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

%%%%%%%%%%Q1
\begin{question}
Find the set of positive integers $n$ for which $n$ does not divide
$(n-1)!.$ Justify your answer. [Note that small values of $n$
may require special consideration.]
\end{question}

%%%%%%%%%%Q2
\begin{question}
Let ${\displaystyle I_{m,n}=\int\cos^{m}x\sin nx\,\mathrm{d}x,}$
where $m$ and $n$ are non-negative integers. Prove that for $m,n\geqslant1,$
\[
(m+n)I_{m,n}=-\cos^{m}\cos nx+mI_{m-1,n-1}.
\]


\begin{questionparts}
\item Show that ${\displaystyle \int_{0}^{\pi}\cos^{m}x\sin nx\,\mathrm{d}x=0}$
whenever $m,n$ are both even or both odd. 
\item Evaluate ${\displaystyle \int_{0}^{\frac{\pi}{2}}\sin^{2}x\sin3x\,\mathrm{d}x.}$
\end{questionparts}

\end{question}

%%%%%%%%% Q3
\begin{question}
\begin{questionparts}
	\item If $z=x+\mathrm{i}y,$ with $x,y$ real,
	show that 
	\[
	\left|x\right|\cos\alpha+\left|y\right|\sin\alpha\leqslant\left|z\right|\leqslant\left|x\right|+\left|y\right|
	\]
	for all real $\alpha.$


	\item By considering $(5-\mathrm{i})^{4}(1+\mathrm{i}),$ show that
	\[
	\frac{\pi}{4}=4\tan^{-1}\left(\frac{1}{5}\right)-\tan^{-1}\left(\frac{1}{239}\right).
	\]
	Prove similarly that 
	\[
	\frac{\pi}{4}=3\tan^{-1}\left(\frac{1}{4}\right)+\tan^{-1}\left(\frac{1}{20}\right)+\tan^{-1}\left(\frac{1}{1985}\right).
	\]
\end{questionparts}
\end{question}


%%%%%% Q4 

\begin{question}$\ $
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Two funnels $A$ and $B$ have surfaces formed by rotating the curves
$y=x^{2}$ and $y=2\sinh^{-1}x$ $(x>0)$ above the $y$-axis. The
bottom of $B$ is one unit lower than the bottom of $A$ and they
are connected by a thin rubber tube with a tap in it. The tap is closed
and $A$ is filled with water to a depth of 4 units. The tap is then
closed. When the water comes to rest, both surfaces are at a height
$h$ above the bottom of $B$, as shown in the diagram. Show that
$h$ satisfies the equation 
\[
h^{2}-3h+\sinh h=15.
\]


\end{question}


%%%%%%%%% Q5
\begin{question}
A secret message consists of the numbers $1,3,7,23,24,37,39,43,43,43,45,47$
arranged in some order as $a_{1},a_{2},\ldots,a_{12}.$ The message
is encoded as $b_{1},b_{2},\ldots,b_{12}$ with $0\leqslant b_{j}\leqslant49$
and 
\begin{alignat*}{1}
b_{2j} & \equiv a_{2j}+n_{0}+j\pmod{50},\\
b_{2j+1} & \equiv a_{2j+1}+n_{1}+j\pmod{50},
\end{alignat*}
for some integers $n_{0}$ and $n_{1}.$ If the coded message is $35,27,2,36,15,35,8,40,40,37,24,48,$
find the original message, explaining your method carefully. 
	\end{question}
	
	%%%%%%%%% Q6
	\begin{question}
The functions $x(t)$ and $y(t)$ satisfy the simultaneous differential
equations 
\begin{alignat*}{1}
\dfrac{\mathrm{d}x}{\mathrm{d}t}+2x-5y & =0\\
\frac{\mathrm{d}y}{\mathrm{d}t}+ax-2y & =2\cos t,
\end{alignat*}
subject to $x=0,$ $\dfrac{\mathrm{d}y}{\mathrm{d}t}=0$ at $t=0.$ 


Solve these equations for $x$ and $y$ in the case when $a=1$. 

Without
solving the equations explicitly, state briefly how the form of the
solutions for $x$ and $y$ if $a>1$ would differ from the form when
$a=1.$ 
	 \end{question}
	 
	 %%%%%%%%% Q7
\begin{question}
Prove that 
\[
\tan^{-1}t=t-\frac{t^{3}}{3}+\frac{t^{5}}{5}-\cdots+\frac{(-1)^{n}t^{2n+1}}{2n+1}+(-1)^{n+1}\int_{0}^{t}\frac{x^{2n+2}}{1+x^{2}}\,\mathrm{d}x.
\]
Hence show that, if $0\leqslant t\leqslant1,$ then 
\[
\frac{t^{2n+3}}{2(2n+3)}\leqslant\left|\tan^{-1}t-\sum_{r=0}^{n}\frac{(-1)^{r}t^{2r+1}}{2r+1}\right|\leqslant\frac{t^{2n+3}}{2n+3}.
\]
Show that, as $n\rightarrow\infty,$ 
\[
4\sum_{r=0}^{n}\frac{(-1)^{r}}{(2r+1)}\rightarrow\pi,
\]
but that the error in approximating $\pi$ by ${\displaystyle 4\sum_{r=0}^{n}\frac{(-1)^{r}}{(2r+1)}}$
is at least $10^{-2}$ if $n$ is less than or equal to $98$. 
	\end{question}
	
	%%%%%%%%% Q8
	\begin{question}
Show that, if the lengths of the diagonals of a parallelogram are
specified, then the parallogram has maximum area when the diagonals
are perpendicular. Show also that the area of a parallelogram is less
than or equal to half the square of the length of its longer diagonal. 


The set $A$ of points $(x,y)$ is given by 
\begin{alignat*}{1}
\left|a_{1}x+b_{1}y-c_{1}\right| & \leqslant\delta,\\
\left|a_{2}x+b_{2}y-c_{2}\right| & \leqslant\delta,
\end{alignat*}
 with $a_{1}b_{2}\neq a_{2}b_{1}.$ Sketch this set and show that
it is possible to find $(x_{1},y_{1}),(x_{2},y_{2})\in A$ with 
\[
(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}\geqslant\frac{8\delta^{2}}{\left|a_{1}b_{2}-a_{2}b_{1}\right|}.
\]
		
		\end{question}
		
		
%%%%%%%%% Q9
		\begin{question}
Let $(G,*)$ and $(H,\circ)$ be two groups and $G\times H$ be the
set of ordered pairs $(g,h)$ with $g\in G$ and $h\in H.$ A multiplication
on $G\times H$ is defined by 
\[
(g_{1},h_{1})(g_{2},h_{2})=(g_{1}*g_{2},h_{1}\circ h_{2})
\]
for all $g_{1},g_{2}\in G$ and $h_{1},h_{2}\in H$. 


Show that, with this multiplication, $G\times H$ is a group. 


State whether the following are true or false and prove your answers. 
\begin{enumerate}
\item[\bf (i)] $G\times H$ is abelian if and only if both $G$ and $H$ are abelian. 
\item[\bf (ii)] $G\times H$ contains a subgroup isomorphic to $G$. 
\item[\bf (iii)] $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ is isomorphic to $\mathbb{Z}_{4}.$
\item[\bf (iv)] $S_{2}\times S_{3}$ is isomorphic to $S_{6}.$
\end{enumerate}

{[}$\mathbb{Z}_{n}$ is the cyclic group of order $n$, and $S_{n}$
is the permutation group on $n$ objects.{]} 
		\end{question}
		
	
%%%%%%%%%% 10
\begin{question}
The \textit{Bernoulli polynomials} $P_{n}(x)$, where $n$ is a non-negative
integer, are defined by $P_{0}(x)=1$ and, for $n\geqslant1$, 
\[
\frac{\mathrm{d}P_{n}}{\mathrm{d}x}=nP_{n-1}(x),\qquad\int_{0}^{1}P_{n}(x)\,\mathrm{d}x=0
\]
Show by induction or otherwise, that 
\[
P_{n}(x+1)-P_{n}(x)=nx^{n-1},\quad\mbox{ for }n\geqslant1.
\]
Deduce that
\[
n\sum_{m=0}^{k}m^{n-1}=P_{n}(k+1)-P_{n}(0)
\]
Hence show that ${\displaystyle \sum_{m=0}^{1000}m^{3}=(500500)^{2}}$
			\end{question}
			
		
		
		
	
\newpage
\section*{Section B: \ \ \ Mechanics}


	
%%%%%%%%%% Q11
\begin{question}
A woman stands in a field at a distance of $a\,\mathrm{m}$ from the
straight bank of a river which flows with negligible speed. She sees
her frightened child clinging to a tree stump standing in the river
$b\,\mathrm{m}$ downstream from where she stands and $c\,\mathrm{m}$
from the bank. She runs at a speed of $u\,\mathrm{ms}^{-1}$ and swims
at $v\,\mathrm{ms}^{-1}$ in straight lines. Find an equation to be
satisfied by $x,$ where $x\,\mathrm{m}$ is the distance upstream from the
stump at which she should enter the river if she is to reach the child
in the shortest possible time. 


Suppose now that the river flows with speed $v$ ms$^{-1}$ and the
stump remains fixed. Show that, in this case, $x$ must satisfy the
equation 
\[
2vx^{2}(b-x)=u(x^{2}-c^{2})[a^{2}+(b-x)^{2}]^{\frac{1}{2}}.
\]
For this second case, draw sketches of the woman's path for the three
possibilities $b>c,$ $b=c$ and $b<c$. 

	\end{question}
	
%%%%%%%%%% Q12
\begin{question}	
A firework consists of a uniform rod of mass $M$ and length $2a$,
pivoted smoothly at one end so that it can rotate in a fixed horizontal
plane, and a rocket attached to the other end. The rocket is a uniform
rod of mass $m(t)$ and length $2l(t)$, with $m(t)=2\alpha l(t)$
and $\alpha$ constant. It is attached to the rod by its front end
and it lies at right angles to the rod in the rod's plane of rotation.
The rocket burns fuel in such a way that $\mathrm{d}m/\mathrm{d}t=-\alpha\beta,$
with $\beta$ constant. The burnt fuel is ejected from the back of
the rocket, with speed $u$ and directly backwards relative to the
rocket. Show that, until the fuel is exhausted, the firework's angular
velocity $\omega$ at time $t$ satisfies 
\[
\frac{\mathrm{d}\omega}{\mathrm{d}t}=\frac{3\alpha\beta au}{2[Ma^{2}+2\alpha l(3a^{2}+l^{2})]}.
\] 
\end{question}

%%%%%%%%%% Q13

\begin{question}
A uniform rod, of mass $3m$ and length $2a,$ is freely hinged at
one end and held by the other end in a horizontal position. A rough
particle, of mass $m$, is placed on the rod at its mid-point. If
the free end is then released, prove that, until the particle begins
to slide on the rod, the inclination $\theta$ of the rod to the horizontal
satisfies the equation 
\[
5a\dot{\theta}^{2}=8g\sin\theta.
\]
The coefficient of friction between the particle and the rod is $\frac{1}{2}.$
Show that, when the particle begins to slide, $\tan\theta=\frac{1}{26}.$
\end{question}
	
%%%%%%%%%% Q14
\begin{question}
It is given that the gravitational force between a disc, of radius
$a,$ thickness $\delta x$ and uniform density $\rho,$ and a particle
of mass $m$ at a distance $b(\geqslant0)$ from the disc on its axis
is 
\[
2\pi mk\rho\delta x\left(1-\frac{b}{(a^{2}+b^{2})^{\frac{1}{2}}}\right),
\]
where $k$ is a constant. Show that the gravitational force on a particle
of mass $m$ at the surface of a uniform sphere of mass $M$ and radius
$r$ is $kmM/r^{2}.$ Deduce that in a spherical cloud of particles
of uniform density, which all attract one another gravitationally,
the radius $r$ and inward velocity $v=-\dfrac{\d r}{\d t}$
of a particle at the surface satisfy the equation 
\[
v\frac{\mathrm{d}v}{\mathrm{d}r}=-\frac{kM}{r^{2}},
\]
where $M$ is the mass of the cloud. 


At time $t=0$, the cloud is instantaneously at rest and has radius
$R$. Show that $r=R\cos^{2}\alpha$ after a time 
\[
\left(\frac{R^{3}}{2kM}\right)^{\frac{1}{2}}(\alpha+\tfrac{1}{2}\sin2\alpha).
\]
\end{question}
	
	\newpage
\section*{Section C: \ \ \ Probability and Statistics}


%%%%%%%%%% Q15
\begin{question}
A patient arrives with blue thumbs at the doctor's surgery. With probability
$p$ the patient is suffering from Fenland fever and requires treatment
costing $\pounds 100.$ With probability $1-p$ he is suffering from
Steppe syndrome and will get better anyway. A test exists which infallibly
gives positive results if the patient is suffering from Fenland fever
but also has probability $q$ of giving positive results if the patient
is not. The test cost $\pounds 10.$ The doctor decides to proceed
as follows. She will give the test repeatedly until \textit{either
}the last test is negative, in which case she dismisses the patient
with kind words, \textit{or }she has given the test $n$ times with
positive results each time, in which case she gives the treatment.
In the case $n=0,$ she treats the patient at once. She wishes to
minimise the expected cost $\pounds E_{n}$ to the National Health
Service. \begin{questionparts}
\item Show that 
\[
E_{n+1}-E_{n}=10p-10(1-p)q^{n}(9-10q),
\]
and deduce that if $p=10^{-4},q=10^{-2},$ she should choose $n=3.$ 
\item Show that if $q$ is larger than some fixed value $q_{0},$ to be
determined explicitly, then whatever the value of $p,$ she should
choose $n=0.$ 
\end{questionparts}
\end{question}

%%%%%%%%%% Q16
\begin{question}
\begin{questionparts} \item $X_{1},X_{2},\ldots,X_{n}$ are independent
identically distributed random variables drawn from a uniform distribution
on $[0,1].$ The random variables $A$ and $B$ are defined by 
\[
A=\min(X_{1},\ldots,X_{n}),\qquad B=\max(X_{1},\ldots,X_{n}).
\]
For any fixed $k$, such that $0<k<\frac{1}{2},$ let 
\[
p_{n}=\mathrm{P}(A\leqslant k\mbox{ and }B\geqslant1-k).
\]
What happens to $p_{n}$ as $n\rightarrow\infty$? Comment briefly
on this result. 


\item Lord Copper, the celebrated and imperious newspaper proprietor,
has decided to run a lottery in which each of the $4,000,000$ readers
of his newspaper will have an equal probability $p$ of winning $\pounds 1,000,000$
and their changes of winning will be independent. He has fixed all
the details leaving to you, his subordinate, only the task of choosing
$p$. If nobody wins $\pounds 1,000,000$, you will be sacked, and
if more than two readers win $\pounds 1,000,000,$ you will also be
sacked. Explaining your reasoning, show that however you choose $p,$
you will have less than a 60\% change of keeping your job. \end{questionparts}
\end{question}
\end{document}
