109. The graph of $y = \cos(\tan^{-1} x)$ and the line $y = mx$, for which set of values of $m$, share exactly one point in common?
(4) $(0, +\infty)$ (3) $(-\infty, 0)$ (2) $(-\infty, +\infty)$ (1) $(-\infty, +\infty) - \{0\}$

Find the number of intersection points of $y = \log x$ and $y = x^2$.

Let $f(x) = x^4 + x^2 + x - 1$. Which of the following is true?
(A) $f$ has exactly two real roots
(B) $f$ has no real roots
(C) $f$ has four real roots
(D) $f$ has exactly two real roots, one of which is $-1$

How many real roots does $x ^ { 4 } + 12 x - 5$ have?

How many real roots does $x ^ { 4 } + 12 x - 5$ have?

The number of real solutions of the equation $x ^ { 2 } = e ^ { x }$ is:
(A) 0
(B) 1
(C) 2
(D) 3 .

The number of distinct real roots of the equation $x \sin ( x ) + \cos ( x ) = x ^ { 2 }$ is
(A) 0
(B) 2
(C) 24
(D) none of the above.

The number of real solutions of $e ^ { x } = \sin ( x )$ is
(A) 0
(B) 1
(C) 2
(D) infinite.

The number of real roots of the polynomial $$p ( x ) = \left( x ^ { 2020 } + 2020 x ^ { 2 } + 2020 \right) \left( x ^ { 3 } - 2020 \right) \left( x ^ { 2 } - 2020 \right)$$ is
(A) 2
(B) 3
(C) 2023
(D) 2025 .

The number of positive solutions to the equation $$e^x \sin x = \log x + e^{\sqrt{x}} + 2$$ is
(A) 0
(B) 1
(C) 2
(D) $\infty$

12. The number of values of $x$ where the function $f ( x ) = \cos x + \cos ( \sqrt { } 2 x )$ attains its maximum is :
(A) 0
(B) 1
(C) 2
(D) infinite

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The real number $s$ lies in the interval
A) $\left( - \frac { 1 } { 4 } , 0 \right)$
B) $\left( - 11 , - \frac { 3 } { 4 } \right)$
C) $\left( - \frac { 3 } { 4 } , - \frac { 1 } { 2 } \right)$
D) $\left( 0 , \frac { 1 } { 4 } \right)$

The number of points in $( - \infty , \infty )$, for which $x ^ { 2 } - x \sin x - \cos x = 0$, is
(A) 6
(B) 4
(C) 2
(D) 0

For every pair of continuous functions $f, g : [0,1] \rightarrow \mathbb{R}$ such that $$\max\{f(x) : x \in [0,1]\} = \max\{g(x) : x \in [0,1]\}$$ the correct statement(s) is(are):
(A) $(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(B) $(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(C) $(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$ for some $c \in [0,1]$
(D) $(f(c))^2 = (g(c))^2$ for some $c \in [0,1]$

Let $f : [0, 4\pi] \rightarrow [0, \pi]$ be defined by $f(x) = \cos^{-1}(\cos x)$. The number of points $x \in [0, 4\pi]$ satisfying the equation $$f(x) = \frac{10 - x}{10}$$ is

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by
$$f ( x ) = \left\{ \begin{array} { c l } x ^ { 2 } \sin \left( \frac { \pi } { x ^ { 2 } } \right) , & \text { if } x \neq 0 \\ 0 , & \text { if } x = 0 \end{array} \right.$$
Then which of the following statements is TRUE?
(A) $f ( x ) = 0$ has infinitely many solutions in the interval $\left[ \frac { 1 } { 10 ^ { 10 } } , \infty \right)$.
(B) $f ( x ) = 0$ has no solutions in the interval $\left[ \frac { 1 } { \pi } , \infty \right)$.
(C) The set of solutions of $f ( x ) = 0$ in the interval $\left( 0 , \frac { 1 } { 10 ^ { 10 } } \right)$ is finite.
(D) $f ( x ) = 0$ has more than 25 solutions in the interval $\left( \frac { 1 } { \pi ^ { 2 } } , \frac { 1 } { \pi } \right)$.

The number of distinct real roots of the equation $x^4 - 4x^3 + 12x^2 + x - 1 = 0$ is: (1) 2 (2) 3 (3) 0 (4) 4

The number of distinct real roots of the equation $x ^ { 7 } - 7 x - 2 = 0$ is
(1) 5
(2) 7
(3) 1
(4) 3

Consider the function $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathrm { R }$ defined by $f ( x ) = 4 \sqrt { 2 } x ^ { 3 } - 3 \sqrt { 2 } x - 1$. Consider the statements (I) The curve $y = f ( x )$ intersects the $x$-axis exactly at one point (II) The curve $y = f ( x )$ intersects the $x$-axis at $x = \cos \frac { \pi } { 12 }$ Then
(1) Only (II) is correct
(2) Both (I) and (II) are incorrect
(3) Only (I) is correct
(4) Both (I) and (II) are correct

If the set of all values of a, for which the equation $5 x ^ { 3 } - 15 x - a = 0$ has three distinct real roots, is the interval $( \alpha , \beta )$, then $\beta - 2 \alpha$ is equal to $\_\_\_\_$

Consider a quadratic function in $x$
$$y = ax^2 + bx + c$$
such that the graph of function (1) passes through the two points $(-1, -1)$ and $(2, 2)$.
(1) When we express $b$ and $c$ in terms of $a$, we have
$$b = \mathbf{A} - a, \quad c = \mathbf{BC}a.$$
(2) Suppose that one of the points of intersection of the graph of function (1) and the $x$-axis is within the interval $0 < x \leqq 1$. Then the range of values of $a$ is [see figure].
(3) When the value of $a$ varies within interval (2), the range of values of $a + bc$ is
$$\frac{\mathbf{GH}}{\square\mathbf{I}} \leqq a + bc \leqq \square.$$

Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Total \hline & & & & & & & \hline \end{tabular}

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.

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

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
[Figure]
(a)
[Figure]
(b)
[Figure]
(c)
[Figure]
(d)
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 $0 \leqslant x \leqslant 1$. [Figure]
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 $b > 1$.

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.

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. $\_\_\_\_$

FOR OFFICE USE ONLY:

\begin{table}[h]
Signature of Invigilator:

Q1	Q2	Q3	Q4	Q5	Q6	Q7

\end{table}

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.

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

A. A sketch of the graph $y = x ^ { 3 } - x ^ { 2 } - x + 1$ appears on which of the following axes?
[Figure]
(a)
[Figure]
(b)
[Figure]
(c)
[Figure]
(d)
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. [Figure]
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: [Figure]
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.

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 $( \sqrt { } )$ 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. A square has centre ( 3,4 ) and one corner at ( 1,5 ). Another corner is at
(a) $( 1,3 )$,
(b) $( 5,5 )$,
(c) $( 4,2 )$,
(d) $( 2,2 )$,
(e) $( 5,2 )$.
B. What is the value of $\int _ { 0 } ^ { 1 } \left( e ^ { x } - x \right) \left( e ^ { x } + x \right) \mathrm { d } x$ ?
(a) $\frac { 3 e ^ { 2 } - 2 } { 6 }$,
(b) $\frac { 3 e ^ { 2 } + 2 } { 6 }$,
(c) $\frac { 2 e ^ { 2 } - 3 } { 6 }$,
(d) $\frac { 3 e ^ { 2 } - 5 } { 6 }$,
(e) $\frac { e ^ { 2 } + 3 } { 6 }$.
C. The sum
$$1 - 4 + 9 - 16 + \cdots + 99 ^ { 2 } - 100 ^ { 2 }$$
equals
(a) - 101
(b) - 1000
(c) -1111
(d) - 4545
(e) $\quad - 5050$.
D. The largest value achieved by $3 \cos ^ { 2 } x + 2 \sin x + 1$ equals
(a) $\frac { 11 } { 5 }$,
(b) $\frac { 13 } { 3 }$,
(c) $\frac { 12 } { 5 }$,
(d) $\frac { 14 } { 9 }$,
(e) $\frac { 12 } { 7 }$.
E. A line is tangent to the parabola $y = x ^ { 2 }$ at the point $\left( a , a ^ { 2 } \right)$ where $a > 0$. The area of the region bounded by the parabola, the tangent line, and the $x$-axis equals
(a) $\frac { a ^ { 2 } } { 3 }$,
(b) $\frac { 2 a ^ { 2 } } { 3 }$,
(c) $\frac { a ^ { 3 } } { 12 }$,
(d) $\frac { 5 a ^ { 3 } } { 6 }$,
(e) $\frac { a ^ { 4 } } { 10 }$. F. Which of the following expressions is equal to $\log _ { 10 } ( 10 \times 9 \times 8 \times \cdots \times 2 \times 1 )$ ?
(a) $1 + 5 \log _ { 10 } 2 + 4 \log _ { 10 } 6$,
(b) $1 + 4 \log _ { 10 } 2 + 2 \log _ { 10 } 6 + \log _ { 10 } 7$,
(c) $2 + 2 \log _ { 10 } 2 + 4 \log _ { 10 } 6 + \log _ { 10 } 7$,
(d) $2 + 6 \log _ { 10 } 2 + 4 \log _ { 10 } 6 + \log _ { 10 } 7$,
(e) $2 + 6 \log _ { 10 } 2 + 4 \log _ { 10 } 6$. G. A cubic has equation $y = x ^ { 3 } + a x ^ { 2 } + b x + c$ and has turning points at $( 1,2 )$ and $( 3 , d )$ for some $d$. What is the value of $d$ ?
(a) - 4 ,
(b) - 2 ,
(c) 0 ,
(d) 2 ,
(e) 4 . H. The following five graphs are, in some order, plots of $y = f ( x ) , y = g ( x ) , y = h ( x )$, $y = \frac { \mathrm { d } f } { \mathrm {~d} x }$ and $y = \frac { \mathrm { d } g } { \mathrm {~d} x }$; that is, three unknown functions and the derivatives of the first two of those functions. Which graph is a plot of $h ( x )$ ? [Figure]
(b) [Figure]
(c) [Figure]
(d) [Figure]
(e) [Figure]
I. In the range $- 90 ^ { \circ } < x < 90 ^ { \circ }$, how many values of $x$ are there for which the sum to infinity
$$\frac { 1 } { \tan x } + \frac { 1 } { \tan ^ { 2 } x } + \frac { 1 } { \tan ^ { 3 } x } + \ldots$$
equals $\tan x$ ?
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4 . J. Consider a square with side length 2 and centre ( 0,0 ), and a circle with radius $r$ and centre $( 0,0 )$. Let $A ( r )$ be the area of the region that is inside the circle but outside the square, and let $B ( r )$ be the area of the region that is inside the square but outside the circle. Which of the following is a sketch of $A ( r ) + B ( r )$ ? [Figure] [Figure] [Figure] [Figure] [Figure]

A strictly increasing linear function $f ( x )$ has the property that if $y > x$ then $f ( y ) > f ( x )$. A student claims that if $f ( x )$ and $g ( x )$ are both strictly increasing linear functions, then so is $f ( x ) \cdot g ( x )$. Is the student correct? If so, prove the student's claim. Otherwise, find a counterexample.