Part IA 2010

• Paper 1, Section I, D

  Analysis I

  Part IA, 2010

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

  Suppose the power series has radius of convergence $R$, with $0<R<\infty$. Show that the sequence $\left|a_{n} z^{n}\right|$ is unbounded if $|z|>R$.

  Find the radius of convergence of $\sum_{n \geqslant 1} z^{n} / n^{3}$.

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• Paper 1, Section I, E

  Analysis I

  Part IA, 2010

  Find the limit of each of the following sequences; justify your answers.

  (i)

  $$\frac{1+2+\ldots+n}{n^{2}}$$

  (ii)

  $$\sqrt[n]{n}$$

  (iii)

  $$\left(a^{n}+b^{n}\right)^{1 / n} \quad \text { with } \quad 0<a \leqslant b$$

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• Paper 2, Section II, A

  Differential Equations

  Part IA, 2010

  (a) Find the general solution of the system of differential equations

  $$\left(\begin{array}{l} \dot{x} \\ \dot{y} \\ \dot{z} \end{array}\right)=\left(\begin{array}{rrr} -1 & 2 & -1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)$$

  (b) Depending on the parameter $\lambda \in \mathbb{R}$, find the general solution of the system of differential equations

  $$\left(\begin{array}{l} \dot{x} \\ \dot{y} \\ \dot{z} \end{array}\right)=\left(\begin{array}{rrr} -1 & 2 & -1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)+2\left(\begin{array}{r} -\lambda \\ 1 \\ \lambda \end{array}\right) e^{2 t},$$

  and explain why $(2)$ has a particular solution of the form $\mathbf{c} e^{2 t}$ with constant vector $\mathbf{c} \in \mathbb{R}^{3}$ for $\lambda=1$ but not for $\lambda \neq 1$.

  [Hint: decompose $\left(\begin{array}{c}-\lambda \\ 1 \\ \lambda\end{array}\right)$ in terms of the eigenbasis of the matrix in (1).]

  (c) For $\lambda=-1$, find the solution of (2) which goes through the point $(0,1,0)$ at $t=0$.

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• Paper 2, Section II, A

  Differential Equations

  Part IA, 2010

  (a) State how the nature of a critical (or stationary) point of a function $f(\mathbf{x})$ with $\mathbf{x} \in \mathbb{R}^{n}$ can be determined by consideration of the eigenvalues of the Hessian matrix $H$ of $f(\mathbf{x})$, assuming $H$ is non-singular.

  (b) Let $f(x, y)=x y(1-x-y)$. Find all the critical points of the function $f(x, y)$ and determine their nature. Determine the zero contour of $f(x, y)$ and sketch a contour plot showing the behaviour of the contours in the neighbourhood of the critical points.

  (c) Now let $g(x, y)=x^{3} y^{2}(1-x-y)$. Show that $(0,1)$ is a critical point of $g(x, y)$ for which the Hessian matrix of $g$ is singular. Find an approximation for $g(x, y)$ to lowest non-trivial order in the neighbourhood of the point $(0,1)$. Does $g$ have a maximum or a minimum at $(0,1)$ ? Justify your answer.

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• Paper 4, Section I, B

  Dynamics and Relativity

  Part IA, 2010

  A particle of mass $m$ and charge $q$ moves with trajectory $\mathbf{r}(t)$ in a constant magnetic field $\mathbf{B}=B \hat{\mathbf{z}}$. Write down the Lorentz force on the particle and use Newton's Second Law to deduce that

  $$\dot{\mathbf{r}}-\omega \mathbf{r} \times \hat{\mathbf{z}}=\mathbf{c}$$

  where $\mathbf{c}$ is a constant vector and $\omega$ is to be determined. Find $\mathbf{c}$ and hence $\mathbf{r}(t)$ for the initial conditions

  $$\mathbf{r}(0)=a \hat{\mathbf{x}} \quad \text { and } \quad \dot{\mathbf{r}}(0)=u \hat{\mathbf{y}}+v \hat{\mathbf{z}}$$

  where $a, u$ and $v$ are constants. Sketch the particle's trajectory in the case $a \omega+u=0$.

  [Unit vectors $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ correspond to a set of Cartesian coordinates. ]

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• Paper 4, Section $I$, B

  Dynamics and Relativity

  Part IA, 2010

  Let $S$ be an inertial frame with coordinates $(t, x)$ in two-dimensional spacetime. Write down the Lorentz transformation giving the coordinates $\left(t^{\prime}, x^{\prime}\right)$ in a second inertial frame $S^{\prime}$ moving with velocity $v$ relative to $S$. If a particle has constant velocity $u$ in $S$, find its velocity $u^{\prime}$ in $S^{\prime}$. Given that $|u|<c$ and $|v|<c$, show that $\left|u^{\prime}\right|<c$.

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• Paper 4 , Section II, B

  Dynamics and Relativity

  Part IA, 2010

  A sphere of uniform density has mass $m$ and radius $a$. Find its moment of inertia about an axis through its centre.

  A marble of uniform density is released from rest on a plane inclined at an angle $\alpha$ to the horizontal. Let the time taken for the marble to travel a distance $\ell$ down the plane be: (i) $t_{1}$ if the plane is perfectly smooth; or (ii) $t_{2}$ if the plane is rough and the marble rolls without slipping.

  Explain, with a clear discussion of the forces acting on the marble, whether or not its energy is conserved in each of the cases (i) and (ii). Show that $t_{1} / t_{2}=\sqrt{5 / 7}$.

  Suppose that the original marble is replaced by a new one with the same mass and radius but with a hollow centre, so that its moment of inertia is $\lambda m a^{2}$ for some constant $\lambda$. What is the new value for $t_{1} / t_{2}$ ?

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• Paper 4, Section II, B

  Dynamics and Relativity

  Part IA, 2010

  A particle of unit mass moves in a plane with polar coordinates $(r, \theta)$ and components of acceleration $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$. The particle experiences a force corresponding to a potential $-Q / r$. Show that

  $$E=\frac{1}{2} \dot{r}^{2}+U(r) \quad \text { and } \quad h=r^{2} \dot{\theta}$$

  are constants of the motion, where

  $$U(r)=\frac{h^{2}}{2 r^{2}}-\frac{Q}{r}$$

  Sketch the graph of $U(r)$ in the cases $Q>0$ and $Q<0$.

  (a) Assuming $Q>0$ and $h>0$, for what range of values of $E$ do bounded orbits exist? Find the minimum and maximum distances from the origin, $r_{\min }$ and $r_{\max }$, on such an orbit and show that

  $$r_{\min }+r_{\max }=\frac{Q}{|E|} .$$

  Prove that the minimum and maximum values of the particle's speed, $v_{\min }$ and $v_{\max }$, obey

  $$v_{\min }+v_{\max }=\frac{2 Q}{h}$$

  (b) Now consider trajectories with $E>0$ and $Q$ of either sign. Find the distance of closest approach, $r_{\min }$, in terms of the impact parameter, $b$, and $v_{\infty}$, the limiting value of the speed as $r \rightarrow \infty$. Deduce that if $b \ll|Q| / v_{\infty}^{2}$ then, to leading order,

  $$r_{\min } \approx \frac{2|Q|}{v_{\infty}^{2}} \text { for } Q<0, \quad r_{\min } \approx \frac{b^{2} v_{\infty}^{2}}{2 Q} \text { for } Q>0$$

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• Paper 4, Section II, B

  Dynamics and Relativity

  Part IA, 2010

  Consider a set of particles with position vectors $\mathbf{r}_{i}(t)$ and masses $m_{i}$, where $i=1,2, \ldots, N$. Particle $i$ experiences an external force $\mathbf{F}_{i}$ and an internal force $\mathbf{F}_{i j}$ from particle $j$, for each $j \neq i$. Stating clearly any assumptions you need, show that

  $$\frac{d \mathbf{P}}{d t}=\mathbf{F} \quad \text { and } \quad \frac{d \mathbf{L}}{d t}=\mathbf{G}$$

  where $\mathbf{P}$ is the total momentum, $\mathbf{F}$ is the total external force, $\mathbf{L}$ is the total angular momentum about a fixed point $\mathbf{a}$, and $\mathbf{G}$ is the total external torque about $\mathbf{a}$.

  Does the result $\frac{d \mathbf{L}}{d t}=\mathbf{G}$ still hold if the fixed point $\mathbf{a}$ is replaced by the centre of mass of the system? Justify your answer.

  Suppose now that the external force on particle $i$ is $-k \frac{d \mathbf{r}_{i}}{d t}$ and that all the particles have the same mass $m$. Show that

  $$\mathbf{L}(t)=\mathbf{L}(0) e^{-k t / m}$$

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• Paper 4, Section II, B

  Dynamics and Relativity

  Part IA, 2010

  A particle $A$ of rest mass $m$ is fired at an identical particle $B$ which is stationary in the laboratory. On impact, $A$ and $B$ annihilate and produce two massless photons whose energies are equal. Assuming conservation of four-momentum, show that the angle $\theta$ between the photon trajectories is given by

  $$\cos \theta=\frac{E-3 m c^{2}}{E+m c^{2}}$$

  where $E$ is the relativistic energy of $A$.

  Let $v$ be the speed of the incident particle $A$. For what value of $v / c$ will the photons move in perpendicular directions? If $v$ is very small compared with $c$, show that

  $$\theta \approx \pi-v / c$$

  [All quantities referred to are measured in the laboratory frame.]

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• Paper 3, Section I, D

  Groups

  Part IA, 2010

  Write down the matrix representing the following transformations of $\mathbb{R}^{3}$ :

  (i) clockwise rotation of $45^{\circ}$ around the $x$ axis,

  (ii) reflection in the plane $x=y$,

  (iii) the result of first doing (i) and then (ii).

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• Paper 3, Section I, D

  Groups

  Part IA, 2010

  Express the element $(123)(234)$ in $S_{5}$ as a product of disjoint cycles. Show that it is in $A_{5}$. Write down the elements of its conjugacy class in $A_{5}$.

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• Paper 1, Section II, E

  Analysis I

  Part IA, 2010

  Determine whether the following series converge or diverge. Any tests that you use should be carefully stated.

  (a)

  $$\sum_{n \geqslant 1} \frac{n !}{n^{n}}$$

  (b)

  $$\sum_{n \geqslant 1} \frac{1}{n+(\log n)^{2}}$$

  (c)

  $$\sum_{n \geqslant 1} \frac{(-1)^{n}}{1+\sqrt{n}}$$

  (d)

  $$\sum_{n \geqslant 1} \frac{(-1)^{n}}{n\left(2+(-1)^{n}\right)}$$

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• Paper 3, Section II, D

  Groups

  Part IA, 2010

  (i) State the orbit-stabilizer theorem.

  Let $G$ be the group of rotations of the cube, $X$ the set of faces. Identify the stabilizer of a face, and hence compute the order of $G$.

  Describe the orbits of $G$ on the set $X \times X$ of pairs of faces.

  (ii) Define what it means for a subgroup $N$ of $G$ to be normal. Show that $G$ has a normal subgroup of order 4 .

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• Paper 3, Section II, D

  Groups

  Part IA, 2010

  State Lagrange's theorem. Let $p$ be a prime number. Prove that every group of order $p$ is cyclic. Prove that every abelian group of order $p^{2}$ is isomorphic to either $C_{p} \times C_{p}$ or $C_{p^{2} \text {. }}$

  Show that $D_{12}$, the dihedral group of order 12 , is not isomorphic to the alternating $\operatorname{group} A_{4}$.

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• Paper 3, Section II, D

  Groups

  Part IA, 2010

  Let $G$ be a group, $X$ a set on which $G$ acts transitively, $B$ the stabilizer of a point $x \in X$.

  Show that if $g \in G$ stabilizes the point $y \in X$, then there exists an $h \in G$ with $h g h^{-1} \in B$.

  Let $G=S L_{2}(\mathbb{C})$, acting on $\mathbb{C} \cup\{\infty\}$ by Möbius transformations. Compute $B=G_{\infty}$, the stabilizer of $\infty$. Given

  $$g=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in G$$

  compute the set of fixed points $\{x \in \mathbb{C} \cup\{\infty\} \mid g x=x\} .$

  Show that every element of $G$ is conjugate to an element of $B$.

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• Paper 3, Section II, D

  Groups

  Part IA, 2010

  Let $G$ be a finite group, $X$ the set of proper subgroups of $G$. Show that conjugation defines an action of $G$ on $X$.

  Let $B$ be a proper subgroup of $G$. Show that the orbit of $G$ on $X$ containing $B$ has size at most the index $|G: B|$. Show that there exists a $g \in G$ which is not conjugate to an element of $B$.

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• Paper 4, Section I, E

  Numbers and Sets

  Part IA, 2010

  (a) Find the smallest residue $x$ which equals $28 ! 13^{28}(\bmod 31)$.

  [You may use any standard theorems provided you state them correctly.]

  (b) Find all integers $x$ which satisfy the system of congruences

  $$\begin{aligned} x & \equiv 1(\bmod 2) \\ 2 x & \equiv 1(\bmod 3) \\ 2 x & \equiv 4(\bmod 10) \\ x & \equiv 10(\bmod 67) \end{aligned}$$

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• Paper 4, Section I, $2 E$

  Numbers and Sets

  Part IA, 2010

  (a) Let $r$ be a real root of the polynomial $f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$, with integer coefficients $a_{i}$ and leading coefficient 1 . Show that if $r$ is rational, then $r$ is an integer.

  (b) Write down a series for $e$. By considering $q ! e$ for every natural number $q$, show that $e$ is irrational.

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• Paper 4, Section II, E

  Numbers and Sets

  Part IA, 2010

  The Fibonacci numbers $F_{n}$ are defined for all natural numbers $n$ by the rules

  $$F_{1}=1, \quad F_{2}=1, \quad F_{n}=F_{n-1}+F_{n-2} \quad \text { for } n \geqslant 3$$

  Prove by induction on $k$ that, for any $n$,

  $$F_{n+k}=F_{k} F_{n+1}+F_{k-1} F_{n} \text { for all } k \geqslant 2 \text {. }$$

  Deduce that

  $$F_{2 n}=F_{n}\left(F_{n+1}+F_{n-1}\right) \quad \text { for all } n \geqslant 2$$

  Put $L_{1}=1$ and $L_{n}=F_{n+1}+F_{n-1}$ for $n>1$. Show that these (Lucas) numbers $L_{n}$ satisfy

  $$L_{1}=1, \quad L_{2}=3, \quad L_{n}=L_{n-1}+L_{n-2} \quad \text { for } n \geqslant 3$$

  Show also that, for all $n$, the greatest common divisor $\left(F_{n}, F_{n+1}\right)$ is 1 , and that the greatest common divisor $\left(F_{n}, L_{n}\right)$ is at most 2 .

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• Paper 4, Section II, $6 \mathrm{E}$

  Numbers and Sets

  Part IA, 2010

  State and prove Fermat's Little Theorem.

  Let $p$ be an odd prime. If $p \neq 5$, show that $p$ divides $10^{n}-1$ for infinitely many natural numbers $n$.

  Hence show that $p$ divides infinitely many of the integers

  $$5,55, \quad 555, \quad 5555, \quad \ldots .$$

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• Paper 4, Section II, $7 \mathrm{E}$

  Numbers and Sets

  Part IA, 2010

  (a) Let $A, B$ be finite non-empty sets, with $|A|=a,|B|=b$. Show that there are $b^{a}$ mappings from $A$ to $B$. How many of these are injective ?

  (b) State the Inclusion-Exclusion principle.

  (c) Prove that the number of surjective mappings from a set of size $n$ onto a set of size $k$ is

  $$\sum_{i=0}^{k}(-1)^{i}\left(\begin{array}{c} k \\ i \end{array}\right)(k-i)^{n} \quad \text { for } n \geqslant k \geqslant 1$$

  Deduce that

  $$n !=\sum_{i=0}^{n}(-1)^{i}\left(\begin{array}{c} n \\ i \end{array}\right)(n-i)^{n}$$

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• Paper 4 , Section II, E

  Numbers and Sets

  Part IA, 2010

  What does it mean for a set to be countable ?

  Show that $\mathbb{Q}$ is countable, but $\mathbb{R}$ is not. Show also that the union of two countable sets is countable.

  A subset $A$ of $\mathbb{R}$ has the property that, given $\epsilon>0$ and $x \in \mathbb{R}$, there exist reals $a, b$ with $a \in A$ and $b \notin A$ with $|x-a|<\epsilon$ and $|x-b|<\epsilon$. Can $A$ be countable ? Can $A$ be uncountable ? Justify your answers.

  A subset $B$ of $\mathbb{R}$ has the property that given $b \in B$ there exists $\epsilon>0$ such that if $0<|b-x|<\epsilon$ for some $x \in \mathbb{R}$, then $x \notin B$. Is $B$ countable ? Justify your answer.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2010

  (a) State and prove Taylor's theorem with the remainder in Lagrange's form.

  (b) Suppose that $e: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $e(0)=1$ and $e^{\prime}(x)=e(x)$ for all $x \in \mathbb{R}$. Use the result of (a) to prove that

  $$e(x)=\sum_{n \geqslant 0} \frac{x^{n}}{n !} \quad \text { for all } \quad x \in \mathbb{R}$$

  [No property of the exponential function may be assumed.]

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• Paper 2, Section I, F

  Probability

  Part IA, 2010

  Jensen's inequality states that for a convex function $f$ and a random variable $X$ with a finite mean, $\mathbb{E} f(X) \geqslant f(\mathbb{E} X)$.

  (a) Suppose that $f(x)=x^{m}$ where $m$ is a positive integer, and $X$ is a random variable taking values $x_{1}, \ldots, x_{N} \geqslant 0$ with equal probabilities, and where the sum $x_{1}+\ldots+x_{N}=1$. Deduce from Jensen's inequality that

  $$\sum_{i=1}^{N} f\left(x_{i}\right) \geqslant N f\left(\frac{1}{N}\right)$$

  (b) $N$ horses take part in $m$ races. The results of different races are independent. The probability for horse $i$ to win any given race is $p_{i} \geqslant 0$, with $p_{1}+\ldots+p_{N}=1$.

  Let $Q$ be the probability that a single horse wins all $m$ races. Express $Q$ as a polynomial of degree $m$ in the variables $p_{1}, \ldots, p_{N}$.

  By using (1) or otherwise, prove that $Q \geqslant N^{1-m}$.

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• Paper 2, Section I, F

  Probability

  Part IA, 2010

  Let $X$ and $Y$ be two non-constant random variables with finite variances. The correlation coefficient $\rho(X, Y)$ is defined by

  $$\rho(X, Y)=\frac{\mathbb{E}[(X-\mathbb{E} X)(Y-\mathbb{E} Y)]}{(\operatorname{Var} X)^{1 / 2}(\operatorname{Var} Y)^{1 / 2}}$$

  (a) Using the Cauchy-Schwarz inequality or otherwise, prove that

  $$-1 \leqslant \rho(X, Y) \leqslant 1$$

  (b) What can be said about the relationship between $X$ and $Y$ when either (i) $\rho(X, Y)=0$ or (ii) $|\rho(X, Y)|=1$. [Proofs are not required.]

  (c) Take $0 \leqslant r \leqslant 1$ and let $X, X^{\prime}$ be independent random variables taking values $\pm 1$ with probabilities $1 / 2$. Set

  $$Y= \begin{cases}X, & \text { with probability } r \\ X^{\prime}, & \text { with probability } 1-r\end{cases}$$

  Find $\rho(X, Y)$.

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• Paper 2, Section II, F

  Probability

  Part IA, 2010

  (a) What does it mean to say that a random variable $X$ with values $n=1,2, \ldots$ has a geometric distribution with a parameter $p$ where $p \in(0,1)$ ?

  An expedition is sent to the Himalayas with the objective of catching a pair of wild yaks for breeding. Assume yaks are loners and roam about the Himalayas at random. The probability $p \in(0,1)$ that a given trapped yak is male is independent of prior outcomes. Let $N$ be the number of yaks that must be caught until a breeding pair is obtained. (b) Find the expected value of $N$. (c) Find the variance of $N$.

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• Paper 2, Section II, F

  Probability

  Part IA, 2010

  The yearly levels of water in the river Camse are independent random variables $X_{1}, X_{2}, \ldots$, with a given continuous distribution function $F(x)=\mathbb{P}\left(X_{i} \leqslant x\right), x \geqslant 0$ and $F(0)=0$. The levels have been observed in years $1, \ldots, n$ and their values $X_{1}, \ldots, X_{n}$ recorded. The local council has decided to construct a dam of height

  $$Y_{n}=\max \left[X_{1}, \ldots, X_{n}\right]$$

  Let $\tau$ be the subsequent time that elapses before the dam overflows:

  $$\tau=\min \left[t \geqslant 1: X_{n+t}>Y_{n}\right]$$

  (a) Find the distribution function $\mathbb{P}\left(Y_{n} \leqslant z\right), z>0$, and show that the mean value $\mathbb{E} Y_{n}=\int_{0}^{\infty}\left[1-F(z)^{n}\right] \mathrm{d} z .$

  (b) Express the conditional probability $\mathbb{P}\left(\tau=k \mid Y_{n}=z\right)$, where $k=1,2, \ldots$ and $z>0$, in terms of $F$.

  (c) Show that the unconditional probability

  $$\mathbb{P}(\tau=k)=\frac{n}{(k+n-1)(k+n)}, \quad k=1,2, \ldots$$

  (d) Determine the mean value $\mathbb{E} \tau$.

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• Paper 2, Section II, F

  Probability

  Part IA, 2010

  In a branching process every individual has probability $p_{k}$ of producing exactly $k$ offspring, $k=0,1, \ldots$, and the individuals of each generation produce offspring independently of each other and of individuals in preceding generations. Let $X_{n}$ represent the size of the $n$th generation. Assume that $X_{0}=1$ and $p_{0}>0$ and let $F_{n}(s)$ be the generating function of $X_{n}$. Thus

  $$F_{1}(s)=\mathbb{E} s^{X_{1}}=\sum_{k=0}^{\infty} p_{k} s^{k},|s| \leqslant 1$$

  (a) Prove that

  $$F_{n+1}(s)=F_{n}\left(F_{1}(s)\right)$$

  (b) State a result in terms of $F_{1}(s)$ about the probability of eventual extinction. [No proofs are required.]

  (c) Suppose the probability that an individual leaves $k$ descendants in the next generation is $p_{k}=1 / 2^{k+1}$, for $k \geqslant 0$. Show from the result you state in (b) that extinction is certain. Prove further that in this case

  $$F_{n}(s)=\frac{n-(n-1) s}{(n+1)-n s}, \quad n \geqslant 1$$

  and deduce the probability that the $n$th generation is empty.

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• Paper 2, Section II, F

  Probability

  Part IA, 2010

  Let $X_{1}, X_{2}$ be bivariate normal random variables, with the joint probability density function

  $$f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right)=\frac{1}{2 \pi \sigma_{1} \sigma_{2} \sqrt{1-\rho^{2}}} \exp \left[-\frac{\varphi\left(x_{1}, x_{2}\right)}{2\left(1-\rho^{2}\right)}\right]$$

  where

  $$\varphi\left(x_{1}, x_{2}\right)=\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}-2 \rho\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)+\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}$$

  and $x_{1}, x_{2} \in \mathbb{R}$.

  (a) Deduce that the marginal probability density function

  $$f_{X_{1}}\left(x_{1}\right)=\frac{1}{\sqrt{2 \pi} \sigma_{1}} \exp \left[-\frac{\left(x_{1}-\mu_{1}\right)^{2}}{2 \sigma_{1}^{2}}\right]$$

  (b) Write down the moment-generating function of $X_{2}$ in terms of $\mu_{2}$ and $\sigma_{2} \cdot[N o$ proofs are required.]

  (c) By considering the ratio $f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right) / f_{X_{2}}\left(x_{2}\right)$ prove that, conditional on $X_{2}=x_{2}$, the distribution of $X_{1}$ is normal, with mean and variance $\mu_{1}+\rho \sigma_{1}\left(x_{2}-\mu_{2}\right) / \sigma_{2}$ and $\sigma_{1}^{2}\left(1-\rho^{2}\right)$, respectively.

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• Paper 3, Section I, C

  Vector Calculus

  Part IA, 2010

  Consider the vector field

  $$\mathbf{F}=\left(-y /\left(x^{2}+y^{2}\right), x /\left(x^{2}+y^{2}\right), 0\right)$$

  defined on all of $\mathbb{R}^{3}$ except the $z$ axis. Compute $\boldsymbol{\nabla} \times \mathbf{F}$ on the region where it is defined.

  Let $\gamma_{1}$ be the closed curve defined by the circle in the $x y$-plane with centre $(2,2,0)$ and radius 1 , and $\gamma_{2}$ be the closed curve defined by the circle in the $x y$-plane with centre $(0,0,0)$ and radius 1 .

  By using your earlier result, or otherwise, evaluate the line integral $\oint_{\gamma_{1}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$.

  By explicit computation, evaluate the line integral $\oint_{\gamma_{2}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$. Is your result consistent with Stokes' theorem? Explain your answer briefly.

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• Paper 3, Section I, C

  Vector Calculus

  Part IA, 2010

  A curve in two dimensions is defined by the parameterised Cartesian coordinates

  $$x(u)=a e^{b u} \cos u, \quad y(u)=a e^{b u} \sin u$$

  where the constants $a, b>0$. Sketch the curve segment corresponding to the range $0 \leqslant u \leqslant 3 \pi$. What is the length of the curve segment between the points $(x(0), y(0))$ and $(x(U), y(U))$, as a function of $U$ ?

  A geometrically sensitive ant walks along the curve with varying speed $\kappa(u)^{-1}$, where $\kappa(u)$ is the curvature at the point corresponding to parameter $u$. Find the time taken by the ant to walk from $(x(2 n \pi), y(2 n \pi))$ to $(x(2(n+1) \pi), y(2(n+1) \pi))$, where $n$ is a positive integer, and hence verify that this time is independent of $n$.

  [You may quote without proof the formula $\kappa(u)=\frac{\left|x^{\prime}(u) y^{\prime \prime}(u)-y^{\prime}(u) x^{\prime \prime}(u)\right|}{\left(\left(x^{\prime}(u)\right)^{2}+\left(y^{\prime}(u)\right)^{2}\right)^{3 / 2}} .$ ]

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• Paper 3, Section II, C

  Vector Calculus

  Part IA, 2010

  (a) Define a rank two tensor and show that if two rank two tensors $A_{i j}$ and $B_{i j}$ are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.

  The quantity $C_{i j}$ has the property that, for every rank two tensor $A_{i j}$, the quantity $C_{i j} A_{i j}$ is a scalar. Is $C_{i j}$ necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.

  (b) Show that, if a tensor $T_{i j}$ is invariant under rotations about the $x_{3}$-axis, then it has the form

  $$\left(\begin{array}{ccc} \alpha & \omega & 0 \\ -\omega & \alpha & 0 \\ 0 & 0 & \beta \end{array}\right)$$

  (c) The inertia tensor about the origin of a rigid body occupying volume $V$ and with variable mass density $\rho(\mathbf{x})$ is defined to be

  $$I_{i j}=\int_{V} \rho(\mathbf{x})\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \mathrm{d} V$$

  The rigid body $B$ has uniform density $\rho$ and occupies the cylinder

  $$\left\{\left(x_{1}, x_{2}, x_{3}\right):-2 \leqslant x_{3} \leqslant 2, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\}$$

  Show that the inertia tensor of $B$ about the origin is diagonal in the $\left(x_{1}, x_{2}, x_{3}\right)$ coordinate system, and calculate its diagonal elements.

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• Paper 3, Section II, C

  Vector Calculus

  Part IA, 2010

  Let $f(x, y)$ be a function of two variables, and $R$ a region in the $x y$-plane. State the rule for evaluating $\int_{R} f(x, y) \mathrm{d} x \mathrm{~d} y$ as an integral with respect to new variables $u(x, y)$ and $v(x, y)$.

  Sketch the region $R$ in the $x y$-plane defined by

  $$R=\left\{(x, y): x^{2}+y^{2} \leqslant 2, x^{2}-y^{2} \geqslant 1, x \geqslant 0, y \geqslant 0\right\}$$

  Sketch the corresponding region in the $u v$-plane, where

  $$u=x^{2}+y^{2}, \quad v=x^{2}-y^{2}$$

  Express the integral

  $$I=\int_{R}\left(x^{5} y-x y^{5}\right) \exp \left(4 x^{2} y^{2}\right) \mathrm{d} x \mathrm{~d} y$$

  as an integral with respect to $u$ and $v$. Hence, or otherwise, calculate $I$.

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• Paper 1, Section II, D

  Analysis I

  Part IA, 2010

  Define what it means for a bounded function $f:[a, \infty) \rightarrow \mathbb{R}$ to be Riemann integrable.

  Show that a monotonic function $f:[a, b] \rightarrow \mathbb{R}$ is Riemann integrable, where $-\infty<a<b<\infty$.

  Prove that if $f:[1, \infty) \rightarrow \mathbb{R}$ is a decreasing function with $f(x) \rightarrow 0$ as $x \rightarrow \infty$, then $\sum_{n \geqslant 1} f(n)$ and $\int_{1}^{\infty} f(x) d x$ either both diverge or both converge.

  Hence determine, for $\alpha \in \mathbb{R}$, when $\sum_{n \geqslant 1} n^{\alpha}$ converges.

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• Paper 3 , Section II, C

  Vector Calculus

  Part IA, 2010

  State the divergence theorem (also known as Gauss' theorem) relating the surface and volume integrals of appropriate fields.

  The surface $S_{1}$ is defined by the equation $z=3-2 x^{2}-2 y^{2}$ for $1 \leqslant z \leqslant 3$; the surface $S_{2}$ is defined by the equation $x^{2}+y^{2}=1$ for $0 \leqslant z \leqslant 1$; the surface $S_{3}$ is defined by the equation $z=0$ for $x, y$ satisfying $x^{2}+y^{2} \leqslant 1$. The surface $S$ is defined to be the union of the surfaces $S_{1}, S_{2}$ and $S_{3}$. Sketch the surfaces $S_{1}, S_{2}, S_{3}$ and (hence) $S$.

  The vector field $\mathbf{F}$ is defined by

  $$\mathbf{F}(x, y, z)=\left(x y+x^{6},-\frac{1}{2} y^{2}+y^{8}, z\right)$$

  Evaluate the integral

  $$\oint_{S} \mathbf{F} \cdot \mathrm{d} \mathbf{S}$$

  where the surface element $\mathrm{d} \mathbf{S}$ points in the direction of the outward normal to $S$.

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• Paper 3, Section II, C

  Vector Calculus

  Part IA, 2010

  Given a spherically symmetric mass distribution with density $\rho$, explain how to obtain the gravitational field $\mathbf{g}=-\nabla \phi$, where the potential $\phi$ satisfies Poisson's equation

  $$\nabla^{2} \phi=4 \pi G \rho$$

  The remarkable planet Geometria has radius 1 and is composed of an infinite number of stratified spherical shells $S_{n}$ labelled by integers $n \geqslant 1$. The shell $S_{n}$ has uniform density $2^{n-1} \rho_{0}$, where $\rho_{0}$ is a constant, and occupies the volume between radius $2^{-n+1}$ and $2^{-n}$.

  Obtain a closed form expression for the mass of Geometria.

  Obtain a closed form expression for the gravitational field $\mathbf{g}$ due to Geometria at a distance $r=2^{-N}$ from its centre of mass, for each positive integer $N \geqslant 1$. What is the potential $\phi(r)$ due to Geometria for $r>1$ ?

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• Paper 1, Section I, $1 A$

  Vectors and Matrices

  Part IA, 2010

  Let $A$ be the matrix representing a linear map $\Phi: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ with respect to the bases $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$ of $\mathbb{R}^{n}$ and $\left\{\mathbf{c}_{1}, \ldots, \mathbf{c}_{m}\right\}$ of $\mathbb{R}^{m}$, so that $\Phi\left(\mathbf{b}_{i}\right)=A_{j i} \mathbf{c}_{j}$. Let $\left\{\mathbf{b}_{1}^{\prime}, \ldots, \mathbf{b}_{n}^{\prime}\right\}$ be another basis of $\mathbb{R}^{n}$ and let $\left\{\mathbf{c}_{1}^{\prime}, \ldots, \mathbf{c}_{m}^{\prime}\right\}$ be another basis of $\mathbb{R}^{m}$. Show that the matrix $A^{\prime}$ representing $\Phi$ with respect to these new bases satisfies $A^{\prime}=C^{-1} A B$ with matrices $B$ and $C$ which should be defined.

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• Paper 1, Section I, C

  Vectors and Matrices

  Part IA, 2010

  (a) The complex numbers $z_{1}$ and $z_{2}$ satisfy the equations

  $$z_{1}^{3}=1, \quad z_{2}^{9}=512 .$$

  What are the possible values of $\left|z_{1}-z_{2}\right|$ ? Justify your answer.

  (b) Show that $\left|z_{1}+z_{2}\right| \leqslant\left|z_{1}\right|+\left|z_{2}\right|$ for all complex numbers $z_{1}$ and $z_{2}$. Does the inequality $\left|z_{1}+z_{2}\right|+\left|z_{1}-z_{2}\right| \leqslant 2 \max \left(\left|z_{1}\right|,\left|z_{2}\right|\right)$ hold for all complex numbers $z_{1}$ and $z_{2}$ ? Justify your answer with a proof or a counterexample.

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• Paper 1, Section II, A

  Vectors and Matrices

  Part IA, 2010

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

  (i) Define the trace of $A, \operatorname{tr}(A)$, and show that $\operatorname{tr}\left(A^{T} B\right)=\operatorname{tr}\left(B^{T} A\right)$.

  (ii) Show that $\operatorname{tr}\left(A^{T} A\right) \geqslant 0$, with $\operatorname{tr}\left(A^{T} A\right)=0$ if and only if $A$ is the zero matrix. Hence show that

  $$\left(\operatorname{tr}\left(A^{T} B\right)\right)^{2} \leqslant \operatorname{tr}\left(A^{T} A\right) \operatorname{tr}\left(B^{T} B\right)$$

  Under what condition on $A$ and $B$ is equality achieved?

  (iii) Find a basis for the subspace of $2 \times 2$ matrices $X$ such that

  $$\begin{gathered} \operatorname{tr}\left(A^{T} X\right)=\operatorname{tr}\left(B^{T} X\right)=\operatorname{tr}\left(C^{T} X\right)=0 \\ \text { where } \quad A=\left(\begin{array}{ll} 1 & 1 \\ 2 & 0 \end{array}\right), \quad B=\left(\begin{array}{rr} 1 & 1 \\ 0 & -2 \end{array}\right), \quad C=\left(\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right) \end{gathered}$$

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• Paper 1, Section II, $6 \mathbf{C}$

  Vectors and Matrices

  Part IA, 2010

  Let $\mathbf{a}_{1}, \mathbf{a}_{2}$ and $\mathbf{a}_{3}$ be vectors in $\mathbb{R}^{3}$. Give a definition of the dot product, $\mathbf{a}_{1} \cdot \mathbf{a}_{2}$, the cross product, $\mathbf{a}_{1} \times \mathbf{a}_{2}$, and the triple product, $\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}$. Explain what it means to say that the three vectors are linearly independent.

  Let $\mathbf{b}_{1}, \mathbf{b}_{2}$ and $\mathbf{b}_{3}$ be vectors in $\mathbb{R}^{3}$. Let $S$ be a $3 \times 3$ matrix with entries $S_{i j}=\mathbf{a}_{i} \cdot \mathbf{b}_{j}$. Show that

  $$\left(\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}\right)\left(\mathbf{b}_{1} \cdot \mathbf{b}_{2} \times \mathbf{b}_{3}\right)=\operatorname{det}(S)$$

  Hence show that $S$ is of maximal rank if and only if the sets of vectors $\left\{\mathbf{a}_{1}, \mathbf{a}_{2}\right.$, $\left.\mathbf{a}_{3}\right\}$ and $\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ are both linearly independent.

  Now let $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ and $\left\{\mathbf{d}_{1}, \mathbf{d}_{2}, \ldots, \mathbf{d}_{n}\right\}$ be sets of vectors in $\mathbb{R}^{n}$, and let $T$ be an $n \times n$ matrix with entries $T_{i j}=\mathbf{c}_{i} \cdot \mathbf{d}_{j}$. Is it the case that $T$ is of maximal rank if and only if the sets of vectors $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ and $\left\{\mathbf{d}_{1}, \mathbf{d}_{2}, \ldots, \mathbf{d}_{n}\right\}$ are both linearly independent? Justify your answer with a proof or a counterexample.

  Given an integer $n>2$, is it always possible to find a set of vectors $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ in $\mathbb{R}^{n}$ with the property that every pair is linearly independent and that every triple is linearly dependent? Justify your answer.

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• Paper 1, Section II, B

  Vectors and Matrices

  Part IA, 2010

  Let $A$ be a complex $n \times n$ matrix with an eigenvalue $\lambda$. Show directly from the definitions that:

  (i) $A^{r}$ has an eigenvalue $\lambda^{r}$ for any integer $r \geqslant 1$; and

  (ii) if $A$ is invertible then $\lambda \neq 0$ and $A^{-1}$ has an eigenvalue $\lambda^{-1}$.

  For any complex $n \times n$ matrix $A$, let $\chi_{A}(t)=\operatorname{det}(A-t I)$. Using standard properties of determinants, show that:

  (iii) $\chi_{A^{2}}\left(t^{2}\right)=\chi_{A}(t) \chi_{A}(-t)$; and

  (iv) if $A$ is invertible,

  $$\chi_{A^{-1}}(t)=(\operatorname{det} A)^{-1}(-1)^{n} t^{n} \chi_{A}\left(t^{-1}\right)$$

  Explain, including justifications, the relationship between the eigenvalues of $A$ and the polynomial $\chi_{A}(t)$.

  If $A^{4}$ has an eigenvalue $\mu$, does it follow that $A$ has an eigenvalue $\lambda$ with $\lambda^{4}=\mu$ ? Give a proof or counterexample.

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• Paper 1, Section II, B

  Vectors and Matrices

  Part IA, 2010

  Let $R$ be a real orthogonal $3 \times 3$ matrix with a real eigenvalue $\lambda$ corresponding to some real eigenvector. Show algebraically that $\lambda=\pm 1$ and interpret this result geometrically.

  Each of the matrices

  $$M=\left(\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right), \quad N=\left(\begin{array}{rrr} 1 & -2 & -2 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right), \quad P=\frac{1}{3}\left(\begin{array}{rrr} 1 & -2 & -2 \\ -2 & 1 & -2 \\ -2 & -2 & 1 \end{array}\right)$$

  has an eigenvalue $\lambda=1$. Confirm this by finding as many independent eigenvectors as possible with this eigenvalue, for each matrix in turn.

  Show that one of the matrices above represents a rotation, and find the axis and angle of rotation. Which of the other matrices represents a reflection, and why?

  State, with brief explanations, whether the matrices $M, N, P$ are diagonalisable (i) over the real numbers; (ii) over the complex numbers.

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• Paper 1, Section II, F

  Analysis I

  Part IA, 2010

  (a) Let $n \geqslant 1$ and $f$ be a function $\mathbb{R} \rightarrow \mathbb{R}$. Define carefully what it means for $f$ to be $n$ times differentiable at a point $x_{0} \in \mathbb{R}$.

  $$\text { Set } \operatorname{sign}(x)= \begin{cases}x /|x|, & x \neq 0 \\ 0, & x=0 .\end{cases}$$

  Consider the function $f(x)$ on the real line, with $f(0)=0$ and

  $$f(x)=x^{2} \operatorname{sign}(x)\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 .$$

  (b) Is $f(x)$ differentiable at $x=0$ ?

  (c) Show that $f(x)$ has points of non-differentiability in any neighbourhood of $x=0$.

  (d) Prove that, in any finite interval $I$, the derivative $f^{\prime}(x)$, at the points $x \in I$ where it exists, is bounded: $\left|f^{\prime}(x)\right| \leqslant C$ where $C$ depends on $I$.

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• Paper 2, Section I, A

  Differential Equations

  Part IA, 2010

  Find the general solutions to the following difference equations for $y_{n}, n \in \mathbb{N}$.

  $$\begin{aligned} \text { (i) } & y_{n+3}-3 y_{n+1}+2 y_{n}=0, \\ \text { (ii) } & y_{n+3}-3 y_{n+1}+2 y_{n}=2^{n} \\ \text { (iii) } & y_{n+3}-3 y_{n+1}+2 y_{n}=(-2)^{n} \\ \text { (iv) } & y_{n+3}-3 y_{n+1}+2 y_{n}=(-2)^{n}+2^{n} . \end{aligned}$$

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• Paper 2, Section I, A

  Differential Equations

  Part IA, 2010

  Let $f(x, y)=g(u, v)$ where the variables $\{x, y\}$ and $\{u, v\}$ are related by a smooth, invertible transformation. State the chain rule expressing the derivatives $\frac{\partial g}{\partial u}$ and $\frac{\partial g}{\partial v}$ in terms of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ and use this to deduce that

  $$\frac{\partial^{2} g}{\partial u \partial v}=\frac{\partial x}{\partial u} \frac{\partial x}{\partial v} \frac{\partial^{2} f}{\partial x^{2}}+\left(\frac{\partial x}{\partial u} \frac{\partial y}{\partial v}+\frac{\partial x}{\partial v} \frac{\partial y}{\partial u}\right) \frac{\partial^{2} f}{\partial x \partial y}+\frac{\partial y}{\partial u} \frac{\partial y}{\partial v} \frac{\partial^{2} f}{\partial y^{2}}+H \frac{\partial f}{\partial x}+K \frac{\partial f}{\partial y}$$

  where $H$ and $K$ are second-order partial derivatives, to be determined.

  Using the transformation $x=u v$ and $y=u / v$ in the above identity, or otherwise, find the general solution of

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

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• Paper 2, Section II, A

  Differential Equations

  Part IA, 2010

  (a) Consider the differential equation

  $$a_{n} \frac{d^{n} y}{d x^{n}}+a_{n-1} \frac{d^{n-1} y}{d x^{n-1}}+\ldots+a_{2} \frac{d^{2} y}{d x^{2}}+a_{1} \frac{d y}{d x}+a_{0} y=0$$

  with $n \in \mathbb{N}$ and $a_{0}, \ldots, a_{n} \in \mathbb{R}$. Show that $y(x)=e^{\lambda x}$ is a solution if and only if $p(\lambda)=0$ where

  $$p(\lambda)=a_{n} \lambda^{n}+a_{n-1} \lambda^{n-1}+\ldots+a_{2} \lambda^{2}+a_{1} \lambda+a_{0}$$

  Show further that $y(x)=x e^{\mu x}$ is also a solution of $(1)$ if $\mu$ is a root of the polynomial $p(\lambda)$ of multiplicity at least 2 .

  (b) By considering $v(t)=\frac{d^{2} u}{d t^{2}}$, or otherwise, find the general real solution for $u(t)$ satisfying

  $$\frac{d^{4} u}{d t^{4}}+2 \frac{d^{2} u}{d t^{2}}=4 t^{2}$$

  By using a substitution of the form $u(t)=y\left(t^{2}\right)$ in $(2)$, or otherwise, find the general real solution for $y(x)$, with $x$ positive, where

  $$4 x^{2} \frac{d^{4} y}{d x^{4}}+12 x \frac{d^{3} y}{d x^{3}}+(3+2 x) \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=x$$

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• Paper 2, Section II, $6 \mathrm{~A}$

  Differential Equations

  Part IA, 2010

  (a) By using a power series of the form

  $$y(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$

  or otherwise, find the general solution of the differential equation

  $$x y^{\prime \prime}-(1-x) y^{\prime}-y=0 .$$

  (b) Define the Wronskian $W(x)$ for a second order linear differential equation

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

  and show that $W^{\prime}+p(x) W=0$. Given a non-trivial solution $y_{1}(x)$ of $(2)$ show that $W(x)$ can be used to find a second solution $y_{2}(x)$ of $(2)$ and give an expression for $y_{2}(x)$ in the form of an integral.

  (c) Consider the equation (2) with

  $$p(x)=-\frac{P(x)}{x} \quad \text { and } \quad q(x)=-\frac{Q(x)}{x}$$

  where $P$ and $Q$ have Taylor expansions

  $$P(x)=P_{0}+P_{1} x+\ldots, \quad Q(x)=Q_{0}+Q_{1} x+\ldots$$

  with $P_{0}$ a positive integer. Find the roots of the indicial equation for (2) with these assumptions. If $y_{1}(x)=1+\beta x+\ldots$ is a solution, use the method of part (b) to find the first two terms in a power series expansion of a linearly independent solution $y_{2}(x)$, expressing the coefficients in terms of $P_{0}, P_{1}$ and $\beta$.

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