\documentclass[a4, 11pt]{report} \pagestyle{myheadings} \markboth{}{Paper III, 2004 \ \ \ \ \ \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} \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\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}} \def\le{\leqslant} \def\ge{\geqslant} \def\var{{\rm Var}\,} \newcommand{\ds}{\displaystyle} \newcommand{\ts}{\textstyle} \def\half{{\textstyle \frac12}} \def\l{\left(} \def\r{\right)} \begin{document} \setcounter{page}{2} \section*{Section A: \ \ \ Pure Mathematics} %%%%%%%%%%Q1 \begin{question} Show that \[ \int_0^a \frac{\sinh x}{2\cosh^2 x -1} \, \mathrm{d} x = \frac{1}{2 \sqrt{2}} \ln \l \frac{\sqrt{2}\cosh a -1}{\sqrt{2}\cosh a +1}\r + \frac{1}{2 \sqrt{2}} \ln \l \frac{\sqrt{2}+1}{\sqrt{2}-1}\r \] and find \[ \int_0^a \frac{\cosh x}{1+2\sinh^2 x} \, \mathrm{d} x \, . \] Hence show that \[ \int_0^\infty \frac{\cosh x - \sinh x}{1+2\sinh^2 x} \, \mathrm{d} x = \frac{\pi}{2\sqrt{2}} - \frac{1}{2 \sqrt{2}} \ln \l \frac{\sqrt{2}+1}{\sqrt{2}-1}\r \, . \] By substituting $u = \e^x$ in this result, or otherwise, find \[ \int_1^\infty \frac{1}{1+u^4} \, \mathrm{d} u \, . \] \end{question} %%%%%%%%%%Q2 \begin{question} The equation of a curve is $y=\f ( x )$ where \[ \f ( x ) = x-4-{16 \l 2x+1 \r^2 \over x^2 \l x - 4 \r} \;. \] \begin{questionparts} \item Write down the equations of the vertical and oblique asymptotes to the curve and show that the oblique asymptote is a tangent to the curve. \item Show that the equation $\f ( x ) =0$ has a double root. \item Sketch the curve. \end{questionparts} \end{question} %%%%%%%%% Q3 \begin{question} Given that $\f''(x) > 0$ when $a \le x \le b\,$, explain with the aid of a sketch why \[ (b-a) \, \f \Big( {a+b \over 2} \Big) < \int^b_a \f(x) \, \mathrm{d}x < (b-a) \, \displaystyle \frac{\f(a) + \f(b)}{2} \;. \] By choosing suitable $a$, $b$ and $\f(x)\,$, show that \[ {4 \over (2n-1)^2} < {1 \over n-1} - {1 \over n} < {1 \over 2} \l {1 \over n^2} + {1 \over (n-1)^2}\r \,, \] where $n$ is an integer greater than 1. Deduce that \[ 4 \l {1 \over 3^2} +{1 \over 5^2} + {1 \over 7^2} + \cdots \r < 1 < {1 \over 2} + \left( {1 \over 2^2} +{1 \over 3^2} + {1 \over 4^2} + \cdots \right)\,. \] Show that \[ {1 \over 2} \l {1 \over 3^2} + {1 \over 4^2} + {1 \over 5^2} + \frac 1 {6^2} + \cdots \right) < {1 \over 3^2} +{1 \over 5^2} + {1 \over 7^2} + \cdots \] and hence show that \[ {3 \over 2} \displaystyle < \sum_{n=1}^\infty {1 \over n^2} <{7 \over 4}\;. \] \end{question} %%%%%% Q4 \begin{question}The triangle $OAB$ is isosceles, with $OA = OB$ and angle $AOB = 2 \alpha$ where $0< \alpha < {\pi \over 2}\,$. The semi-circle $\mathrm{C}_0$ has its centre at the midpoint of the base $AB$ of the triangle, and the sides $OA$ and $OB$ of the triangle are both tangent to the semi-circle. $\mathrm{C}_1, \mathrm{C}_2, \mathrm{C}_3, \ldots$ are circles such that $\mathrm{C}_n$ is tangent to $\mathrm{C}_{n-1}$ and to sides $OA$ and $OB$ of the triangle. Let $r_n$ be the radius of $\mathrm{C}_n\,$. Show that \[ \frac{r_{n+1}}{r_n} = \frac{1-\sin\alpha}{1+\sin\alpha}\;. \] Let $S$ be the total area of the semi-circle $\mathrm{C}_0$ and the circles $\mathrm{C}_1$, $\mathrm{C}_2$, $\mathrm{C}_3$, $\ldots\;$. Show that \[ S = {1 + \sin^2 \alpha \over 4 \sin \alpha} \, \pi r_0^2 \;. \] Show that there are values of $\alpha$ for which $S$ is more than four fifths of the area of triangle~$OAB$. \end{question} %%%%%%%%% Q5 \begin{question} Show that if $\, \cos(x - \alpha) = \cos \beta \,$ then either $\, \tan x = \tan ( \alpha + \beta)\,$ or $\; \tan x = \tan ( \alpha - \beta)\,$. By choosing suitable values of $x$, $\alpha$ and $\beta\,$, give an example to show that if $\,\tan x = \tan ( \alpha + \beta)\,$, then $\,\cos(x - \alpha) \, $ need not equal $ \cos \beta \,$. Let $\omega$ be the acute angle such that $\tan \omega = \frac 43\,$. \begin{questionparts} \item For $0 \le x \le 2 \pi$, solve the equation \[ \cos x -7 \sin x = 5 \] giving both solutions in terms of $\omega\,$. \item For $0 \le x \le 2 \pi$, solve the equation \[ 2\cos x + 11 \sin x = 10 \] showing that one solution is twice the other and giving both in terms of $\omega\,$. \end{questionparts} \end{question} %%%%%%%%% Q6 \begin{question} Given a sequence $w_0$, $w_1$, $w_2$, $\ldots\,$, the sequence $F_1$, $F_2$, $\ldots$ is defined by $$F_n = w_n^2 + w_{n-1}^2 - 4w_nw_{n-1} \,.$$ Show that $\; F_{n}-F_{n-1} = \l w_n-w_{n-2} \r \l w_n+w_{n-2}-4w_{n-1} \r \; $ for $n \ge 2\,$. \begin{questionparts} \item The sequence $u_0$, $u_1$, $u_2$, $\ldots$ has $u_0 = 1$, and $u_1 = 2$ and satisfies \[ u_n = 4u_{n-1} -u_{n-2} \quad (n \ge 2)\;. \] Prove that \ $ u_n^2 + u_{n-1}^2 = 4u_nu_{n-1}-3 \; $ for $n \ge 1\,$. \item A sequence $v_0$, $v_1$, $v_2$, $\ldots\,$ has $v_0=1$ and satisfies \begin{equation*} v_n^2 + v_{n-1}^2 = 4v_nv_{n-1}-3 \quad (n \ge 1). \tag{$\ast$} \end{equation*} \makebox[7mm]{(a) \hfill}Find $v_1$ and prove that, for each $n\ge2\,$, either $v_n= 4v_{n-1} -v_{n-2}$ or $v_n=v_{n-2}\,$. \makebox[7mm]{(b) \hfill}Show that the sequence, with period 2, defined by \begin{equation*} v_n = \begin{cases} 1 & \mbox{for $n$ even} \\ 2 & \mbox{for $n$ odd} \end{cases} \end{equation*} \makebox[7mm]{\hfill}satisfies $(\ast)$. \makebox[7mm]{(c) \hfill}Find a sequence $v_n$ with period 4 which has $v_0=1\,$, and satisfies~$(\ast)$. \end{questionparts} \end{question} %%%%%%%%% Q7 \begin{question} For $n=1$, $2$, $3$, $\ldots\,$, let \[ I_n = \int_0^1 {t^{n-1} \over \l t+1 \r^n} \, \mathrm{d} t \, . \] By considering the greatest value taken by $\ds {t \over t+1}$ for $0 \le t \le 1$ show that $I_{n+1} < {1 \over 2} I_{n}\,$. Show also that $\; \ds I_{n+1}= - \frac 1{\; n\, 2^n} + I_{n}\,$. Deduce that $\; \ds I_n < \frac1 {\; n \, 2^{n-1}}\,$. Prove that \[ \ln 2 = \sum_{r=1}^n {1 \over \; r\, 2^r} + I_{n+1} \] and hence show that \hspace{2 pt} ${2 \over 3} < \ln 2 < {17 \over 24}\,$. \end{question} %%%%%%%%% Q8 \begin{question} Show that if \[ {\mathrm{d}y \over \mathrm{d} x}=\f(x)y + {\g(x) \over y} \] then the substitution $u = y^2$ gives a linear differential equation for $u(x)\,$. Hence or otherwise solve the differential equation \[ {\mathrm{d}y \over \mathrm{d} x}={y \over x} - {1 \over y}\;. \] Determine the solution curves of this equation which pass through $(1 \,, 1)\,$, $(2\, , 2)$ and $(4 \, , 4)$ and sketch graphs of all three curves on the same axes. \end{question} \newpage \section*{Section B: \ \ \ Mechanics} %%%%%%%%%% Q9 \begin{question} A circular hoop of radius $a$ is free to rotate about a fixed horizontal axis passing through a point $P$ on its circumference. The plane of the hoop is perpendicular to this axis. The hoop hangs in equilibrium with its centre, $O$, vertically below $P$. The point $A$ on the hoop is vertically below $O$, so that $POA$ is a diameter of the hoop. A mouse $M$ runs at constant speed $u$ round the rough inner surface of the lower part of the hoop. Show that the mouse can choose its speed so that the hoop remains in equilibrium with diameter $POA$ vertical. Describe what happens to the hoop when the mouse passes the point at which angle $AOM = 2 \arctan \mu\,$, where $\mu$ is the coefficient of friction between mouse and hoop. \end{question} %%%%%%%%%% Q10 \begin{question} A particle $P$ of mass $m$ is attached to points $A$ and $B$, where $A$ is a distance $9a$ vertically above $B$, by elastic strings, each of which has modulus of elasticity $6mg$. The string $AP$ has natural length $6a$ and the string $BP$ has natural length $2a$. Let $x$ be the distance $AP$. The system is released from rest with $P$ on the vertical line $AB$ and $x = 6a$. Show that the acceleration $\ddot{x}$ of $P$ is $\ds{4g \over a}(7a - x)$ for $6ax_1\,$. Each particle is subject to a repulsive force from the other of magnitude $\displaystyle {2 \over z^3}$, where $z = x_2-x_1 \,$. Initially, $x_1=0$, $x_2 = 1$, $Q$ is at rest and $P$ moves towards $Q$ with speed 1. Show that $z$ obeys the equation $\displaystyle {\mathrm{d}^2 z \over \mathrm{d}t^2} = {3 \over z^3}$. By first writing $\displaystyle {\mathrm{d}^2 z \over \mathrm{d}t^2} = v {\mathrm{d}v \over \mathrm{d}z} \,$, where $\displaystyle v={\mathrm{d}z \over \mathrm{d}t}\,$, show that $z=\sqrt{4t^2-2t+1}\,$. By considering the equation satisfied by $2x_1+x_2\,$, find $x_1$ and $x_2$ in terms of $t \,$. \end{question} \newpage \section*{Section C: \ \ \ Probability and Statistics} %%%%%%%%%% Q12 \begin{question} A team of $m$ players, numbered from $1$ to $m$, puts on a set of a $m$ shirts, similarly numbered from $1$ to $m$. The players change in a hurry, so that the shirts are assigned to them randomly, one to each player. Let $C_i$ be the random variable that takes the value $1$ if player $i$ is wearing shirt $i$, and 0 otherwise. Show that $\mathrm{E}\left(C_1\right)={1 \over m}$ and find $\var \left(C_1\right)$ and $\mathrm{Cov}\left(C_1 \, , \; C_2 \right) \,$. Let $\, N = C_1 + C_2 + \cdots + C_m \,$ be the random variable whose value is the number of players who are wearing the correct shirt. Show that $\mathrm{E}\left(N\right)= \var \left(N\right) = 1 \,$. Explain why a Normal approximation to $N$ is not likely to be appropriate for any $m$, but that a Poisson approximation might be reasonable. In the case $m = 4$, find, by listing equally likely possibilities or otherwise, the probability that no player is wearing the correct shirt and verify that an appropriate Poisson approximation to $N$ gives this probability with a relative error of about $2\%$. [Use $\e \approx 2\frac{72}{100} \,$.] \end{question} %%%%%%%%%% Q13 \begin{question} A men's endurance competition has an unlimited number of rounds. In each round, a competitor has, independently, a probability $p$ of making it through the round; otherwise, he fails the round. Once a competitor fails a round, he drops out of the competition; before he drops out, he takes part in every round. The grand prize is awarded to any competitor who makes it through a round which all the other remaining competitors fail; if all the remaining competitors fail at the same round the grand prize is not awarded. If the competition begins with three competitors, find the probability that: \begin{questionparts} \item all three drop out in the same round; \item two of them drop out in round $r$ (with $r \ge 2$) and the third in an earlier round; \item the grand prize is awarded. \end{questionparts} \end{question} %%%%%%%%%% Q14 \begin{question} \textit{In this question, $\Phi(z)$ is the cumulative distribution function of a standard normal random variable.} A random variable is known to have a Normal distribution with mean $\mu$ and standard deviation either $\sigma_0$ or $\sigma_1$, where $\sigma_0 < \sigma_1\,$. The mean, $\overline{X}$, of a random sample of $n$ values of $X$ is to be used to test the hypothesis $\mathrm{H}_0: \sigma = \sigma_0$ against the alternative $\mathrm{H}_1: \sigma = \sigma_1\,$. Explain carefully why it is appropriate to use a two sided test of the form: accept $\mathrm{H}_0$ if \phantom{} $\mu - c < \overline{X} < \mu+c\,$, otherwise accept $\mathrm{H}_1$. Given that the probability of accepting $\mathrm{H}_1$ when $\mathrm{H}_0$ is true is $\alpha$, determine $c$ in terms of $n$, $\sigma_0$ and $z_{\alpha}$, where $z_\alpha $ is defined by $\ds\Phi(z_{\alpha}) = 1 - \tfrac{1}{2}\alpha$. The probability of accepting $\mathrm{H}_0$ when $\mathrm{H}_1$ is true is denoted by $\beta$. Show that $\beta$ is independent of $n$. Given that $\Phi(1.960)\approx 0.975$ and that $\Phi(0.063) \approx 0.525\,$, determine, approximately, the minimum value of $\ds \frac{\sigma_1}{\sigma_0}$ if $\alpha$ and $\beta$ are both to be less than $0.05\,$. \end{question} \end{document}