\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.

Let $Q$ denote the quarter-disc of points $( x , y )$ such that $x \geqslant 0 , y \geqslant 0$ and $x ^ { 2 } + y ^ { 2 } \leqslant 1$ as drawn in Figures A and B below.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-12_741_750_766_262}
\captionsetup{labelformat=empty}
\caption{Figure A}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-12_741_751_766_1039}
\captionsetup{labelformat=empty}
\caption{Figure B}
\end{center}
\end{figure}

(i) On the axes in Figure A, sketch the graphs of

$$x + y = \frac { 1 } { 2 } , \quad x + y = 1 , \quad x + y = \frac { 3 } { 2 } .$$

What is the largest value of $x + y$ achieved at points $( x , y )$ in $Q$ ? Justify your answer.\\
(ii) On the axes in Figure B, sketch the graphs of

$$x y = \frac { 1 } { 4 } , \quad x y = 1 , \quad x y = 2 .$$

What is the largest value of $x ^ { 2 } + y ^ { 2 } + 4 x y$ achieved at points $( x , y )$ in $Q$ ?\\
What is the largest value of $x ^ { 2 } + y ^ { 2 } - 6 x y$ achieved at points $( x , y )$ in $Q$ ?\\
(iii) Describe the curve

$$x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = k$$

where $k > - 5$.\\
What is the smallest value of $x ^ { 2 } + y ^ { 2 } - 4 x - 2 y$ achieved at points ( $x , y$ ) in $Q$ ?