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.

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\section*{MATHEMATICS ADMISSIONS TEST }
\section*{Thursday 2 November 2017}
\section*{Time Allowed: $\mathbf { 2 } \frac { \mathbf { 1 } } { \mathbf { 2 } }$ hours}
Please complete these details below in block capitals.

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Centre Number &  &  &  &  &  \\
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Candidate Number & $\mathbf { M }$ &  &  &  &  &  \\
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UCAS Number (if known) &  &  &  \\
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Date of Birth &  &  \\
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Please tick the appropriate box:\\
□ I have attempted Questions $\mathbf { 1 , 2 , 3 , 4 , 5 }$\\
□ I have attempted Questions 1,2,3,5,6\\
□ I have attempted Questions 1,2,5,6,7

FOR OFFICE USE ONLY

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Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 \\
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\section*{1. For ALL APPLICANTS.}
For each part of the question on pages 3-7 you will be given five possible answers, just one of which is correct. Indicate for each part $\mathbf { A } - \mathbf { J }$ which answer (a), (b), (c), (d), or (e) 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. Let

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

For what values of the real number $k$ does the graph $y = f ( x )$ have two distinct real stationary points?\\
(a) $- 2 \sqrt { 3 } < k < 2 \sqrt { 3 }$\\
(b) $k < - 2 \sqrt { 3 }$ or $2 \sqrt { 3 } < k$\\
(c) $k < - \sqrt { 21 } - 3$ or $\sqrt { 21 } - 3 < k$\\
(d) $- \sqrt { 21 } - 3 < k < \sqrt { 21 } - 3$\\
(e) all values of $k$.\\
B. The minimum value achieved by the function

$$f ( x ) = 9 \cos ^ { 4 } x - 12 \cos ^ { 2 } x + 7$$

equals\\
(a) 3\\
(b) 4\\
(c) 5\\
(d) 6\\
(e) 7 .\\
C. A sequence $\left( a _ { n } \right)$ has the property that

$$a _ { n + 1 } = \frac { a _ { n } } { a _ { n - 1 } }$$

for every $n \geqslant 2$. Given that $a _ { 1 } = 2$ and $a _ { 2 } = 6$, what is $a _ { 2017 }$ ?\\
(a) $\frac { 1 } { 6 }$\\
(b) $\frac { 1 } { 3 }$\\
(c) $\frac { 1 } { 2 }$\\
(d) 2\\
(e) 3 .\\
D. The diagram below shows the graph of $y = f ( x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{4f9b4c7b-cd2d-4862-9649-23a1574ede43-06_382_447_1327_797}

The graph of the function $y = - f ( - x )$ is drawn in which of the following diagrams?\\
(a)\\
\includegraphics[max width=\textwidth, alt={}, center]{4f9b4c7b-cd2d-4862-9649-23a1574ede43-06_381_450_1783_335}\\
(b)\\
\includegraphics[max width=\textwidth, alt={}, center]{4f9b4c7b-cd2d-4862-9649-23a1574ede43-06_381_450_1783_949}\\
(c)\\
\includegraphics[max width=\textwidth, alt={}, center]{4f9b4c7b-cd2d-4862-9649-23a1574ede43-06_379_445_1783_1564}\\
(d)\\
\includegraphics[max width=\textwidth, alt={}, center]{4f9b4c7b-cd2d-4862-9649-23a1574ede43-06_380_446_2169_337}\\
(e)\\
\includegraphics[max width=\textwidth, alt={}, center]{4f9b4c7b-cd2d-4862-9649-23a1574ede43-06_379_446_2172_949}\\
E. Let $a$ and $b$ be positive integers such that $a + b = 20$. What is the maximum value that $a ^ { 2 } b$ can take?\\
(a) 1000\\
(b) 1152\\
(c) 1176\\
(d) 1183\\
(e) 1196 .\\
F. The picture below shows the unit circle, where each point has coordinates $( \cos x , \sin x )$ for some $x$. Which of the marked arcs corresponds to

$$\tan x < \cos x < \sin x ?$$

\includegraphics[max width=\textwidth, alt={}, center]{4f9b4c7b-cd2d-4862-9649-23a1574ede43-07_962_1015_1299_509}\\
(a) $A$\\
(b) $B$\\
(c) $C$\\
(d) $D$\\
(e) $E$.\\
G. For all $\theta$ in the range $0 \leqslant \theta < 2 \pi$ the line

$$( y - 1 ) \cos \theta = ( x + 1 ) \sin \theta$$

divides the disc $x ^ { 2 } + y ^ { 2 } \leqslant 4$ into two regions. Let $A ( \theta )$ denote the area of the larger region.

Then $A ( \theta )$ achieves its maximum value at\\
(a) one value of $\theta$\\
(b) two values of $\theta$\\
(c) three values of $\theta$\\
(d) four values of $\theta$\\
(e) all values of $\theta$.\\
H. In this question $a$ and $b$ are real numbers, and $a$ is non-zero.

When the polynomial $x ^ { 2 } - 2 a x + a ^ { 4 }$ is divided by $x + b$ the remainder is 1 .\\
The polynomial $b x ^ { 2 } + x + 1$ has $a x - 1$ as a factor.\\
It follows that $b$ equals\\
(a) 1 only\\
(b) 0 or - 2\\
(c) 1 or 2\\
(d) 1 or 3\\
(e) - 1 or 2 .\\
I. Let $a , b , c > 0$ and $a \neq 1$. The equation

$$\log _ { b } \left( \left( b ^ { x } \right) ^ { x } \right) + \log _ { a } \left( \frac { c ^ { x } } { b ^ { x } } \right) + \log _ { a } \left( \frac { 1 } { b } \right) \log _ { a } ( c ) = 0$$

has a repeated root when\\
(a) $b ^ { 2 } = 4 a c$\\
(b) $b = \frac { 1 } { a }$\\
(c) $c = \frac { b } { a }$\\
(d) $\quad c = \frac { 1 } { b }$\\
(e) $a = b = c$.\\
J. Which of these integrals has the largest value? You are not expected to calculate the exact value of any of these.\\
(a) $\quad \int _ { 0 } ^ { 2 } \left( x ^ { 2 } - 4 \right) \sin ^ { 8 } ( \pi x ) \mathrm { d } x$\\
(b) $\quad \int _ { 0 } ^ { 2 \pi } ( 2 + \cos x ) ^ { 3 } \mathrm {~d} x$\\
(c) $\quad \int _ { 0 } ^ { \pi } \sin ^ { 100 } x \mathrm {~d} x$\\
(d) $\quad \int _ { 0 } ^ { \pi } ( 3 - \sin x ) ^ { 6 } \mathrm {~d} x$\\
(e) $\quad \int _ { 0 } ^ { 8 \pi } 108 \left( \sin ^ { 3 } x - 1 \right) d x$.