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\section*{Section A: \ \ \ Pure Mathematics}
%%%%%%%%%%Q1
\begin{question}
Show that the equation of any circle passing through the points of
intersection of the ellipse $(x+2)^2 +2y^2 =18$ and the ellipse
$9(x-1)^2 +16y^2 = 25$ can be written in the form
\[
x^2-2ax +y^2 =5-4a\;.
\]
\end{question}
%%%%%%%%%%Q2
\begin{question}
Let $\f(x) = x^m(x-1)^n$, where $m$ and $n$ are both integers greater than $1$.
Show that the curve $y=\f(x)$ has a stationary point with $00\,$. Prove
that there are no further
intersections between the line and the curve. Draw the line
on your sketch.
By considering the area under the curve for $0\le x\le1$, show that $\pi>3\,$.
Show also, by considering the volume formed by rotating the curve about
the $y$ axis, that $\ln 2 >2/3\,$.
\vspace{2mm}
[\hspace{1pt}
{\bf Note}: $\displaystyle \int_0^ 1 \frac1 {1+x^2}\, \d x = \frac\pi 4\,.\;$]
\end{question}
%%%%%%%%% Q5
\begin{question}
Let
\[
\f(x) = x^n + a_1 x^{n-1} + \cdots + a_n\;,
\]
where $a_1$, $a_2$, $\ldots$\,, $a_n$ are given numbers.
It is given that $\f(x)$ can be written in the form
\[
\f(x) = (x+k_1)(x+k_2)\cdots(x+k_n)\;.
\]
By considering
$\f(0)$, or otherwise, show that $k_1k_2 \ldots k_n =a_n$.
Show also that $$(k_1+1)(k_2+1)\cdots(k_n+1)= 1+a_1+a_2+\cdots+a_n$$ and give a
corresponding result for $(k_1-1)(k_2-1)\cdots(k_n-1)$.
Find the roots of the equation
\[
x^4 +22x^3 +172x^2 +552x+576=0\;,
\]
given that they are all integers.
\end{question}
%%%%%%%%% Q6
\begin{question}
A pyramid stands on horizontal ground. Its base is an equilateral triangle with sides of length~$a$, the other three
sides of the pyramid are of length $b$ and its volume is $V$. Given that the
formula for the volume of any pyramid is
$
\textstyle
\frac13 \times \mbox{area of base} \times \mbox {height} \,,
$
show that
\[
V= \frac1{12} {a^2(3b^2-a^2)}^{\frac12}\;.
\]
The pyramid is then placed so that a non-equilateral face lies on the ground.
Show that the new height, $h$, of the pyramid is given by
\[
h^2 = \frac{a^2(3b^2-a^2)}{4b^2-a^2}\;.
\]
Find, in terms of $a$ and $b\,$, the angle between the
equilateral triangle and the horizontal.
\end{question}
%%%%%%%%% Q7
\begin{question}
Let
\[
I= \int_0^a \frac {\cos x}{\sin x + \cos x} \; \d x \,
\mbox{ \ \ \ \ and \ \ \ \ }
J= \int_0^a \frac {\sin x}{\sin x + \cos x} \; \d x \;,
\]
where $0\le a <\frac{3}{4}\pi\,$.
By considering $I+J$ and $I-J$, show that
$
2I= a + \ln (\sin a +\cos a)\;.
$
Find also:
\begin{questionparts}
\item
$\displaystyle \int_0^{\frac{1}{2}\pi} \frac {\cos x}{p\sin x + q\cos x} \; \d x \,$, where $p$ and $q$
are positive numbers;
%\item [(ii)]
%$\displaystyle \int_0^{\frac{1}{2}\pi/2} \frac {\cos x}{\sin (x+k)} \; \d x \,$, where $0}(3,4)(9,4)
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A lorry of weight $W$ stands on a plane inclined at an angle $\alpha$ to the
horizontal. Its wheels are a distance $2d$ apart, and its centre of
gravity $G$ is at a distance $h$ from the plane, and halfway between the sides
of the lorry. A horizontal force $P$ acts on the lorry
through $G\,$, as shown.
\begin{questionparts}
\item If the normal reactions on the lower and
higher wheels of the lorry are equal,
show that the sum of the frictional forces between the wheels and the ground is zero.
\item If $P$ is such that the lorry does not tip
over (but the normal reactions on the lower and
higher wheels of the lorry need not be equal), show that
\[
W\tan(\alpha - \beta)
\le P
\le
W\tan(\alpha+\beta)\;,
\]
where $\tan\beta = d/h\,$.
\end{questionparts}
\end{question}
%%%%%%%%%% Q10
\begin{question}
A bicycle pump consists of a cylinder and a piston. The piston is pushed
in with steady speed~$u$. A particle of air moves to and fro between the
piston and the end of the cylinder, colliding perfectly elastically with the piston
and the end of the cylinder, and always moving parallel with the axis of the cylinder.
Initially, the particle is moving towards the piston at speed $v$.
Show that the speed, $v_n$, of the particle just after the
$n$th collision with the piston is given by $v_n=v+2nu$.
Let $d_n$ be the distance between the piston and the end of the cylinder
at the $n$th collision, and let $t_n$ be the time between the
$n$th and $(n+1)$th collisions. Express $d_n - d_{n+1}$ in terms
of $u$ and $t_n$, and show that
\[
d_{n+1} = \frac{v+(2n-1)u}{v+(2n+1)u} \, d_n \;.
\]
Express $d_n$ in terms of $d_1$, $u$, $v$ and $n$.
In the case $v=u$, show that $ut_n = \displaystyle \frac {d_1} {n(n+1)}$.
%%%%%Verify that $\sum\limits_1^\infty t_n = d/u$.
\end{question}
%%%%%%%%%% Q11
\begin{question}$\,$
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A particle $P_1$ of mass $m$ collides with a particle $P_2$
of mass $km$ which is at rest. No energy is lost in the collision.
The direction of motion
of $P_1$ and $P_2$ after the collision make
non-zero
angles of $\theta$ and $\phi$, respectively, with the direction of motion
of $P_1$ before the collision, as shown. Show that
\[
\sin^2\theta + k\sin^2\phi = k\sin^2(\theta+\phi) \;.
\]
Show that, if the angle between the particles after the collision is a right angle,
then $k=1\,$.
\end{question}
\newpage
\section*{Section C: \ \ \ Probability and Statistics}
%%%%%%%%%% Q12
\begin{question}
Harry the Calculating Horse will do any mathematical problem I set him,
providing the answer is 1, 2, 3 or 4. When I set him a problem,
he places a hoof on
a large grid consisting of unit squares and his answer is the
number of squares partly covered by his hoof.
Harry has circular hoofs, of radius $1/4$ unit.
After many years of collaboration, I suspect that Harry no longer bothers to
do the calculations, instead merely placing his hoof on the grid completely at random.
I often ask him to divide 4 by 4, but only about $1/4$ of his answers are right;
I often ask him to add 2 and 2, but disappointingly only about
$\pi/16$ of his answers are right. Is this consistent with my suspicions?
I decide to investigate further by setting Harry many problems, the answers to which
are 1, 2, 3, or 4 with equal frequency. If Harry is placing his hoof at random,
find the expected value of his answers. The average of Harry's answers turns out to
be 2. Should I get a new horse?
\end{question}
%%%%%%%%%% Q13
\begin{question}
The random variable $U$ takes the values $+1$, $0$ and $-1\,$, each with probability
$\frac13\,$. The random variable $V$ takes the values $+1$ and $-1$ as follows:
\begin{center}
\begin{tabular}{ll}
if $U=1\,$,&then $\P(V=1)= \frac13$ and $\P(V=-1)=\frac23\,$;\\[2mm]
if $U=0\,$,&then $\P(V=1)= \frac12$ and $\P(V=-1)=\frac12\,$;\\[2mm]
if $U=-1\,$,&then $\P(V=1)= \frac23$ and $\P(V=-1)=\frac13\,$.
\end{tabular}
\end{center}
\begin{questionparts}
\item Show that the probability that
both roots of the equation $x^2+Ux+V=0$ are real is $\frac12\;$.
\item Find the expected value of the larger root of the equation
$x^2+Ux+V=0\,$, given that both roots are real.
\item
Find the probability that the roots of the equation
$$x^3+(U-2V)x^2+(1-2UV)x + U=0$$ are all positive.
\end{questionparts}
\end{question}
%%%%%%%%%% Q14
\begin{question}
In order to get money from a cash dispenser
I have to punch in an
identification number. I have forgotten my identification number,
but I do know that it is equally likely to be any one of the
integers $1$, $2$, \ldots , $n$.
I plan to punch in integers in order until I get the right
one. I can do this at the rate of $r$ integers per minute.
As soon as I punch in the first wrong number, the police will be alerted.
The probability that they will arrive within a time $t$ minutes
is $1-\e^{-\lambda t}$, where $\lambda$ is a positive constant.
If I follow my plan, show that the probability of the police arriving
before I get my money is
\[
\sum_{k=1}^n \frac{1-\e^{-\lambda(k-1)/r}}n\;.
\]
Simplify the sum.
On past experience, I know that I will be so flustered that I will
just punch in possible integers at random, without noticing which I have
already tried. Show that the probability of the police arriving before
I get my money is
\[
1-\frac1{n-(n-1)\e^{-\lambda/r}} \;.
\]
\end{question}
\end{document}