Computer Science and Computer Science \& Philosophy applicants should turn to page 14. Let $0 < k < 2$. Below is sketched a graph of $y = f _ { k } ( x )$ where $f _ { k } ( x ) = x ( x - k ) ( x - 2 )$. Let $A ( k )$ denote the area of the shaded region. [Figure] (i) Without evaluating them, write down an expression for $A ( k )$ in terms of two integrals. (ii) Explain why $A ( k )$ is a polynomial in $k$ of degree 4 or less. [You are not required to calculate $A ( k )$ explicitly.] (iii) Verify that $f _ { k } ( 1 + t ) = - f _ { 2 - k } ( 1 - t )$ for any $t$. (iv) How can the graph of $y = f _ { k } ( x )$ be transformed to the graph of $y = f _ { 2 - k } ( x )$ ? Deduce that $A ( k ) = A ( 2 - k )$. (v) Explain why there are constants $a , b , c$ such that $$A ( k ) = a ( k - 1 ) ^ { 4 } + b ( k - 1 ) ^ { 2 } + c .$$ [You are not required to calculate $a , b , c$ explicitly.]
(i) [2 marks] Clearly
\section*{3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.}
Computer Science and Computer Science \& Philosophy applicants should turn to page 14.
Let $0 < k < 2$. Below is sketched a graph of $y = f _ { k } ( x )$ where $f _ { k } ( x ) = x ( x - k ) ( x - 2 )$. Let $A ( k )$ denote the area of the shaded region.\\
\includegraphics[max width=\textwidth, alt={}, center]{73653fd2-ae8e-477f-a6a6-0d193579bbd9-12_503_757_870_646}\\
(i) Without evaluating them, write down an expression for $A ( k )$ in terms of two integrals.\\
(ii) Explain why $A ( k )$ is a polynomial in $k$ of degree 4 or less. [You are not required to calculate $A ( k )$ explicitly.]\\
(iii) Verify that $f _ { k } ( 1 + t ) = - f _ { 2 - k } ( 1 - t )$ for any $t$.\\
(iv) How can the graph of $y = f _ { k } ( x )$ be transformed to the graph of $y = f _ { 2 - k } ( x )$ ?
Deduce that $A ( k ) = A ( 2 - k )$.\\
(v) Explain why there are constants $a , b , c$ such that
$$A ( k ) = a ( k - 1 ) ^ { 4 } + b ( k - 1 ) ^ { 2 } + c .$$
[You are not required to calculate $a , b , c$ explicitly.]