2gh$ then the horizontal range for a given height $h$ and initial speed $u$ is less than or equal to \[ \frac{u\sqrt{u^{2}-2gh}}{g}. \] Show that there is always an angle of firing for which this value is attained. \end{question} %%%%%%%%%% Q10 \begin{question} One end $A$ of a light elastic string of natural length $l$ and modulus of elasticity $\lambda$ is fixed and a particle of mass $m$ is attached to the other end $B$. The particle moves in a horizontal circle with centre on the vertical through $A$ with angular velocity $\omega.$ If $\theta$ is the angle $AB$ makes with the downward vertical, find an expression for $\cos\theta$ in terms of $m,g,l,\lambda$ and $\omega.$ Show that the motion described is possible only if \[ \frac{g\lambda}{l(\lambda+mg)}<\omega^{2}<\frac{\lambda}{ml}. \] \end{question} %%%%%%%%%% Q11 \begin{question} $\,$ \begin{center} \psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.3pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-2.2,-0.26)(7.1,4.4) \pscircle[fillcolor=black,fillstyle=solid,opacity=0.4](5,4){0.19} \psline(-2,0)(7,0) \psline(0,0)(5,4) \psline(5,4)(5,0) \psline(-0.08,0.26)(4.88,4.15) \rput[tl](-0.4,0.98){$A$} \rput[tl](5.59,3){$B$} \psline(5.18,4.02)(5.18,3) \begin{scriptsize} \psdots[dotsize=13pt 0,dotstyle=*](-0.08,0.23) \psdots[dotsize=11pt 0,dotstyle=*](5.18,3) \end{scriptsize} \end{pspicture*} \par \end{center} The diagram shows a small railway wagon $A$ of mass $m$ standing at the bottom of a smooth railway track of length $d$ inclined at an angle $\theta$ to the horizontal. A light inextensible string, also of length $d$, is connected to the wagon and passes over a light frictionless pulley at the top of the incline. On the other end of the string is a ball $B$ of mass $M$ which hangs freely. The system is initially at rest and is then released. \begin{questionparts} \item Find the condition which $m,M$ and $\theta$ must satisfy to ensure that the ball will fall to the ground. Assuming that this condition is satisfied, show that the velocity $v$ of the ball when it hits the ground satisfies \[ v^{2}=\frac{2g(M-m\sin\theta)d\sin\theta}{M+m}. \] \item Find the condition which $m,M$ and $\theta$ must satisfy if the wagon is not to collide with the pulley at the top of the incline. \end{questionparts} \end{question} \newpage \section*{Section C: \ \ \ Probability and Statistics} %%%%%%%%%% Q12 \begin{question} There are 28 colleges in Cambridge, of which two (New Hall and Newnham) are for women only; the others admit both men and women. Seven women, Anya, Betty, Celia, Doreen, Emily, Fariza and Georgina, are all applying to Cambridge. Each has picked three colleges at random to enter on her application form. \begin{itemize}[indent] \setlength{\itemsep}{3mm} \item[\bf (i)] What is the probability that Anya's first choice college is single-sex? \item[\bf (ii)] What is the probability that Betty has picked Newnham? \item[\bf (iii)] What is the probability that Celia has picked at least one single-sex college? \item[\bf (iv)] Doreen's first choice is Newnham. What is the probability that one of her other two choices is New Hall? \item[\bf (v)] Emily has picked Newnham. What is the probability that she has also picked New Hall? \item[\bf (vi)] Fariza's first choice college is single-sex. What is the probability that she has also chosen the other single-sex college? \item[\bf (vii)] One of Georgina's choices is a single-sex college. What is the probability that she has also picked the other single-sex college? \end{itemize} \end{question} %%%%%%%%%% Q13 \begin{question} I have a bag containing $M$ tokens, $m$ of which are red. I remove $n$ tokens from the bag at random without replacement. Let \[ X_{i}=\begin{cases} 1 & \mbox{ if the \ensuremath{i}th token I remove is red;}\\ 0 & \mbox{ otherwise.} \end{cases} \] Let $X$ be the total number of red tokens I remove. \begin{itemize}[indent] \setlength{\itemsep}{3mm} \item[\bf (i)] Explain briefly why $X=X_{1}+X_{2}+\cdots+X_{n}.$ \item[\bf (ii)] Find the expectation $\mathrm{E(}X_{i}).$ \item[\bf (iii)] Show that $\mathrm{E}(X)=mn/M$. \item[\bf (iv)] Find $\mathrm{P}(X=k)$ for $k=0,1,2,\ldots,n$. \item[\bf (v)] Deduce that \[ \sum_{k=1}^{n}k\binom{m}{k}\binom{M-m}{n-k}=m\binom{M-1}{n-1}. \] \end{itemize} \end{question} %%%%%%%%%% Q14 \begin{question} Each of my $n$ students has to hand in an essay to me. Let $T_{i}$ be the time at which the $i$th essay is handed in and suppose that $T_{1},T_{2},\ldots,T_{n}$ are independent, each with probability density function $\lambda\mathrm{e}^{-\lambda t}$ ($t\geqslant0$). Let $T$ be the time I receive the first essay to be handed in and let $U$ be the time I receive the last one. \begin{questionparts} \item Find the mean and variance of $T_{i}.$ \item Show that $\mathrm{P}(U\leqslant u)=(1-\mathrm{e}^{-\lambda u})^{n}$ for $u\geqslant0,$ and hence find the probability density function of $U$. \item Obtain $\mathrm{P}(T>t),$ and hence find the probability density function of $T$. \item Write down the mean and variance of $T$. \end{questionparts} \end{question} \end{document}