When $0 \leq \theta < 2 \pi$, the range of all values of $\theta$ such that the quadratic equation in $x$ $$6 x ^ { 2 } + ( 4 \cos \theta ) x + \sin \theta = 0$$ has no real roots is $\alpha < \theta < \beta$. What is the value of $3 \alpha + \beta$? [3 points]
(1) $\frac { 5 } { 6 } \pi$
(2) $\pi$
(3) $\frac { 7 } { 6 } \pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$

The set of values of $m$ for which $mx^2 - 6mx + 5m + 1 > 0$ for all real $x$ is
(A) $m < \frac{1}{4}$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac{1}{4}$
(D) $0 \leq m < \frac{1}{4}$

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(a) $m < \frac { 1 } { 4 }$
(b) $m \geq 0$
(c) $0 \leq m \leq \frac { 1 } { 4 }$
(d) $0 \leq m < \frac { 1 } { 4 }$.

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(a) $m < \frac { 1 } { 4 }$
(b) $m \geq 0$
(c) $0 \leq m \leq \frac { 1 } { 4 }$
(d) $0 \leq m < \frac { 1 } { 4 }$.

The set of values of $m$ for which $mx^2 - 6mx + 5m + 1 > 0$ for all real $x$ is
(A) $m < \frac{1}{4}$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac{1}{4}$
(D) $0 \leq m < \frac{1}{4}$

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(A) $m < \frac { 1 } { 4 }$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac { 1 } { 4 }$
(D) $0 \leq m < \frac { 1 } { 4 }$

Let $\mathbb { R } ^ { 2 }$ denote $\mathbb { R } \times \mathbb { R }$. Let
$$S = \left\{ ( a , b , c ) : a , b , c \in \mathbb { R } \text { and } a x ^ { 2 } + 2 b x y + c y ^ { 2 } > 0 \text { for all } ( x , y ) \in \mathbb { R } ^ { 2 } - \{ ( 0,0 ) \} \right\}$$
Then which of the following statements is (are) TRUE?
(A) $\left( 2 , \frac { 7 } { 2 } , 6 \right) \in S$
(B) If $\left( 3 , b , \frac { 1 } { 12 } \right) \in S$, then $| 2 b | < 1$.
(C) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & a x + b y = 1 \\ & b x + c y = - 1 \end{aligned}$$
has a unique solution.
(D) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & ( a + 1 ) x + b y = 0 \\ & b x + ( c + 1 ) y = 0 \end{aligned}$$
has a unique solution.

The number of integral values of $m$ for which the quadratic expression $( 1 + 2 m ) x ^ { 2 } - 2 ( 1 + 3 m ) x + 4 ( 1 + m ) , x \in R$ is always positive, is
(1) 7
(2) 3
(3) 6
(4) 8

The number of integral values of $m$ for which the equation, $1 + m ^ { 2 } x ^ { 2 } - 21 + 3 m x + 1 + 8 m = 0$ has no real root, is
(1) 2
(2) 3
(3) Infinitely many
(4) 1

The integer $k$, for which the inequality $x ^ { 2 } - 2 ( 3 k - 1 ) x + 8 k ^ { 2 } - 7 > 0$ is valid for every $x$ in $R$ is:
(1) 4
(2) 2
(3) 3
(4) 0

If the set of all $\mathrm { a } \in \mathbf { R }$, for which the equation $2 x ^ { 2 } + ( a - 5 ) x + 15 = 3 \mathrm { a }$ has no real root, is the interval $( \alpha , \beta )$, and $X = \{ x \in Z : \alpha < x < \beta \}$, then $\sum _ { x \in X } x ^ { 2 }$ is equal to :
(1) 2109
(2) 2129
(3) 2119
(4) 2139

Consider the two functions
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + 4 a - 3 \\ & g ( x ) = 2 x + 1 \end{aligned}$$
We are to find the condition on $a$ for which $f ( x ) \geqq g ( x )$ for all $x$ and also find the range of values of the minimum of $f ( x )$ under this condition.
We must find the condition under which
$$x ^ { 2 } + \mathbf { A } ( a - \mathbf { A } ) x + \mathbf { A C } a - \mathbf { A D } \geq 0$$
for all $x$.
For each of $\mathbf { E } \sim \mathbf{ H }$ in the following questions, choose the correct answer from among (0) $\sim$ (7) below each question.
(1) The required condition is that $a$ satisfy the quadratic inequality $\mathbf{E}$. Hence $a$ is in the range $\mathbf { F }$. (0) $a ^ { 2 } - 5 a + 4 \geqq 0$
(1) $a ^ { 2 } - 6 a + 5 \geqq 0$
(2) $a ^ { 2 } - 5 a + 4 \leqq 0$
(3) $a ^ { 2 } - 6 a + 5 \leqq 0$
(4) $a \leqq 1$ or $5 \leqq a$
(5) $1 \leqq a \leqq 5$ (6) $1 \leqq a \leqq 4$ (7) $a \leqq 1$ or $4 \leqq a$
(2) Let $m$ be the minimum value of $f ( x )$. Then, since $m = \mathbf { G }$, the range of values which $m$ can take under the condition in (1) is $\mathbf { H }$. (0) $a ^ { 2 } + 4 a - 3$
(1) $4 a ^ { 2 } + 4 a - 3$
(2) $- a ^ { 2 } + 4 a - 3$
(3) $2 a ^ { 2 } - 4 a + 3$
(4) $- 5 \leqq m \leqq 1$
(5) $- 8 \leqq m \leqq 1$ (6) $- 8 \leqq m \leqq - 1$ (7) $- 5 \leqq m \leqq - 1$

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.
This page will be detached and not marked

MATHEMATICS ADMISSIONS TEST

Thursday 2 November 2017

Time Allowed: $\mathbf { 2 } \frac { \mathbf { 1 } } { \mathbf { 2 } }$ hours

Please complete these details below in block capitals.

Centre Number

Candidate Number

$\mathbf { M }$

UCAS Number (if known)

$-$$-$

\multicolumn{3}{c}{$d$}
Date of Birth

$-$$-$
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

Q1	Q2	Q3	Q4	Q5	Q6	Q7

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.

	(a)	(b)	(c)	(d)	(e)
A
B
C
D
E
F
G
H
I
J

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 $y = f ( x )$. [Figure]
The graph of the function $y = - f ( - x )$ is drawn in which of the following diagrams?
(a) [Figure]
(b) [Figure]
(c) [Figure]
(d) [Figure]
(e) [Figure]
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
$$\tan x < \cos x < \sin x ?$$
[Figure]
(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) $\quad \int _ { 0 } ^ { 8 \pi } 108 \left( \sin ^ { 3 } x - 1 \right) d x$.

17. For what values of the non-zero real number $a$ does the quadratic equation $a x ^ { 2 } + ( a - 2 ) x = 2$ have real distinct roots?
A All values of $a$
B $\quad a = - 2$
C $\quad a > - 2$
D $\quad a \neq - 2$
E No values of $a$

Find the complete set of values of the real constant $k$ for which the expression
$$x^2 + kx + 2x + 1 - 2k$$
is positive for all real values of $x$.

The graph of the quadratic
$$y = p x ^ { 2 } + q x + p$$
where $p > 0$, intersects the $x$-axis at two distinct points.
In which one of the following graphs does the shaded region show the complete set of possible values that $p$ and $q$ could take?