Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Total \\
\hline
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\hline
\end{tabular}
\end{center}

\section*{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.

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
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 & (a) & (b) & (c) & (d) \\
\hline
A &  &  &  &  \\
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B &  &  &  &  \\
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C &  &  &  &  \\
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D &  &  &  &  \\
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E &  &  &  &  \\
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F &  &  &  &  \\
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G &  &  &  &  \\
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H &  &  &  &  \\
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I &  &  &  &  \\
\hline
J &  &  &  &  \\
\hline
\end{tabular}
\end{center}

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 $y = f ( x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{43f3ba53-7486-4997-ada9-58d9009d4293-03_522_513_824_769}

The graph of the function $y = - f ( x + 1 )$ is drawn in which of the following diagrams?\\
\includegraphics[max width=\textwidth, alt={}, center]{43f3ba53-7486-4997-ada9-58d9009d4293-03_472_467_1482_541}\\
(a)\\
\includegraphics[max width=\textwidth, alt={}, center]{43f3ba53-7486-4997-ada9-58d9009d4293-03_464_465_2010_543}\\
(c)\\
\includegraphics[max width=\textwidth, alt={}, center]{43f3ba53-7486-4997-ada9-58d9009d4293-03_470_469_1484_1041}\\
(b)\\
\includegraphics[max width=\textwidth, alt={}, center]{43f3ba53-7486-4997-ada9-58d9009d4293-03_467_465_2010_1043}\\
(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