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. As shown in the diagram below: $C$ is the parabola with equation $y = x ^ { 2 } ; P$ is the point $( 0,1 ) ; Q$ is the point ( $a , a ^ { 2 }$ ) on $C ; L$ is the normal to $C$ which passes through $Q$. [Figure] (i) Find the equation of $L$. (ii) For what values of $a$ does $L$ pass through $P$ ? (iii) Determine $| Q P | ^ { 2 }$ as a function of $a$, where $| Q P |$ denotes the distance from $P$ to $Q$. (iv) Find the values of $a$ for which $| Q P |$ is smallest. (v) Find a point $R$, in the $x y$-plane but not on $C$, such that $| R Q |$ is smallest for a unique value of $a$. Briefly justify your answer.
( 3 n - 1 ) ( n + B ) & \geqslant 3 ( 4 n - 1 ) n \text { for } n \geqslant 1 ;
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.
As shown in the diagram below: $C$ is the parabola with equation $y = x ^ { 2 } ; P$ is the point $( 0,1 ) ; Q$ is the point ( $a , a ^ { 2 }$ ) on $C ; L$ is the normal to $C$ which passes through $Q$.\\
\includegraphics[max width=\textwidth, alt={}, center]{373b6bde-f147-4e16-aaa9-78f86989fc96-12_636_967_756_541}\\
(i) Find the equation of $L$.\\
(ii) For what values of $a$ does $L$ pass through $P$ ?\\
(iii) Determine $| Q P | ^ { 2 }$ as a function of $a$, where $| Q P |$ denotes the distance from $P$ to $Q$.\\
(iv) Find the values of $a$ for which $| Q P |$ is smallest.\\
(v) Find a point $R$, in the $x y$-plane but not on $C$, such that $| R Q |$ is smallest for a unique value of $a$. Briefly justify your answer.