Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Total \\
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\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) \\
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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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H &  &  &  &  \\
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I &  &  &  &  \\
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J &  &  &  &  \\
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\end{tabular}
\end{center}

A. The area of the region bounded by the curves $y = x ^ { 2 }$ and $y = x + 2$ equals\\
(a) $\frac { 7 } { 3 }$\\
(b) $\frac { 7 } { 2 }$\\
(c) $\frac { 9 } { 2 }$\\
(d) $\frac { 11 } { 2 }$\\
B. The smallest value of the function

$$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x + 3$$

in the range $0 \leq x \leq 2$ is\\
(a) 1\\
(b) 3\\
(c) 5\\
(d) 7\\
C. What is the reflection of the point $( 3,4 )$ in the line $3 x + 4 y = 50$ ?\\
(a) $( 9,12 )$\\
(b) $( 6,8 )$\\
(c) $( 12,16 )$\\
(d) $( 16,12 )$\\
D. The equation $x ^ { 3 } - 30 x ^ { 2 } + 108 x - 104 = 0$\\
(a) no real roots;\\
(b) exactly one real root;\\
(c) three distinct real roots;\\
(d) a repeated root.\\
E. The fact that

$$6 \times 7 = 42$$

is a counter-example to which of the following statements?\\
(a) the product of any two odd integers is odd;\\
(b) if the product of two integers is not a multiple of 4 then the integers are not consecutive;\\
(c) if the product of two integers is a multiple of 4 then the integers are not consecutive;\\
(d) any even integer can be written as the product of two even integers.\\
F. How many values of $x$ satisfy the equation

$$2 \cos ^ { 2 } x + 5 \sin x = 4$$

in the range $0 \leqslant x < 2 \pi$ ?\\
(a) 2\\
(b) 4\\
(c) 6\\
(d) 8\\
G. The inequalities $x ^ { 2 } + 3 x + 2 > 0$ and $x ^ { 2 } + x < 2$, are met by all $x$ in the region:\\
(a) $x < - 2$;\\
(b) $- 1 < x < 1$;\\
(c) $x > - 1$;\\
(d) $x > - 2$.\\
H. Given that

$$\log _ { 10 } 2 = 0.3010 \text { to } 4 \text { d.p. and that } 10 ^ { 0.2 } < 2$$

it is possible to deduce that\\
(a) $2 ^ { 100 }$ begins in a 1 and is 30 digits long;\\
(b) $2 ^ { 100 }$ begins in a 2 and is 30 digits long;\\
(c) $2 ^ { 100 }$ begins in a 1 and is 31 digits long;\\
(d) $2 ^ { 100 }$ begins in a 2 and is 31 digits long.\\
I. The power of $x$ which has the greatest coefficient in the expansion of $\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }$ is\\
(a) $x ^ { 2 }$\\
(b) $x ^ { 3 }$\\
(c) $x ^ { 5 }$\\
(d) $x ^ { 10 }$\\
J. A sketch of the curve with equation $x ^ { 2 } y ^ { 2 } ( x + y ) = 1$ is drawn in which of the following diagrams?\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-07_522_517_1080_411}\\
(a)\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-07_533_522_1731_406}\\
(c)\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-07_533_531_1073_1110}\\
(b)\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-07_528_524_1736_1114}\\
(d)