For each of A $\sim$ D in questions (1)$\sim$(4) below, choose the appropriate answer from among (0) $\sim$ (3) of each question. For $\mathbf { E } \sim \mathbf { G }$ in question (5), put the correct number.
Suppose that $a , b$ and $c$ are integers, and $a > 0$. Also, suppose that the graph of a quadratic function $y = a x ^ { 2 } - 2 b x + c$ intersects the $x$-axis and all points of intersection are in the interval $0 < x < 1$.
(1) The relationship between $a$ and $b$ is A. (0) $a > b$
(1) $a < b$
(2) $a = b$
(3) indeterminate
(2) The conditions on $b$ and $c$ are $\mathbf { B }$. (0) $b < 0 , c < 0$
(1) $b < 0 , c > 0$
(2) $b > 0 , c < 0$
(3) $b > 0 , c > 0$
(3) The relationship between $2 b$ and $a + c$ is $\mathbf { C }$. (0) $2 b > a + c$
(1) $2 b < a + c$
(2) $2 b = a + c$
(3) indeterminate
(4) The relationship between $b$ and $c$ is $\mathbf { D }$. (0) $b > c$
(1) $b < c$
(2) $b = c$
(3) indeterminate
(5) The smallest integer which $a$ can take is $\mathbf { E }$. In this case, the value of $b$ is $\mathbf { F }$, and the value of $c$ is $\mathbf { G }$.

Q1 The quadratic function $f ( x ) = 2 x ^ { 2 } + a x - 1$ in $x$ satisfies
$$f ( - 1 ) \geqq - 3 , \quad f ( 2 ) \geqq 3 .$$
Let us consider the minimum value $m$ of $f ( x )$.
(1) $\quad m$ can be expressed in terms of $a$ as
$$m = -\frac{\mathbf{A}}{\mathbf{B}} a^2 - \mathbf{C}$$
(2) The range of the values of $a$ such that $f ( x )$ satisfies condition (1) is
$$\mathbf { D E } \leqq a \leqq \mathbf { F } .$$
(3) The value of $m$ is maximized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { G }$, and then the value of $m$ is $\mathbf { H I }$.
(4) The value of $m$ is minimized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { J K }$, and then the value of $m$ is $\mathbf { L M }$.

Let us throw one dice three times, and let the number that comes up on the first throw be $a$, on the second throw be $b$, and on the third throw be $c$. Using these $a , b$ and $c$, we consider the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$.
(1) The probability that $b = 4$ and that the quadratic equation $f ( x ) = 0$ has two different real solutions is $\frac { \mathbf { N } } { \mathbf { O P Q } }$.
(2) Let us find the probability that $f ( 10 ) > 453$.
The number of the cases of $( a , b , c )$ such that $f ( 10 ) > 453$ is as follows: when $a = 4$ and $b = 5$, it is $\mathbf { R }$; when $a = 4$ and $b = 6$, it is $\mathbf{S}$; when $a = 5$, it is $\mathbf { T U }$; when $a = 6$, it is $\mathbf{VW}$. Hence, the probability that $f ( 10 ) > 453$ is $\frac { \mathbf { X } } { \mathbf { Y } }$.

Let us consider the three quadratic functions
$$f ( x ) = - x ^ { 2 } - 2 x + 1 , \quad g ( x ) = - x ^ { 2 } + 4 x , \quad h ( x ) = 2 x ^ { 2 } + a x + b$$
(1) When we denote the discriminant of the quadratic equation $h ( x ) - f ( x ) = 0$ by $D _ { 1 }$ and the discriminant of the quadratic equation $h ( x ) - g ( x ) = 0$ by $D _ { 2 }$, we have
$$D _ { 1 } = \mathbf { N } , \quad D _ { 2 } = \mathbf { O }$$
(for N and O, choose the correct answers from among choices (0) $\sim$ (5) below). (0) $a ^ { 2 } + 4 a - 3 b + 7$
(1) $a ^ { 2 } - 8 a - 12 b + 16$
(2) $a ^ { 2 } + 4 a - 12 b + 16$
(3) $a ^ { 2 } + 8 a + 12 b + 16$
(4) $a ^ { 2 } - 4 a + 12 b + 16$
(5) $a ^ { 2 } - 8 a - 3 b + 7$
(2) The values of $a$ and $b$ such that both of the two equations $f ( x ) = h ( x )$ and $g ( x ) = h ( x )$ have only one real solution are
$$a = \mathbf { P } , \quad b = \frac { \mathbf { Q } } { \mathbf{4} } .$$
In this case, the solution of $f ( x ) = h ( x )$ is $x = - \frac { \mathbf { S } } { \mathbf{T} }$ and the solution of $g ( x ) = h ( x )$ is $x = \frac { \mathbf { U } } { \mathbf{4} }$.
(3) Let $b = 3$. Then the range of the values of $a$ such that both $f ( x ) < h ( x )$ and $g ( x ) < h ( x )$ hold for any $x$ is $\square$ W (for $\square$ W, choose the correct answer from among choices (0) $\sim$ (5) below). (0) $- 2 - 2 \sqrt { 6 } < a < 10$
(1) $a < - 2 - 2 \sqrt { 6 } , 10 < a$
(2) $a < - 1 - \sqrt { 6 } , 10 < a$
(3) $- 2 < a < - 1 + \sqrt { 6 }$
(4) $- 2 < a < - 2 + 2 \sqrt { 6 }$
(5) $- 1 - \sqrt { 6 } < a < 10$

Let us throw one dice three times, and let the number that comes up on the first throw be $a$, on the second throw be $b$, and on the third throw be $c$. Using these $a , b$ and $c$, we consider the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$.
(1) The probability that $b = 4$ and that the quadratic equation $f ( x ) = 0$ has two different real solutions is $\frac { \mathbf { N } } { \mathbf { O } \mathbf { P Q } }$.
(2) Let us find the probability that $f ( 10 ) > 453$.
The number of the cases of $( a , b , c )$ such that $f ( 10 ) > 453$ is as follows: when $a = 4$ and $b = 5$, it is $\mathbf { R }$; when $a = 4$ and $b = 6$, it is $\mathbf{S}$; when $a = 5$, it is $\mathbf{TU}$; when $a = 6$, it is $\mathbf{VW}$. Hence, the probability that $f ( 10 ) > 453$ is $\frac { \mathbf { X } } { \mathbf { Y } }$.

Q2 Let $a$ be a real number. For the two quadratic functions in $x$
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + a ^ { 2 } - a , \\ & g ( x ) = 4 - x ^ { 2 } , \end{aligned}$$
answer the following questions.
(1) The range of the values of $a$ such that the equation $f ( x ) = g ( x )$ has two different solutions is
$$- \mathbf { K } < a < \mathbf { L } .$$
(2) In the case of (1), the parabolas $y = f ( x )$ and $y = g ( x )$ intersect at two points. We are to find the range of the values of $a$ such that both of the $y$ coordinates of these points of intersection are positive.
First, let $h ( x ) = f ( x ) - g ( x )$. Since the solutions of the equation $f ( x ) = g ( x )$ are the $x$ coordinates of the points of intersection of parabolas $y = f ( x )$ and $y = g ( x )$, the solutions of $h ( x ) = 0$ have to be between $- \mathbf { M }$ and $\mathbf { N }$. Accordingly, we have
$$\begin{aligned} & h \left( - \mathbf { M } \right) = a ^ { 2 } - \mathbf { O } a + \mathbf { P } > 0 , \quad \cdots \cdots \cdots (1)\\ & h ( \mathbf { N } ) = a ^ { 2 } + \mathbf { Q } a + \mathbf { R } > 0 . \quad \cdots \cdots \cdots (2) \end{aligned}$$
Also, from the position of the axis of the parabola $y = h ( x )$ we have that
$$- \mathbf { S } < a < \mathbf { T } . \quad \cdots \cdots \cdots (3)$$
Therefore, from (1), (2), (3) and (4) we obtain
$$- \mathbf { U } < a < \mathbf{V}.$$

2. (a) For what values of the constant $k$ does the quadratic equation
$$x ^ { 2 } - 2 x - 1 = k$$
have:
(i) no real solutions;
(ii) one real solution;
(iii) two real solutions.
(b) Showing your working, express $\left( x ^ { 2 } - 2 x - 1 \right) ^ { 2 }$ as a polynomial of degree 4 in $x$.
(c) Show that the quartic equation
$$x ^ { 4 } - 4 x ^ { 3 } + 2 x ^ { 2 } + 4 x + 1 = h$$
has exactly two real solutions if either $h = 0$ or $h > 4$. Show that there is no value of $h$ such that the above quartic equation has just one real solution.

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) $\quad \int _ { 0 } ^ { 8 \pi } 108 \left( \sin ^ { 3 } x - 1 \right) d x$.

Let $f ( x )$ be a quadratic polynomial function with real coefficients such that $f ( x ) = 0$ has no real roots. Select the correct options.
(1) $f ( 0 ) > 0$
(2) $f ( 1 ) f ( 2 ) > 0$
(3) If $f ( x ) - 1 = 0$ has real roots, then $f ( x ) - 2 = 0$ has real roots
(4) If $f ( x ) - 1 = 0$ has a double root, then $f ( x ) - \frac { 1 } { 2 } = 0$ has no real roots
(5) If $f ( x ) - 1 = 0$ has two distinct real roots, then $f ( x ) - \frac { 1 } { 2 } = 0$ has real roots

Consider the real coefficient polynomial $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + a x + b$. It is known that the equation $f ( x ) = 0$ has a complex root $1 + 2 i$ (where $i = \sqrt { - 1 }$). Select the correct options.
(1) $1 - 2i$ is also a root of $f ( x ) = 0$
(2) Both $a$ and $b$ are positive numbers
(3) $f ^ { \prime } ( 2.1 ) < 0$
(4) The function $y = f ( x )$ has a local minimum at $x = 1$
(5) The $x$-coordinates of all inflection points of the graph $y = f ( x )$ are greater than 0

Let $b$ and $c$ be real numbers. The quadratic equation $x ^ { 2 } + b x + c = 0$ has real roots, but the quadratic equation $x ^ { 2 } + ( b + 2 ) x + c = 0$ has no real roots. Select the correct options.
(1) $c < 0$
(2) $b < 0$
(3) $x ^ { 2 } + ( b + 1 ) x + c = 0$ has real roots
(4) $x ^ { 2 } + ( b + 2 ) x - c = 0$ has real roots
(5) $x ^ { 2 } + ( b - 2 ) x + c = 0$ has real roots

9. The roots of the equation $2 x ^ { 2 } - 11 x + c = 0$ differ by 2 . The value of $c$ is
A $\frac { 105 } { 8 }$
B $\frac { 113 } { 8 }$
C $\frac { 117 } { 8 }$
D $\frac { 119 } { 8 }$

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 $a$

Consider the quadratic $f ( x ) = x ^ { 2 } - 2 p x + q$ and the statement:
$\left( ^ { * } \right) f ( x ) = 0$ has two real roots whose difference is greater than 2 and less than 4.
Which one of the following statements is true if and only if (*) is true?

Which one of the following is a sufficient condition for the equation $x ^ { 3 } - 3 x ^ { 2 } + a = 0$, where $a$ is a constant, to have exactly one real root?
A $a > 0$
B $a \leqslant 0$
C $\quad a \geqslant 4$
D $a < 4$
$\mathbf { E } \quad | a | > 4$
$\mathbf { F } \quad | a | \leqslant 4$
G $\quad a = \frac { 9 } { 4 }$
$\mathbf { H } \quad | a | = \frac { 3 } { 2 }$

Consider the simultaneous equations
$$\begin{array} { r } 3 x ^ { 2 } + 2 x y = 4 \\ x + y = a \end{array}$$
where $a$ is a real constant.
Find the complete set of values of $a$ for which the equations have two distinct real solutions for $x$.
A There are no values of $a$.
B $- 2 < a < 2$
C $- 1 < a < 1$
D $a = 0$
E $a < - 1$ or $a > 1$
F $a < - 2$ or $a > 2$
G All real values of $a$

Find the complete set of values of the constant $c$ for which the cubic equation
$$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + c = 0$$
has three distinct real solutions.
A $- 20 < c < 7$
B $- 7 < c < 20$
C $c > 7$
D $c > - 7$
E $c < 20$
F $c < - 20$

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?

Find the complete set of values of $m$ in terms of $c$ such that the graphs of $y = mx + c$ and $y = \sqrt{x}$ have two points of intersection.
A $0 < m < \frac{1}{4c}$
B $0 < m < 4c^2$
C $m > \frac{1}{4c}$
D $m < \frac{1}{4c}$
E $m > 4c^2$
F $m < 4c^2$

For how many values of $a$ is the equation
$$(x - a)(x^2 - x + a) = 0$$
satisfied by exactly two distinct values of $x$?
A $0$
B $1$
C $2$
D $3$
E $4$
F more than 4

The line $y = 2 x + 3$ meets the curve $y = x ^ { 2 } + b x + c$ at exactly one point. The line $y = 4 x - 2$ also meets the curve $y = x ^ { 2 } + b x + c$ at exactly one point. What is the value of $b - c$ ?
A - 9 B - 5.5 C - 1 D 5 E 6 F 14

The graphs of $y = x ^ { 2 } + 5 x + 6$ and $y = m x - 3$, where $m$ is a constant, are plotted on the same set of axes.
Given that the graphs do not meet, what is the complete range of possible values of $m$ ?

The equation $x ^ { 4 } + b x ^ { 2 } + c = 0$ has four distinct real roots if and only if which of the following conditions is satisfied?
A $b ^ { 2 } > 4 c$ B $b ^ { 2 } < 4 c$ C $c > 0$ and $b > 2 \sqrt { c }$ D $c > 0$ and $b < - 2 \sqrt { c }$ E $\quad c < 0$ and $b < 0$ F $\quad c < 0$ and $b > 0$

$$\begin{aligned} & P ( x ) = x ^ { 2 } - 2 x + m \\ & Q ( x ) = x ^ { 2 } + 3 x + n \end{aligned}$$
polynomials are given. These two polynomials have a common root and the roots of the polynomial $P(x)$ are equal, so what is the sum $m + n$?
A) $-5$
B) $-3$
C) 2
D) 4
E) 5