Part IA, 2017

Analysis I

• Paper 1, Section I, $4 \mathrm{E}$

  Analysis I | Part IA, 2017

  Show that if the power series $\sum_{n=0}^{\infty} a_{n} z^{n}(z \in \mathbb{C})$ converges for some fixed $z=z_{0}$, then it converges absolutely for every $z$ satisfying $|z|<\left|z_{0}\right|$.

  Define the radius of convergence of a power series.

  Give an example of $v \in \mathbb{C}$ and an example of $w \in \mathbb{C}$ such that $|v|=|w|=1, \sum_{n=1}^{\infty} \frac{v^{n}}{n}$ converges and $\sum_{n=1}^{\infty} \frac{w^{n}}{n}$ diverges. [You may assume results about standard series without proof.] Use this to find the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{z^{n}}{n}$.

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• Paper 1, Section I, F

  Analysis I | Part IA, 2017

  Given an increasing sequence of non-negative real numbers $\left(a_{n}\right)_{n=1}^{\infty}$, let

  $$s_{n}=\frac{1}{n} \sum_{k=1}^{n} a_{k}$$

  Prove that if $s_{n} \rightarrow x$ as $n \rightarrow \infty$ for some $x \in \mathbb{R}$ then also $a_{n} \rightarrow x$ as $n \rightarrow \infty$

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• Paper 1, Section II, D

  Analysis I | Part IA, 2017

  Let $a, b \in \mathbb{R}$ with $a<b$ and let $f:(a, b) \rightarrow \mathbb{R}$.

  (a) Define what it means for $f$ to be continuous at $y_{0} \in(a, b)$.

  $f$ is said to have a local minimum at $c \in(a, b)$ if there is some $\varepsilon>0$ such that $f(c) \leqslant f(x)$ whenever $x \in(a, b)$ and $|x-c|<\varepsilon$.

  If $f$ has a local minimum at $c \in(a, b)$ and $f$ is differentiable at $c$, show that $f^{\prime}(c)=0$.

  (b) $f$ is said to be convex if

  $$f(\lambda x+(1-\lambda) y) \leqslant \lambda f(x)+(1-\lambda) f(y)$$

  for every $x, y \in(a, b)$ and $\lambda \in[0,1]$. If $f$ is convex, $r \in \mathbb{R}$ and $\left[y_{0}-|r|, y_{0}+|r|\right] \subset(a, b)$, prove that

  $$(1+\lambda) f\left(y_{0}\right)-\lambda f\left(y_{0}-r\right) \leqslant f\left(y_{0}+\lambda r\right) \leqslant(1-\lambda) f\left(y_{0}\right)+\lambda f\left(y_{0}+r\right)$$

  for every $\lambda \in[0,1]$.

  Deduce that if $f$ is convex then $f$ is continuous.

  If $f$ is convex and has a local minimum at $c \in(a, b)$, prove that $f$ has a global minimum at $c$, i.e., that $f(x) \geqslant f(c)$ for every $x \in(a, b)$. [Hint: argue by contradiction.] Must $f$ be differentiable at $c$ ? Justify your answer.

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• Paper 1, Section II, D

  Analysis I | Part IA, 2017

  (a) State the Intermediate Value Theorem.

  (b) Define what it means for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ to be differentiable at a point $a \in \mathbb{R}$. If $f$ is differentiable everywhere on $\mathbb{R}$, must $f^{\prime}$ be continuous everywhere? Justify your answer.

  State the Mean Value Theorem.

  (c) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable everywhere. Let $a, b \in \mathbb{R}$ with $a<b$.

  If $f^{\prime}(a) \leqslant y \leqslant f^{\prime}(b)$, prove that there exists $c \in[a, b]$ such that $f^{\prime}(c)=y$. [Hint: consider the function $g$ defined by

  $$g(x)=\frac{f(x)-f(a)}{x-a}$$

  if $x \neq a$ and $\left.g(a)=f^{\prime}(a) .\right]$

  If additionally $f(a) \leqslant 0 \leqslant f(b)$, deduce that there exists $d \in[a, b]$ such that $f^{\prime}(d)+f(d)=y$.

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• Paper 1, Section II, E

  Analysis I | Part IA, 2017

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function defined on the closed, bounded interval $[a, b]$ of $\mathbb{R}$. Suppose that for every $\varepsilon>0$ there is a dissection $\mathcal{D}$ of $[a, b]$ such that $S_{\mathcal{D}}(f)-s_{\mathcal{D}}(f)<\varepsilon$, where $s_{\mathcal{D}}(f)$ and $S_{\mathcal{D}}(f)$ denote the lower and upper Riemann sums of $f$ for the dissection $\mathcal{D}$. Deduce that $f$ is Riemann integrable. [You may assume without proof that $s_{\mathcal{D}}(f) \leqslant S_{\mathcal{D}^{\prime}}(f)$ for all dissections $\mathcal{D}$ and $\mathcal{D}^{\prime}$ of $\left.[a, b] .\right]$

  Prove that if $f:[a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ is Riemann integrable.

  Let $g:(0,1] \rightarrow \mathbb{R}$ be a bounded continuous function. Show that for any $\lambda \in \mathbb{R}$, the function $f:[0,1] \rightarrow \mathbb{R}$ defined by

  $$f(x)= \begin{cases}g(x) & \text { if } 0<x \leqslant 1 \\ \lambda & \text { if } x=0\end{cases}$$

  is Riemann integrable.

  Let $f:[a, b] \rightarrow \mathbb{R}$ be a differentiable function with one-sided derivatives at the endpoints. Suppose that the derivative $f^{\prime}$ is (bounded and) Riemann integrable. Show that

  $$\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)$$

  [You may use the Mean Value Theorem without proof.]

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• Paper 1, Section II, F

  Analysis I | Part IA, 2017

  (a) Let $\left(x_{n}\right)_{n=1}^{\infty}$ be a non-negative and decreasing sequence of real numbers. Prove that $\sum_{n=1}^{\infty} x_{n}$ converges if and only if $\sum_{k=0}^{\infty} 2^{k} x_{2^{k}}$ converges.

  (b) For $s \in \mathbb{R}$, prove that $\sum_{n=1}^{\infty} n^{-s}$ converges if and only if $s>1$.

  (c) For any $k \in \mathbb{N}$, prove that

  $$\lim _{n \rightarrow \infty} 2^{-n} n^{k}=0$$

  (d) The sequence $\left(a_{n}\right)_{n=0}^{\infty}$ is defined by $a_{0}=1$ and $a_{n+1}=2^{a_{n}}$ for $n \geqslant 0$. For any $k \in \mathbb{N}$, prove that

  $$\lim _{n \rightarrow \infty} \frac{2^{n^{k}}}{a_{n}}=0$$

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Differential Equations

• Paper 2, Section I, $2 \mathrm{C}$

  Differential Equations | Part IA, 2017

  Consider the function

  $$f(x, y)=\frac{x}{y}+\frac{y}{x}-\frac{(x-y)^{2}}{a^{2}}$$

  defined for $x>0$ and $y>0$, where $a$ is a non-zero real constant. Show that $(\lambda, \lambda)$ is a stationary point of $f$ for each $\lambda>0$. Compute the Hessian and its eigenvalues at $(\lambda, \lambda)$.

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• Paper 2, Section I, C

  Differential Equations | Part IA, 2017

  (a) The numbers $z_{1}, z_{2}, \ldots$ satisfy

  $$z_{n+1}=z_{n}+c_{n} \quad(n \geqslant 1),$$

  where $c_{1}, c_{2}, \ldots$ are given constants. Find $z_{n+1}$ in terms of $c_{1}, c_{2}, \ldots, c_{n}$ and $z_{1}$.

  (b) The numbers $x_{1}, x_{2}, \ldots$ satisfy

  $$x_{n+1}=a_{n} x_{n}+b_{n} \quad(n \geqslant 1),$$

  where $a_{1}, a_{2}, \ldots$ are given non-zero constants and $b_{1}, b_{2}, \ldots$ are given constants. Let $z_{1}=x_{1}$ and $z_{n+1}=x_{n+1} / U_{n}$, where $U_{n}=a_{1} a_{2} \cdots a_{n}$. Calculate $z_{n+1}-z_{n}$, and hence find $x_{n+1}$ in terms of $x_{1}, b_{1}, \ldots, b_{n}$ and $U_{1}, \ldots, U_{n}$.

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• Paper 2, Section II, $7 \mathrm{C}$

  Differential Equations | Part IA, 2017

  Let $y_{1}$ and $y_{2}$ be two solutions of the differential equation

  $$y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)=0, \quad-\infty<x<\infty$$

  where $p$ and $q$ are given. Show, using the Wronskian, that

  • either there exist $\alpha$ and $\beta$, not both zero, such that $\alpha y_{1}(x)+\beta y_{2}(x)$ vanishes for all $x$,

  • or given $x_{0}, A$ and $B$, there exist $a$ and $b$ such that $y(x)=a y_{1}(x)+b y_{2}(x)$ satisfies the conditions $y\left(x_{0}\right)=A$ and $y^{\prime}\left(x_{0}\right)=B$.

  Find power series $y_{1}$ and $y_{2}$ such that an arbitrary solution of the equation

  $$y^{\prime \prime}(x)=x y(x)$$

  can be written as a linear combination of $y_{1}$ and $y_{2}$.

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• Paper 2, Section II, C

  Differential Equations | Part IA, 2017

  (a) Consider the system

  $$\begin{aligned} &\frac{d x}{d t}=x(1-x)-x y \\ &\frac{d y}{d t}=\frac{1}{8} y(4 x-1) \end{aligned}$$

  for $x(t) \geqslant 0, y(t) \geqslant 0$. Find the critical points, determine their type and explain, with the help of a diagram, the behaviour of solutions for large positive times $t$.

  (b) Consider the system

  $$\begin{aligned} &\frac{d x}{d t}=y+\left(1-x^{2}-y^{2}\right) x \\ &\frac{d y}{d t}=-x+\left(1-x^{2}-y^{2}\right) y \end{aligned}$$

  for $(x(t), y(t)) \in \mathbb{R}^{2}$. Rewrite the system in polar coordinates by setting $x(t)=$ $r(t) \cos \theta(t)$ and $y(t)=r(t) \sin \theta(t)$, and hence describe the behaviour of solutions for large positive and large negative times.

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• Paper 2, Section II, C

  Differential Equations | Part IA, 2017

  The current $I(t)$ at time $t$ in an electrical circuit subject to an applied voltage $V(t)$ obeys the equation

  $$L \frac{d^{2} I}{d t^{2}}+R \frac{d I}{d t}+\frac{1}{C} I=\frac{d V}{d t}$$

  where $R, L$ and $C$ are the constant resistance, inductance and capacitance of the circuit with $R \geqslant 0, L>0$ and $C>0$.

  (a) In the case $R=0$ and $V(t)=0$, show that there exist time-periodic solutions of frequency $\omega_{0}$, which you should find.

  (b) In the case $V(t)=H(t)$, the Heaviside function, calculate, subject to the condition

  $$R^{2}>\frac{4 L}{C}$$

  the current for $t \geqslant 0$, assuming it is zero for $t<0$.

  (c) If $R>0$ and $V(t)=\sin \omega_{0} t$, where $\omega_{0}$ is as in part (a), show that there is a timeperiodic solution $I_{0}(t)$ of period $T=2 \pi / \omega_{0}$ and calculate its maximum value $I_{M}$.

  (i) Calculate the energy dissipated in each period, i.e., the quantity

  $$D=\int_{0}^{T} R I_{0}(t)^{2} d t$$

  Show that the quantity defined by

  $$Q=\frac{2 \pi}{D} \times \frac{L I_{M}^{2}}{2}$$

  satisfies $Q \omega_{0} R C=1$.

  (ii) Write down explicitly the general solution $I(t)$ for all $R>0$, and discuss the relevance of $I_{0}(t)$ to the large time behaviour of $I(t)$.

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• Paper 2, Section II, C

  Differential Equations | Part IA, 2017

  (a) Solve $\frac{d z}{d t}=z^{2}$ subject to $z(0)=z_{0}$. For which $z_{0}$ is the solution finite for all $t \in \mathbb{R}$ ?

  Let $a$ be a positive constant. By considering the lines $y=a\left(x-x_{0}\right)$ for constant $x_{0}$, or otherwise, show that any solution of the equation

  $$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=0$$

  is of the form $f(x, y)=F(y-a x)$ for some function $F$.

  Solve the equation

  $$\frac{\partial f}{\partial x}+a \frac{\partial f}{\partial y}=f^{2}$$

  subject to $f(0, y)=g(y)$ for a given function $g$. For which $g$ is the solution bounded on $\mathbb{R}^{2}$ ?

  (b) By means of the change of variables $X=\alpha x+\beta y$ and $T=\gamma x+\delta y$ for appropriate real numbers $\alpha, \beta, \gamma, \delta$, show that the equation

  $$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial x \partial y}=0$$

  can be transformed into the wave equation

  $$\frac{1}{c^{2}} \frac{\partial^{2} F}{\partial T^{2}}-\frac{\partial^{2} F}{\partial X^{2}}=0$$

  where $F$ is defined by $f(x, y)=F(\alpha x+\beta y, \gamma x+\delta y)$. Hence write down the general solution of $(*)$.

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Dynamics and Relativity

• Paper 4, Section I, A

  Dynamics and Relativity | Part IA, 2017

  A tennis ball of mass $m$ is projected vertically upwards with initial speed $u_{0}$ and reaches its highest point at time $T$. In addition to uniform gravity, the ball experiences air resistance, which produces a frictional force of magnitude $\alpha v$, where $v$ is the ball's speed and $\alpha$ is a positive constant. Show by dimensional analysis that $T$ can be written in the form

  $$T=\frac{m}{\alpha} f(\lambda)$$

  for some function $f$ of a dimensionless quantity $\lambda$.

  Use the equation of motion of the ball to find $f(\lambda)$.

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• Paper 4, Section I, A

  Dynamics and Relativity | Part IA, 2017

  Consider a system of particles with masses $m_{i}$ and position vectors $\mathbf{x}_{i}$. Write down the definition of the position of the centre of mass $\mathbf{R}$ of the system. Let $T$ be the total kinetic energy of the system. Show that

  $$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} \sum_{i} m_{i} \dot{\mathbf{y}}_{i} \cdot \dot{\mathbf{y}}_{i}$$

  where $M$ is the total mass and $\mathbf{y}_{i}$ is the position vector of particle $i$ with respect to $\mathbf{R}$.

  The particles are connected to form a rigid body which rotates with angular speed $\omega$ about an axis $\mathbf{n}$ through $\mathbf{R}$, where $\mathbf{n} \cdot \mathbf{n}=1$. Show that

  $$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} I \omega^{2},$$

  where $I=\sum_{i} I_{i}$ and $I_{i}$ is the moment of inertia of particle $i$ about $\mathbf{n}$.

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• Paper 4, Section II, A

  Dynamics and Relativity | Part IA, 2017

  (a) A rocket moves in a straight line with speed $v(t)$ and is subject to no external forces. The rocket is composed of a body of mass $M$ and fuel of mass $m(t)$, which is burnt at constant rate $\alpha$ and the exhaust is ejected with constant speed $u$ relative to the rocket. Show that

  $$(M+m) \frac{d v}{d t}-\alpha u=0 .$$

  Show that the speed of the rocket when all its fuel is burnt is

  $$v_{0}+u \log \left(1+\frac{m_{0}}{M}\right)$$

  where $v_{0}$ and $m_{0}$ are the speed of the rocket and the mass of the fuel at $t=0$.

  (b) A two-stage rocket moves in a straight line and is subject to no external forces. The rocket is initially at rest. The masses of the bodies of the two stages are $k M$ and $(1-k) M$, with $0 \leqslant k \leqslant 1$, and they initially carry masses $k m_{0}$ and $(1-k) m_{0}$ of fuel. Both stages burn fuel at a constant rate $\alpha$ when operating and the exhaust is ejected with constant speed $u$ relative to the rocket. The first stage operates first, until all its fuel is burnt. The body of the first stage is then detached with negligible force and the second stage ignites.

  Find the speed of the second stage when all its fuel is burnt. For $0 \leqslant k<1$ compare it with the speed of the rocket in part (a) in the case $v_{0}=0$. Comment on the case $k=1$.

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• Paper 4, Section II, A

  Dynamics and Relativity | Part IA, 2017

  (a) Consider an inertial frame $S$, and a frame $S^{\prime}$ which rotates with constant angular velocity $\boldsymbol{\omega}$ relative to $S$. The two frames share a common origin. Identify each term in the equation

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

  (b) A small bead $P$ of unit mass can slide without friction on a circular hoop of radius $a$. The hoop is horizontal and rotating with constant angular speed $\omega$ about a fixed vertical axis through a point $O$ on its circumference.

  (i) Using Cartesian axes in the rotating frame $S^{\prime}$, with origin at $O$ and $x^{\prime}$-axis along the diameter of the hoop through $O$, write down the position vector of $P$ in terms of $a$ and the angle $\theta$ shown in the diagram $\left(-\frac{1}{2} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\right)$.

  

  (ii) Working again in the rotating frame, find, in terms of $a, \theta, \dot{\theta}$ and $\omega$, an expression for the horizontal component of the force exerted by the hoop on the bead.

  (iii) For what value of $\theta$ is the bead in stable equilibrium? Find the frequency of small oscillations of the bead about that point.

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• Paper 4, Section II, A

  Dynamics and Relativity | Part IA, 2017

  A particle of unit mass moves under the influence of a central force. By considering the components of the acceleration in polar coordinates $(r, \theta)$ prove that the magnitude of the angular momentum is conserved. [You may use $\ddot{\mathbf{r}}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+(2 \dot{r} \dot{\theta}+r \ddot{\theta}) \hat{\boldsymbol{\theta}}$. ]

  Now suppose that the central force is derived from the potential $k / r$, where $k$ is a constant.

  (a) Show that the total energy of the particle can be written in the form

  $$E=\frac{1}{2} \dot{r}^{2}+V_{\mathrm{eff}}(r)$$

  Sketch $V_{\text {eff }}(r)$ in the cases $k>0$ and $k<0$.

  (b) The particle is projected from a very large distance from the origin with speed $v$ and impact parameter $b$. [The impact parameter is the distance of closest approach to the origin in absence of any force.]

  (i) In the case $k<0$, sketch the particle's trajectory and find the shortest distance $p$ between the particle and the origin, and the speed $u$ of the particle when $r=p$.

  (ii) In the case $k>0$, sketch the particle's trajectory and find the corresponding shortest distance $\widetilde{p}$ between the particle and the origin, and the speed $\widetilde{u}$of the particle when $r=\widetilde{p}$.

  (iii) Find $p \tilde{p}$ and $u \tilde{u}$ in terms of $b$ and $v$. [In answering part (iii) you should assume that $|k|$ is the same in parts (i) and (ii).]

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• Paper 4, Section II, A

  Dynamics and Relativity | Part IA, 2017

  (a) A photon with energy $E_{1}$ in the laboratory frame collides with an electron of rest mass $m$ that is initially at rest in the laboratory frame. As a result of the collision the photon is deflected through an angle $\theta$ as measured in the laboratory frame and its energy changes to $E_{2}$.

  Derive an expression for $\frac{1}{E_{2}}-\frac{1}{E_{1}}$ in terms of $\theta, m$ and $c$.

  (b) A deuterium atom with rest mass $m_{1}$ and energy $E_{1}$ in the laboratory frame collides with another deuterium atom that is initially at rest in the laboratory frame. The result of this collision is a proton of rest mass $m_{2}$ and energy $E_{2}$, and a tritium atom of rest mass $m_{3}$. Show that, if the proton is emitted perpendicular to the incoming trajectory of the deuterium atom as measured in the laboratory frame, then

  $$m_{3}^{2}=m_{2}^{2}+2\left(m_{1}+\frac{E_{1}}{c^{2}}\right)\left(m_{1}-\frac{E_{2}}{c^{2}}\right)$$

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Groups

• Paper 3 , Section II, E

  Groups | Part IA, 2017

  State Lagrange's theorem. Show that the order of an element $x$ in a finite group $G$ is finite and divides the order of $G$.

  State Cauchy's theorem.

  List all groups of order 8 up to isomorphism. Carefully justify that the groups on your list are pairwise non-isomorphic and that any group of order 8 is isomorphic to one on your list. [You may use without proof the Direct Product Theorem and the description of standard groups in terms of generators satisfying certain relations.]

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• Paper 3, Section I, E

  Groups | Part IA, 2017

  What does it mean to say that $H$ is a normal subgroup of the group $G$ ? For a normal subgroup $H$ of $G$ define the quotient group $G / H$. [You do not need to verify that $G / H$ is a group.]

  State the Isomorphism Theorem.

  Let

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

  be the group of $2 \times 2$ invertible upper-triangular real matrices. By considering a suitable homomorphism, show that the subset

  $$H=\left\{\left(\begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \mid b \in \mathbb{R}\right\}$$

  of $G$ is a normal subgroup of $G$ and identify the quotient $G / H$.

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• Paper 3, Section I, E

  Groups | Part IA, 2017

  Let $w_{1}, w_{2}, w_{3}$ be distinct elements of $\mathbb{C} \cup\{\infty\}$. Write down the Möbius map $f$ that sends $w_{1}, w_{2}, w_{3}$ to $\infty, 0,1$, respectively. [Hint: You need to consider four cases.]

  Now let $w_{4}$ be another element of $\mathbb{C} \cup\{\infty\}$ distinct from $w_{1}, w_{2}, w_{3}$. Define the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ in terms of $f$.

  Prove that there is a circle or line through $w_{1}, w_{2}, w_{3}$ and $w_{4}$ if and only if the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ is real.

  [You may assume without proof that Möbius maps map circles and lines to circles and lines and also that there is a unique circle or line through any three distinct points of $\mathbb{C} \cup\{\infty\} .]$

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• Paper 3, Section II, $7 \mathrm{E}$

  Groups | Part IA, 2017

  (a) Let $G$ be a finite group acting on a finite set $X$. State the Orbit-Stabiliser theorem. [Define the terms used.] Prove that

  $$\sum_{x \in X}|\operatorname{Stab}(x)|=n|G|$$

  where $n$ is the number of distinct orbits of $X$ under the action of $G$.

  Let $S=\{(g, x) \in G \times X: g \cdot x=x\}$, and for $g \in G$, let $\operatorname{Fix}(g)=\{x \in X: g \cdot x=x\}$.

  Show that

  $$|S|=\sum_{x \in X}|\operatorname{Stab}(x)|=\sum_{g \in G}|\operatorname{Fix}(g)|$$

  and deduce that

  $$n=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$$

  (b) Let $H$ be the group of rotational symmetries of the cube. Show that $H$ has 24 elements. [If your proof involves calculating stabilisers, then you must carefully verify such calculations.]

  Using $(*)$, find the number of distinct ways of colouring the faces of the cube red, green and blue, where two colourings are distinct if one cannot be obtained from the other by a rotation of the cube. [A colouring need not use all three colours.]

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• Paper 3, Section II, E

  Groups | Part IA, 2017

  Prove that every element of the symmetric group $S_{n}$ is a product of transpositions. [You may assume without proof that every permutation is the product of disjoint cycles.]

  (a) Define the sign of a permutation in $S_{n}$, and prove that it is well defined. Define the alternating group $A_{n}$.

  (b) Show that $S_{n}$ is generated by the set $\left\{\left(\begin{array}{lll}1 & 2\end{array}\right),\left(\begin{array}{llll}1 & 2 & 3 & \ldots\end{array}\right)\right\}$.

  Given $1 \leqslant k<n$, prove that the set $\left\{\left(\begin{array}{ll}1 & \left.1+k),\left(\begin{array}{lll}1 & 2 & 3\end{array} \ldots n\right)\right\} \text { generates } S_{n} \text { if and only }\end{array}\right.\right.$ if $k$ and $n$ are coprime.

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• Paper 3, Section II, E

  Groups | Part IA, 2017

  Let $N$ be a normal subgroup of a finite group $G$ of prime index $p=|G: N|$.

  By considering a suitable homomorphism, show that if $H$ is a subgroup of $G$ that is not contained in $N$, then $H \cap N$ is a normal subgroup of $H$ of index $p$.

  Let $C$ be a conjugacy class of $G$ that is contained in $N$. Prove that $C$ is either a conjugacy class in $N$ or is the disjoint union of $p$ conjugacy classes in $N$.

  [You may use standard theorems without proof.]

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Numbers and Sets

• Paper 4, Section I, D

  Numbers and Sets | Part IA, 2017

  (a) Give the definitions of relation and equivalence relation on a set $S$.

  (b) Let $\Sigma$ be the set of ordered pairs $(A, f)$ where $A$ is a non-empty subset of $\mathbb{R}$ and $f: A \rightarrow \mathbb{R}$. Let $\mathcal{R}$ be the relation on $\Sigma$ defined by requiring $(A, f) \mathcal{R}(B, g)$ if the following two conditions hold:

  (i) $(A \backslash B) \cup(B \backslash A)$ is finite and

  (ii) there is a finite set $F \subset A \cap B$ such that $f(x)=g(x)$ for all $x \in A \cap B \backslash F$.

  Show that $\mathcal{R}$ is an equivalence relation on $\Sigma$.

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• Paper 4, Section I, D

  Numbers and Sets | Part IA, 2017

  (a) Show that for all positive integers $z$ and $n$, either $z^{2 n} \equiv 0(\bmod 3)$ or $z^{2 n} \equiv 1(\bmod 3)$.

  (b) If the positive integers $x, y, z$ satisfy $x^{2}+y^{2}=z^{2}$, show that at least one of $x$ and $y$ must be divisible by 3 . Can both $x$ and $y$ be odd?

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• Paper 4, Section II, 7D

  Numbers and Sets | Part IA, 2017

  (a) For positive integers $n, m, k$ with $k \leqslant n$, show that

  $$\left(\begin{array}{l} n \\ k \end{array}\right)\left(\frac{k}{n}\right)^{m}=\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right) \sum_{\ell=0}^{m-1} a_{n, m, \ell}\left(\frac{k-1}{n-1}\right)^{m-1-\ell}$$

  giving an explicit formula for $a_{n, m, \ell}$. [You may wish to consider the expansion of $\left.\left(\frac{k-1}{n-1}+\frac{1}{n-1}\right)^{m-1} .\right]$

  (b) For a function $f:[0,1] \rightarrow \mathbb{R}$ and each integer $n \geqslant 1$, the function $B_{n}(f):[0,1] \rightarrow \mathbb{R}$ is defined by

  $$B_{n}(f)(x)=\sum_{k=0}^{n} f\left(\frac{k}{n}\right)\left(\begin{array}{l} n \\ k \end{array}\right) x^{k}(1-x)^{n-k}$$

  For any integer $m \geqslant 0$ let $f_{m}(x)=x^{m}$. Show that $B_{n}\left(f_{0}\right)(x)=1$ and $B_{n}\left(f_{1}\right)(x)=x$ for all $n \geqslant 1$ and $x \in[0,1]$.

  Show that for each integer $m \geqslant 0$ and each $x \in[0,1]$,

  $$B_{n}\left(f_{m}\right)(x) \rightarrow f_{m}(x) \text { as } n \rightarrow \infty$$

  Deduce that for each integer $m \geqslant 0$,

  $$\lim _{n \rightarrow \infty} \frac{1}{4^{n}} \sum_{k=0}^{2 n}\left(\frac{k}{n}\right)^{m}\left(\begin{array}{c} 2 n \\ k \end{array}\right)=1$$

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• Paper 4, Section II, D

  Numbers and Sets | Part IA, 2017

  Let $\left(a_{k}\right)_{k=1}^{\infty}$ be a sequence of real numbers.

  (a) Define what it means for $\left(a_{k}\right)_{k=1}^{\infty}$ to converge. Define what it means for the series $\sum_{k=1}^{\infty} a_{k}$ to converge.

  Show that if $\sum_{k=1}^{\infty} a_{k}$ converges, then $\left(a_{k}\right)_{k=1}^{\infty}$ converges to 0 .

  If $\left(a_{k}\right)_{k=1}^{\infty}$ converges to $a \in \mathbb{R}$, show that

  $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} a_{k}=a$$

  (b) Suppose $a_{k}>0$ for every $k$. Let $u_{n}=\sum_{k=1}^{n}\left(a_{k}+\frac{1}{a_{k}}\right)$ and $v_{n}=\sum_{k=1}^{n}\left(a_{k}-\frac{1}{a_{k}}\right)$.

  Show that $\left(u_{n}\right)_{n=1}^{\infty}$ does not converge.

  Give an example of a sequence $\left(a_{k}\right)_{k=1}^{\infty}$ with $a_{k}>0$ and $a_{k} \neq 1$ for every $k$ such that $\left(v_{n}\right)_{n=1}^{\infty}$ converges.

  If $\left(v_{n}\right)_{n=1}^{\infty}$ converges, show that $\frac{u_{n}}{n} \rightarrow 2$.

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• Paper 4, Section II, D

  Numbers and Sets | Part IA, 2017

  (a) Define what it means for a set to be countable.

  (b) Let $A$ be an infinite subset of the set of natural numbers $\mathbb{N}=\{0,1,2, \ldots\}$. Prove that there is a bijection $f: \mathbb{N} \rightarrow A$.

  (c) Let $A_{n}$ be the set of natural numbers whose decimal representation ends with exactly $n-1$ zeros. For example, $71 \in A_{1}, 70 \in A_{2}$ and $15000 \in A_{4}$. By applying the result of part (b) with $A=A_{n}$, construct a bijection $g: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$. Deduce that the set of rationals is countable.

  (d) Let $A$ be an infinite set of positive real numbers. If every sequence $\left(a_{j}\right)_{j=1}^{\infty}$ of distinct elements with $a_{j} \in A$ for each $j$ has the property that

  $$\lim _{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^{N} a_{j}=0$$

  prove that $A$ is countable.

  [You may assume without proof that a countable union of countable sets is countable.]

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• Paper 4, Section II, D

  Numbers and Sets | Part IA, 2017

  (a) State and prove the Fermat-Euler Theorem. Deduce Fermat's Little Theorem. State Wilson's Theorem.

  (b) Let $p$ be an odd prime. Prove that $X^{2} \equiv-1(\bmod p)$ is solvable if and only if $p \equiv 1(\bmod 4)$.

  (c) Let $p$ be prime. If $h$ and $k$ are non-negative integers with $h+k=p-1$, prove that $h ! k !+(-1)^{h} \equiv 0(\bmod p) .$

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Probability

• Paper 2, Section I, F

  Probability | Part IA, 2017

  Let $X$ and $Y$ be real-valued random variables with joint density function

  $$f(x, y)= \begin{cases}x e^{-x(y+1)} & \text { if } x \geqslant 0 \text { and } y \geqslant 0 \\ 0 & \text { otherwise. }\end{cases}$$

  (i) Find the conditional probability density function of $Y$ given $X$.

  (ii) Find the expectation of $Y$ given $X$.

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• Paper 2, Section I, F

  Probability | Part IA, 2017

  Let $X$ be a non-negative integer-valued random variable such that $0<\mathbb{E}\left(X^{2}\right)<\infty$.

  Prove that

  $$\frac{\mathbb{E}(X)^{2}}{\mathbb{E}\left(X^{2}\right)} \leqslant \mathbb{P}(X>0) \leqslant \mathbb{E}(X)$$

  [You may use any standard inequality.]

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• Paper 2, Section II, 10F

  Probability | Part IA, 2017

  (a) For any random variable $X$ and $\lambda>0$ and $t>0$, show that

  $$\mathbb{P}(X>t) \leqslant \mathbb{E}\left(e^{\lambda X}\right) e^{-\lambda t}$$

  For a standard normal random variable $X$, compute $\mathbb{E}\left(e^{\lambda X}\right)$ and deduce that

  $$\mathbb{P}(X>t) \leqslant e^{-\frac{1}{2} t^{2}}$$

  (b) Let $\mu, \lambda>0, \mu \neq \lambda$. For independent random variables $X$ and $Y$ with distributions $\operatorname{Exp}(\lambda)$ and $\operatorname{Exp}(\mu)$, respectively, compute the probability density functions of $X+Y$ and $\min \{X, Y\}$.

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• Paper 2, Section II, 12F

  Probability | Part IA, 2017

  (a) Let $k \in\{1,2, \ldots\}$. For $j \in\{0, \ldots, k+1\}$, let $D_{j}$ be the first time at which a simple symmetric random walk on $\mathbb{Z}$ with initial position $j$ at time 0 hits 0 or $k+1$. Show $\mathbb{E}\left(D_{j}\right)=j(k+1-j)$. [If you use a recursion relation, you do not need to prove that its solution is unique.]

  (b) Let $\left(S_{n}\right)$ be a simple symmetric random walk on $\mathbb{Z}$ starting at 0 at time $n=0$. For $k \in\{1,2, \ldots\}$, let $T_{k}$ be the first time at which $\left(S_{n}\right)$ has visited $k$ distinct vertices. In particular, $T_{1}=0$. Show $\mathbb{E}\left(T_{k+1}-T_{k}\right)=k$ for $k \geqslant 1$. [You may use without proof that, conditional on $S_{T_{k}}=i$, the random variables $\left(S_{T_{k}+n}\right)_{n \geqslant 0}$ have the distribution of a simple symmetric random walk starting at $i$.]

  (c) For $n \geqslant 3$, let $\mathbb{Z}_{n}$ be the circle graph consisting of vertices $0, \ldots, n-1$ and edges between $k$ and $k+1$ where $n$ is identified with 0 . Let $\left(Y_{i}\right)$ be a simple random walk on $\mathbb{Z}_{n}$ starting at time 0 from 0 . Thus $Y_{0}=0$ and conditional on $Y_{i}$ the random variable $Y_{i+1}$ is $Y_{i} \pm 1$ with equal probability (identifying $k+n$ with $k$ ).

  The cover time $T$ of the simple random walk on $\mathbb{Z}_{n}$ is the first time at which the random walk has visited all vertices. Show that $\mathbb{E}(T)=n(n-1) / 2$.

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• Paper 2, Section II, F

  Probability | Part IA, 2017

  Let $\beta>0$. The Curie-Weiss Model of ferromagnetism is the probability distribution defined as follows. For $n \in \mathbb{N}$, define random variables $S_{1}, \ldots, S_{n}$ with values in $\{\pm 1\}$ such that the probabilities are given by

  $$\mathbb{P}\left(S_{1}=s_{1}, \ldots, S_{n}=s_{n}\right)=\frac{1}{Z_{n, \beta}} \exp \left(\frac{\beta}{2 n} \sum_{i=1}^{n} \sum_{j=1}^{n} s_{i} s_{j}\right)$$

  where $Z_{n, \beta}$ is the normalisation constant

  $$Z_{n, \beta}=\sum_{s_{1} \in\{\pm 1\}} \cdots \sum_{s_{n} \in\{\pm 1\}} \exp \left(\frac{\beta}{2 n} \sum_{i=1}^{n} \sum_{j=1}^{n} s_{i} s_{j}\right)$$

  (a) Show that $\mathbb{E}\left(S_{i}\right)=0$ for any $i$.

  (b) Show that $\mathbb{P}\left(S_{2}=+1 \mid S_{1}=+1\right) \geqslant \mathbb{P}\left(S_{2}=+1\right)$. [You may use $\mathbb{E}\left(S_{i} S_{j}\right) \geqslant 0$ for all $i, j$ without proof. ]

  (c) Let $M=\frac{1}{n} \sum_{i=1}^{n} S_{i}$. Show that $M$ takes values in $E_{n}=\left\{-1+\frac{2 k}{n}: k=0, \ldots, n\right\}$, and that for each $m \in E_{n}$ the number of possible values of $\left(S_{1}, \ldots, S_{n}\right)$ such that $M=m$ is

  $$\frac{n !}{\left(\frac{1+m}{2} n\right) !\left(\frac{1-m}{2} n\right) !}$$

  Find $\mathbb{P}(M=m)$ for any $m \in E_{n}$.

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• Paper 2, Section II, F

  Probability | Part IA, 2017

  For a positive integer $N, p \in[0,1]$, and $k \in\{0,1, \ldots, N\}$, let

  $$p_{k}(N, p)=\left(\begin{array}{c} N \\ k \end{array}\right) p^{k}(1-p)^{N-k}$$

  (a) For fixed $N$ and $p$, show that $p_{k}(N, p)$ is a probability mass function on $\{0,1, \ldots, N\}$ and that the corresponding probability distribution has mean $N p$ and variance $N p(1-p)$.

  (b) Let $\lambda>0$. Show that, for any $k \in\{0,1,2, \ldots\}$,

  $$\lim _{N \rightarrow \infty} p_{k}(N, \lambda / N)=\frac{e^{-\lambda} \lambda^{k}}{k !}$$

  Show that the right-hand side of $(*)$ is a probability mass function on $\{0,1,2, \ldots\}$.

  (c) Let $p \in(0,1)$ and let $a, b \in \mathbb{R}$ with $a<b$. For all $N$, find integers $k_{a}(N)$ and $k_{b}(N)$ such that

  $$\sum_{k=k_{a}(N)}^{k_{b}(N)} p_{k}(N, p) \rightarrow \frac{1}{\sqrt{2 \pi}} \int_{a}^{b} e^{-\frac{1}{2} x^{2}} d x \quad \text { as } N \rightarrow \infty$$

  [You may use the Central Limit Theorem.]

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Vector Calculus

• Paper 3, Section $I$, B

  Vector Calculus | Part IA, 2017

  (a) The two sets of basis vectors $\mathbf{e}_{i}$ and $\mathbf{e}_{i}^{\prime}$ (where $i=1,2,3$ ) are related by

  $$\mathbf{e}_{i}^{\prime}=R_{i j} \mathbf{e}_{j}$$

  where $R_{i j}$ are the entries of a rotation matrix. The components of a vector $\mathbf{v}$ with respect to the two bases are given by

  $$\mathbf{v}=v_{i} \mathbf{e}_{i}=v_{i}^{\prime} \mathbf{e}_{i}^{\prime}$$

  Derive the relationship between $v_{i}$ and $v_{i}^{\prime}$.

  (b) Let $\mathbf{T}$ be a $3 \times 3$ array defined in each (right-handed orthonormal) basis. Using part (a), state and prove the quotient theorem as applied to $\mathbf{T}$.

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• Paper 3, Section I, B

  Vector Calculus | Part IA, 2017

  Use the change of variables $x=r \cosh \theta, y=r \sinh \theta$ to evaluate

  $$\int_{A} y d x d y$$

  where $A$ is the region of the $x y$-plane bounded by the two line segments:

  $$\begin{aligned} &y=0, \quad 0 \leqslant x \leqslant 1 \\ &5 y=3 x, \quad 0 \leqslant x \leqslant \frac{5}{4} \end{aligned}$$

  and the curve

  $$x^{2}-y^{2}=1, \quad x \geqslant 1$$

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2017

  Let $S$ be a piecewise smooth closed surface in $\mathbb{R}^{3}$ which is the boundary of a volume $V$.

  (a) The smooth functions $\phi$ and $\phi_{1}$ defined on $\mathbb{R}^{3}$ satisfy

  $$\nabla^{2} \phi=\nabla^{2} \phi_{1}=0$$

  in $V$ and $\phi(\mathbf{x})=\phi_{1}(\mathbf{x})=f(\mathbf{x})$ on $S$. By considering an integral of $\boldsymbol{\nabla} \psi \cdot \boldsymbol{\nabla} \psi$, where $\psi=\phi-\phi_{1}$, show that $\phi_{1}=\phi$.

  (b) The smooth function $u$ defined on $\mathbb{R}^{3}$ satisfies $u(\mathbf{x})=f(\mathbf{x})+C$ on $S$, where $f$ is the function in part (a) and $C$ is constant. Show that

  $$\int_{V} \nabla u \cdot \nabla u d V \geqslant \int_{V} \nabla \phi \cdot \nabla \phi d V$$

  where $\phi$ is the function in part (a). When does equality hold?

  (c) The smooth function $w(\mathbf{x}, t)$ satisfies

  $$\nabla^{2} w=\frac{\partial w}{\partial t}$$

  in $V$ and $\frac{\partial w}{\partial t}=0$ on $S$ for all $t$. Show that

  $$\frac{d}{d t} \int_{V} \nabla w \cdot \nabla w d V \leqslant 0$$

  with equality only if $\nabla^{2} w=0$ in $V$.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2017

  (a) Let $\mathbf{x}=\mathbf{r}(s)$ be a smooth curve parametrised by arc length $s$. Explain the meaning of the terms in the equation

  $$\frac{d \mathbf{t}}{d s}=\kappa \mathbf{n},$$

  where $\kappa(s)$ is the curvature of the curve.

  Now let $\mathbf{b}=\mathbf{t} \times \mathbf{n}$. Show that there is a scalar $\tau(s)$ (the torsion) such that

  $$\frac{d \mathbf{b}}{d s}=-\tau \mathbf{n}$$

  and derive an expression involving $\kappa$ and $\tau$ for $\frac{d \mathbf{n}}{d s}$.

  (b) Given a (nowhere zero) vector field $\mathbf{F}$, the field lines, or integral curves, of $\mathbf{F}$ are the curves parallel to $\mathbf{F}(\mathbf{x})$ at each point $\mathbf{x}$. Show that the curvature $\kappa$ of the field lines of $\mathbf{F}$ satisfies

  $$\frac{\mathbf{F} \times(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{F}}{F^{3}}=\pm \kappa \mathbf{b}$$

  where $F=|\mathbf{F}|$.

  (c) Use $(*)$ to find an expression for the curvature at the point $(x, y, z)$ of the field lines of $\mathbf{F}(x, y, z)=(x, y,-z)$.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2017

  By a suitable choice of $\mathbf{u}$ in the divergence theorem

  $$\int_{V} \nabla \cdot \mathbf{u} d V=\int_{S} \mathbf{u} \cdot d \mathbf{S}$$

  show that

  $$\int_{V} \nabla \phi d V=\int_{S} \phi d \mathbf{S}$$

  for any continuously differentiable function $\phi$.

  For the curved surface of the cone

  $$\mathbf{x}=(r \cos \theta, r \sin \theta, \sqrt{3} r), \quad 0 \leqslant \sqrt{3} r \leqslant 1, \quad 0 \leqslant \theta \leqslant 2 \pi$$

  show that $d \mathbf{S}=(\sqrt{3} \cos \theta, \sqrt{3} \sin \theta,-1) r d r d \theta$.

  Verify that $(*)$ holds for this cone and $\phi(x, y, z)=z^{2}$.

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• Paper 3, Section II, B

  Vector Calculus | Part IA, 2017

  (a) The time-dependent vector field $\mathbf{F}$ is related to the vector field $\mathbf{B}$ by

  $$\mathbf{F}(\mathbf{x}, t)=\mathbf{B}(\mathbf{z})$$

  where $\mathbf{z}=t \mathbf{x}$. Show that

  $$(\mathbf{x} \cdot \nabla) \mathbf{F}=t \frac{\partial \mathbf{F}}{\partial t} \text {. }$$

  (b) The vector fields $\mathbf{B}$ and $\mathbf{A}$ satisfy $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$.

  (c) The vector field $\mathbf{B}$ satisfies $\boldsymbol{\nabla} \cdot \mathbf{B}=0$. Show that

  $$\mathbf{B}(\mathbf{x})=\nabla \times(\mathbf{D}(\mathbf{x}) \times \mathbf{x})$$

  where

  $$\mathbf{D}(\mathbf{x})=\int_{0}^{1} t \mathbf{B}(t \mathbf{x}) d t$$

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Vectors and Matrices

• Paper 1, Section I, A

  Vectors and Matrices | Part IA, 2017

  Consider $z \in \mathbb{C}$ with $|z|=1$ and $\arg z=\theta$, where $\theta \in[0, \pi)$.

  (a) Prove algebraically that the modulus of $1+z$ is $2 \cos \frac{1}{2} \theta$ and that the argument is $\frac{1}{2} \theta$. Obtain these results geometrically using the Argand diagram.

  (b) Obtain corresponding results algebraically and geometrically for $1-z$.

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• Paper 1, Section I, C

  Vectors and Matrices | Part IA, 2017

  Let $A$ and $B$ be real $n \times n$ matrices.

  Show that $(A B)^{T}=B^{T} A^{T}$.

  For any square matrix, the matrix exponential is defined by the series

  $$e^{A}=I+\sum_{k=1}^{\infty} \frac{A^{k}}{k !}$$

  Show that $\left(e^{A}\right)^{T}=e^{A^{T}}$. [You are not required to consider issues of convergence.]

  Calculate, in terms of $A$