4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.
Mathematics \& Computer Science and Computer Science applicants should turn to page 14. In the diagram below is sketched the circle with centre ( 1,1 ) and radius 1 and a line $L$. The line $L$ is tangential to the circle at $Q$; further $L$ meets the $y$-axis at $R$ and the $x$-axis at $P$ in such a way that the angle $O P Q$ equals $\theta$ where $0 < \theta < \pi / 2$. [Figure] (i) Show that the co-ordinates of $Q$ are $$( 1 + \sin \theta , 1 + \cos \theta ) ,$$ and that the gradient of $P Q R$ is $- \tan \theta$. Write down the equation of the line $P Q R$ and so find the co-ordinates of $P$. (ii) The region bounded by the circle, the $x$-axis and $P Q$ has area $A ( \theta )$; the region bounded by the circle, the $y$-axis and $Q R$ has area $B ( \theta )$. (See diagram.) Explain why $$A ( \theta ) = B ( \pi / 2 - \theta )$$ for any $\theta$. Calculate $A ( \pi / 4 )$. (iii) Show that $$A \left( \frac { \pi } { 3 } \right) = \sqrt { 3 } - \frac { \pi } { 3 } .$$
(i) [7 marks] Let $C = ( 1,1 )$ denote the centre of the circle. then $C Q$ makes angle $\theta$ with the vertical and is of length 1. So
\section*{4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
Mathematics \& Computer Science and Computer Science applicants should turn to page 14.
In the diagram below is sketched the circle with centre ( 1,1 ) and radius 1 and a line $L$. The line $L$ is tangential to the circle at $Q$; further $L$ meets the $y$-axis at $R$ and the $x$-axis at $P$ in such a way that the angle $O P Q$ equals $\theta$ where $0 < \theta < \pi / 2$.\\
\includegraphics[max width=\textwidth, alt={}, center]{88ab9917-8b40-4dbf-a806-1a546164dcee-12_719_888_865_625}\\
(i) Show that the co-ordinates of $Q$ are
$$( 1 + \sin \theta , 1 + \cos \theta ) ,$$
and that the gradient of $P Q R$ is $- \tan \theta$.\\
Write down the equation of the line $P Q R$ and so find the co-ordinates of $P$.\\
(ii) The region bounded by the circle, the $x$-axis and $P Q$ has area $A ( \theta )$; the region bounded by the circle, the $y$-axis and $Q R$ has area $B ( \theta )$. (See diagram.)
Explain why
$$A ( \theta ) = B ( \pi / 2 - \theta )$$
for any $\theta$.\\
Calculate $A ( \pi / 4 )$.\\
(iii) Show that
$$A \left( \frac { \pi } { 3 } \right) = \sqrt { 3 } - \frac { \pi } { 3 } .$$