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\begin{document}
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\section*{Section A: \ \ \ Pure Mathematics}
%%%%%%%%%%%%% Q1
\begin{question}
Show that, if $k$ is a root of
the quartic equation
\[
x^4 + ax^3 + bx^2 + ax + 1 = 0\,,
\tag{$*$}
\]
then $k^{-1}$ is a root.
You are now given that $a$ and $b$ in $(*)$ are both
real and are such that the roots are all real.
\begin{questionparts}
\item Write down all
the values of $a$ and $b$ for which $(*)$ has only one
distinct root.
\item
Given that $(*)$ has exactly three distinct roots, show that
either $b=2a-2$ or \mbox{$b=-2a-2\,$}.
\item
Solve $(*)$ in the case
$b= 2 a -2\,$,
giving your solutions in terms of $a$.
\end{questionparts}
Given that $a$ and $b$ are both real and that the roots of $(*)$
are all real,
find
necessary and sufficient conditions, in terms of $a$ and $b$,
for $(*)$ to have
exactly three distinct real roots.
\end{question}
%%%%%%%%%%%%%Q2
\begin{question}
A function $\f(x)$ is said to be {\em concave} for
$a< x < b$
if
\[
\ t\,\f(x_1) +(1-t)\,\f(x_2)
\le
\f\big(tx_1+ (1-t)x_2\big)
\,
,\]
for $a< x_1 < b\,$, \
$a< x_2< b$ and $0\le t \le 1\,$.
Illustrate this definition by means of
a sketch,
showing the chord joining the points
$\big(x_1, \f(x_1)\big) $
and
$\big(x_2, \f(x_2)\big) $, in the case $x_1 (4n)!\,$.
\item [2.] For every positive integer
$n$, there is a prime number between $2n$ and $4n$.
\end{enumerate}
Find all pairs of positive integers $(n,m)$ that satisfy
\[
1! \times 3! \times \cdots \times (2n-1)! = m! \,.
\]
\end{questionparts}
\end{question}
%%%%%%%%%%%%%% Q7
\begin{question}
The points $O$, $A$ and $B$ are the vertices
of an acute-angled triangle. The points $M$ and $N$
lie on the sides $OA$ and $OB$ respectively, and the lines
$AN$ and $BM$
intersect at $Q$. The position vector of $A$ with respect to
$O$ is $\bf a$, and the position vectors of the
other points are labelled similarly.
Given that $\vert MQ \vert = \mu \vert QB\vert $, and that
$\vert NQ \vert = \nu \vert QA\vert $, where $\mu$ and $\nu$ are positive and
$\mu \nu <1$, show that
\[
{\bf m} =
\frac {(1+\mu)\nu}{1+\nu} \,
{\bf a}
\,.
\]
The point $L$ lies
on
the
side $OB$, and $\vert OL \vert = \lambda \vert OB \vert \,$.
Given that $ML$ is parallel to $AN$,
express~$\lambda$ in terms of $\mu$ and $\nu$.
What is the geometrical significance of the condition $\mu\nu<1\,$?
\end{question}
%%%%%%%%%%%%%% Q8
\begin{question}
\begin{questionparts}
\item Use the substitution $v= \sqrt y$
to solve the differential equation
\[
\frac{\d y}{\d t} = \alpha y^{\frac12} - \beta y
\ \ \ \ \ \ \ \ \ \ (y\ge0, \ \ t\ge0)
\,,
\]
where $\alpha$ and $\beta$ are positive constants.
Find the non-constant solution
$y_1(x)$
that satisfies $y_1(0)=0\,$.
\item
Solve the differential equation
\[
\frac{\d y}{\d t} = \alpha y^{\frac23} - \beta y
\ \ \ \ \ \ \ \ \ \ (y\ge0, \ \ t\ge0)
\,,
\]
where $\alpha$ and $\beta$ are positive constants.
Find the non-constant solution
$y_2(x)$
that satisfies
$y_2(0)=0\,$.
\item In the case $\alpha=\beta$, sketch
$y_1(x)$ and $y_2(x)$
on the same
axes, indicating clearly which is
$y_1(x)$ and which is $y_2(x)$.
You should explain how you determined the positions of the
curves relative to each other.
\end{questionparts}
\end{question}
\newpage
\section*{Section B: \ \ \ Mechanics}
%%%%%%%%%%%%%% Q9
\begin{question} Two small beads, $A$ and $B$,
of the same mass, are threaded onto a
vertical wire on which they slide without friction,
and which is fixed to the ground at $P$.
They are released simultaneously from rest, $A$
from a height of $8h$ above $P$ and $B$ from
a height of $17h$ above~$P$.
When $A$ reaches the ground
for the first time, it is
moving with speed $ V$. It then
rebounds with coefficient of restitution~$\frac{1}{2}$
and subsequently collides with $B$ at height $H$ above~$P$.
Show that $H= \frac{15}8h$ and find, in terms of $g$ and $h$, the speeds $u_A$ and
$u_B$ of the two beads just before the collision.
When $A$ reaches the ground for the second time,
it is again moving with speed $ V$.
Determine the coefficient of restitution between the two beads.
\end{question}
%%%%%%%%%%%%%% Q10
\begin{question}
A uniform elastic string lies on a smooth horizontal table.
One end of the string
is attached to a fixed peg,
and the other
end is pulled at constant speed $u$. At time
$t=0$, the string is
taut and its length is $a$. Obtain an expression for
the speed, at time $t$,
of the point on the string
which is a distance
$x$ from the peg at time~$t$.
An ant walks along the string starting at $t=0$ at the peg.
The ant walks at constant speed~$v$ along the string (so that
its speed relative to the peg is the sum of $v$ and the speed of the
point on the string beneath the ant).
At time $t$, the ant is a distance $x$ from the
peg.
Write down
a first order differential equation
for $x$, and
verify
that
\[
\frac{\d }{\d t} \left( \frac x {a+ut}\right) = \frac v {a+ut} \,.
\]
Show that the time $T$ taken for the ant to
reach the end of the string is given by
\[uT = a(\e^k-1)\,,\]
where $k=u/v$.
On reaching the end of the string, the ant turns round and walks back to the
peg. Find in terms of $T$ and $k$
the time taken for the journey
back.
\end{question}
%%%%%%%%%%%%%% Q11
\begin{question}
The axles of the wheels of a motorbike of mass $m$
are a distance $b$ apart. Its centre of
mass is a horizontal distance of $d$ from the front axle, where
$d