Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Total \hline & & & & & & & & & & & & & & \hline \end{tabular}
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 ( ✓ ) in the corresponding column in the table below. Please show any rough working in the space provided between the parts.
A. The point lying between $P ( 2,3 )$ and $Q ( 8 , - 3 )$ which divides the line $P Q$ in the ratio $1 : 2$ has co-ordinates
(a) $( 4 , - 1 )$
(b) $( 6 , - 2 )$
(c) $\left( \frac { 14 } { 3 } , 2 \right)$
(d) $( 4,1 )$
B. The diagram below shows the graph of the function $y = f ( x )$.
[Figure]The graph of the function $y = - f ( x + 1 )$ is drawn in which of the following diagrams?
[Figure](a)
[Figure](c)
[Figure](b)
[Figure](d)
C. Which of the following numbers is largest in value? (All angles are given in radians.)
(a) $\tan \left( \frac { 5 \pi } { 4 } \right)$
(b) $\sin ^ { 2 } \left( \frac { 5 \pi } { 4 } \right)$
(c) $\log _ { 10 } \left( \frac { 5 \pi } { 4 } \right)$
(d) $\log _ { 2 } \left( \frac { 5 \pi } { 4 } \right)$
D. The numbers $x$ and $y$ satisfy the following inequalities
$$\begin{aligned}
2 x + 3 y & \leqslant 23 \\
x + 2 & \leqslant 3 y \\
3 y + 1 & \leqslant 4 x
\end{aligned}$$
The largest possible value of $x$ is
(a) 6
(b) 7
(c) 8
(d) .9
E. In the range $0 \leqslant x < 2 \pi$ the equation
$$\cos ( \sin x ) = \frac { 1 } { 2 }$$
has
(a) no solutions;
(b) one solution;
(c) two solutions;
(d) three solutions. F. The turning point of the parabola
$$y = x ^ { 2 } - 2 a x + 1$$
is closest to the origin when
(a) $a = 0$
(b) $a = \pm 1$
(c) $a = \pm \frac { 1 } { \sqrt { 2 } }$ or $a = 0$
(d) $a = \pm \frac { 1 } { \sqrt { 2 } }$. G. The four digit number 2652 is such that any two consecutive digits from it make a multiple of 13 . Another number $N$ has this same property, is 100 digits long, and begins in a 9 . What is the last digit of $N$ ?
(a) 2
(b) 3
(c) 6
(d) 9 H. The equation
$$\left( x ^ { 2 } + 1 \right) ^ { 10 } = 2 x - x ^ { 2 } - 2$$
(a) has $x = 2$ as a solution;
(b) has no real solutions;
(c) has an odd number of real solutions;
(d) has twenty real solutions. I. .Observe that $2 ^ { 3 } = 8,2 ^ { 5 } = 32,3 ^ { 2 } = 9$ and $3 ^ { 3 } = 27$. From these facts, we can deduce that $\log _ { 2 } 3$, the logarithm of 3 to base 2 , is
(a) between $1 \frac { 1 } { 3 }$ and $1 \frac { 1 } { 2 }$;
(b) between $1 \frac { 1 } { 2 }$ and $1 \frac { 2 } { 3 }$;
(c) between $1 \frac { 2 } { 3 }$ and 2;
(d) between 2 and 3. J. Into how many regions is the plane divided when the following three parabolas are drawn?
$$\begin{aligned}
& y = x ^ { 2 } \\
& y = x ^ { 2 } - 2 x \\
& y = x ^ { 2 } + 2 x + 2
\end{aligned}$$
(a) 4
(b) 5
(c) 6
(d) 7