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\begin{document}
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\section*{Section A: \ \ \ Pure Mathematics}
%%%%%%%%%%%%%% Q1
\begin{question}
{\bf Note: } In this question
you may use without proof the result
$ \dfrac{\d \ }{\d x}\big(\!\arctan x \big) = \dfrac 1 {1+x^2}\,$.
\vspace{3mm}
Let
\[
I_n = \int_0^1 x^n \arctan x \, \d x \;,
\]
where $n=0$, 1, 2, 3, $\ldots$ .
\begin{questionparts}
\item
Show that, for $n\ge0\,$,
\[
(n+1) I_n = \frac \pi 4 -
\int _0^1 \frac {x^{n+1}}{1+x^2} \, \d x
\,
\]
and evaluate $I_0$.
\item
Find an expression, in terms of $n$,
for $(n+3)I_{n+2}+(n+1)I_{n}\,$.
Use this result to evaluate $I_4$.
\item
Prove by induction that, for $n\ge1$,
\[
(4n+1) I_{4n}
=A - \frac12 \sum_{r=1}^{2n} (-1)^r \frac 1 {r}
\,,
\]
where $A$ is a constant to be determined.
\end{questionparts}
\end{question}
%%%%%%%%Q2
\begin{question}
The sequence of numbers $x_0$, $x_1$, $x_2$, $\ldots$ satisfies
\[
x_{n+1} = \frac{ax_n-1}{x_n+b}
\,.
\]
(You may assume that $a$, $b$ and $x_0$ are such that $x_n+b\ne0\,$.)
Find an expression for $x_{n+2}$ in terms of $a$, $b$ and $x_n$.
\begin{questionparts}
\item
Show that $a+b=0$ is a necessary condition for
the sequence to be periodic with period~2.
{\bf Note: } The sequence is said to be
periodic with period $k$ if $x_{n+k} = x_n$ for all $n$,
and there is no integer $m$ with $0 0\,$,
\[
\frac {\e^t -1}{\e^t+1} \le \frac t 2
\,.
\]
\item
By setting $ \f(x)= x $
in $(*)$,
and choosing $ \g(x)$ suitably,
show that
\[
\int_0^1\e^{-\frac12 x^2}\d x \ge 12 \big(1-\e^{-\frac14})^2
\,.
\]
\item
Use $(*)$ to show that
\[
\frac {64}{25\pi} \le \int_0^{\frac12\pi}
\!\!
{\textstyle \sqrt{\, \sin x\, } }
\, \d x
\le \sqrt{\frac \pi 2 }
\,.
\]
\end{questionparts}
\end{question}
%%%%%%%%%%%%%% Q5
\begin{question}
A curve $C$ is determined by the parametric equations
\[
x=at^2 \, , \; y = 2at\,,
\]
where $a>0$\,.
\begin{questionparts}
\item Show that the normal to
$C$ at a point $P$, with non-zero parameter $p$,
meets $C$
again at a point $N$, with parameter $n$, where
\[
n= - \left( p + \frac{2}{p} \right).
\]
\item
Show that the distance $\left| PN \right|$ is given by
\[
\vert PN\vert^2 = 16a^2\frac{(p^2+1)^3}{p^4}
\]
and that this is minimised when $p^2=2\,$.
\item The point $Q$, with parameter $q$,
is the point at which the circle with
diameter $PN$ cuts~$C$ again.
By considering the gradients of $QP$ and $QN$,
show that
\[
2 = p^2-q^2 + \frac{2q}p
\,.
\]
Deduce that $\left| PN \right|$ is at its minimum when
$Q$ is at the origin.
\end{questionparts}
\end{question}
%%%%%%%%%%%%%% Q6
\begin{question}
Let
\[
S_n = \sum_{r=1}^n \frac 1 {\sqrt r \ }
\,,
\]
where $n$ is a positive integer.
\begin{questionparts}
\item
Prove by induction that
\[
S_n \le 2\sqrt n -1\,
.
\]
\item
Show that $(4k+1)\sqrt{k+1} > (4k+3)\sqrt k\,$ for $k\ge0\,$.
Determine the smallest number $C$ such that
\[
S_n \ge 2\sqrt n + \frac 1 {2\sqrt n} -C
\,.\]
\end{questionparts}
\end{question}
%%%%%Q7
\begin{question}
%In this question,
%the definition of $a^b$ (for $a>0$) is
%$
%a^b = \e^{b \ln a} \,.
%$
%\\
The functions $\f$ and $\g$ are defined, for $x>0$, by
\[
\f(x) = x^x\,, \ \ \ \ \ \g(x) = x^{\f(x)}\,.
\]
\begin{questionparts}
\item
By taking logarithms,
or otherwise, show that $\f(x)> x$ for $01 \,$.
\item
Find the value of
$x$ for which $\f'(x)=0\,$.
\item
Use the result $x\ln x \to 0$ as $x\to 0$
to
find $\lim\limits_{x\to0}\f(x)$,
and write down $\lim\limits_{x\to0}\g(x)\,$.
\item
Show that $ x^{-1}+\, \ln x \ge 1\,$ for $x>0$.
\\[5pt]
Using this result, or otherwise,
show
that~$\g'(x) >0\,$.
\end{questionparts}
\vspace{3pt}
Sketch the graphs, for $x>0$, of $y=x$, \ $y=\f(x)$ and $y=\g(x)$
on the same axes.
\end{question}
%%%%%%%%%%%%%% Q8
\begin{question}
All vectors in this question lie in the same plane.
The vertices of the non-right-angled triangle $ABC$ have position
vectors $\bf a$, $\bf b$ and $\bf c$, respectively.
The non-zero vectors $\bf u$ and $\bf v$ are
perpendicular to $BC$ and $CA$, respectively.
Write down the vector equation of the line through $A$
perpendicular to $BC$, in terms of
$\bf u$,~$\bf a$
and a parameter $\lambda $.
The line through $A$ perpendicular
to $BC$ intersects the line through $B$ perpendicular to $CA$ at $P$.
Find
the position
vector of $P$
in terms of $\bf a$,~$\bf b$, $\bf c$ and $\bf u$.
Hence show that
the line $CP$ is perpendicular to the line $AB$.
\end{question}
\section*{Section B: \ \ \ Mechanics}
%%%%%%%%%%%%%% Q9
\begin{question}
Two identical rough cylinders of radius
$r$
and weight
$W$
rest,
not
touching each other
but a
negligible distance apart,
on a horizontal floor.
A thin flat rough plank of width $2a$, where
$a < r$,
and weight
$kW$
rests symmetrically and horizontally
on the cylinders, with its length parallel
to the axes of the cylinders and its faces horizontal.
A vertical cross-section is shown in the diagram below.
\vspace{1.1cm}
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\vspace{-1.5cm}
The coefficient of friction at all four
contacts is
$\frac12$.
The system is in equilibrium.
\begin{questionparts}
\item
Let $F$ be the frictional force between one cylinder and the floor,
and let $R$ be the normal reaction between the plank
and one cylinder.
Show that
\[
R\sin\theta = F(1+\cos\theta)\,,
\]
where $\theta$ is the acute angle between the plank and the tangent to
the cylinder at the point of contact.
Deduce that $2\sin\theta \le 1+\cos\theta\,$.
\item
Show that
\[
N=
\left( 1+\frac2 k\right)\left(\frac{1+\cos\theta}{\sin\theta} \right) F
\,,
\]
where $N$ is the normal reaction between the
floor and one cylinder.
Write down the condition
that the cylinder does
not slip on the floor and
show that it is satisfied with no extra restrictions on $\theta$.
\item
Show that $\sin\theta\le\frac45\,$ and hence that $r\le5a\,$.
\end{questionparts}
\end{question}
%%%%%%%%%%%%%% Q10
\begin{question}
A car of mass $m$ makes a journey of distance $2d$
in a straight line.
It experiences air resistance and rolling resistance
so that the total resistance to motion when it is moving with
speed $v$ is $Av^2 +R$, where $A$ and $R$ are constants.
The car starts from rest and
moves with constant acceleration $a$ for a distance $d$.
Show that the work done by the engine
for this half of the journey
is
\[
\int_0^d (ma+R+Av^2) \, \d x
\]
and that it can be written
in the form
\[
\int_0^w \frac {(ma+R+Av^2)v}a\; \d v
\,,
\]
where $w =\sqrt {2ad\,}\,$.
For the second half of the journey, the acceleration of
the car is $-a$.
\begin{questionparts}
\item
In the case $R>ma$,
show that the work done by the
engine for the whole journey~is
\[
2Aad^2 + 2Rd
\,.
\]
\item
In the case $ma-2Aad< R< ma$, show that at a certain speed
the driving
force required to maintain the constant acceleration
falls to zero.
Thereafter, the engine does no work
(and the driver applies the brakes to maintain
the constant acceleration).
Show that the work done by the engine for the whole journey~is
\[
2Aad^2 + 2 Rd
+ \frac{(ma-R)^2}{4Aa}
\,
.\]
\end{questionparts}
\end{question}
%%%%%%%%%%%%%% Q11
\begin{question}
Two thin vertical parallel
walls, each of height $2a$, stand a distance $a$ apart
on horizontal ground.
The projectiles in this
question move in a plane perpendicular to the walls.
\begin{questionparts}
\item
A particle is projected with speed $\sqrt{5ag}$
towards the two
walls
from a point $ A$ at ground level.
It just clears the first wall. By considering the energy
of the particle, find its
speed when it passes over the first wall.
Given that it just clears the second wall,
show that
the angle
its trajectory makes with the horizontal when it
passes over the first wall
is $45^\circ\,$.
Find the distance of $A$ from the foot of the first wall.
\item
A second particle is projected with speed $\sqrt{5ag}$
from a point $B$ at ground level towards the two
walls. It passes a distance $h$ above the first wall, where $h>0$.
Show that it does not clear the second wall.
\end{questionparts}
\end{question}
\newpage
\section*{Section C: \ \ \ Probability and Statistics}
%%%%%%%%%%%%%% Q12
\begin{question}
Adam
and Eve are
catching fish.
The number of fish, $X$,
that Adam catches in any time interval is Poisson distributed
with parameter $\lambda t$, where $\lambda$ is a constant and $t$ is the
length of the time interval.
The number of fish, $Y$,
that Eve catches in any time interval is Poisson distributed
with parameter $\mu t$, where $\mu$ is a constant and $t$ is the
length of the time interval
The two Poisson variables are independent.
You may assume that
that expected time between
Adam catching a fish and Adam catching his next fish is $\lambda^{-1}$,
and similarly for Eve.
\begin{questionparts}
\item
By considering $\P( X + Y = r)$, show that the total number of fish caught
by Adam and Eve in time $T$ also has a Poisson distribution.
\item
Given that Adam and Eve catch a
total of $k$ fish in time $T$,
where $k$ is fixed, show that the number caught by
Adam has a binomial distribution.
\item
Given that
Adam and Eve start fishing at the same time, find
the probability that the first fish is caught by Adam.
\item
Find the expected time from the moment Adam and Eve
start fishing until they have each caught at least one fish.
\end{questionparts}
\noindent
[{\bf Note }
This question has been redrafted to make the meaning clearer.]
\end{question}
%%%%%%%%%%%%%% Q13
\begin{question}
In a television game show, a contestant has to open a door
using a key. The contestant is given a
bag containing $n$ keys,
where $n\ge2$. Only one key in the bag will open the door.
There are three versions
of the game. In each version, the contestant starts by choosing a
key at random from the bag.
\begin{questionparts}
\item
In version 1, after each failed attempt at opening the door
the key that has
been tried is put back into the bag and the contestant again
selects a key at random from the bag.
By considering the binomial expansion of $( 1 - q)^{-2}$,
or otherwise, find the expected number of attempts required to open
the door.
\item
In
version 2, after each failed attempt at opening the door
the key that has been tried is put aside and the contestant
selects another key
at random
from the bag. Find the expected number of
attempts required to open the door.
\item
In version
3, after each failed attempt at opening the door the key that has
been~tried is put back into the bag and another incorrect key is added to
the bag.
The contestant then selects a key at random from the bag.
Show that the probability that the contestant draws the
correct key at the $k$th attempt is
\[
\frac{n-1}{(n+k-1)(n+k-2)}
\,.
\]
Show also, using partial fractions, that
the expected number of attempts required to open the door is infinite.
You may use without proof the result that
$
\displaystyle
\sum_{m=1}^N \dfrac 1 m \to \infty
\,
$
as $N\to \infty\,$.
\end{questionparts}
\end{question}
\end{document}