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.

\section*{For Test Supervisors Use Only:}
[ ] Tick here if special arrangements were made for the test. Please either include details below or securely attach to the test script a letter with the details.\\
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\section*{FOR OFFICE USE ONLY:}
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\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.

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A. A sketch of the graph $y = x ^ { 3 } - x ^ { 2 } - x + 1$ appears on which of the following axes?

\begin{figure}[h]
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  \includegraphics[alt={},max width=\textwidth]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-03_427_656_388_363}
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\caption{(a)}
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\begin{figure}[h]
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  \includegraphics[alt={},max width=\textwidth]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-03_424_656_391_1035}
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\caption{(b)}
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\begin{figure}[h]
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  \includegraphics[alt={},max width=\textwidth]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-03_425_656_880_363}
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\caption{(c)}
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\begin{figure}[h]
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  \includegraphics[alt={},max width=\textwidth]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-03_425_652_880_1035}
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\caption{(d)}
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B. A rectangle has perimeter $P$ and area $A$. The values $P$ and $A$ must satisfy\\
(a) $P ^ { 3 } > A$,\\
(b) $A ^ { 2 } > 2 P + 1$,\\
(c) $P ^ { 2 } \geqslant 16 \mathrm {~A}$,\\
(d) $P A \geqslant A + P$.\\
C. The sequence $x _ { n }$ is given by the formula

$$x _ { n } = n ^ { 3 } - 9 n ^ { 2 } + 631 .$$

The largest value of $n$ for which $x _ { n } > x _ { n + 1 }$ is\\
(a) 5 ,\\
(b) 7,\\
(c) 11,\\
(d) 17 .\\
D. The fraction of the interval $0 \leqslant x \leqslant 2 \pi$, for which one (or both) of the inequalities

$$\sin x \geqslant \frac { 1 } { 2 } , \quad \sin 2 x \geqslant \frac { 1 } { 2 }$$

is true, equals\\
(a) $\frac { 1 } { 3 }$,\\
(b) $\frac { 13 } { 24 }$,\\
(c) $\frac { 7 } { 12 }$,\\
(d) $\frac { 5 } { 8 }$.\\
E. The circle in the diagram has centre $C$. Three angles $\alpha , \beta , \gamma$ are also indicated.\\
\includegraphics[max width=\textwidth, alt={}, center]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-05_549_742_388_657}

The angles $\alpha , \beta , \gamma$ are related by the equation:\\
(a) $\cos \alpha = \sin ( \beta + \gamma )$;\\
(b) $\sin \beta = \sin \alpha \sin \gamma$;\\
(c) $\sin \beta ( 1 - \cos \alpha ) = \sin \gamma$;\\
(d) $\sin ( \alpha + \beta ) = \cos \gamma \sin \alpha$.\\
F. Given $\theta$ in the range $0 \leqslant \theta < \pi$, the equation

$$x ^ { 2 } + y ^ { 2 } + 4 x \cos \theta + 8 y \sin \theta + 10 = 0$$

represents a circle for\\
(a) $0 < \theta < \frac { \pi } { 3 }$,\\
(b) $\frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$,\\
(c) $0 < \theta < \frac { \pi } { 2 }$,\\
(d) all values of $\theta$.\\
G. A graph of the function $y = f ( x )$ is sketched on the axes below:\\
\includegraphics[max width=\textwidth, alt={}, center]{fc6f49e6-218d-48e2-a799-b90a6c6181d7-06_370_1086_397_483}

The value of $\int _ { - 1 } ^ { 1 } f \left( x ^ { 2 } - 1 \right) \mathrm { d } x$ equals\\
(a) $\frac { 1 } { 4 }$,\\
(b) $\frac { 1 } { 3 }$,\\
(c) $\frac { 3 } { 5 }$,\\
(d) $\frac { 2 } { 3 }$.\\
H. The number of positive values $x$ which satisfy the equation

$$x = 8 ^ { \log _ { 2 } x } - 9 ^ { \log _ { 3 } x } - 4 ^ { \log _ { 2 } x } + \log _ { 0.5 } 0.25$$

is\\
(a) 0 ,\\
(b) 1,\\
(c) 2 ,\\
(d) 3 .\\
I. In the range $0 \leqslant x < 2 \pi$ the equation

$$\sin ^ { 8 } x + \cos ^ { 6 } x = 1$$

has\\
(a) 3 solutions ,\\
(b) 4 solutions,\\
(c) 6 solutions,\\
(d) 8 solutions.\\
J. The function $f ( n )$ is defined for positive integers $n$ according to the rules\\
$f ( 1 ) = 1$,\\
$f ( 2 n ) = f ( n )$,\\
$f ( 2 n + 1 ) = ( f ( n ) ) ^ { 2 } - 2$.

The value of $f ( 1 ) + f ( 2 ) + f ( 3 ) + \cdots + f ( 100 )$ is\\
(a) -86,\\
(b) -31,\\
(c) 23 ,\\
(d) 58.