\documentclass[a4, 11pt]{report} \pagestyle{myheadings} \markboth{}{Paper III, 2015 \ \ \ \ \ \today } \RequirePackage{amssymb} \RequirePackage{amsmath} \RequirePackage{graphicx} \RequirePackage{color} \RequirePackage[flushleft]{paralist}[2013/06/09] \RequirePackage{geometry} \geometry{% a4paper, lmargin=2cm, rmargin=2.5cm, tmargin=3.5cm, bmargin=2.5cm, footskip=12pt, headheight=24pt} \makeatletter \newcommand{\raisemath}[1]{\mathpalette{\raisem@th{#1}}} \newcommand{\raisem@th}[3]{\raisebox{#1}{$#2#3$}} \makeatother \newcommand{\comment}[1]{{\bf Comment} {\it #1}} %\renewcommand{\comment}[1]{} \newcommand{\bluecomment}[1]{{\color{blue}#1}} %\renewcommand{\comment}[1]{} \newcommand{\redcomment}[1]{{\color{red}#1}} \usepackage{epsfig} \usepackage{pstricks-add} \usepackage{tgheros} %% changes sans-serif font to TeX Gyre Heros (tex-gyre) \renewcommand{\familydefault}{\sfdefault} %% changes font to sans-serif %\usepackage{sfmath} %%%% this makes equation sans-serif %\input RexFigs \setlength{\parskip}{10pt} \setlength{\parindent}{0pt} \newlength{\qspace} \setlength{\qspace}{20pt} \newcounter{qnumber} \setcounter{qnumber}{0} \newenvironment{question}% {\vspace{\qspace} \begin{enumerate}[\bfseries 1\quad][10]% \setcounter{enumi}{\value{qnumber}}% \item% } { \end{enumerate} \filbreak \stepcounter{qnumber} } \newenvironment{questionparts}[1][1]% { \begin{enumerate}[\bfseries (i)]% \setcounter{enumii}{#1} \addtocounter{enumii}{-1} \setlength{\itemsep}{5mm} \setlength{\parskip}{8pt} } { \end{enumerate} } \DeclareMathOperator{\cosec}{cosec} \DeclareMathOperator{\Var}{Var} \def\d{{\mathrm d}} \def\e{{\mathrm e}} \def\g{{\mathrm g}} \def\h{{\mathrm h}} \def\f{{\mathrm f}} \def\p{{\mathrm p}} \def\s{{\mathrm s}} \def\t{{\mathrm t}} \def\i{{\mathrm i}} \def\A{{\mathrm A}} \def\B{{\mathrm B}} \def\E{{\mathrm E}} \def\F{{\mathrm F}} \def\G{{\mathrm G}} \def\H{{\mathrm H}} \def\P{{\mathrm P}} \def\bb{\mathbf b} \def \bc{\mathbf c} \def\bx {\mathbf x} \def\bn {\mathbf n} \newcommand{\low}{^{\vphantom{()}}} %%%%% to lower suffices: $X\low_1$ etc \newcommand{\subone}{ {\vphantom{\dot A}1}} \newcommand{\subtwo}{ {\vphantom{\dot A}2}} \DeclareMathOperator{\sech}{sech} \def\le{\leqslant} \def\ge{\geqslant} \def\arcosh{{\rm arcosh}\,} \def\var{{\rm Var}\,} \newcommand{\ds}{\displaystyle} \newcommand{\ts}{\textstyle} \def\half{{\textstyle \frac12}} \def\l{\left(} \def\r{\right)} \renewcommand{\.}[1]{\ensuremath{\mathrm{#1}}} \newcommand{\+}[1]{\ensuremath{\mathbf{#1}}} \newcommand{\ud}{\mathop{}\!\mathrm{d}} \begin{document} \setcounter{page}{2} \section*{Section A: \ \ \ Pure Mathematics} %%%%%%%%%%Q1 \begin{question} \begin{questionparts} \item Let \[ I_n= \int_0^\infty \frac 1 {(1+u^2)^n}\, \d u \,, \] where $n$ is a positive integer. Show that \[ I_n - I_{n+1} = \frac 1 {2n} I_n \] and deduce that \[ I_{n+1} = \frac{(2n)!\, \pi}{2^{2n+1}(n!)^2} \,. \] \item Let \[ J = \int_0^\infty \f\big( (x- x^{-1})^2\big ) \, \d x \,, \] where $\f$ is any function for which the integral exists. Show that \[ J = \int_0^\infty x^{-2} \f\big( (x- x^{-1})^2\big) \, \d x \, = \frac12 \int_0^\infty (1 + x^{-2}) \f\big( (x- x^{-1})^2\big ) \, \d x \, = \int_0^\infty \f\big(u^2\big) \,\d u \,. \] \item Hence evaluate \[ \int_0^\infty \frac {x^{2n-2}}{(x^4-x^2+1)^n} \, \d x \,, \] where $n$ is a positive integer. \end{questionparts} \end{question} %%%%%%%%%%Q2 \begin{question} If $s_1$, $s_2$, $s_3$, $\ldots$ and $t_1$, $t_2$, $t_3$, $\ldots$ are sequences of positive numbers, we write \[ (s_n)\le (t_n) \] to mean \begin{center} "there exists a positive integer $m$ such that $s_n \le t_n$ whenever $n\ge m$". \end{center} Determine whether each of the following statements is true or false. In the case of a true statement, you should give a proof which includes an explicit determination of an appropriate $m$; in the case of a false statement, you should give a counterexample. \begin{questionparts} \item $(1000n) \le (n^2)\,$. \item If it is not the case that $(s_n)\le (t_n)$, then it is the case that $(t_n)\le (s_n)\,$. \item If $(s_n)\le (t_n)$ and $(t_n) \le (u_n)$, then $(s_n)\le (u_n)\,$. \item $(n^2)\le (2^n)\,$. \end{questionparts} \end{question} %%%%%%%%% Q3 \begin{question} In this question, $r$ and $\theta$ are polar coordinates with $r \ge0$ and $- \pi <\theta\le \pi$, and $a$ and $b$ are positive constants. Let $L$ be a fixed line and let $A$ be a fixed point not lying on $L$. Then the locus of points that are a fixed distance (call it $d$) from $L$ measured along lines through $A$ is called a {\em conchoid of Nicomedes}. \begin{questionparts} \item Show that if \[ \vert r- a \sec\theta \vert = b\,, \tag{$*$} \] where $a>b$, then $\sec\theta >0$. Show that all points with coordinates satisfying ($*$) lie on a certain conchoid of Nicomedes (you should identify $L$, $d$ and $A$). Sketch the locus of these points. \item In the case $a0$ and $-\pi < \theta_k <\pi$, satisfy \[ \textstyle \sum\limits _{k=1}^3 r_k \sin (\theta_k) = 0\,, \ \ \ \ \textstyle \sum\limits _{k=1}^3 r_k^2 \sin (2\theta_k) = 0\,, \ \ \ \ \ \textstyle \sum\limits _{k=1}^3 r_k^3 \sin (3\theta_k) = 0\, . \] Show that $\theta_k=0$ for at least one value of $k$. Show further that if $\theta_1=0$ then $\theta_2 = - \theta_3\,$. \end{questionparts} \end{question} %%%%%%%%% Q5 \begin{question} \begin{questionparts} \item In the following argument to show that $\sqrt2$ is irrational, give proofs appropriate for steps 3, 5 and~6. 1. Assume that $\sqrt2$ is rational. 2. Define the set $S$ to be the set of positive integers with the following property: \begin{center} $n$ is in $S$ if and only if $n \sqrt2$ is an integer. \end{center} 3. Show that the set $S$ contains at least one positive integer. 4. Define the integer $k$ to be the smallest positive integer in $S$. 5. Show that $(\sqrt2-1)k$ is in $S$. 6. Show that steps 4 and 5 are contradictory and hence that $\sqrt2$ is irrational. \item Prove that $2^{\raisemath{2pt}{\frac13}} $ is rational if and only if $2^{\raisemath{2pt}{\frac23}}$ is rational. Use an argument similar to that of part (i) to prove that $2^{\raisemath{2pt}{\frac13}} $ and $2^{\raisemath{2pt}{\frac23}}$ are irrational. \end{questionparts} \end{question} %%%%%%%%% Q6 \begin{question} \begin{questionparts} \item Let $w$ and $z$ be complex numbers, and let $u= w+z$ and $v=w^2+z^2$. Prove that $w$ and $z$ are real if and only if $u$ and $v$ are real and $u^2\le2v$. \item The complex numbers $u$, $w$ and $z$ satisfy the equations \begin{align*} w+z-u&=0 \\ w^2+z^2 -u^2 &= - \tfrac 23 \\ w^3+z^3 -\lambda u &= -\lambda\, \end{align*} where $\lambda $ is a positive real number. Show that for all values of $\lambda$ except one (which you should find) there are three possible values of $u$, all real. Are $w$ and $z$ necessarily real? Give a proof or counterexample. \end{questionparts} \end{question} %%%%%%%%% Q7 \begin{question} An operator $\rm D$ is defined, for any function $\f$, by \[ {\rm D}\f(x) = x\frac{\d\f(x)}{\d x} .\] The notation ${\rm D}^n$ means that $\rm D$ is applied $n$ times; for example \[ \displaystyle {\rm D}^2\f(x) = x\frac{\d\ }{\d x}\left( x\frac{\d\f(x)}{\d x} \right) \,. \] Show that, for any constant $a$, ${\rm D}^2 x^a = a^2 x^a\,$. \begin{questionparts} \item Show that if $\P(x)$ is a polynomial of degree $r$ (where $r\ge1$) then, for any positive integer $n$, ${\rm D}^n\P(x)$ is also a polynomial of degree $r$. \item Show that if $n$ and $m$ are positive integers with $n0$, the differential equation \[ (y+x)\frac{\d y}{\d x} = y-x \] becomes \[ \frac{\d r}{\d\theta} + r=0 \,. \] Sketch a solution in the $x$-$y$ plane. \item Show that the solutions of \[ \left( y+x -x(x^2+y^2) \right) \, \frac{\d y }{\d x} = y-x - y(x^2+y^2) \] can be written in the form \\ \[ r^2 = \dfrac 1 {1+A\e^{2\theta}}\, \]\\ and sketch the different forms of solution that arise according to the value of $A$. \end{questionparts} \end{question} \newpage \section*{Section B: \ \ \ Mechanics} %%%%%%%%%% Q9 \begin{question} A particle $P$ of mass $m$ moves on a smooth fixed straight horizontal rail and is attached to a fixed peg $Q$ by a light elastic string of natural length $a$ and modulus $\lambda$. The peg $Q$ is a distance $a$ from the rail. Initially $P$ is at rest with $PQ=a$. An impulse imparts to $P$ a speed $v$ along the rail. Let $x$ be the displacement at time $t$ of $P$ from its initial position. Obtain the equation \[ \dot x^2 = v^2 - k^2 \left( \sqrt{x^2+a^2} -a\right)^{\!2} \] where $ k^2 = \lambda/(ma)$, $k>0$ and the dot denotes differentiation with respect to $t$. Find, in terms of $k$, $a$ and $v$, the greatest value, $x_0$, attained by $x$. Find also the acceleration of $P$ at $x=x_0$. Obtain, in the form of an integral, an expression for the period of the motion. Show that in the case $v\ll ka$ (that is, $v$ is much less than $ka$), this is approximately \[ \sqrt {\frac {32a}{kv}} \int_0^1 \frac 1 {\sqrt{1-u^4}} \, \d u \, . \] \end{question} %%%%%%%%%% Q10 \begin{question} A light rod of length $2a$ has a particle of mass $m$ attached to each end and it moves in a vertical plane. The midpoint of the rod has coordinates $(x,y)$, where the $x$-axis is horizontal (within the plane of motion) and $y$ is the height above a horizontal table. Initially, the rod is vertical, and at time $t$ later it is inclined at an angle $\theta$ to the vertical. Show that the velocity of one particle can be written in the form \[ \begin{pmatrix} \dot x + a \dot\theta \cos\theta \\ \dot y - a \dot\theta \sin\theta \end{pmatrix} \] and that \[ m\begin{pmatrix} \ddot x + a\ddot\theta \cos\theta - a \dot\theta^2 \sin\theta \\ \ddot y- a\ddot\theta \sin\theta - a \dot\theta^2 \cos\theta \end{pmatrix} =-T\begin{pmatrix} \sin\theta \\ \cos\theta \end{pmatrix} -mg \begin{pmatrix} 0 \\ 1 \end{pmatrix} \] where the dots denote differentiation with respect to time $t$ and $T$ is the tension in the rod. Obtain the corresponding equations for the other particle. Deduce that $\ddot x =0$, $\ddot y = -g$ and $\ddot\theta =0$. Initially, the midpoint of the rod is a height $h$ above the table, the velocity of the higher particle is $\Big(\begin{matrix} \, u \, \\ v \end{matrix}\Big)$, and the velocity of the lower particle is $\Big(\begin{matrix}\, 0 \, \\ v\end{matrix}\Big)$. Given that the two particles hit the table for the first time simultaneously, when the rod has rotated by $\frac12\pi$, show that \[ 2hu^2 = \pi^2a^2 g - 2\pi uva \,. \] \end{question} %%%%%%%%%% Q11 \begin{question} \begin{questionparts} \item A horizontal disc of radius $r$ rotates about a vertical axis through its centre with angular speed $\omega$. One end of a light rod is fixed by a smooth hinge to the edge of the disc so that it can rotate freely in a vertical plane through the centre of the disc. A particle $P$ of mass $m$ is attached to the rod at a distance $d$ from the hinge. The rod makes a constant angle~$\alpha$ with the upward vertical, as shown in the diagram, and $d\sin\alpha