every integer $n \geq 1, n^{3}+5 n$ is divisible by 6 .

4.II.5E

commentShow that the set of all subsets of $\mathbb{N}$ is uncountable, and that the set of all finite subsets of $\mathbb{N}$ is countable.

Let $X$ be the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$, and let $Y \subset X$ be the set

$Y=\{f \in X \mid \text { for all but finitely many } n \in \mathbb{N}, f(n)=n\}$

Show that $X$ is uncountable, but that $Y$ is countable.

4.II.6E

commentProve Fermat's Theorem: if $p$ is prime and $(x, p)=1$ then $x^{p-1} \equiv 1(\bmod p)$.

Let $n$ and $x$ be positive integers with $(x, n)=1$. Show that if $n=m p$ where $p$ is prime and $(m, p)=1$, then

$x^{n-1} \equiv 1 \quad(\bmod p) \quad \text { if and only if } \quad x^{m-1} \equiv 1 \quad(\bmod p)$

Now assume that $n$ is a product of distinct primes. Show that $x^{n-1} \equiv 1(\bmod n)$ if and only if, for every prime divisor $p$ of $n$,

$x^{(n / p)-1} \equiv 1 \quad(\bmod p) .$

Deduce that if every prime divisor $p$ of $n$ satisfies $(p-1) \mid(n-1)$, then for every $x$ with $(x, n)=1$, the congruence

$x^{n-1} \equiv 1 \quad(\bmod n)$

holds.

4.II.7E

commentPolynomials $P_{r}(X)$ for $r \geq 0$ are defined by

$\begin{aligned} &P_{0}(X)=1 \\ &P_{r}(X)=\frac{X(X-1) \cdots(X-r+1)}{r !}=\prod_{i=1}^{r} \frac{X-i+1}{i} \quad \text { for } r \geq 1 \end{aligned}$

Show that $P_{r}(n) \in \mathbb{Z}$ for every $n \in \mathbb{Z}$, and that if $r \geq 1$ then $P_{r}(X)-P_{r}(X-1)=$ $P_{r-1}(X-1)$.

Prove that if $F$ is any polynomial of degree $d$ with rational coefficients, then there are unique rational numbers $c_{r}(F)(0 \leq r \leq d)$ for which

$F(X)=\sum_{r=0}^{d} c_{r}(F) P_{r}(X)$

Let $\Delta F(X)=F(X+1)-F(X)$. Show that

$\Delta F(X)=\sum_{r=0}^{d-1} c_{r+1}(F) P_{r}(X)$

Show also that, if $F$ and $G$ are polynomials such that $\Delta F=\Delta G$, then $F-G$ is a constant.

By induction on the degree of $F$, or otherwise, show that if $F(n) \in \mathbb{Z}$ for every $n \in \mathbb{Z}$, then $c_{r}(F) \in \mathbb{Z}$ for all $r$.

4.II.8E

commentLet $X$ be a finite set, $X_{1}, \ldots, X_{m}$ subsets of $X$ and $Y=X \backslash \bigcup X_{i}$. Let $g_{i}$ be the characteristic function of $X_{i}$, so that

$g_{i}(x)= \begin{cases}1 & \text { if } x \in X_{i} \\ 0 & \text { otherwise }\end{cases}$

Let $f: X \rightarrow \mathbb{R}$ be any function. By considering the expression

$\sum_{x \in X} f(x) \prod_{i=1}^{m}\left(1-g_{i}(x)\right)$

or otherwise, prove the Inclusion-Exclusion Principle in the form

$\sum_{x \in Y} f(x)=\sum_{r \geq 0}(-1)^{r} \sum_{i_{1}<\cdots<i_{r}}\left(\sum_{x \in X_{i_{1}} \cap \cdots \cap X_{i_{r}}} f(x)\right)$

Let $n>1$ be an integer. For an integer $m$ dividing $n$ let

$X_{m}=\{0 \leq x<n \mid x \equiv 0(\bmod m)\}$

By considering the sets $X_{p}$ for prime divisors $p$ of $n$, show that

$\phi(n)=n \prod_{p \mid n}\left(1-\frac{1}{p}\right)$

(where $\phi$ is Euler's function) and

$\sum_{\substack{0<x<n \\(x, n)=1}} x=\frac{n^{2}}{2} \prod_{p \mid n}\left(1-\frac{1}{p}\right) .$

2.I.3F

commentDefine the covariance, $\operatorname{cov}(X, Y)$, of two random variables $X$ and $Y$.

Prove, or give a counterexample to, each of the following statements.

(a) For any random variables $X, Y, Z$

$\operatorname{cov}(X+Y, Z)=\operatorname{cov}(X, Z)+\operatorname{cov}(Y, Z)$

(b) If $X$ and $Y$ are identically distributed, not necessarily independent, random variables then

$\operatorname{cov}(X+Y, X-Y)=0$

2.I.4F

commentThe random variable $X$ has probability density function

$f(x)= \begin{cases}c x(1-x) & \text { if } 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise }\end{cases}$

Determine $c$, and the mean and variance of $X$.

2.II.10F

commentDefine the conditional probability $P(A \mid B)$ of the event $A$ given the event $B$.

A bag contains four coins, each of which when tossed is equally likely to land on either of its two faces. One of the coins shows a head on each of its two sides, while each of the other three coins shows a head on only one side. A coin is chosen at random, and tossed three times in succession. If heads turn up each time, what is the probability that if the coin is tossed once more it will turn up heads again? Describe the sample space you use and explain carefully your calculations.

2.II.11F

commentThe random variables $X_{1}$ and $X_{2}$ are independent, and each has an exponential distribution with parameter $\lambda$. Find the joint density function of

$Y_{1}=X_{1}+X_{2}, \quad Y_{2}=X_{1} / X_{2}$

and show that $Y_{1}$ and $Y_{2}$ are independent. What is the density of $Y_{2}$ ?

2.II.12F

commentLet $A_{1}, A_{2}, \ldots, A_{r}$ be events such that $A_{i} \cap A_{j}=\emptyset$ for $i \neq j$. Show that the number $N$ of events that occur satisfies

$P(N=0)=1-\sum_{i=1}^{r} P\left(A_{i}\right)$

Planet Zog is a sphere with centre $O$. A number $N$ of spaceships land at random on its surface, their positions being independent, each uniformly distributed over the surface. A spaceship at $A$ is in direct radio contact with another point $B$ on the surface if $\angle A O B<\frac{\pi}{2}$. Calculate the probability that every point on the surface of the planet is in direct radio contact with at least one of the $N$ spaceships.

[Hint: The intersection of the surface of a sphere with a plane through the centre of the sphere is called a great circle. You may find it helpful to use the fact that $N$ random great circles partition the surface of a sphere into $N(N-1)+2$ disjoint regions with probability one.]

2.II.9F

commentLet $X$ be a positive-integer valued random variable. Define its probability generating function $p_{X}$. Show that if $X$ and $Y$ are independent positive-integer valued random variables, then $p_{X+Y}=p_{X} p_{Y}$.

A non-standard pair of dice is a pair of six-sided unbiased dice whose faces are numbered with strictly positive integers in a non-standard way (for example, $(2,2,2,3,5,7$ ) and $(1,1,5,6,7,8))$. Show that there exists a non-standard pair of dice $A$ and $B$ such that when thrown

$P\{$ total shown by $A$ and $B$ is $n\}=P\{$ total shown by pair of ordinary dice is $n\}$

for all $2 \leqslant n \leqslant 12$.

[Hint: $\left.\left(x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}\right)=x(1+x)\left(1+x^{2}+x^{4}\right)=x\left(1+x+x^{2}\right)\left(1+x^{3}\right) .\right]$

3.I.3C

commentIf $\mathbf{F}$ and $\mathbf{G}$ are differentiable vector fields, show that

(i) $\boldsymbol{\nabla} \times(\mathbf{F} \times \mathbf{G})=\mathbf{F}(\boldsymbol{\nabla} \cdot \mathbf{G})-\mathbf{G}(\boldsymbol{\nabla} \cdot \mathbf{F})+(\mathbf{G} \cdot \boldsymbol{\nabla}) \mathbf{F}-(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{G}$,

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

3.I.4C

commentDefine the curvature, $\kappa$, of a curve in $\mathbb{R}^{3}$.

The curve $C$ is parametrised by

$\mathbf{x}(t)=\left(\frac{1}{2} e^{t} \cos t, \frac{1}{2} e^{t} \sin t, \frac{1}{\sqrt{2}} e^{t}\right) \quad \text { for }-\infty<t<\infty$

Obtain a parametrisation of the curve in terms of its arc length, $s$, measured from the origin. Hence obtain its curvature, $\kappa(s)$, as a function of $s$.

3.II.10C

commentExplain what is meant by an exact differential. The three-dimensional vector field $\mathbf{F}$ is defined by

$\mathbf{F}=\left(e^{x} z^{3}+3 x^{2}\left(e^{y}-e^{z}\right), e^{y}\left(x^{3}-z^{3}\right), 3 z^{2}\left(e^{x}-e^{y}\right)-e^{z} x^{3}\right)$

Find the most general function that has $\mathbf{F} \cdot \mathbf{d} \mathbf{x}$ as its differential.

Hence show that the line integral

$\int_{P_{1}}^{P_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$

along any path in $\mathbb{R}^{3}$ between points $P_{1}=(0, a, 0)$ and $P_{2}=(b, b, b)$ vanishes for any values of $a$ and $b$.

The two-dimensional vector field $\mathbf{G}$ is defined at all points in $\mathbb{R}^{2}$ except $(0,0)$ by

$\mathbf{G}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right)$

$(\mathbf{G}$ is not defined at $(0,0)$.) Show that

$\oint_{C} \mathbf{G} \cdot \mathbf{d} \mathbf{x}=2 \pi$

for any closed curve $C$ in $\mathbb{R}^{2}$ that goes around $(0,0)$ anticlockwise precisely once without passing through $(0,0)$.

3.II.11C

commentLet $S_{1}$ be the 3 -dimensional sphere of radius 1 centred at $(0,0,0), S_{2}$ be the sphere of radius $\frac{1}{2}$ centred at $\left(\frac{1}{2}, 0,0\right)$ and $S_{3}$ be the sphere of radius $\frac{1}{4}$ centred at $\left(\frac{-1}{4}, 0,0\right)$. The eccentrically shaped planet Zog is composed of rock of uniform density $\rho$ occupying the region within $S_{1}$ and outside $S_{2}$ and $S_{3}$. The regions inside $S_{2}$ and $S_{3}$ are empty. Give an expression for Zog's gravitational potential at a general coordinate $\mathbf{x}$ that is outside $S_{1}$. Is there a point in the interior of $S_{3}$ where a test particle would remain stably at rest? Justify your answer.

3.II.12C

commentState (without proof) the divergence theorem for a vector field $\mathbf{F}$ with continuous first-order partial derivatives throughout a volume $V$ enclosed by a bounded oriented piecewise-smooth non-self-intersecting surface $S$.

By calculating the relevant volume and surface integrals explicitly, verify the divergence theorem for the vector field

$\mathbf{F}=\left(x^{3}+2 x y^{2}, y^{3}+2 y z^{2}, z^{3}+2 z x^{2}\right)$

defined within a sphere of radius $R$ centred at the origin.

Suppose that functions $\phi, \psi$ are continuous and that their first and second partial derivatives are all also continuous in a region $V$ bounded by a smooth surface $S$.

Show that

$\begin{aligned} \int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S} \\ \int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d \tau &=\int_{S} \phi \boldsymbol{\nabla} \psi \cdot \mathbf{d} \mathbf{S}-\int_{S} \psi \boldsymbol{\nabla} \phi \cdot \mathbf{d} \mathbf{S} \end{aligned}$

Hence show that if $\rho(\mathbf{x})$ is a continuous function on $V$ and $g(\mathbf{x})$ a continuous function on $S$ and $\phi_{1}$ and $\phi_{2}$ are two continuous functions such that

$\begin{aligned} \nabla^{2} \phi_{1}(\mathbf{x}) &=\nabla^{2} \phi_{2}(\mathbf{x})=\rho(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { in } V, \text { and } \\ \phi_{1}(\mathbf{x}) &=\phi_{2}(\mathbf{x})=g(\mathbf{x}) \quad \text { for all } \mathbf{x} \text { on } S \end{aligned}$

then $\phi_{1}(\mathbf{x})=\phi_{2}(\mathbf{x})$ for all $\mathbf{x}$ in $V$.

3.II.9C

commentFor a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ state if the following implications are true or false. (No justification is required.)

(i) $f$ is differentiable $\Rightarrow f$ is continuous.

(ii) $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist $\Rightarrow f$ is continuous.

(iii) directional derivatives $\frac{\partial f}{\partial \mathbf{n}}$ exist for all unit vectors $\mathbf{n} \in \mathbb{R}^{2} \Rightarrow f$ is differentiable.

(iv) $f$ is differentiable $\Rightarrow \frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are continuous.

(v) all second order partial derivatives of $f$ exist $\Rightarrow \frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial^{2} f}{\partial y \partial x}$.

Now let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be defined by

$f(x, y)= \begin{cases}\frac{x y\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

Show that $f$ is continuous at $(0,0)$ and find the partial derivatives $\frac{\partial f}{\partial x}(0, y)$ and $\frac{\partial f}{\partial y}(x, 0)$. Then show that $f$ is differentiable at $(0,0)$ and find its derivative. Investigate whether the second order partial derivatives $\frac{\partial^{2} f}{\partial x \partial y}(0,0)$ and $\frac{\partial^{2} f}{\partial y \partial x}(0,0)$ are the same. Are the second order partial derivatives of $f$ at $(0,0)$ continuous? Justify your answer.