\documentclass[a4, 11pt]{report} \pagestyle{myheadings} \markboth{}{Paper I, 2012 \ \ \ \ \ \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\arcosh{{\rm arcosh}\,} \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} The line $L$ has equation $y=c-mx$, with $m>0$ and $c>0$. It passes through the point $R(a,b)$ and cuts the axes at the points $P(p,0)$ and $Q(0,q)$, where $a$, $b$, $p$ and $q$ are all positive. Find $p$ and $q$ in terms of $a$, $b$ and $m$. As $L$ varies with $R$ remaining fixed, show that the minimum value of the sum of the distances of $P$ and $Q$ from the origin is $(a^{\frac12} + b^{\frac12})^2$, and find in a similar form the minimum distance between $P$ and $Q$. (You may assume that any stationary values of these distances are minima.) \end{question} %%%%%%%%%%Q2 \begin{question} \begin{questionparts} \item Sketch the curve $ y= x^4-6x^2+9 $ giving the coordinates of the stationary points. Let $n$ be the number of distinct real values of $x$ for which \[ x^4-6x^2 +b=0. \] State the values of $b$, if any, for which \ (a) $n=0\,$; \ (b) $n=1\,$; \ (c) $n=2\,$; \ (d) $n=3\,$; \ (e) $n=4\,$. \item For which values of $a$ does the curve $y= x^4-6x^2 +ax +b$ have a point at which both $\dfrac{\d y}{\d x}=0$ and $\dfrac{\d^2y}{\d x^2}=0\,$? For these values of $a$, find the number of distinct real values of $x$ for which $\vphantom{\dfrac{A}{B}}$ \[ x^4-6x^2 +ax +b=0\,, \] in the different cases that arise according to the value of $b$. \item Sketch the curve $y= x^4-6x^2 +ax$ in the case $a>8\,$. \end{questionparts} \end{question} %%%%%%%%% Q3 \begin{question} \begin{questionparts} \item Sketch the curve $y=\sin x$ for $0\le x \le \tfrac12 \pi$ and add to your diagram the tangent to the curve at the origin and the chord joining the origin to the point $(b, \sin b)$, where $01$, show that \[ \frac{2(a-1)}{a+1} < \ln a < -1 + \sqrt{2a-1\,}\,. \] [{\bf Hint}: You may wish to write $a^x$ as $\e^{x\ln a}$.] \end{questionparts} \end{question} %%%%%% Q4 \begin{question} The curve $C$ has equation $xy = \frac12$. The tangents to $C$ at the distinct points $ P\big(p, \frac1 { \rule[0pt]{0pt}{2.7mm} \mbox{\fontsize{10pt}{13}\selectfont$2p$} } \big) $ and $ Q\big(q, \frac1 { \rule[0pt]{0pt}{2.7mm} \mbox{\fontsize{10pt}{13}\selectfont$2q$} } \big), $ where $p$ and $q$ are positive, intersect at $T$ and the normals to $C$ at these points intersect at~$N$. Show that $T$ is the point \[ \left( \frac{2pq}{p+q}\,,\, \frac 1 {p+q}\right)\!. \] In the case $pq=\frac12$, find the coordinates of $N$. Show (in this case) that $T$ and $N$ lie on the line $y=x$ and are such that the product of their distances from the origin is constant. \end{question} %%%%%%%%% Q5 \begin{question} Show that \[ \int_0^{\frac14\pi} \sin (2x) \ln(\cos x)\, \d x = \frac14(\ln 2 -1)\,, \] and that \[ \int_0^{\frac14\pi} \cos (2x) \ln(\cos x)\, \d x = \frac18(\pi -\ln 4-2)\,. \] Hence evaluate \[ \int_{\frac14\pi}^{\frac12\pi} \big ( \cos(2x) + \sin (2x)\big) \, \ln \big( \cos x + \sin x\big)\, \d x\,. \] \end{question} %%%%%%%%% Q6 \begin{question} A thin circular path with diameter $AB$ is laid on horizontal ground. A vertical flagpole is erected with its base at a point $D$ on the diameter $AB$. The angles of elevation of the top of the flagpole from $A$ and $B$ are $\alpha$ and $\beta$ respectively (both are acute). The point $C$ lies on the circular path with $DC$ perpendicular to $AB$ and the angle of elevation of the top of the flagpole from $C$ is $\phi$. Show that $\cot\alpha\cot \beta = \cot^2\phi$. Show that, for any $p$ and $q$, \[ \cos p \cos q \sin^2\tfrac12(p+q) - \sin p\sin q \cos^2 \tfrac12 (p+q) = \tfrac12 \cos(p+q) -\tfrac12 \cos(p+q)\cos(p-q) .\] Deduce that, if $p$ and $q$ are positive and $ p+q \le \tfrac12 \pi$, then \[ \cot p\cot q\, \ge \cot^2 \tfrac12(p+q) \, \] and hence show that $\phi \le \tfrac12(\alpha+\beta)$ when $ \alpha +\beta \le \tfrac12 \pi\,$. \end{question} %%%%%%%%% Q7 \begin{question} A sequence of numbers $t_0$, $t_1$, $t_2$, $\ldots\,$ satisfies \[ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t_{n+2} = p t_{n+1}+qt_{n} \ \ \ \ \ \ \ \ \ \ (n\ge0), \] where $p$ and $q$ are real. Throughout this question, $x$, $y$ and $z$ are non-zero real numbers. \begin{questionparts} \item Show that, if $t_n=x$ for all values of $n$, then $p+q=1$ and $x$ can be any (non-zero) real number. \item Show that, if $t_{2n} = x$ and $t_{2n+1}=y$ for all values of $n$, then $q\pm p=1$. Deduce that either $x=y$ or $x=-y$, unless $p$ and $q$ take certain values that you should identify. \item Show that, if $t_{3n} = x$, $t_{3n+1}=y$ and $t_{3n+2}=z$ for all values of $n$, then \[ p^3+q^3 +3pq-1=0\,. \] Deduce that either $p+q=1$ or $(p-q)^2 +(p+1)^2+(q+1)^2=0$. Hence show that either $x=y=z$ or $x+y+z=0$. \end{questionparts} \end{question} %%%%%%%%% Q8 \begin{question} \begin{questionparts} \item Show that substituting $y=xv$, where $v$ is a function of $x$, in the differential equation \[ \hphantom{(x\ne0)\hspace*{2cm}} xy \frac{\d y}{\d x} +y^2- 2x^2 =0 {\hspace*{2cm}(x\ne0)} \] leads to the differential equation \[ xv\frac{\d v}{\d x} +2v^2 -2=0\,. \] Hence show that the general solution can be written in the form \[ x^2(y^2 -x^2) = C \,, \] where $C$ is a constant. \item Find the general solution of the differential equation \[ \hphantom{(x\ne0)\hspace*{2cm}} y \frac{\d y}{\d x} +6x +5y=0\, {\hspace*{2cm}(x\ne0)} . \] \end{questionparts} \end{question} \newpage \section*{Section B: \ \ \ Mechanics} %%%%%%%%%% Q9 \begin{question} A tall shot-putter projects a small shot from a point $2.5\,$m above the ground, which is horizontal. The speed of projection is $10\,$m\,s$^{- 1}$ and the angle of projection is $\theta$ above the horizontal. Taking the acceleration due to gravity to be $10\,$m\,s$^{-2}$, show that the time, in seconds, that elapses before the shot hits the ground is \[ \frac1{\sqrt2}\left ( \sqrt{1-c}+ \sqrt{2-c}\right), \] where $c = \cos2\theta$. Find an expression for the range in terms of $c$ and show that it is greatest when $c= \frac15\,$. Show that the extra distance attained by projecting the shot at this angle rather than at an angle of $45^\circ$ is $5(\sqrt6 -\sqrt2 -1)\,$m. \end{question} %%%%%%%%%% Q10 \begin{question} I stand at the top of a vertical well. The depth of the well, from the top to the surface of the water, is $D$. I drop a stone from the top of the well and measure the time that elapses between the release of the stone and the moment when I hear the splash of the stone entering the water. In order to gauge the depth of the well, I climb a distance $\delta$ down into the well and drop a stone from my new position. The time until I hear the splash is $t$ less than the previous time. Show that \[ t = \sqrt{\frac{2D}g} - \sqrt{\frac{2(D-\delta)}g} + \frac \delta u\,, \] where $u$ is the (constant) speed of sound. Hence show that \[ D = \tfrac12 gT^2\,, \] where $T= \dfrac12 \beta + \dfrac \delta{\beta g}$ and $\beta = t - \dfrac \delta u\,$. Taking $u=300\,$m\,s$^{-1}$ and $g=10\,$m\,s$^{-2}$, show that if $t= \frac 15\,$s and $\delta=10\,$m, the well is approximately $185\,$m deep. \end{question} %%%%%%%%%% Q11 \begin{question} The diagram shows two particles, $A$ of mass $5m$ and $B$ of mass $3m$, connected by a light inextensible string which passes over two smooth, light, fixed pulleys, $Q$ and $R$, and under a smooth pulley $P$ which has mass $M$ and is free to move vertically. Particles $A$ and $B$ lie on fixed rough planes inclined to the horizontal at angles of $\arctan \frac 7{24}$ and $\arctan\frac43$ respectively. The segments $AQ$ and $RB$ of the string are parallel to their respective planes, and segments $QP$ and $PR$ are vertical. The coefficient of friction between each particle and its plane is $\mu$. \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=3pt 0,linewidth=0.3pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-1.07,-0.16)(9.6,4.76) \pscircle[fillcolor=black,fillstyle=solid,opacity=0.15](5,4){0.25} \pscircle[fillcolor=black,fillstyle=solid,opacity=0.15](6.5,4){0.25} \pspolygon[linewidth=0pt,fillcolor=black,fillstyle=solid,opacity=0.3](-1,0)(5,0)(5,4) \pspolygon[linewidth=0pt,fillcolor=black,fillstyle=solid,opacity=0.3](9.5,0)(6.5,0)(6.5,4) \psline(1.63,2.12)(4.85,4.2) \psline(8.29,2.04)(6.712,4.13) \psline(5.25,4)(5.25,1.5) \psline(6.25,3.99)(6.25,1.5) \rput[tl](4.85,4.68){$Q$} \rput[tl](6.38,4.64){$R$} \rput[tl](8.29,2.72){$B$} \rput[tl](5.6,0.85){$P$} \rput[tl](1.54,2.85){$A$} \begin{scriptsize} \psdots[dotsize=28.5pt,dotstyle=*](5.75,1.5) \psdots[dotsize=17pt 0,dotstyle=*](1.63,2.12) \psdots[dotsize=15pt 0,dotstyle=*](8.29,2.04) \end{scriptsize} \end{pspicture*} \end{center} \begin{questionparts} \item Given that the system is in equilibrium, with both $A$ and $B$ on the point of moving up their planes, determine the value of $\mu$ and show that $M = 6m$. \item In the case when $M = 9m$, determine the initial accelerations of $A$, $B$ and $P$ in terms of $g$. \end{questionparts} \end{question} \newpage \section*{Section C: \ \ \ Probability and Statistics} %%%%%%%%%% Q12 \begin{question} Fire extinguishers may become faulty at any time after manufacture and are tested annually on the anniversary of manufacture. The time $T$ years after manufacture until a fire extinguisher becomes faulty is modelled by the continuous probability density function \[ \f(t) = \begin{cases} \dfrac{2t}{(1+t^2)^2}& \text{for $t\ge0$}\,,\\[4mm] \ \ \ \ 0& \text{otherwise}. \end{cases} \] A faulty fire extinguisher will fail an annual test with probability $p$, in which case it is destroyed immediately. A non-faulty fire extinguisher will always pass the test. All of the annual tests are independent. Show that the probability that a randomly chosen fire extinguisher will be destroyed exactly three years after its manufacture is $p(5p^2-13p +9)/10$. Find the probability that a randomly chosen fire extinguisher that was destroyed exactly three years after its manufacture was faulty 18 months after its manufacture. \end{question} %%%%%%%%%% Q13 \begin{question} I choose at random an integer in the range 10000 to 99999, all choices being equally likely. Given that my choice does not contain the digits 0, 6, 7, 8 or 9, show that the expected number of different digits in my choice is 3.3616. \end{question} \end{document}