\section*{4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 14.\\
(i) Let $a > 0$. On the axes opposite, sketch the graph of

$$y = \frac { a + x } { a - x } \quad \text { for } \quad - a < x < a .$$

(ii) Let $0 < \theta < \pi / 2$. In the diagram below is the half-disc given by $x ^ { 2 } + y ^ { 2 } \leqslant 1$ and $y \geqslant 0$. The shaded region $A$ consists of those points with $- \cos \theta \leqslant x \leqslant \sin \theta$. The region $B$ is the remainder of the half-disc.

Find the area of $A$.\\
\includegraphics[max width=\textwidth, alt={}, center]{73653fd2-ae8e-477f-a6a6-0d193579bbd9-14_515_809_1101_612}\\
(iii) Assuming only that $\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = 1$, show that $\sin \theta \cos \theta \leqslant 1 / 2$.\\
(iv) What is the largest that the ratio

$$\frac { \text { area of } A } { \text { area of } B }$$

can be, as $\theta$ varies?\\
\includegraphics[max width=\textwidth, alt={}, center]{73653fd2-ae8e-477f-a6a6-0d193579bbd9-15_757_892_351_577}