\documentclass[a4, 11pt]{report}
\pagestyle{myheadings}
\markboth{}{Paper II, 2005
\ \ \ \ \
\today
}
\RequirePackage{amssymb}
\RequirePackage{amsmath}
\RequirePackage{graphicx}
\RequirePackage{color}
\RequirePackage[flushleft]{paralist}[2013/06/09]
\RequirePackage{geometry}
\geometry{%
a4paper,
lmargin=2cm,
rmargin=2.5cm,
tmargin=3.5cm,
bmargin=2.5cm,
footskip=12pt,
headheight=24pt}
\newcommand{\comment}[1]{{\bf Comment} {\it #1}}
%\renewcommand{\comment}[1]{}
\newcommand{\bluecomment}[1]{{\color{blue}#1}}
%\renewcommand{\comment}[1]{}
\newcommand{\redcomment}[1]{{\color{red}#1}}
\usepackage{epsfig}
\usepackage{pstricks-add}
\usepackage{tgheros} %% changes sans-serif font to TeX Gyre Heros (tex-gyre)
\renewcommand{\familydefault}{\sfdefault} %% changes font to sans-serif
%\usepackage{sfmath} %%%% this makes equation sans-serif
%\input RexFigs
\setlength{\parskip}{10pt}
\setlength{\parindent}{0pt}
\newlength{\qspace}
\setlength{\qspace}{20pt}
\newcounter{qnumber}
\setcounter{qnumber}{0}
\newenvironment{question}%
{\vspace{\qspace}
\begin{enumerate}[\bfseries 1\quad][10]%
\setcounter{enumi}{\value{qnumber}}%
\item%
}
{
\end{enumerate}
\filbreak
\stepcounter{qnumber}
}
\newenvironment{questionparts}[1][1]%
{
\begin{enumerate}[\bfseries (i)]%
\setcounter{enumii}{#1}
\addtocounter{enumii}{-1}
\setlength{\itemsep}{5mm}
\setlength{\parskip}{8pt}
}
{
\end{enumerate}
}
\DeclareMathOperator{\cosec}{cosec}
\DeclareMathOperator{\Var}{Var}
\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\g{{\mathrm g}}
\def\h{{\mathrm h}}
\def\f{{\mathrm f}}
\def\p{{\mathrm p}}
\def\s{{\mathrm s}}
\def\t{{\mathrm t}}
\def\A{{\mathrm A}}
\def\B{{\mathrm B}}
\def\E{{\mathrm E}}
\def\F{{\mathrm F}}
\def\G{{\mathrm G}}
\def\H{{\mathrm H}}
\def\P{{\mathrm P}}
\def\bb{\mathbf b}
\def \bc{\mathbf c}
\def\bx {\mathbf x}
\def\bn {\mathbf n}
\newcommand{\low}{^{\vphantom{()}}}
%%%%% to lower suffices: $X\low_1$ etc
\newcommand{\subone}{ {\vphantom{\dot A}1}}
\newcommand{\subtwo}{ {\vphantom{\dot A}2}}
\def\le{\leqslant}
\def\ge{\geqslant}
\def\var{{\rm Var}\,}
\newcommand{\ds}{\displaystyle}
\newcommand{\ts}{\textstyle}
\def\half{{\textstyle \frac12}}
\def\l{\left(}
\def\r{\right)}
\begin{document}
\setcounter{page}{2}
\section*{Section A: \ \ \ Pure Mathematics}
%%%%%%%%%%Q1
\begin{question}
Find the three values of $x$ for which the derivative
of $x^2 \e^{-x^2}$ is zero.
Given that $a$ and $b$ are
distinct positive numbers,
find a polynomial $\P(x)$ such that the derivative of
$\P(x)\e^{-x^2}$ is zero for
$x=0$, $x=\pm a$ and $x=\pm b\,$, but for no other values of $x$.
\end{question}
%%%%%%%%%%Q2
\begin{question}
For any positive integer $N$, the function $\f(N)$ is defined by
\[
\f(N) = N\Big(1-\frac1{p_1}\Big)\Big(1-\frac1{p_2}\Big)
\cdots\Big(1-\frac1{p_k}\Big)
\]
where $p_1$, $p_2$, $\dots$ , $p_k$ are the only prime numbers that are factors
of $N$.
\\
Thus $\f(80)=80(1-\frac12)(1-\frac15)\,$.
\begin{questionparts}
\item
\textbf{(a)} \ \ Evaluate $\f(12)$ and $\f(180)$.
\textbf{(b)} \
Show that $\f(N)$ is an integer for all $N$.
\item Prove, or disprove by means of a counterexample,
each of the following:
\textbf{(a)} \ \ $\f(m) \f(n) = \f(mn)\,$;
\textbf{(b)} \ $\f(p) \f(q) = \f(pq)$ if $p$ and $q$ are distinct prime numbers;
\textbf{(c)}
$\f(p) \f(q) = \f(pq)$ only if $p$ and $q$ are distinct prime numbers.
\item Find
a positive integer $m$ and a prime number $p$ such that
$\f(p^m) = 146410\,$.
\end{questionparts}
\end{question}
%%%%%%%%% Q3
\begin{question}
Give a sketch, for
$0 \le x \le \frac{1}{2}\pi$, of the curve
$$
y = (\sin x - x\cos x)\;,
$$
and show that $0\le y \le 1\,$.
Show that:
\begin{questionparts}
\item \ \ $\displaystyle
\int_0^{\frac{1}{2}\pi}\,y\;\d x = 2 -\frac \pi 2 \;;
$
\item \ \ $\displaystyle
\int_0^{\frac{1}{2}\pi}\,y^2\,\d x = \frac{\pi^3}{48}-\frac \pi 8 \;.
$
\end{questionparts}
Deduce that
$\pi^3 +18 \pi< 96\,$.
\end{question}
%%%%%% Q4
\begin{question}
The positive numbers $a$, $b$ and $c$ satisfy
$bc=a^2+1$. Prove that
$$
\arctan\left(\frac1 {a+b}\right)+
\arctan\left(\frac1 {a+c}\right)=
\arctan\left(\frac1 a \right).
$$
The positive numbers $p$, $q$, $r$, $s$, $t$, $u$ and $v$ satisfy
$$
st = (p+q)^2 + 1 \;, \ \ \ \ \ \ uv=(p+r)^2 + 1 \;, \ \ \ \ \ \
qr = p^2+1\;.
$$
Prove that
$$
\arctan \! \!\left(\!\frac1 {p+q+s}\!\right) +
\arctan \! \!\left(\!\frac 1{p+q+t}\!\right) +
\arctan \! \!\left(\!\frac 1 {p+r+u}\!\right) +
\arctan \! \!\left(\!\frac1 {p+r+v}\!\right)
=\arctan \! \!\left( \! \frac1 p \! \right) .
$$
Hence show that
$$
\arctan\left(\frac1 {13}\right)
+\arctan\left(\frac1 {21}\right)
+\arctan\left(\frac1 {82}\right)
+\arctan\left(\frac1 {187}\right)
=\arctan\left(\frac1 {7}\right).
$$
[\,Note that $\arctan x$ is another notation for $ \tan^{-1}x \,.\,$]
\end{question}
%%%%%%%%% Q5
\begin{question}
The angle $A$ of triangle $ABC$ is a right angle and the sides
$BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$,
respectively.
Each side of the triangle is tangent to the circle $S_1$ which is
of radius $r$.
Show that $2r = b+c-a$.
Each vertex of the triangle
lies on the circle~$S_2$.
The ratio of the area of the region between~$S_1$
and the triangle to the area of $S_2$ is denoted by $R\,$.
Show that
$$
\pi R = -(\pi-1)q^2 + 2\pi q -(\pi+1) \;,
$$
where $q=\dfrac{b+c}a\,$.
Deduce that
$$
R\le \frac1 {\pi( \pi - 1)} \;.
$$
\end{question}
%%%%%%%%% Q6
\begin{question}
\begin{questionparts}
\item Write down the general term in the expansion
in powers of $x$
of $(1-x)^{-1}$, $(1-x)^{-2}$ and $(1-x)^{-3}$, where
$\vert x \vert <1\,$.
Evaluate \
$\ds \sum_{n=1}^\infty n 2^{-n}\;$ \
and \ \
$\ds \sum_{n=1}^\infty n^22^{-n}\;$.
\item Show that
\
$\ds (1-x)^{-\frac12} = \sum_{n=0}^\infty \frac{(2n)!}{(n!)^2} \,
\frac{x^n}{2^{2n}}\,$, \ for $\vert x \vert <1\,$.
Evaluate \
$\ds \sum_{n=0}^\infty \frac{(2n)!} {(n!)^2 2^{2n}3^{n}}\; $ \ and \ \
$\ds \sum_{n=1}^\infty \frac{n(2n)!} {(n!)^2 2^{2n}3^{n}}\; $.
\end{questionparts}
\end{question}
%%%%%%%%% Q7
\begin{question}
The position vectors, relative to an origin $O$,
at time $t$ of the particles $P$ and $Q$ are
$$\cos t \; {\bf i} + \sin t\;{\bf j} + 0 \; {\bf k}
\hbox{ \ \ \ \ \ \ and \ \ \ \ \ \ }
\cos (t+\tfrac14{\pi})\, \big[{\tfrac32}{\bf i} +
{ \tfrac {3\sqrt{3}}2} {\bf k}\big]
+
3\sin(t+\tfrac14{\pi}) \; {\bf j}\;,$$
respectively, where $0\le t \le 2\pi\,$.
\begin{questionparts}
\item
Give a geometrical description of the motion of $P$ and $Q$.
\item
Let $\theta$ be the angle $POQ$ at time $t$ that satisfies
$0\le\theta\le\pi\,$. Show that
\[
\cos\theta = \tfrac{3\surd2}{8} -\tfrac14 \cos( 2t +\tfrac14 \pi)\;.
\]
\item Show that
the total time for which
$\theta \ge \frac14 \pi$ is $\tfrac32 \pi\,$.
\end{questionparts}
\end{question}
%%%%%%%%% Q8
\begin{question}
For $x \ge 0$ the curve $C$ is defined by
$$
{\frac{\d y} {\d x}} = \frac{x^3y^2}{(1 + x^2)^{5/2}}
$$
with $y = 1$ when $x=0\,$. Show that
\[
\frac 1 y = \frac {2+3x^2}{3(1+x^2)^{3/2}} +\frac13
\]
and hence
that for large positive $x$
$$
y \approx 3 - \frac 9 x\;.
$$
Draw a sketch of $C$.
On a separate diagram draw a sketch of the two curves
defined for $x \ge 0$ by
$$
\frac {\d z} {\d x} = \frac{x^3z^3}{2(1 + x^2)^{5/2}}
$$
with $z = 1$ at $x=0$ on one curve, and
$z = -1$ at $x=0$ on the other.
\end{question}
\newpage
\section*{Section B: \ \ \ Mechanics}
%%%%%%%%%% Q9
\begin{question}
Two particles, $A$ and $B$, of masses $m$ and $2m$,
respectively, are placed on a line of greatest slope, $\ell$, of a
rough inclined plane which makes
an angle of $30^{\circ}$ with the horizontal. The coefficient
of friction between $A$ and the plane is $\frac16\sqrt{3}$
and the coefficient of
friction between $B$ and the plane is $\frac13 \sqrt{3}$.
The particles are at rest with
$B$ higher up $\ell$ than $A$ and are connected by a light inextensible string
which is taut. A force $P$ is applied to $B$.
\begin{questionparts}
\item Show that the least magnitude of $P$ for which
the two particles move upwards along $\ell$ is
$\frac{11}8 \sqrt{3}\, mg$ and give, in this case,
the direction in which $P$ acts.
\item Find the least magnitude of $P$ for which the particles
do not slip downwards along~$\ell$.
\end{questionparts}
\end{question}
%%%%%%%%%% Q10
\begin{question}
The points $A$ and $B$ are $180$ metres apart and lie
on horizontal ground.
A missile is launched from $A$
at speed of $100\,$m\,s$^{-1}$ and at an
acute angle of elevation to the
line $AB$ of $\arcsin \frac35$. A time $T$ seconds later,
an anti-missile missile is launched from $B$,
at speed of $200\,$m\,s$^{-1}$
and at an acute
angle of elevation to the line $BA$ of $\arcsin \frac45$.
The motion of both missiles
takes place in the vertical plane containing $A$ and $B$, and the
missiles collide.
Taking $g =10\,$m\,s$^{-2}$ and ignoring air resistance,
find $T$.
\noindent
[Note that $\arcsin \frac35$ is another notation for $\sin^{-1} \frac35\,$.]
\end{question}
%%%%%%%%%% Q11
\begin{question}
A plane is inclined at an
angle $\arctan \frac34$ to the horizontal and
a small, smooth, light pulley~$P$
is fixed to the top of the plane. A string, $APB$, passes over the pulley.
A particle of mass~$m_1$
is attached to the string at $A$ and rests on the inclined plane with $AP$
parallel to a line of greatest slope in the plane.
A particle of mass $m_2$, where $m_2>m_1$,
is attached to the string at $B$
and hangs freely with $BP$
vertical. The coefficient of
friction between the particle at $A$
and the plane is $ \frac{1}{2}$.
The system is released from rest with the string taut.
Show that the acceleration of the
particles is $\ds \frac{m_2-m_1}{m_2+m_1}g$.
At a time $T$ after release, the string breaks.
Given that the particle at $A$
does not reach the pulley at any point in its motion,
find an expression in terms of $T$ for the time
after release at which the particle at $A$
reaches its maximum height. It is found that, regardless
of when the string broke, this time is equal to the time
taken by the particle at $A$ to descend
from its point of maximum height to the point
at which it was released. Find the ratio $m_1 : m_2$.
\noindent
[Note that $\arctan \frac34$ is another notation for $\tan^{-1} \frac34\,$.]
\end{question}
\newpage
\section*{Section C: \ \ \ Probability and Statistics}
%%%%%%%%%% Q12
\begin{question}
The twins Anna and Bella share a computer and never sign their e-mails.
When I e-mail them, only the twin
currently online responds. The
probability that it is Anna who is online is $p$ and she answers each
question I ask her truthfully with probability $a$, independently of all her
other answers, even if a question is repeated. The probability that it is
Bella who is online is~$q$, where $q=1-p$, and she answers each question
truthfully with probability $b$, independently of all her other answers,
even if a question is repeated.
\begin{questionparts}
\item
I send the twins the e-mail:
`Toss a fair coin and answer the following question.
Did the coin come down heads?'. I receive the answer `yes'.
Show that the probability that the coin
did come down heads is $\frac{1}{2}$ if and
only if $2(ap+bq)=1$.
\item
I send the twins the e-mail:
`Toss a fair coin and answer the following question.
Did the coin come down heads?'. I receive the answer `yes'.
I then send the e-mail: `Did the coin come down heads?' and I receive
the answer `no'. Show that the probability (taking into
account these answers) that the coin did come down heads is $\frac{1}{2}\,$.
\item
I send the twins the e-mail: `Toss a fair coin and answer the following
question. Did the coin come down heads?'. I receive the answer `yes'.
I then send the e-mail: `Did the coin come down heads?' and I receive
the answer `yes'. Show that, if $2(ap+bq)=1$,
the probability (taking into account these answers) that the coin did
come down heads is $\frac{1}{2}\,$.
\end{questionparts}
\end{question}
%%%%%%%%%% Q13
\begin{question}
The number of printing errors on any page of a large book of $N$ pages is
modelled by a
Poisson variate with parameter $\lambda$ and is statistically
independent of the number of printing errors on any other page. The number
of pages in a random sample of $n$ pages (where $n$ is much smaller than $N$
and $n\ge2$)
which contain fewer than two errors is denoted by $Y$.
Show that $\P(Y=k) = \binom n k p^kq^{n-k}$ where
$p=(1+\lambda)e^{-\lambda}$ and $q=1-p\,$.
Show also that, if $\lambda$ is sufficiently small,
\begin{questionparts}
\item $q\approx \frac12 \lambda^2\,$;
\item the largest value of
$n$ for which $\P(Y=n)\ge 1-\lambda$ is approximately $2/\lambda\,$;
\item $
\P(Y>1 \;\vert\; Y>0) \approx 1-n(\lambda^2/2)^{n-1}\;.$
\end{questionparts}
\end{question}
%%%%%%%%%% Q14
\begin{question}
The probability density function $\f(x)$
of the random variable $X$ is given by
$$
\f(x) = k\left[{\phi}(x) + {\lambda}\g(x)\right],\,\,\,\,
$$
where
${\phi}(x)$ is the probability density function
of a normal variate with mean~0 and variance~1,
$\lambda $ is a positive constant, and $\g(x)$ is a probability density function defined by
\[
\g(x)=
\begin{cases}
1/\lambda & \mbox{for $0 \le x \le {\lambda}$}\,;\\
0& \mbox{otherwise} .
\end{cases}
\]
Find $\mu$, the mean of $X$, in terms of $\lambda$, and prove that
$\sigma$, the standard deviation of $X$, satisfies.
$$
\sigma^2 = \frac{\lambda^4 +4{\lambda}^3+12{\lambda}+12}
{12(1 + \lambda )^2}\;.
$$
In the case $\lambda=2$:
\begin{questionparts}
\item draw a sketch of the curve $y=\f(x)$;
\item express the cumulative distribution function of $X$ in terms of $\Phi(x)$,
the cumulative distribution function corresponding to $\phi(x)$;
\item evaluate $\P(0