\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 $a , b , m$ be positive numbers with $0 < a < b$. In the diagram below are sketched the parabola with equation $y = ( x - a ) ( b - x )$ and the line $y = m x$. The line is tangential to the parabola.\\
$R$ is the region bounded by the $x$-axis, the line and the parabola. $S$ is the region bounded by the parabola and the $x$-axis.\\
\includegraphics[max width=\textwidth, alt={}, center]{9b5230b6-1357-4a2a-9c24-53c6f69fdc21-12_504_1038_1073_475}\\
(i) For $c > 0$, evaluate

$$\int _ { 0 } ^ { c } x ( c - x ) \mathrm { d } x$$

Without further calculation, explain why the area of region $S$ equals $\frac { ( b - a ) ^ { 3 } } { 6 }$.\\
(ii) The line $y = m x$ meets the parabola tangentially as drawn in the diagram. Show that $m = ( \sqrt { b } - \sqrt { a } ) ^ { 2 }$.\\
(iii) Assume now that $a = 1$ and write $b = \beta ^ { 2 }$ where $\beta > 1$. Given that the area of $R$ equals $( 2 \beta + 1 ) ( \beta - 1 ) ^ { 2 } / 6$, show that the areas of regions $R$ and $S$ are equal precisely when

$$( \beta - 1 ) ^ { 2 } \left( \beta ^ { 4 } + 2 \beta ^ { 3 } - 4 \beta - 2 \right) = 0$$

Explain why there is a solution $\beta$ to ( $*$ ) in the range $\beta > 1$.

Without further calculation, deduce that for any $a > 0$ there exists $b > a$ such that the area of region $S$ equals the area of region $R$.