Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Total \\
\hline
 &  &  &  &  &  &  &  \\
\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 $( \checkmark )$ 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|}
\hline
 & (a) & (b) & (c) & (d) \\
\hline
A &  &  &  &  \\
\hline
B &  &  &  &  \\
\hline
C &  &  &  &  \\
\hline
D &  &  &  &  \\
\hline
E &  &  &  &  \\
\hline
F &  &  &  &  \\
\hline
G &  &  &  &  \\
\hline
H &  &  &  &  \\
\hline
I &  &  &  &  \\
\hline
J &  &  &  &  \\
\hline
\end{tabular}
\end{center}

A. The values of $k$ for which the line $y = k x$ intersects the parabola $y = ( x - 1 ) ^ { 2 }$ are precisely\\
(a) $k \leqslant 0$,\\
(b) $k \geqslant - 4$,\\
(c) $k \geqslant 0$ or $k \leqslant - 4$,\\
(d) $- 4 \leqslant k \leqslant 0$.\\
B. The sum of the first $2 n$ terms of

$$1,1,2 , \frac { 1 } { 2 } , 4 , \frac { 1 } { 4 } , 8 , \frac { 1 } { 8 } , 16 , \frac { 1 } { 16 } , \ldots$$

is\\
(a) $2 ^ { n } + 1 - 2 ^ { 1 - n }$,\\
(b) $\quad 2 ^ { n } + 2 ^ { - n }$,\\
(c) $2 ^ { 2 n } - 2 ^ { 3 - 2 n }$,\\
(d) $\frac { 2 ^ { n } - 2 ^ { - n } } { 3 }$.\\
C. In the range $0 \leqslant x < 2 \pi$, the equation

$$\sin ^ { 2 } x + 3 \sin x \cos x + 2 \cos ^ { 2 } x = 0$$

has\\
(a) 1 solution,\\
(b) 2 solutions,\\
(c) 3 solutions,\\
(d) 4 solutions.\\
D. The graph of $y = \sin ^ { 2 } \sqrt { x }$ is drawn in

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f229588a-5602-44a5-acb0-6efec65f41af-04_410_634_1384_331}
\captionsetup{labelformat=empty}
\caption{(a)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f229588a-5602-44a5-acb0-6efec65f41af-04_412_636_1382_1078}
\captionsetup{labelformat=empty}
\caption{(b)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f229588a-5602-44a5-acb0-6efec65f41af-04_412_634_1905_331}
\captionsetup{labelformat=empty}
\caption{(c)}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f229588a-5602-44a5-acb0-6efec65f41af-04_412_650_1905_1071}
\captionsetup{labelformat=empty}
\caption{(d)}
\end{center}
\end{figure}

E. Which is the largest of the following four numbers?\\
(a) $\quad \log _ { 2 } 3$,\\
(b) $\quad \log _ { 4 } 8$,\\
(c) $\quad \log _ { 3 } 2$,\\
(d) $\quad \log _ { 5 } 10$.\\
F. The graph $y = f ( x )$ of a function is drawn below for $0 \leqslant x \leqslant 1$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f229588a-5602-44a5-acb0-6efec65f41af-05_695_1081_1199_484}

The trapezium rule is then used to estimate

$$\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x$$

by dividing $0 \leqslant x \leqslant 1$ into $n$ equal intervals. The estimate calculated will equal the actual integral when\\
(a) $n$ is a multiple of 4 ;\\
(b) $n$ is a multiple of 6 ;\\
(c) $n$ is a multiple of 8 ;\\
(d) $n$ is a multiple of 12 .

Turn Over\\
G. The function $f$, defined for whole positive numbers, satisfies $f ( 1 ) = 1$ and also the rules

$$\begin{aligned}
f ( 2 n ) & = 2 f ( n ) , \\
f ( 2 n + 1 ) & = 4 f ( n ) ,
\end{aligned}$$

for all values of $n$. How many numbers $n$ satisfy $f ( n ) = 16$ ?\\
(a) 3 ,\\
(b) 4,\\
(c) 5 ,\\
(d) 6 .\\
H. Given a positive integer $n$ and a real number $k$, consider the following equation in $x$,

$$( x - 1 ) ( x - 2 ) ( x - 3 ) \times \cdots \times ( x - n ) = k$$

Which of the following statements about this equation is true?\\
(a) If $n = 3$, then the equation has no real solution $x$ for some values of $k$.\\
(b) If $n$ is even, then the equation has a real solution $x$ for any given value of $k$.\\
(c) If $k \geqslant 0$ then the equation has (at least) one real solution $x$.\\
(d) The equation never has a repeated solution $x$ for any given values of $k$ and $n$.\\
I. For a positive number $a$, let

$$I ( a ) = \int _ { 0 } ^ { a } \left( 4 - 2 ^ { x ^ { 2 } } \right) \mathrm { d } x$$

Then $\mathrm { d } I / \mathrm { d } a = 0$ when $a$ equals\\
(a) $\frac { 1 + \sqrt { 5 } } { 2 }$,\\
(b) $\sqrt { 2 }$,\\
(c) $\frac { \sqrt { 5 } - 1 } { 2 }$,\\
(d) 1 .\\
J. Let $a , b , c$ be positive numbers. There are finitely many positive whole numbers $x , y$ which satisfy the inequality

$$a ^ { x } > c b ^ { y }$$

if\\
(a) $a > 1$ or $b < 1$.\\
(b) $a < 1$ or $b < 1$.\\
(c) $a < 1$ and $b < 1$.\\
(d) $a < 1$ and $b > 1$.