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

%%%%%%%%%%Q1
\begin{question}
Given that 
\[
\mathrm{f}(x)=\frac{3x^{2}+2(a+b)x+ab}{x^{3}+(a+b)x^{2}+abx},\qquad\mbox{ where }a\mbox{ and }b\mbox{ are non-zero}
\]
express $\mathrm{f}(x)$ in partial fractions, considering any special
case which may arrise. 


If $x,a$ and $b$ are positive integers, show that $\mathrm{f}(x)$
takes the value 1 for only a finite number of values of $x,a$ and
$b$. 
\end{question}

%%%%%%%%%%Q2
\begin{question}
	Given that $x,y$ and $z$ satisfy the equations 
\begin{alignat*}{1}
x^{2}-yz & =a,\\
y^{2}-zx & =b,\\
z^{2}-xy & =c,
\end{alignat*}
 where $a,b$ and $c$ are positive distinct real numbers, show that
\[
\frac{y-z}{b-c}=\frac{z-x}{c-a}=\frac{x-y}{a-b}=\frac{1}{x+y+z}.
\]
By considering 
\[
(y-z)^{2}+(z-x)^{2}+(x-y)^{2},
\]
or otherwise, show that 
\[
x+y+z=\triangle,
\]
where 
\[
\triangle^{2}=\frac{a^{2}+b^{2}+c^{2}-bc-ca-ab}{a+b+c}.
\]
Hence solve the given equations for $x,y$ and $z$.
\end{question}

%%%%%%%%% Q3
\begin{question}
	Prove de Moivre's theorem, that 
\[
(\cos\theta+\mathrm{i}\sin\theta)^{n}=\cos n\theta+\mathrm{i}\sin n\theta,
\]
where $n$ is a positive integer. 


Find all real numbers $x$ and $y$ which satisfy 
\begin{alignat*}{1}
x^{3}\cos3y+2x^{2}\cos2y+2x\cos y & =-1,\\ 
x^{3}\sin3y+2x^{2}\sin2y+2x\sin y & =0.
\end{alignat*}
\end{question}

%%%%%% Q4 

\begin{question}
\begin{questionparts}
\item Show that 
\[
\sin^{-1}(\tanh x)=\tan^{-1}(\sinh x),
\]
when principal values only are considered. 


\item Show that 
\[
\sinh^{-1}(\tan y)=\tanh^{-1}(\sin y),
\]
when $-\frac{1}{2}\pi<y<\frac{1}{2}\pi.$ 


Sketch the graphs of $\sinh^{-1}(\tan y)$ and $\tanh^{-1}(\sin y)$
in the interval $-\pi<y<\pi$ and find the relationship between the
two expressions when $\frac{1}{2}\pi<y<\pi.$

\end{questionparts}


\end{question}


%%%%%%%%% Q5
\begin{question}
	Explain, by means of a sketch, or otherwise, why 
	\[
	\sum_{r=n}^{\infty}\frac{1}{r^{2}}>\int_{n}^{\infty}\frac{\mathrm{d}x}{x^{2}}>\sum_{r=n+1}^{\infty}\frac{1}{r^{2}}.
	\]
	Deduce that 
	\[
	\frac{1}{n}>A-\sum_{r=1}^{n}\frac{1}{r^{2}}>\frac{1}{n+1},\qquad\mbox{ where }{\displaystyle A=\sum_{r=1}^{\infty}\frac{1}{r^{2}}.}
	\]
	Find the smallest value of $n$ for which ${\displaystyle \sum_{r=1}^{n}\frac{1}{r^{2}}}$
	approximates $A$ with an error of less than $10^{-4}.$ Show that,
	for this $n$, 
	\[
	\frac{1}{n+1}+\sum_{r=1}^{n}\frac{1}{r^{2}}
	\]
	approximates $A$ with an error of less than $10^{-8}.$
	\end{question}
	
	%%%%%%%%% Q6
	\begin{question}
	\textit{In this question, standard properties of exponential, logarithmic
	and trigonometric functions should not be used.}


	A function $\mathrm{f}$ satisfies 
	\[
	\frac{\mathrm{d}}{\mathrm{d}x}[\mathrm{f}(x)]=\mathrm{f}(x)
	\]
	with $\mathrm{f}(0)=1.$ For any fixed number $a$, show that 
	\[
	\frac{\mathrm{d}}{\mathrm{d}x}[\mathrm{f}(a-x)\mathrm{f}(x)]=0,
	\]
	and deduce that $\mathrm{f}(x)\mathrm{f}(y)=\mathrm{f}(x+y)$ for
	all $x$ and $y$. 


	Functions $\mathrm{c}$ and $\mathrm{s}$ satisfy 
	\[
	\frac{\mathrm{d}}{\mathrm{d}x}[\mathrm{c}(x)]=-\mathrm{s}(x)\quad\mbox{ and }\quad\frac{\mathrm{d}}{\mathrm{d}x}[\mathrm{s}(x)]=\mathrm{c}(x),
	\]
	with $\mathrm{s}(0)=0$ and $\mathrm{c}(0)=1$. Show that 
	\[
	\mathrm{c}(x+y)=\mathrm{c}(x)\mathrm{c}(y)-\mathrm{s}(x)\mathrm{s}(y).
	\]
	
	 \end{question}
	 
	 %%%%%%%%% Q7
\begin{question}
	Let 
	\[
	I=\int_{0}^{\ln K}[\mathrm{e}^{x}]\,\mathrm{d}x,
	\]
	where the notation $[y]$ means the largest integer less than or equal
	to $y$. Show that 
	\[
	I=N\ln K-\ln(N!),
	\]
	where $N=[K].$
	
	
	\end{question}
	
	%%%%%%%%% Q8
	\begin{question}
		Let $S$ be the set of consecutive integers $1,2,\ldots,(N-1),$ where
		$N\geqslant3,$ and let $G$ be a subset of $S$ which forms a group
		under multiplication modulo $N.$ Show that if $(N-1)\in G$, then
		the order of $(N-1)$ is $2$. 


		Let $m$ and $n$ be elements of $G$, with orders $p$ and $q$ respectively,
		such that $m+n=N$. Explain your reasoning carefully, show that 
		
		\begin{questionparts}
		\item if $p$ and $q$ are both even, then $p=q$, 
		\item if $p$ is even and $q$ is off, then $p=2q$, 
		\item it is impossible for both $p$ and $q$ to be odd. 
		\end{questionparts}

		Now suppose that 
		\[
		G=\{1,2,4,5,8,10,11,13,16,17,19,20\},
		\]
		 which may be assumed to form a group under multiplication modulo
		21. Calculate the order of the elements $2$ and $5$ of this group.
		By making deductions about the orders of all other elements of $G$,
		or otherwise, prove that $G$ is not isomorphic to the cyclic group
		of order 12.
		
		\end{question}
		
		
%%%%%%%%% Q9
		\begin{question}
		\begin{questionparts}
			\item Let $\mathbf{a}$ and $\mathbf{b}$ be given
		vectors with $\mathbf{b\neq0}$, and let $\mathbf{x}$ be a position
		vector. Find the condition for the sphere $\left|\mathbf{x}\right|=R$,
		where $R>0$, and the plane $(\mathbf{x-a})\cdot\mathbf{b}=0$ to
		intersect. 


		When this condition is satisfied, find the radius and the position
		vector of the centre of the circle in which the plane and sphere intersect. 


		\item Let $\mathbf{c}$ be a given vector, with $\mathbf{c\neq0}.$
		The vector $\mathbf{x}'$ is related to the vector $\mathbf{x}$ by
		\[
		\mathbf{x}'=\mathbf{x}-\frac{2(\mathbf{x}\cdot\mathbf{c})\mathbf{c}}{\left|\mathbf{c}\right|^{2}}.
		\]
		Interpret this relation geometrically. 
	\end{questionparts}
		\end{question}
		
	
%%%%%%%%%% 10
\begin{question}
	The distinct island of Amphibia is populated by speaking frogs and
			toads. They spend much of their time in small groups, making statements
			about themselves. Toads always tell the truth and frogs always lie.
			In each of the following four scenes from Amphibian life, decide which
			characters mentioned are frogs and which are toads, explaining your
			reasoning carefully: 
			


\begin{questionparts}
			\item $A$: ``Both $B$ and myself are frogs''
			\item $C$: ``At least one of $D$ and myself is a frog.''
			\item $E$: ``Both $G$ and $H$ are toads.''
			
			$G$: ``This is true.''
			
			$H$: ``No, that is not true.''

			\item $I$ and $J$ talking about $I,J$ and $K$: 
			
			$I$: ``All of us are frogs.''
			
			$J$: ``Exactly one of us is a toad.''

			\end{questionparts}
			\end{question}
			
		
		
		
	
\newpage
\section*{Section B: \ \ \ Mechanics}


	
%%%%%%%%%% Q11
\begin{question}
	Two points $A$ and $B$ are at a distance $a$ apart on a horizontal
	plane. A particle of mass $m$ is projected from $A$ with speed $V$,
	at an angle of elevation of $45^{\circ}$ to the line $AB.$ Another
	particle, also of mass $m$, is projected from $B$ with speed $U$
	at an angle of elevation of $30^{\circ}$ to the line $BA$ so that
	the two particles collide at the instant when each particle is at
	the highest point of its trajectory. 


	Show that $U^{2}=2V^{2}$ and that 
	\[
	a=\frac{V^{2}}{2g}(1+\sqrt{3}).
	\]
	At impact the two particles coalesce. When the combined particle strikes
	the horizontal plane the velocity of the particle is inclined at an
	angle $\phi$ to the horizontal. Show that $\tan\phi=1+\sqrt{3}.$
	\end{question}
	
%%%%%%%%%% Q12
\begin{question}	
A thin smooth wire in the form of a circle, of radius $a$ and centre
$O$, is fixed in a horizontal plane. Two small beads $A$ and $B$,
each of mass $m$, are threaded on the wire and are connected by a
light straight spring of natural length $2a\sin\alpha$ and modulus
$\lambda,$ where $0<\alpha<\frac{1}{4}\pi.$ The spring is compressed
so that the angle $AOB$ is $2\beta$ and the beads are then released
from rest. Show that in the ensuing motion 
\[
ma\dot{\theta}^{2}\sin\alpha+\lambda(\sin\theta-\sin\alpha)^{2}=\lambda(\sin\beta-\sin\alpha)^{2}
\]
where $2\theta$ denotes the angle $AOB$ at time $t$ after release. 

\begin{questionparts}
\item If $\beta-\alpha$ is small, show that $T$, the preiod of oscillations,
is given approximately by 
\[
T=2\pi\sqrt{\frac{ma\sin\alpha}{\lambda\cos^{2}\alpha}}.
\]

\item If $\beta-\alpha$ is not small, write down an expression, in the
form of a definite integral, for the exact period of oscillations,
in each of the two cases (a) $\sin\beta>2\sin\alpha-1$ and (b) $\sin\beta<2\sin\alpha-1$. 
\end{questionparts}
\end{question}

%%%%%%%%%% Q13

\begin{question}
A chocolate orange consists of a solid sphere of uniform chocolate
of mass $M$ and radius $a$, sliced into segments by planes through
its axis. It stands on a horizontal table with its axis vertical,
and it is held together only by a narrow ribbon round its equator. 


Show that the tension in the ribbon is at least $\frac{3}{32}Mg$. 


{[}\textit{You may assume that the centre of mass of a segment of
angle $2\theta$ is at a distance $\dfrac{3\pi a\sin\theta}{16\theta}$
from the axis.}{]} 
\end{question}
	
%%%%%%%%%% Q14
\begin{question}
	A uniform disc of mass $M$ and radius $a$ is free to rotate in a
	horizontal plane about a fixed vertical axis through the centre, $O$,
	of the disc. A particle of mass $\frac{1}{2}M$ is attached by a light
	straight wire of length $a/2$ to the vertical axis at $O$, so that
	the particle can rotate freely about the vertical axis. The particle,
	initially at rest, is placed gently on the disc at time $t=0$, when
	the disc is spinning with angular speed $\Omega.$ Relative motion
	between the particle and disc is opposed by a frictional force of
	magnitude $Mak(\omega_{1}-\omega_{2}),$ where, at time $t$, $\omega_{1}$
	is the angular speed of the disc, $\omega_{2}$ is the angular speed
	of the wire, and $k$ is a constant. Derive equations for the rate
	of change of $\omega_{1}$ and $\omega_{2},$ and show that 
	\[
	4\omega_{1}+\omega_{2}=4\Omega.
	\]
	Show further that 
	\[
	\omega_{1}=\frac{\Omega}{5}(4+\mathrm{e}^{-5kt}).
	\]
\end{question}
	
	\newpage
\section*{Section C: \ \ \ Probability and Statistics}


%%%%%%%%%% Q15
\begin{question}
The King of Smorgasbrod wishes to raise as much money as possible
by fining people who sell underweight cartons of kippers. The weight
of a kipper is normally distributed with mean 200 grams and standard
deviation 10 grams. Kippers are packed in cartons of 625, and vast
quantities of them are sold. 


Every day a carton is to be selected at random from each vendor of
Kippers. Three schemes for determining the fines are proposed: 
\begin{questionparts}
\item Weight the entire carton, and find the vendor 1500 crowns if the average
weight of a kipper is less than 199 grams. 
\item Weigh 25 kippers selected at random from the carton and fine the vendor
100 crowns if the average weight of a kipper is less than 198 grams. 
\item Remove kippers one at a time and at random from the carton until a
kipper has been found which weighs \textit{more }than 200 grams and
fine the vendor $3n(n-1)$ crowns, where $n$ is the number of kippers
removed. 
\end{questionparts}

Determine which scheme the king should select, justifying your answer. 
\end{question}

%%%%%%%%%% Q16
\begin{question}
A tennis tournament is arranged for $2^{n}$ players. It is organised
as a knockout tournament so that only the winners in any given round
proceed to the next round. Opponents in each round except the final
are drawn at random, and in any match either player has a probability
of $\frac{1}{2}$ of winning. Two players are chosen at random before
the draw for the first round. Find the probabilities that they play
each other: 

\begin{questionparts}
\item in the first round, 
\item in the final, 
\item in the tournament. \end{questionparts}
\end{question}
\end{document}
