\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 20.

Charlie is trying to cut a cake. The cake is a square with side length 2 , and its corners are at $( 0,0 ) , ( 2,0 ) , ( 2,2 )$, and $( 0,2 )$. Charlie's first cut is a straight line segment from the point $( x , y )$ to $( x , 0 )$, where $0 \leqslant x \leqslant 2$ and $0 \leqslant y \leqslant 2$.

Charlie plans to make a second straight cut from the point $( x , y )$ to a point $( 0 , k )$ somewhere on the left-hand edge of the cake. This will make a slice of cake which is bounded to the left of the first cut and bounded below the second cut.\\
\includegraphics[max width=\textwidth, alt={}, center]{c3a5e72c-19e2-41de-8833-ff60948250a1-16_490_535_1059_733}\\
(i) Find the area of the slice of cake in terms of $x , y$, and $k$. Check your expression by verifying that if $x = 1$ and $y = 1$, then choosing $k = 1$ gives a slice of cake with area 1 .\\
(ii) Find another point ( $x , y$ ) on the cake such that choosing $k = 1$ gives a slice of cake with area 1 .\\
(iii) Show that it is only possible to choose a value of $k$ that gives a slice of cake with area 1 if both $x y \leqslant 2$ and $x ( 2 + y ) \geqslant 2$.\\
(iv) Sketch the region $R$ of the cake for which both inequalities in part (iii) hold, indicating any relevant points on the edges of the cake.\\
(v) Charlie may instead plan to make the second straight cut from $( x , y )$ to a point $( m , 2 )$ on the top edge of the cake in order to make a slice bounded to the left of the two cuts. Find two necessary and sufficient inequalities for $x$ and $y$ which must both hold in order for this to give a slice of area 1 for some value of $m$. Sketch the region of the cake for which both inequalities hold.



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