\documentclass[a4, 11pt]{report} \pagestyle{myheadings} \markboth{}{Paper II, 2015 \ \ \ \ \ \today } \RequirePackage{amssymb} \RequirePackage{amsmath} \RequirePackage{graphicx} \RequirePackage{color} \RequirePackage[flushleft]{paralist}[2013/06/09] \RequirePackage{asymptote} \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\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}} \begin{asydef} import markers; import geometry; import graph; usepackage("amsmath"); \end{asydef} \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 By use of calculus, show that $x- \ln(1+x)$ is positive for all positive $x$. Use this result to show that \[ \sum_{k=1}^n \frac 1 k > \ln (n+1) \,. \] \item By considering $ x+\ln (1-x)$, show that \[ \sum_{k=1}^\infty \frac 1 {k^2} <1+ \ln 2 \,. \] \end{questionparts} \end{question} %%%%%%%%%%Q2 \begin{question} In the triangle $ABC$, angle $BAC = \alpha$ and angle $CBA= 2\alpha$, where $2\alpha$ is acute, and $BC= x$. Show that $AB = (3-4 \sin^2\alpha)x$. The point $D$ is the midpoint of $AB$ and the point $E$ is the foot of the perpendicular from $C$ to $AB$. Find an expression for $DE$ in terms of $x$. The point $F$ lies on the perpendicular bisector of $AB$ and is a distance $x$ from $C$. The points $F$ and $B$ lie on the same side of the line through $A$ and $C$. Show that the line $FC$ trisects the angle $ACB$. \end{question} %%%%%%%%% Q3 \begin{question} Three rods have lengths $a$, $b$ and $c$, where $a< b< c$. The three rods can be made into a triangle (possibly of zero area) if $a+b\ge c$. Let $T_{n}$ be the number of triangles that can be made with three rods chosen from $n$ rods of lengths $1$, $2$, $3$, $\ldots$ , $n$ (where $n\ge3$). Show that $T_8-T_7 = 2+4+6$ and evaluate $T_8 -T_6$. Write down expressions for $T_{2m}-T_{2m-1}$ and $T_{2m} - T_{2m-2}$. Prove by induction that $T_{2m}=\frac 16 m (m-1)(4m+1)\,$, and find the corresponding result for an odd number of rods. \end{question} %%%%%% Q4 \begin{question} \begin{questionparts} \item The continuous function $\f$ is defined by \[ \tan \f(x) = x \ \ \ \ \ (-\infty < x <\infty) \] and $\f(0)=\pi$. Sketch the curve $y=\f(x)$\,. \item The continuous function $\g$ is defined by \[ \tan \g(x) = \frac x {1+x^2} \ \ \ \ \ \ (-\infty < x <\infty) \] and $\g(0)=\pi$. Sketch the curves $y= \dfrac x {1+x^2} \ $ and $y=\g(x)$\,. \item The continuous function $\h $ is defined by $\h (0)=\pi$ and \[ \tan \h (x)= \frac x {1-x^2}\, \ \ \ \ \ (x \ne \pm 1) \,. \] (The values of $\h (x)$ at $x=\pm1$ are such that $\h (x)$ is continuous at these points.) Sketch the curves $y= \dfrac x {1-x^2} \ $ and $y=\h (x)$. %\item The continuous functions $\h_1$ and $\h_2$ are % defined by: $\h_1(0)=\h_2(0)=\pi $, %\[ %\tan \h_1(x) = \frac {x+x^4} {1+x^2+x^4} %\ \ \ \ \ \text{and} \ \ \ \ \ \ %\tan \h_2(x) = \frac {4x-x^3} {1-x^4} %\,. %\] %for values of $x$ at which the right hand sides are defined. %Find $\lim\limits_{x\to\infty}\h_1(x)$ and $\lim\limits_{x\to\infty}\h_2(x)\,$. \end{questionparts} \end{question} %%%%%%%%% Q5 \begin{question} In this question, the $\mathrm{arctan}$ function satisfies $0\le \arctan x <\frac12 \pi$ for $x\ge0\,$. \begin{questionparts} \item Let \[ S_n= \sum_{m=1}^n \arctan \left(\frac1 {2m^2}\right) \,, \] for $n=1$, 2, 3, $\ldots$ . Prove by induction that \[ \tan S_n = \frac n {n+1} \,. \] Prove also that \[ S_n = \arctan \frac n {n+1} \,. \] \item In a triangle $ABC$, the lengths of the sides $AB$ and $BC$ are $4n^2$ and $4n^4-1$, respectively, and the angle at $B$ is a right angle. Let $\angle BCA = 2\alpha_n$. Show that \[ \sum_{n=1}^\infty \alpha_n = \tfrac14\pi \,. \] \end{questionparts} \end{question} %%%%%%%%% Q6 \begin{question} \begin{questionparts} \item Show that \[ \mathrm{sec}^2\left(\tfrac14\pi-\tfrac12 x\right)=\frac{2}{1+\sin x} \,. \] Hence integrate $\dfrac{1}{1+\sin x}$ with respect to $x$. \item By means of the substitution $y=\pi -x$, show that \[ \int_0^\pi x \f (\sin x)\, \d x = \frac \pi 2 \int_0^\pi \f(\sin x) \, \d x ,\] where $\mathrm{f}$ is any function for which these integrals exist. Hence evaluate \[ \int_0^\pi \frac x {1+\sin x} \, \d x \,. \] \item Evaluate \[ \int_0^\pi\frac{ 2x^3 -3\pi x^2}{(1+\sin x)^2}\, \d x .\] \end{questionparts} \end{question} %%%%%%%%% Q7 \begin{question} A circle $C$ is said to be {\em bisected} by a curve $X$ if $X$ meets $C$ in exactly two points and these points are diametrically opposite each other on $C$. \begin{questionparts} \item Let $C$ be the circle of radius $a$ in the $x$-$y$ plane with centre at the origin. Show, by giving its equation, that it is possible to find a circle of given radius $r$ that bisects $C$ provided $r>a$. Show that no circle of radius $r$ bisects $C$ if $r\le a\,$. \item Let $C_1$ and $C_2$ be circles with centres at $(-d,0)$ and $(d,0)$ and radii $a_1$ and $a_2$, respectively, where $d>a_1$ and $d>a_2$. Let $D$ be a circle of radius~$r$ that bisects both $C_1$ and $C_2$. Show that the $x$-coordinate of the centre of $D$ is $\dfrac{a_2^2 - a_1^2}{4d}$. Obtain an expression in terms of $d$, $r$, $a_1$ and $a_2$ for the $y$-coordinate of the centre of~$D$, and deduce that $r$ must satisfy \[ 16r^2d^2 \ge \big (4d^2 +(a_2-a_1)^2\big) \, \big (4d^2 +(a_2+a_1)^2\big) \,. \] \end{questionparts} \end{question} %%%%%%%%% Q8 \begin{question} \noindent \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*}(-2.94,-1.87)(7.07,3.86) \pscircle(0,1){1.25} \pscircle(3,0){0.55} \rput[tl](5.33,-0.41){$P$} \psline(-2.44,-0.03)(6.18,-0.85) \psline(-2.04,3.71)(6.55,-1.48) \rput[tl](-0.18,1.1){$C_1$} \rput[tl](2.85 ,0.15){$C_2$} \rput[tl](-0.65,3.29){$L'$} \rput[tl](-1.5,-0.34){$L$} \end{pspicture*} \end{center} The diagram above shows two non-overlapping circles $C_1$ and $C_2$ of different sizes. The lines $L$ and $L'$ are the two common tangents to $C_1$ and $C_2$ such that the two circles lie on the same side of each of the tangents. The lines $L$ and $L'$ intersect at the point $P$ which is called the {\em focus} of $C_1$ and $C_2$. \begin{questionparts} \item Let ${\bf x}_1$ and ${\bf x}_2$ be the position vectors of the centres of $C_1$ and $C_2$, respectively. Show that the position vector of $P$ is \[ \frac{r_1 {\bf x}_2- r_2 {\bf x}_1}{r_1-r_2} \,, \] where $r_1$ and $r_2$ are the radii of $C_1$ and $C_2$, respectively. \item The circle $C_3$ does not overlap either $C_1$ or $C_2$ and its radius, $r_3$, satisfies $r_1 \ne r_3 \ne r_2$. The focus of $C_1$ and $C_3$ is $Q$, and the focus of $C_2$ and $C_3$ is $R$. Show that $P$, $Q$ and~$R$ lie on the same straight line. \item Find a condition on $r_1$, $r_2$ and $r_3$ for $Q$ to lie half-way between $P$ and $R$. \end{questionparts} \end{question} \newpage \section*{Section B: \ \ \ Mechanics} %%%%%%%%%% Q9 \begin{question} An equilateral triangle $ABC$ is made of three light rods each of length $a$. It is free to rotate in a vertical plane about a horizontal axis through $A$. Particles of mass $3m$ and $5m$ are attached to $B$ and $C$ respectively. Initially, the system hangs in equilibrium with $BC$ below~$A$. \begin{questionparts} \item Show that, initially, the angle $\theta$ that $BC$ makes with the horizontal is given by $\sin\theta = \frac17$. \item The triangle receives an impulse that imparts a speed $v$ to the particle $B$. Find the minimum speed $v_0$ such that the system will perform complete rotations if $v>v_0$. \end{questionparts} \end{question} %%%%%%%%%% Q10 \begin{question} A particle of mass $m$ is pulled along the floor of a room in a straight line by a light string which is pulled at constant speed $V$ through a hole in the ceiling. The floor is smooth and horizontal, and the height of the room is $h$. Find, in terms of $V$ and $\theta$, the speed of the particle when the string makes an angle of $\theta$ with the vertical (and the particle is still in contact with the floor). Find also the acceleration, in terms of $V$, $h$ and $\theta$. Find the tension in the string and hence show that the particle will leave the floor when \[ \tan^4\theta = \frac{V^2}{gh}\,. \] \end{question} %%%%%%%%%% Q11 \begin{question} Three particles, $A$, $B$ and $C$, each of mass $m$, lie on a smooth horizontal table. Particles $A$ and $C$ are attached to the two ends of a light inextensible string of length $2a$ and particle~$B$ is attached to the midpoint of the string. Initially, $A$, $B$ and $C$ are at rest at points $(0,a)$, $(0,0)$ and $(0,-a)$, respectively. An impulse is delivered to $B$, imparting to it a speed $u$ in the positive $x$ direction. The string remains taut throughout the subsequent motion. \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=3pt 0,linewidth=0.5pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-2.18,-3.26)(6.26,3.18) \psline[linewidth=2pt](2.,1.5)(3.,0.) \psline[linewidth=2pt](3.,0.)(2.,-1.5) \psline(-2.,0.)(6.,0.) \psline(0.,3.)(0.,-3.) \parametricplot{2.1587989303424644}{3.1415926535897936}{0.7*cos(t)+3.|0.7*sin(t)+0.} \rput[tl](-0.25,2.96){$y$} \rput[tl](5.76,-0.1){$x$} \rput[tl](2.5,0.38){$\theta$} \rput[tl](2.1,1.94){$A$} \rput[tl](3.2,0.4){$B$} \rput[tl](2.16,-1.46){$C$} \begin{scriptsize} \psdots[dotsize=8pt 0,dotstyle=*](2.,1.5) \psdots[dotsize=8pt 0,dotstyle=*](3.,0.) \psdots[dotsize=8pt 0,dotstyle=*](2.,-1.5) \end{scriptsize} \end{pspicture*} \end{center} \begin{questionparts} \item At time $t$, the angle between the $x$-axis and the string joining $A$ and $B$ is $\theta$, as shown in the diagram, and $B$ is at $(x,0)$. Write down the coordinates of $A$ in terms of $x,a$ and $\theta$. Given that the velocity of $B$ is $(v,0)$, show that the velocity of $A$ is $(\dot x + a\sin\theta \,\dot \theta\,,\, a\cos\theta\, \dot\theta)$, where the dot denotes differentiation with respect to time. \item Show that, before particles $A$ and $C$ first collide, \[ 3\dot x + 2a \dot\theta \sin\theta =v \text{ \ \ \ \ \ \ and \ \ \ \ \ \ } \dot \theta^2 = \frac{v^2}{a^2(3-2\sin^2\theta)} \,. \] \item When $A$ and $C$ collide, the collision is elastic (no energy is lost). At what value of $\theta$ does the second collision between particles $A$ and $C$ occur? (You should justify your answer.) \item When $v=0$, what are the possible values of $\theta$? Is $v =0$ whenever $\theta$ takes these values? \end{questionparts} \end{question} \newpage \section*{Section C: \ \ \ Probability and Statistics} %%%%%%%%%% Q12 \begin{question} Four players $A$, $B$, $C$ and $D$ play a coin-tossing game with a fair coin. Each player chooses a sequence of heads and tails, as follows: \noindent Player A: HHT; \ \ Player B: THH; \ \ Player C: TTH; \ \ Player D: HTT. \noindent The coin is then tossed until one of these sequences occurs, in which case the corresponding player is the winner. \begin{questionparts} \item Show that, if only $A$ and $B$ play, then $A$ has a probability of $\frac14$ of winning. \item If all four players play together, find the probabilities of each one winning. \item Only $B$ and $C$ play. What is the probability of $C$ winning if the first two tosses are~TT? Let the probabilities of $C$ winning if the first two tosses are HT, TH and HH be $p$, $q$ and $r$, respectively. Show that $p=\frac12 +\frac12q$. Find the probability that $C$ wins. \end{questionparts} \end{question} %%%%%%%%%% Q13 \begin{question} The maximum height $X$ of flood water each year on a certain river is a random variable with probability density function $\f$ given by \[ \f(x) = \begin{cases} \lambda \e^{-\lambda x} & \text{for $x\ge0$}\,, \\ 0 & \text{otherwise,} \end{cases} \] where $\lambda$ is a positive constant. It costs $ky$ pounds each year to prepare for flood water of height $y$ or less, where $k$ is a positive constant and $y\ge0$. If $X \le y$ no further costs are incurred but if $X> y$ the additional cost of flood damage is $a(X - y )$ pounds where $a$ is a positive constant. \begin{questionparts} \item Let $C$ be the total cost of dealing with the floods in the year. Show that the expectation of $C$ is given by \[\mathrm{E}(C)=ky+\frac{a}{\lambda}\mathrm{e}^{-\lambda y} \, . \] How should $y$ be chosen in order to minimise $\mathrm{E}(C)$, in the different cases that arise according to the value of $a/k$? \item Find the variance of $C$, and show that the more that is spent on preparing for flood water in advance the smaller this variance. \end{questionparts} \end{question} \end{document}