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. Consider two circles $S _ { 1 }$ and $S _ { 2 }$ centred at $A$ and $B$ and with radii $\sqrt { 6 }$ and $\sqrt { 3 } - 1$, respectively. Suppose that the two circles intersect at two distinct points $C$ and $D$. Suppose further that the two centres $A$ and $B$ are of distance 2 apart. The sketch below is not to scale. [Figure] (i) Find the angle $\angle C B A$, and deduce that $A$ and $B$ lie on the same side of the line $C D$. (ii) Show that $C D$ has length $3 - \sqrt { 3 }$ and hence calculate the angle $\angle C A D$. (iii) Show that the area of the region lying inside the circle $S _ { 2 }$ and outside of the circle $S _ { 1 }$ (that is the shaded region in the picture) is equal to $$\frac { \pi } { 6 } ( 5 - 4 \sqrt { 3 } ) + 3 - \sqrt { 3 } .$$ (iv) Suppose that a line through $C$ is drawn such that the total area covered by $S _ { 1 }$ and $S _ { 2 }$ is split into two equal areas. Let $E$ be the intersection of this line with $S _ { 1 }$ and $x$ denote the angle $\angle C A E$. You may assume that $E$ lies on the larger $\operatorname { arc } C D$ of $S _ { 1 }$. Write down an equation which $x$ satisfies and explain why there is a unique solution $x$.
\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.
Consider two circles $S _ { 1 }$ and $S _ { 2 }$ centred at $A$ and $B$ and with radii $\sqrt { 6 }$ and $\sqrt { 3 } - 1$, respectively. Suppose that the two circles intersect at two distinct points $C$ and $D$. Suppose further that the two centres $A$ and $B$ are of distance 2 apart. The sketch below is not to scale.\\
\includegraphics[max width=\textwidth, alt={}, center]{df72dfe2-9f61-49a0-a6b9-a4ad637d3c86-14_825_920_950_529}\\
(i) Find the angle $\angle C B A$, and deduce that $A$ and $B$ lie on the same side of the line $C D$.\\
(ii) Show that $C D$ has length $3 - \sqrt { 3 }$ and hence calculate the angle $\angle C A D$.\\
(iii) Show that the area of the region lying inside the circle $S _ { 2 }$ and outside of the circle $S _ { 1 }$ (that is the shaded region in the picture) is equal to
$$\frac { \pi } { 6 } ( 5 - 4 \sqrt { 3 } ) + 3 - \sqrt { 3 } .$$
(iv) Suppose that a line through $C$ is drawn such that the total area covered by $S _ { 1 }$ and $S _ { 2 }$ is split into two equal areas. Let $E$ be the intersection of this line with $S _ { 1 }$ and $x$ denote the angle $\angle C A E$. You may assume that $E$ lies on the larger $\operatorname { arc } C D$ of $S _ { 1 }$. Write down an equation which $x$ satisfies and explain why there is a unique solution $x$.