
\documentclass[a4, 11pt]{report}


\pagestyle{myheadings}
\markboth{}{Paper I, Specimen
\ \ \ \ \ 
\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{{\rm d}}
\def\e{{\rm e}}
\def\g{{\rm g}}
\def\h{{\rm h}}
\def\f{{\rm f}}
\def\p{{\rm p}}
\def\s{{\rm s}}
\def\t{{\rm t}}


\def\A{{\rm A}}
\def\B{{\rm B}}
\def\E{{\rm E}}
\def\F{{\rm F}}
\def\G{{\rm G}}
\def\H{{\rm H}}
\def\P{{\rm 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}




\begin{document}
\setcounter{page}{2}

 
\section*{Section A: \ \ \ Pure Mathematics}

%%%%%%%%%%Q1
\begin{question}
\begin{questionparts}
\item The real numbers $x$ and $y$ satisfy the
equation $4x^{2}+16xy+y^{2}+24x=0.$ Prove that either $x\leqslant0$
or $x\geqslant\frac{2}{5},$ and, similarly, find restrictions on
the values of $y$. 


\item Find the coefficient of $x^{n}$ in the expansion, in ascending
powers of $x$, of 
\[
\frac{9}{(2-x)^{2}(1+x)}.
\]
State the set of values of $x$ for which this expansion is valid.
\end{questionparts}
\end{question}

%%%%%%%%%%Q2
\begin{question}
In the figure, the angle $PAC$ is a right angle. $AB=h,BC=d,AP=x$
and the angle $BPC$ is $\theta$. Express $\tan\theta$ in terms
of $h,d$ and $x$. 


\noindent \begin{center}
\psset{xunit=0.6cm,yunit=0.6cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.5pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-2.26,-5.36)(6.04,3.78) \pspolygon(-1,2.58)(-0.58,2.58)(-0.58,3)(-1,3) \psline(-1,3)(5,3) \psline(-1,3)(-1,-5) \psline(-1,-4)(5,3) \psline(-1,-4)(2.56,3) \pscustom{\parametricplot{0.8621700546672264}{1.1003151111843517}{1.5*cos(t)+-1|1.5*sin(t)+-4}\lineto(-1,-4)\closepath} \rput[tl](-1.46,3.62){$A$} \rput[tl](-1.48,0.4){$x$} \rput[tl](-1.55,-3.64){$P$} \rput[tl](-0.1,-2.2){$\theta$} \rput[tl](0.62,3.66){$h$} \rput[tl](2.48,3.7){$B$} \rput[tl](5.24,3.6){$C$} \rput[tl](3.76,3.66){$d$} \end{pspicture*}
\par\end{center}


Given that $x$ may be varied, deduce, or find otherwise, the value
of $x$ (in terms of the constants $d$ and $h$) for which $\theta$
has its maximum value. 
\end{question}

%%%%%%%%% Q3
\begin{question}
Using the substitution $z=\dfrac{\mathrm{d}y}{\mathrm{d}x}-y,$ or
otherwise, solve the equation 
\[
\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+2\frac{\mathrm{d}y}{\mathrm{d}x}-3y=2\mathrm{e}^{x},
\]
given that 
\[
y=1\mbox{ and }\frac{\mathrm{d}y}{\mathrm{d}x}=2\quad\mbox{ when }x=0.
\]
\end{question}


%%%%%% Q4 

\begin{question}Evaluate 
\[
\int_{1}^{\infty}\frac{1}{(x+1)\sqrt{x^{2}+2x-2}}\,\mathrm{d}x.
\]

\end{question}


%%%%%%%%% Q5
\begin{question}
\begin{questionparts} 
\item You are writing down all the positive integers
in increasing order, starting from 1, until you have written 1000
digits, at which point you stop (even possibly in the middle of a
number). How many times have you used the digit 7? Explain your reasoning
briefly but clearly. 


\item In how many zeros does the number $365!$ terminate, when written
in base 10? Justify your answer. 
\end{questionparts}
\end{question}
	
	%%%%%%%%% Q6
	\begin{question}
$\,$\vspace{-0.5cm}



\noindent \begin{center}
\psset{xunit=0.65cm,yunit=0.65cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.5pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-3.91,-3.36)(4.84,4.8)
 
\psbezier(-2.05, 2.74)(-1.44, 3.3)(-0.17, 3.92)(0.58, 4.07)
\psbezier(0.58, 4.07)(1.5,4)(2.8,3.2)(3.32,2.75)
\psbezier(3.32,2.75)(3.73,2.05)(4.06,1.07)(4.02,-0.1)
\psbezier(4.02,-0.1)(3.82, -0.68)(3.1, -1.8)(2.17, -2.48)
\psbezier(2.17, -2.48)(1.46, -2.7)(0.3, -2.75)(-0.75, -2.5)
\psbezier(-0.75, -2.5)(-1.5,-2)(-2.13, -1.17)(-2.66, -0.16)
\psbezier(-2.66, -0.16)(-2.69, 0.39)(-2.61, 1.71)(-2.05, 2.74)

\rput[tl](-3.26,-0.13){$A$} \rput[tl](-2.47,3.31){$B$} \rput[tl](0.51,4.69){$C$} \rput[tl](3.34,3.3){$D$} \rput[tl](4.17,0.01){$E$} \rput[tl](2.44,-2.52){$F$} \rput[tl](-1.06,-2.66){$G$} \end{pspicture*}
\par\end{center}


The diagram shows a cross-section, parallel to its faces, of a British
50 pence coin. The seven arcs $AB,BC,\ldots,FG,GA$ are all of equal
length and each arc is formed from the circle having its centre at
the vertex diametrically opposite the mid-point of the arc. Given
that the radius of each of these circles is $a$, show that the area
of a face of the coin is 
\[
\frac{a^{2}}{2}\left(\pi-7\tan\frac{\pi}{14}\right).
\]
	 \end{question}
	 
%%%%%%%%% Q7
\begin{question}
Find the modulus and argument of $1+\mathrm{e}^{2\mathrm{i}\alpha}$
where $-\frac{1}{2}\pi<\alpha<\frac{1}{2}\pi.$


By using de Moivre's theorem, or otherwise, sum the series 
\[
\sum_{r=0}^{n}\frac{n!}{r!(n-r)!}\sin(2r+1)\alpha.
\]
	\end{question}
	
	%%%%%%%%% Q8
	\begin{question}
Let $(a,b)$ be a fixed point, and $(x,y)$ a variable point, on the
curve $y=\mathrm{f}(x),$ $(x\geqslant a,\mathrm{f}'(x)\geqslant0).$
The curve divides the rectangle with vertices $(a,b),(a,y),(x,y)$
and $(x,b)$ into two portions, the lower of which has always half
the area of the upper. Show that the curve is an arc of a parabola
with its vertex at $(a,b)$. 
		
		\end{question}
		
		
%%%%%%%%% Q9
\begin{question}
\begin{questionparts} 
	\item Find the set of values of $x$ which satisfy
the inequality 
\[
\left|x+\frac{x-1}{x+1}\right|<2,
\]
expressing the set in terms of intervals whose end points are numbers
of the form \phantom{  } $a+b\sqrt{3},$ where $a$ and $b$ are integers. 


\item Evaluate ${\displaystyle \int_{0}^{\pi}\left|\cos x-\sin x\right|\,\mathrm{d}x.}$
\end{questionparts}
\end{question}

\newpage
\section*{Section B: \ \ \ Mechanics}


	
%%%%%%%%%% Q10
\begin{question}
An engine of mass $m$ moves along a straight level track against
a resistance which at any instant has magnitude $mkv,$ where $v$
is the speed of the engine and $k$ is a constant. The engine is working
at a constant rate $9mku^{2}.$ Given that the engine starts from
rest, state the maximum possible speed of the engine, and explain
briefly whether this speed can be obtained in a finite time. Show
that the speed of the engine increases from $u$ to $2u$ in time
\[
\frac{1}{2k}\ln\left(\frac{8}{5}\right).
\]
When the engine has attained a speed $2u$, the power is cut off,
and an additional resistance to motion, of constant magnitude $mkB,$
is applied so that the engine is brought to rest. Find the distance
travelled by the engine with the power cut off. 
	\end{question}
	
%%%%%%%%%% Q11
\begin{question}	
$\,$


\noindent \begin{center}
\psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.5pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-5.46,-0.32)(4.34,3.58) \pspolygon[linewidth=0pt,linecolor=white,fillcolor=black,fillstyle=solid,opacity=0.05](-4.87,0.54)(-1.87,0.54)(-1.87,2.54) \psline(-5,0)(4,0) \psline(0,0)(3,3) \pscustom{\parametricplot{0.0}{0.7853981633974483}{1.16*cos(t)+0|1.16*sin(t)+0}\lineto(0,0)\closepath} \pscircle(-0.58,1.3){1.3} \psline(-4.87,0.54)(-1.87,0.54) \psline(-1.87,0.54)(-1.87,2.54) \psline{->}(-1.87,1.32)(-1.12,1.32) \rput[tl](-1.68,1.77){$P$} \rput[tl](0.52,0.48){$45^{\circ}$} \end{pspicture*}
\par\end{center}


A uniform cylinder of mass $M$ rests on a horizontal floor touching
a loading ramp at $45^{\circ}$ to the horizontal, as shown in the
diagram. The cylinder is pushed from the side with a force of magnitude
$P$ by the vertical face of a piece of moving equipment. The coefficient
of friction between the cylinder and the vertical face is $\mu$ and
the coefficient of friction between the cylinder and the ramp is $v$.
The value of $P$ is such that the cylinder is just about to roll
up the ramp. State on which surface the friction must be limiting,
and hence show that 
\[
P=\frac{Mg}{1-\mu(1+\sqrt{2})}.
\]
Show further that $\mu<\sqrt{2}-1$ and $v\geqslant\mu/(\sqrt{2}-\mu)$.  
\end{question}

%%%%%%%%%% Q12

\begin{question}
A game consists of a player sliding a small uniform disc $A$, initially
at a point $O$, across horizontal ice. The disc collides with an
identical disc $B$ at rest at a distance $d$ away from $O.$ After
the collision, $B$ comes to rest at a target $C$ distant $2d$ away
from $O$. All motion is along the line $OC$ and the discs may be
treated as point masses. Show that the speed of $B$ immediately after
the collision is $\sqrt{2\mu gd}$, where $\mu$ is the coefficient
of friction between the discs and the ice. Deduce that $A$ is started
with speed $U$ given by 
\[
U=\frac{\sqrt{2\mu gd}}{1+e}(5+2e+e^{2})^{\frac{1}{2}},
\]
where $e$ is the coefficient of restitution between the discs. 
\end{question}
	
%%%%%%%%%% Q13
\begin{question}
A particle of mass $m$ is attached to one end of a light elastic
string of natural length $a,$ and the other end of the string is
attached to a fixed point $A$. When at rest and hanging vertically
the string has length $2a.$ The particle is set in motion so that
it moves in a horizontal circle below the level of $A$. The vertical
plane through $A$ containing the string rotates with constant angular
speed $\omega.$ Show that for this motion to be possible, the string
must be stretched to a length greater than $2a$ and $\omega$ must
satisfy 
\[
\frac{g}{2a}<\omega^{2}<\frac{g}{a}.
\]
\end{question}
	
	\newpage
\section*{Section C: \ \ \ Probability and Statistics}


%%%%%%%%%% Q14
\begin{question}
$A$ and $B$ play the following game. $A$ throws two unbiased four-faced
dice onto a table (the four faces of each die are numbered $1,2,3,4$
respectively). The total score is the sum of the numbers on the faces
in contact with the table. $B$ tries to guess this score, and guesses
$x$. If his guess is right he wins $100x^{2}$ pence, and if his
guess is wrong he loses $50x$ pence. 

\begin{questionparts}
\item Show that $B$'s expected gain if he guesses 8 is 25 pence. 
\item Which value of $x$ would you advise $B$ to choose, and what is his
expected gain in this case? 
\end{questionparts}


\end{question}

%%%%%%%%%% Q15
\begin{question}
The random variable $C$ takes integral values in the interval $-5$
to $5,$ with probabilities 
\begin{alignat*}{1}
\mathrm{P}(C=-5) & =\mathrm{P}(C=5)=\frac{1}{20},\\
\mathrm{P}(C=i) & =\frac{1}{10},\quad\mbox{ for }-4\leqslant i\leqslant4.
\end{alignat*}
Calculate the expectation and variance of $C$. 


A shopper buys 36 items at random in a supermarket and, instead of
adding up her bill exactly, she first rounds the cost of each item
to the nearest 10 pence, rounding up or down with equal probability
if there is an odd amount of 5 pence. Should she suspect a mistake
if the cashier asks her for 20 pence more than she estimated? Explain
your reasoning briefly but clearly. 
\end{question}

%%%%%%%%%% Q16
\begin{question}
The length in minutes of a telephone call made by a man from a public
call-box is a random variable denoted by $T$. The probability density
function of $T$ is given by: 
\[
\mathrm{f}(t)=\begin{cases}
0 & t<0\\
\tfrac{1}{2} & 0\leqslant t<1,\\
k\mathrm{e}^{-2t} & t\geqslant1,
\end{cases}
\]
where $k$ is constant. Show that the expected length of a call is
one minute. Find the cumulative distribution function of $T$. 


To pay for a call, the man inserts into the coin box a 10 pence coin
at the beginning of the call, and then another 10 pence coin after
each half-minute of the call has elapsed. The random variable $C$
is the cost in pence of the call. Prove that 
\[
\mathrm{E}(C)=22\tfrac{1}{2}+\frac{5}{\mathrm{e}-1}.
\]
Comment \textit{briefly} on why this value differs from the expected
length of a call multiplied by the charging rate of 20 pence per minute.

	\end{question}
\end{document}
