$\lim _ { x \rightarrow \frac { \pi } { 2 } } \left( \frac { \int _ { x ^ { 3 } } ^ { ( \pi / 2 ) ^ { 3 } } \left( \sin \left( 2 t ^ { 1 / 3 } \right) + \cos \left( t ^ { 1 / 3 } \right) \right) d t } { \left( x - \frac { \pi } { 2 } \right) ^ { 2 } } \right)$ is equal to
(1) $\frac { 5 \pi ^ { 2 } } { 9 }$
(2) $\frac { 9 \pi ^ { 2 } } { 8 }$
(3) $\frac { 11 \pi ^ { 2 } } { 10 }$
(4) $\frac { 3 \pi ^ { 2 } } { 2 }$

Let $f ( x ) = \int _ { 0 } ^ { x } t \left( t ^ { 2 } - 9 t + 20 \right) d t , 1 \leq x \leq 5$. If the range of $f$ is $[ \alpha , \beta ]$, then $4 ( \alpha + \beta )$ equals:
(1) 253
(2) 154
(3) 125
(4) 157

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 applicants should turn to page 14. Let
$$f ( x ) = \left\{ \begin{array} { c c } x + 1 & \text { for } 0 \leqslant x \leqslant 1 ; \\ 2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leqslant x \leqslant 2 . \end{array} \right.$$
(i) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leqslant x \leqslant 2$, labelling any turning points and the values attained at $x = 0,1,2$.
(ii) For $1 \leqslant t \leqslant 2$, define
$$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$
Express $g ( t )$ as a cubic in $t$.
(iii) Calculate and factorize $g ^ { \prime } ( t )$.
(iv) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leqslant t \leqslant 2$ ? [Figure]