\documentclass[a4, 11pt]{report} \pagestyle{myheadings} \markboth{}{Paper II, 2008 \ \ \ \ \ \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} A sequence of points $(x_1,y_1)$, $(x_2,y_2)$, $\ldots$ in the cartesian plane is generated by first choosing $(x_1,y_1)$ then applying the rule, for $n=1$, $2$, $\ldots$, \[ (x_{n+1}, y_{n+1}) = (x_n^2-y_n^2 +a, \; 2x_ny_n+b+2)\,, \] where $a$ and $b$ are given real constants. \begin{questionparts} \item In the case $a=1$ and $b=-1$, find the values of $(x_1,y_1)$ for which the sequence is constant. \item Given that $(x_1,y_1) = (-1,1)$, find the values of $a$ and $b$ for which the sequence has period 2. \end{questionparts} \end{question} %%%%%%%%%%Q2 \begin{question} Let $a_n$ be the coefficient of $x^n$ in the series expansion, in ascending powers of $x$, of \[\displaystyle \frac{1+x}{(1-x)^2(1+x^2)} \,, \] where $\vert x \vert <1\,$. Show, using partial fractions, that either $a_n =n+1$ or $a_n = n+2$ according to the value of $n$. Hence find a decimal approximation, to nine significant figures, for the fraction $ \displaystyle \frac{11\,000}{8181}$. \newline [You are not required to justify the accuracy of your approximation.] \end{question} %%%%%%%%% Q3 \begin{question} \begin{questionparts} \item Find the coordinates of the turning points of the curve $y=27x^3-27x^2+4$. Sketch the curve and deduce that $x^2(1-x)\le 4/27$ for all $x\ge0\,$. Given that each of the numbers $a$, $b$ and $c$ lies between $0$ and $1$, prove by contradiction that at least one of the numbers $bc(1-a)$, $ca(1-b)$ and $ab(1-c)$ is less than or equal to $4/27$. \item Given that each of the numbers $p$ and $q$ lies between $0$ and $1$, prove that at least one of the numbers $p(1-q)$ and $q(1-p)$ is less than or equal to $1/4$. \end{questionparts} \end{question} %%%%%% Q4 \begin{question} A curve is given by \[x^2+y^2 +2axy = 1,\] where $a$ is a constant satisfying $0 1.4$. Deduce that $ 2^{\sqrt2} > 1+ \sqrt2\, $. Use this result to determine which of the above integrals is greater. \end{question} %%%%%%%%% Q6 \begin{question} A curve has the equation $y=\f(x)$, where \[ \f(x) = \cos \Big( 2x+ \frac \pi 3\Big) + \sin \Big ( \frac{3x}2 - \frac \pi 4\Big). \] \begin{questionparts} \item Find the period of $\f(x)$. \item Determine all values of $x$ in the interval $-\pi\le x \le \pi$ for which $\f(x)=0$. Find a value of~$x$ in this interval at which the curve touches the $x$-axis without crossing it. \item Find the value or values of $x$ in the interval $0\le x \le 2\pi$ for which $\f(x)=2\,$. \end{questionparts} \end{question} %%%%%%%%% Q7 \begin{question} \begin{questionparts} \item By writing $y=u{(1+x^2)\vphantom{\dot A}}^{\frac12}$, where $u$ is a function of $x$, find the solution of the equation \[ \frac 1 y \frac{\d y} {\d x} = xy + \frac x {1+x^2} \] for which $y=1$ when $x=0$. \item Find the solution of the equation \[ \frac 1 y \frac{\d y} {\d x} = x^2y + \frac {x^2 } {1+x^3} \] for which $y=1$ when $x=0$. \item Give, without proof, a conjecture for the solution of the equation \[ \frac 1 y \frac{\d y} {\d x} = x^{n-1}y + \frac {x^{n-1} } {1+x^n} \] for which $y=1$ when $x=0$, where $n$ is an integer greater than 1. \end{questionparts} \end{question} %%%%%%%%% Q8 \begin{question} The points $A$ and $B$ have position vectors $\bf a$ and $\bf b$, respectively, relative to the origin $O$. The points $A$, $B$ and $O$ are not collinear. The point $P$ lies on $AB$ between $A$ and $B$ such that \[ AP : PB = (1-\lambda):\lambda\,. \] Write down the position vector of $P$ in terms of $\bf a$, $\bf b$ and $\lambda$. Given that $OP$ bisects $\angle AOB$, determine $\lambda$ in terms of $a$ and $b$, where $a=\vert \bf a\vert$ and $b=\vert \bb\vert$. The point $Q$ also lies on $AB$ between $A$ and $B$, and is such that $AP=BQ$. Prove that $$OQ^2-OP^2=(b-a)^2\,.$$ \end{question} \newpage \section*{Section B: \ \ \ Mechanics} %%%%%%%%%% Q9 \begin{question} In this question, use $g=10\,$m\,s$^{-2}$. In cricket, a fast bowler projects a ball at $40\,$m\,s$^{-1}$ from a point $h\,$m above the ground, which is horizontal, and at an angle $\alpha$ above the horizontal. The trajectory is such that the ball will strike the stumps at ground level a horizontal distance of $20\,$m from the point of projection. \begin{questionparts} \item Determine, in terms of $h$, the two possible values of $\tan\alpha$. Explain which of these two values is the more appropriate one, and deduce that the ball hits the stumps after approximately half a second. \item State the range of values of $h$ for which the bowler projects the ball below the horizontal. \item In the case $h=2.5$, give an approximate value in degrees, correct to two significant figures, for $\alpha$. You need not justify the accuracy of your approximation. \end{questionparts} [You may use the small-angle approximations $\cos\theta \approx 1$ and $\sin\theta\approx \theta$.] \end{question} %%%%%%%%%% Q10 \begin{question} The lengths of the sides of a rectangular billiards table $ABCD$ are given by $AB = DC = a$ and \mbox{$AD=BC = 2b$}. There are small pockets at the midpoints $M$ and $N$ of the sides $AD$ and $BC$, respectively. The sides of the table may be taken as smooth vertical walls. A small ball is projected along the table from the corner $A$. It strikes the side $BC$ at $X$, then the side $DC$ at $Y$ and then goes directly into the pocket at $M$. The angles $BAX$, $CXY$ and $DY\!M$ are $\alpha$, $\beta$ and $\gamma$ respectively. On each stage of its path, the ball moves with constant speed in a straight line, the speeds being $u$, $v$ and $w$ respectively. The coefficient of restitution between the ball and the sides is $e$, where $e>0$. \begin{questionparts} \item Show that $\tan\alpha \tan \beta = e$ and find $\gamma$ in terms of $\alpha$. \item Show that $\displaystyle \tan\alpha = \frac {(1+2e)b} {(1+e)a}$ and deduce that the shot is possible whatever the value of~$e$. \item Find an expression in terms of $e$ for the fraction of the kinetic energy of the ball that is lost during the motion. \end{questionparts} \end{question} %%%%%%%%%% Q11 \begin{question} A wedge of mass $km$ has the shape (in cross-section) of a right-angled triangle. It stands on a smooth horizontal surface with one face vertical. The inclined face makes an angle $\theta$ with the horizontal surface. A particle $P$, of mass $m$, is placed on the inclined face and released from rest. The horizontal face of the wedge is smooth, but the inclined face is rough and the coefficient of friction between $P$ and this face is $\mu$. \begin{questionparts} \item When $P$ is released, it slides down the inclined plane at an acceleration $a$ relative to the wedge. Show that the acceleration of the wedge is \[ \frac {a \cos\theta}{k+1}\,. \] To a stationary observer, $P$ appears to descend along a straight line inclined at an angle~$45^\circ$ to the horizontal. Show that \[ \tan\theta = \frac k {k+1}\,. \] In the case $k=3$, find an expression for $a$ in terms of $g$ and $\mu$. \item What happens when $P$ is released if $\tan\theta \le \mu$? \end{questionparts} \end{question} \newpage \section*{Section C: \ \ \ Probability and Statistics} %%%%%%%%%% Q12 \begin{question} In the High Court of Farnia, the outcome of each case is determined by three judges: the ass, the beaver and the centaur. Each judge decides its verdict independently. Being simple creatures, they make their decisions entirely at random. Past verdicts show that the ass gives a guilty verdict with probability $p$, the beaver gives a guilty verdict with probability $p/3$ and the centaur gives a guilty verdict with probability $p^2$. Let $X$ be the number of guilty verdicts given by the three judges in a case. Given that $\E(X)= 4/3$, find the value of $p$. The probability that a defendant brought to trial is guilty is $t$. The King pronounces that the defendant is guilty if at least two of the judges give a guilty verdict; otherwise, he pronounces the defendant not guilty. Find the value of $t$ such that the probability that the King pronounces correctly is $1/2$. \end{question} %%%%%%%%%% Q13 \begin{question} Bag $P$ and bag $Q$ each contain $n$ counters, where $n\ge2$. The counters are identical in shape and size, but coloured either black or white. First, $k$ counters ($0\le k\le n$) are drawn at random from bag $P$ and placed in bag $Q$. Then, $k$ counters are drawn at random from bag $Q$ and placed in bag $P$. \begin{questionparts} \item If initially $n-1$ counters in bag $P$ are white and one is black, and all $n$ counters in bag~$Q$ are white, find the probability in terms of $n$ and $k$ that the black counter ends up in bag $P$. Find the value or values of $k$ for which this probability is maximised. \item If initially $n-1$ counters in bag~$P$ are white and one is black, and $n-1$ counters in bag~$Q$ are white and one is black, find the probability in terms of $n$ and $k$ that the black counters end up in the same bag. Find the value or values of $k$ for which this probability is maximised. \end{questionparts} \end{question} \end{document}