Question 1 is a multiple choice question with ten parts. Marks are given solely for correct answers but any rough working should be shown in the space between parts. Answer Question 1 on the grid on Page 2. Each part is worth 4 marks.
Answers to questions 2-7 should be written in the space provided, continuing on to the blank pages at the end of this booklet if necessary. Each of Questions 2-7 is worth 15 marks.
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1. For ALL APPLICANTS.
For each part of the question on pages $3 - 7$ you will be given four possible answers, just one of which is correct. Indicate for each part $\mathbf { A } - \mathbf { J }$ which answer (a), (b), (c), or (d) you think is correct with a tick $( \checkmark )$ in the corresponding column in the table below. Please show any rough working in the space provided between the parts.
A. Which of the following lines is a tangent to the circle with equation
$$x ^ { 2 } + y ^ { 2 } = 4 ?$$
(a) $x + y = 2$;
(b) $y = x - 2 \sqrt { 2 }$;
(c) $x = \sqrt { 2 }$;
(d) $y = \sqrt { 2 } - x$.
B. Let $N = 2 ^ { k } \times 4 ^ { m } \times 8 ^ { n }$ where $k , m , n$ are positive whole numbers. Then $N$ will definitely be a square number whenever
(a) $k$ is even;
(b) $k + n$ is odd;
(c) $k$ is odd but $m + n$ is even;
(d) $k + n$ is even.
C. Which is the smallest of the following numbers?
(a) $( \sqrt { 3 } ) ^ { 3 }$,
(b) $\quad \log _ { 3 } \left( 9 ^ { 2 } \right)$,
(c) $\quad \left( 3 \sin \frac { \pi } { 3 } \right) ^ { 2 }$,
(d) $\quad \log _ { 2 } \left( \log _ { 2 } \left( 8 ^ { 5 } \right) \right)$.
D. Shown below is a diagram of the square with vertices $( 0,0 ) , ( 0,1 ) , ( 1,1 ) , ( 1,0 )$ and the line $y = x + c$. The shaded region is the region of the square which lies below the line; this shaded region has area $A ( c )$.
[Figure]Which of the following graphs shows $A ( c )$ as $c$ varies?
[Figure](a)
[Figure](b)
[Figure](c)
[Figure](d)
E. Which one of the following equations could possibly have the graph given below?
(a) $y = ( 3 - x ) ^ { 2 } ( 3 + x ) ^ { 2 } ( 1 - x )$;
(b) $y = - x ^ { 2 } ( x - 9 ) \left( x ^ { 2 } - 3 \right)$;
(c) $y = ( x - 6 ) ( x - 2 ) ^ { 2 } ( x + 2 ) ^ { 2 }$;
(d) $y = \left( x ^ { 2 } - 1 \right) ^ { 2 } ( 3 - x )$.
[Figure] F. Let
$$T = \left( \int _ { - \pi / 2 } ^ { \pi / 2 } \cos x \mathrm {~d} x \right) \times \left( \int _ { \pi } ^ { 2 \pi } \sin x \mathrm {~d} x \right) \times \left( \int _ { 0 } ^ { \pi / 8 } \frac { \mathrm {~d} x } { \cos 3 x } \right)$$
Which of the following is true?
(a) $\quad T = 0$;
(b) $T < 0$;
(c) $T > 0$;
(d) $T$ is not defined. G. There are positive real numbers $x$ and $y$ which solve the equations
$$2 x + k y = 4 , \quad x + y = k$$
for
(a) all values of $k$;
(b) no values of $k$;
(c) $k = 2$ only;
(d) only $k > - 2$. H. In the region $0 < x \leqslant 2 \pi$, the equation
$$\int _ { 0 } ^ { x } \sin ( \sin t ) d t = 0$$
has
(a) no solution;
(b) one solution;
(c) two solutions;
(d) three solutions. I. The vertices of an equilateral triangle are labelled $X , Y$ and $Z$. The points $X , Y$ and $Z$ lie on a circle of circumference 10 units. Let $P$ and $A$ be the numerical values of the triangle's perimeter and area, respectively. Which of the following is true?
(a) $\frac { A } { P } = \frac { 5 } { 4 \pi } ;$
(b) $P < A$;
(c) $\frac { P } { A } = \frac { 10 } { 3 \pi }$;
(d) $P ^ { 2 }$ is rational. J. If two chords $Q P$ and $R P$ on a circle of radius 1 meet in an angle $\theta$ at $P$, for example as drawn in the diagram below,
[Figure] then the largest possible area of the shaded region $R P Q$ is
(a) $\theta \left( 1 + \cos \left( \frac { \theta } { 2 } \right) \right) ;$
(b) $\theta + \sin \theta$;
(c) $\frac { \pi } { 2 } ( 1 - \cos \theta )$;
(d) $\quad \theta$.