\documentclass[a4, 11pt]{report} \pagestyle{myheadings} \markboth{}{Paper II, 2005 \ \ \ \ \ \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} Find the three values of $x$ for which the derivative of $x^2 \e^{-x^2}$ is zero. Given that $a$ and $b$ are distinct positive numbers, find a polynomial $\P(x)$ such that the derivative of $\P(x)\e^{-x^2}$ is zero for $x=0$, $x=\pm a$ and $x=\pm b\,$, but for no other values of $x$. \end{question} %%%%%%%%%%Q2 \begin{question} For any positive integer $N$, the function $\f(N)$ is defined by \[ \f(N) = N\Big(1-\frac1{p_1}\Big)\Big(1-\frac1{p_2}\Big) \cdots\Big(1-\frac1{p_k}\Big) \] where $p_1$, $p_2$, $\dots$ , $p_k$ are the only prime numbers that are factors of $N$. \\ Thus $\f(80)=80(1-\frac12)(1-\frac15)\,$. \begin{questionparts} \item \textbf{(a)} \ \ Evaluate $\f(12)$ and $\f(180)$. \textbf{(b)} \ Show that $\f(N)$ is an integer for all $N$. \item Prove, or disprove by means of a counterexample, each of the following: \textbf{(a)} \ \ $\f(m) \f(n) = \f(mn)\,$; \textbf{(b)} \ $\f(p) \f(q) = \f(pq)$ if $p$ and $q$ are distinct prime numbers; \textbf{(c)} $\f(p) \f(q) = \f(pq)$ only if $p$ and $q$ are distinct prime numbers. \item Find a positive integer $m$ and a prime number $p$ such that $\f(p^m) = 146410\,$. \end{questionparts} \end{question} %%%%%%%%% Q3 \begin{question} Give a sketch, for $0 \le x \le \frac{1}{2}\pi$, of the curve $$ y = (\sin x - x\cos x)\;, $$ and show that $0\le y \le 1\,$. Show that: \begin{questionparts} \item \ \ $\displaystyle \int_0^{\frac{1}{2}\pi}\,y\;\d x = 2 -\frac \pi 2 \;; $ \item \ \ $\displaystyle \int_0^{\frac{1}{2}\pi}\,y^2\,\d x = \frac{\pi^3}{48}-\frac \pi 8 \;. $ \end{questionparts} Deduce that $\pi^3 +18 \pi< 96\,$. \end{question} %%%%%% Q4 \begin{question} The positive numbers $a$, $b$ and $c$ satisfy $bc=a^2+1$. Prove that $$ \arctan\left(\frac1 {a+b}\right)+ \arctan\left(\frac1 {a+c}\right)= \arctan\left(\frac1 a \right). $$ The positive numbers $p$, $q$, $r$, $s$, $t$, $u$ and $v$ satisfy $$ st = (p+q)^2 + 1 \;, \ \ \ \ \ \ uv=(p+r)^2 + 1 \;, \ \ \ \ \ \ qr = p^2+1\;. $$ Prove that $$ \arctan \! \!\left(\!\frac1 {p+q+s}\!\right) + \arctan \! \!\left(\!\frac 1{p+q+t}\!\right) + \arctan \! \!\left(\!\frac 1 {p+r+u}\!\right) + \arctan \! \!\left(\!\frac1 {p+r+v}\!\right) =\arctan \! \!\left( \! \frac1 p \! \right) . $$ Hence show that $$ \arctan\left(\frac1 {13}\right) +\arctan\left(\frac1 {21}\right) +\arctan\left(\frac1 {82}\right) +\arctan\left(\frac1 {187}\right) =\arctan\left(\frac1 {7}\right). $$ [\,Note that $\arctan x$ is another notation for $ \tan^{-1}x \,.\,$] \end{question} %%%%%%%%% Q5 \begin{question} The angle $A$ of triangle $ABC$ is a right angle and the sides $BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$, respectively. Each side of the triangle is tangent to the circle $S_1$ which is of radius $r$. Show that $2r = b+c-a$. Each vertex of the triangle lies on the circle~$S_2$. The ratio of the area of the region between~$S_1$ and the triangle to the area of $S_2$ is denoted by $R\,$. Show that $$ \pi R = -(\pi-1)q^2 + 2\pi q -(\pi+1) \;, $$ where $q=\dfrac{b+c}a\,$. Deduce that $$ R\le \frac1 {\pi( \pi - 1)} \;. $$ \end{question} %%%%%%%%% Q6 \begin{question} \begin{questionparts} \item Write down the general term in the expansion in powers of $x$ of $(1-x)^{-1}$, $(1-x)^{-2}$ and $(1-x)^{-3}$, where $\vert x \vert <1\,$. Evaluate \ $\ds \sum_{n=1}^\infty n 2^{-n}\;$ \ and \ \ $\ds \sum_{n=1}^\infty n^22^{-n}\;$. \item Show that \ $\ds (1-x)^{-\frac12} = \sum_{n=0}^\infty \frac{(2n)!}{(n!)^2} \, \frac{x^n}{2^{2n}}\,$, \ for $\vert x \vert <1\,$. Evaluate \ $\ds \sum_{n=0}^\infty \frac{(2n)!} {(n!)^2 2^{2n}3^{n}}\; $ \ and \ \ $\ds \sum_{n=1}^\infty \frac{n(2n)!} {(n!)^2 2^{2n}3^{n}}\; $. \end{questionparts} \end{question} %%%%%%%%% Q7 \begin{question} The position vectors, relative to an origin $O$, at time $t$ of the particles $P$ and $Q$ are $$\cos t \; {\bf i} + \sin t\;{\bf j} + 0 \; {\bf k} \hbox{ \ \ \ \ \ \ and \ \ \ \ \ \ } \cos (t+\tfrac14{\pi})\, \big[{\tfrac32}{\bf i} + { \tfrac {3\sqrt{3}}2} {\bf k}\big] + 3\sin(t+\tfrac14{\pi}) \; {\bf j}\;,$$ respectively, where $0\le t \le 2\pi\,$. \begin{questionparts} \item Give a geometrical description of the motion of $P$ and $Q$. \item Let $\theta$ be the angle $POQ$ at time $t$ that satisfies $0\le\theta\le\pi\,$. Show that \[ \cos\theta = \tfrac{3\surd2}{8} -\tfrac14 \cos( 2t +\tfrac14 \pi)\;. \] \item Show that the total time for which $\theta \ge \frac14 \pi$ is $\tfrac32 \pi\,$. \end{questionparts} \end{question} %%%%%%%%% Q8 \begin{question} For $x \ge 0$ the curve $C$ is defined by $$ {\frac{\d y} {\d x}} = \frac{x^3y^2}{(1 + x^2)^{5/2}} $$ with $y = 1$ when $x=0\,$. Show that \[ \frac 1 y = \frac {2+3x^2}{3(1+x^2)^{3/2}} +\frac13 \] and hence that for large positive $x$ $$ y \approx 3 - \frac 9 x\;. $$ Draw a sketch of $C$. On a separate diagram draw a sketch of the two curves defined for $x \ge 0$ by $$ \frac {\d z} {\d x} = \frac{x^3z^3}{2(1 + x^2)^{5/2}} $$ with $z = 1$ at $x=0$ on one curve, and $z = -1$ at $x=0$ on the other. \end{question} \newpage \section*{Section B: \ \ \ Mechanics} %%%%%%%%%% Q9 \begin{question} Two particles, $A$ and $B$, of masses $m$ and $2m$, respectively, are placed on a line of greatest slope, $\ell$, of a rough inclined plane which makes an angle of $30^{\circ}$ with the horizontal. The coefficient of friction between $A$ and the plane is $\frac16\sqrt{3}$ and the coefficient of friction between $B$ and the plane is $\frac13 \sqrt{3}$. The particles are at rest with $B$ higher up $\ell$ than $A$ and are connected by a light inextensible string which is taut. A force $P$ is applied to $B$. \begin{questionparts} \item Show that the least magnitude of $P$ for which the two particles move upwards along $\ell$ is $\frac{11}8 \sqrt{3}\, mg$ and give, in this case, the direction in which $P$ acts. \item Find the least magnitude of $P$ for which the particles do not slip downwards along~$\ell$. \end{questionparts} \end{question} %%%%%%%%%% Q10 \begin{question} The points $A$ and $B$ are $180$ metres apart and lie on horizontal ground. A missile is launched from $A$ at speed of $100\,$m\,s$^{-1}$ and at an acute angle of elevation to the line $AB$ of $\arcsin \frac35$. A time $T$ seconds later, an anti-missile missile is launched from $B$, at speed of $200\,$m\,s$^{-1}$ and at an acute angle of elevation to the line $BA$ of $\arcsin \frac45$. The motion of both missiles takes place in the vertical plane containing $A$ and $B$, and the missiles collide. Taking $g =10\,$m\,s$^{-2}$ and ignoring air resistance, find $T$. \noindent [Note that $\arcsin \frac35$ is another notation for $\sin^{-1} \frac35\,$.] \end{question} %%%%%%%%%% Q11 \begin{question} A plane is inclined at an angle $\arctan \frac34$ to the horizontal and a small, smooth, light pulley~$P$ is fixed to the top of the plane. A string, $APB$, passes over the pulley. A particle of mass~$m_1$ is attached to the string at $A$ and rests on the inclined plane with $AP$ parallel to a line of greatest slope in the plane. A particle of mass $m_2$, where $m_2>m_1$, is attached to the string at $B$ and hangs freely with $BP$ vertical. The coefficient of friction between the particle at $A$ and the plane is $ \frac{1}{2}$. The system is released from rest with the string taut. Show that the acceleration of the particles is $\ds \frac{m_2-m_1}{m_2+m_1}g$. At a time $T$ after release, the string breaks. Given that the particle at $A$ does not reach the pulley at any point in its motion, find an expression in terms of $T$ for the time after release at which the particle at $A$ reaches its maximum height. It is found that, regardless of when the string broke, this time is equal to the time taken by the particle at $A$ to descend from its point of maximum height to the point at which it was released. Find the ratio $m_1 : m_2$. \noindent [Note that $\arctan \frac34$ is another notation for $\tan^{-1} \frac34\,$.] \end{question} \newpage \section*{Section C: \ \ \ Probability and Statistics} %%%%%%%%%% Q12 \begin{question} The twins Anna and Bella share a computer and never sign their e-mails. When I e-mail them, only the twin currently online responds. The probability that it is Anna who is online is $p$ and she answers each question I ask her truthfully with probability $a$, independently of all her other answers, even if a question is repeated. The probability that it is Bella who is online is~$q$, where $q=1-p$, and she answers each question truthfully with probability $b$, independently of all her other answers, even if a question is repeated. \begin{questionparts} \item I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. Show that the probability that the coin did come down heads is $\frac{1}{2}$ if and only if $2(ap+bq)=1$. \item I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. I then send the e-mail: `Did the coin come down heads?' and I receive the answer `no'. Show that the probability (taking into account these answers) that the coin did come down heads is $\frac{1}{2}\,$. \item I send the twins the e-mail: `Toss a fair coin and answer the following question. Did the coin come down heads?'. I receive the answer `yes'. I then send the e-mail: `Did the coin come down heads?' and I receive the answer `yes'. Show that, if $2(ap+bq)=1$, the probability (taking into account these answers) that the coin did come down heads is $\frac{1}{2}\,$. \end{questionparts} \end{question} %%%%%%%%%% Q13 \begin{question} The number of printing errors on any page of a large book of $N$ pages is modelled by a Poisson variate with parameter $\lambda$ and is statistically independent of the number of printing errors on any other page. The number of pages in a random sample of $n$ pages (where $n$ is much smaller than $N$ and $n\ge2$) which contain fewer than two errors is denoted by $Y$. Show that $\P(Y=k) = \binom n k p^kq^{n-k}$ where $p=(1+\lambda)e^{-\lambda}$ and $q=1-p\,$. Show also that, if $\lambda$ is sufficiently small, \begin{questionparts} \item $q\approx \frac12 \lambda^2\,$; \item the largest value of $n$ for which $\P(Y=n)\ge 1-\lambda$ is approximately $2/\lambda\,$; \item $ \P(Y>1 \;\vert\; Y>0) \approx 1-n(\lambda^2/2)^{n-1}\;.$ \end{questionparts} \end{question} %%%%%%%%%% Q14 \begin{question} The probability density function $\f(x)$ of the random variable $X$ is given by $$ \f(x) = k\left[{\phi}(x) + {\lambda}\g(x)\right],\,\,\,\, $$ where ${\phi}(x)$ is the probability density function of a normal variate with mean~0 and variance~1, $\lambda $ is a positive constant, and $\g(x)$ is a probability density function defined by \[ \g(x)= \begin{cases} 1/\lambda & \mbox{for $0 \le x \le {\lambda}$}\,;\\ 0& \mbox{otherwise} . \end{cases} \] Find $\mu$, the mean of $X$, in terms of $\lambda$, and prove that $\sigma$, the standard deviation of $X$, satisfies. $$ \sigma^2 = \frac{\lambda^4 +4{\lambda}^3+12{\lambda}+12} {12(1 + \lambda )^2}\;. $$ In the case $\lambda=2$: \begin{questionparts} \item draw a sketch of the curve $y=\f(x)$; \item express the cumulative distribution function of $X$ in terms of $\Phi(x)$, the cumulative distribution function corresponding to $\phi(x)$; \item evaluate $\P(0