\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.

In the diagram below is sketched a semicircle with centre $B$ and radius 1 . Three points $A , C , D$ lie on the semicircle as shown with $\alpha$ denoting angle $C A B$ and $\beta$ denoting angle $D A B$. The triangles $A B C$ and $A B D$ intersect in a triangle $A B X$.

Throughout the question we shall consider the value of $\alpha$ fixed. Assume for now that $0 < \alpha \leqslant \beta \leqslant \pi / 2$.\\
\includegraphics[max width=\textwidth, alt={}, center]{3e018931-0964-46fa-8f09-e829097e009e-14_467_757_928_638}\\
(i) Show that the area of the triangle $A B C$ equals

$$\frac { 1 } { 2 } \sin ( 2 \alpha ) .$$

(ii) Let

$$F = \frac { \text { area of triangle } A B X } { \text { area of triangle } A B C }$$

Without calculation, explain why, for every $k$ in the range $0 \leqslant k \leqslant 1$, there is a unique value of $\beta$ such that $F = k$.\\
(iii) Find the value of $\beta$ such that $F = 1 / 2$.\\
(iv) Show that

$$F = \frac { \sin ( 2 \beta ) \sin \alpha } { \sin ( 2 \beta - \alpha ) \sin ( 2 \alpha ) }$$

(v) Suppose now that $0 < \beta < \alpha \leqslant \pi / 2$. Write down, without further calculation, an expression for the area of $A B X$ and hence a formula for $F$.