1. For ALL APPLICANTS.
For each part of the question on pages $3 - 7$ you will be given five possible answers, just one of which is correct. Indicate for each part $\mathbf { A } - \mathbf { J }$ which answer (a), (b), (c), (d), or (e) you think is correct with a tick $( \sqrt { } )$ in the corresponding column in the table below. Please show any rough working in the space provided between the parts.
| (a) | (b) | (c) | (d) | (e) |
| A | | | | | |
| B | | | | | |
| C | | | | | |
| D | | | | | |
| E | | | | | |
| F | | | | | |
| G | | | | | |
| H | | | | | |
| I | | | | | |
| J | | | | | |
A. A square has centre ( 3,4 ) and one corner at ( 1,5 ). Another corner is at
(a) $( 1,3 )$,
(b) $( 5,5 )$,
(c) $( 4,2 )$,
(d) $( 2,2 )$,
(e) $( 5,2 )$.
B. What is the value of $\int _ { 0 } ^ { 1 } \left( e ^ { x } - x \right) \left( e ^ { x } + x \right) \mathrm { d } x$ ?
(a) $\frac { 3 e ^ { 2 } - 2 } { 6 }$,
(b) $\frac { 3 e ^ { 2 } + 2 } { 6 }$,
(c) $\frac { 2 e ^ { 2 } - 3 } { 6 }$,
(d) $\frac { 3 e ^ { 2 } - 5 } { 6 }$,
(e) $\frac { e ^ { 2 } + 3 } { 6 }$.
C. The sum
$$1 - 4 + 9 - 16 + \cdots + 99 ^ { 2 } - 100 ^ { 2 }$$
equals
(a) - 101
(b) - 1000
(c) -1111
(d) - 4545
(e) $\quad - 5050$.
D. The largest value achieved by $3 \cos ^ { 2 } x + 2 \sin x + 1$ equals
(a) $\frac { 11 } { 5 }$,
(b) $\frac { 13 } { 3 }$,
(c) $\frac { 12 } { 5 }$,
(d) $\frac { 14 } { 9 }$,
(e) $\frac { 12 } { 7 }$.
E. A line is tangent to the parabola $y = x ^ { 2 }$ at the point $\left( a , a ^ { 2 } \right)$ where $a > 0$. The area of the region bounded by the parabola, the tangent line, and the $x$-axis equals
(a) $\frac { a ^ { 2 } } { 3 }$,
(b) $\frac { 2 a ^ { 2 } } { 3 }$,
(c) $\frac { a ^ { 3 } } { 12 }$,
(d) $\frac { 5 a ^ { 3 } } { 6 }$,
(e) $\frac { a ^ { 4 } } { 10 }$. F. Which of the following expressions is equal to $\log _ { 10 } ( 10 \times 9 \times 8 \times \cdots \times 2 \times 1 )$ ?
(a) $1 + 5 \log _ { 10 } 2 + 4 \log _ { 10 } 6$,
(b) $1 + 4 \log _ { 10 } 2 + 2 \log _ { 10 } 6 + \log _ { 10 } 7$,
(c) $2 + 2 \log _ { 10 } 2 + 4 \log _ { 10 } 6 + \log _ { 10 } 7$,
(d) $2 + 6 \log _ { 10 } 2 + 4 \log _ { 10 } 6 + \log _ { 10 } 7$,
(e) $2 + 6 \log _ { 10 } 2 + 4 \log _ { 10 } 6$. G. A cubic has equation $y = x ^ { 3 } + a x ^ { 2 } + b x + c$ and has turning points at $( 1,2 )$ and $( 3 , d )$ for some $d$. What is the value of $d$ ?
(a) - 4 ,
(b) - 2 ,
(c) 0 ,
(d) 2 ,
(e) 4 . H. The following five graphs are, in some order, plots of $y = f ( x ) , y = g ( x ) , y = h ( x )$, $y = \frac { \mathrm { d } f } { \mathrm {~d} x }$ and $y = \frac { \mathrm { d } g } { \mathrm {~d} x }$; that is, three unknown functions and the derivatives of the first two of those functions. Which graph is a plot of $h ( x )$ ?
[Figure](b) [Figure](c) [Figure](d) [Figure](e) [Figure]I. In the range $- 90 ^ { \circ } < x < 90 ^ { \circ }$, how many values of $x$ are there for which the sum to infinity
$$\frac { 1 } { \tan x } + \frac { 1 } { \tan ^ { 2 } x } + \frac { 1 } { \tan ^ { 3 } x } + \ldots$$
equals $\tan x$ ?
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4 . J. Consider a square with side length 2 and centre ( 0,0 ), and a circle with radius $r$ and centre $( 0,0 )$. Let $A ( r )$ be the area of the region that is inside the circle but outside the square, and let $B ( r )$ be the area of the region that is inside the square but outside the circle. Which of the following is a sketch of $A ( r ) + B ( r )$ ?
[Figure] [Figure] [Figure] [Figure] [Figure]