Determine the number of stationary points on the curve with equation
$$y = 3 x ^ { 4 } + 4 x ^ { 3 } + 6 x ^ { 2 } - 5$$
A 0
B 1
C 2
D 3
E 4

Find the coefficient of the $x ^ { 5 }$ term in the expansion of
$$( 1 + x ) ^ { 5 } \times \sum _ { i = 0 } ^ { 5 } x ^ { i }$$
A 1
B 5
C 16
D 25
E 32

Consider the following statement about the positive integer $n$ if $n$ is prime, then $n ^ { 2 } + 2$ is not prime
Which of the following is a counterexample to this statement? I $n = 2$ II $n = 3$ III $n = 4$
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

The point $P$ has coordinates $( p , q )$, and the equation of a circle is
$$x ^ { 2 } + 2 f x + y ^ { 2 } + 2 g y + h = 0$$
where $f , g , h , p$ and $q$ are all real constants. Let $L$ be the distance between the centre of the circle and the point $P$. Which one of the following is sufficient on its own to be able to calculate $L$ ?
A the values of $f , g$ and $h$
B the values of $f , g , p$ and $q$
C the values of $f , h , p$ and $q$
D the values of $g , h , p$ and $q$
E none of the options A-D is sufficient on its own

A straight line $L$ passes through $( 1,2 )$. Let P be the statement if the $y$-intercept of $L$ is negative, then the $x$-intercept of $L$ is positive. Which of the following statements must be true? I P II the converse of P III the contrapositive of P
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

A list consists of $n$ integers. Consider the following statements: P: $\quad n$ is odd. Q: The median of the list is one of the numbers in the list. Which one of the following is true?
A P is necessary and sufficient for Q.
B P is necessary but not sufficient for Q.
C P is sufficient but not necessary for Q.
D P is not necessary and not sufficient for Q.

Consider the following claim: The difference between two consecutive positive cube numbers is always prime. Here is an attempted proof of this claim:
$$\text { I } \quad ( x + 1 ) ^ { 3 } = x ^ { 3 } + 3 x ^ { 2 } + 3 x + 1$$
II Taking $x$ to be a positive integer, the difference between two consecutive cube numbers can be expressed as $( x + 1 ) ^ { 3 } - x ^ { 3 } = 3 x ^ { 2 } + 3 x + 1$
III It is impossible to factorise $3 x ^ { 2 } + 3 x + 1$ into two linear factors with integer coefficients because its discriminant is negative.
IV Therefore for every positive integer value of $x$ the integer $3 x ^ { 2 } + 3 x + 1$ cannot be factorised.
V Hence, the difference between two consecutive cube numbers will always be prime. Which of the following best describes this proof?
A The proof is completely correct, and the claim is true.
B The proof is completely correct, but there are counterexamples to the claim.
C The proof is wrong, and the first error occurs on line I.
D The proof is wrong, and the first error occurs on line II.
E The proof is wrong, and the first error occurs on line III. F The proof is wrong, and the first error occurs on line IV. G The proof is wrong, and the first error occurs on line V.

A selection, $S$, of $n$ terms is taken from the arithmetic sequence $1,4,7,10 , \ldots , 70$. Consider the following statement: (*) There are two distinct terms in $S$ whose sum is 74 .
What is the smallest value of $n$ for which (*) is necessarily true?
A 12
B 13
C 14
D 21
E 22 F 23

Consider the following statement: () For all real numbers $x$, if $x < k$ then $x ^ { 2 } < k$ What is the complete set of values of $k$ for which () is true?
A no real numbers
B $k > 0$
C $k < 1$
D $k \leq 1$
E $\quad 0 < k < 1$ F $0 < k \leq 1$ G all real numbers

Which of the following statements is/are true?
I For all real numbers $x$ and for all positive integers $n , x < n$ II For all real numbers $x$, there exists a positive integer $n$ such that $x < n$ III There exists a real number $x$ such that for all positive integers $n , x < n$
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

The diagram shows a kite $P Q R S$ whose diagonals meet at $O$.
$$\begin{aligned} & O P = x \\ & O Q = y \\ & O R = x \\ & O S = z \end{aligned}$$
Which of the following is necessary and sufficient for angle $S P Q$ to be a right angle?
A $x = y = z$
B $2 x = y + z$
C $\quad x ^ { 2 } = y z$
D $y = z$
E $y ^ { 2 } = x ^ { 2 } + z ^ { 2 }$

Place the following integrals in order of size, starting with the smallest.
$$\begin{aligned} & P = \int _ { 0 } ^ { 1 } 2 ^ { \sqrt { x } } \mathrm {~d} x \\ & Q = \int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x \\ & R = \int _ { 0 } ^ { 1 } ( \sqrt { 2 } ) ^ { x } \mathrm {~d} x \end{aligned}$$
A $P < Q < R$
B $P < R < Q$
C $Q < P < R$
D $Q < R < P$
E $\quad R < P < Q$ F $R < Q < P$

Consider the statement () about a real number $x$ : () There exists a real number $y$ such that $x - x y + y$ is negative.
For how many real values of $x$ is (*) true?
A no values of $x$
B exactly one value of $x$
C exactly two values of $x$
D all except exactly two values of $x$
E all except exactly one value of $x$ F all values of $x$

Consider the two inequalities:
$$\begin{aligned} & | x + 5 | < | x + 11 | \\ & | x + 11 | < | x + 1 | \end{aligned}$$
Which one of the following is correct?
A There is no real number for which both inequalities are true.
B There is exactly one real number for which both inequalities are true.
C The real numbers for which both inequalities are true form an interval of length 1 .
D The real numbers for which both inequalities are true form an interval of length 2 .
E The real numbers for which both inequalities are true form an interval of length 3 .
F The real numbers for which both inequalities are true form an interval of length 4 .
G The real numbers for which both inequalities are true form an interval of length 5 .

The real numbers $x , y$ and $z$ are all greater than 1 , and satisfy the equations
$$\log _ { x } y = z \quad \text { and } \quad \log _ { y } z = x$$
Which one of the following equations for $\log _ { z } x$ must be true?
A $\quad \log _ { z } x = y$
B $\quad \log _ { z } x = \frac { 1 } { y }$
C $\log _ { z } x = x y$
D $\log _ { z } x = \frac { 1 } { x y }$
E $\quad \log _ { z } x = x z$ F $\log _ { z } x = \frac { 1 } { x z }$ G $\log _ { z } x = y z$ H $\log _ { z } x = \frac { 1 } { y z }$

In this question, $a _ { 1 } , \ldots , a _ { 100 }$ and $b _ { 1 } , \ldots , b _ { 100 }$ and $c _ { 1 } , \ldots , c _ { 100 }$ are three sequences of integers such that
$$a _ { n } \leq b _ { n } + c _ { n }$$
for each $n$. Which of the following statements must be true? I (minimum of $\left. a _ { 1 } , \ldots , a _ { 100 } \right) \leq$ (minimum of $\left. b _ { 1 } , \ldots , b _ { 100 } \right) + \left( \right.$ minimum of $\left. c _ { 1 } , \ldots , c _ { 100 } \right)$ II (minimum of $\left. a _ { 1 } , \ldots , a _ { 100 } \right) \geq$ (minimum of $\left. b _ { 1 } , \ldots , b _ { 100 } \right) +$ (minimum of $c _ { 1 } , \ldots , c _ { 100 }$ ) III (maximum of $\left. a _ { 1 } , \ldots , a _ { 100 } \right) \leq$ (maximum of $\left. b _ { 1 } , \ldots , b _ { 100 } \right) +$ (maximum of $c _ { 1 } , \ldots , c _ { 100 }$ )
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

A student answered the following question: $a$ and $b$ are non-zero real numbers. Prove that the equation $x ^ { 3 } + a x ^ { 2 } + b = 0$ has three distinct real roots if $27 b \left( b + \frac { 4 a ^ { 3 } } { 27 } \right) < 0$
Here is the student's solution: I We differentiate $y = x ^ { 3 } + a x ^ { 2 } + b$ to get $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 2 a x = x ( 3 x + 2 a )$ Solving $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ shows that the stationary points are at $( 0 , b )$ and $\left( - \frac { 2 a } { 3 } , b + \frac { 4 a ^ { 3 } } { 27 } \right)$
II If $27 b \left( b + \frac { 4 a ^ { 3 } } { 27 } \right) < 0$, then $b$ and $b + \frac { 4 a ^ { 3 } } { 27 }$ must have opposite signs, and so one of the stationary points is above the $x$-axis and one is below.
III If the cubic has three distinct real roots, then one of the stationary points is above the $x$-axis and one is below.
IV Hence if $27 b \left( b + \frac { 4 a ^ { 3 } } { 27 } \right) < 0$, then the equation has three distinct real roots.
Which one of the following options best describes the student's solution?
A It is a completely correct solution.
B The student has instead proved the converse of the statement in the question.
C The solution is wrong, because the student should have stated step II after step III.
D The solution is wrong, because the student should have shown the converse of the result in step II.
E The solution is wrong, because the student should have shown the converse of the result in step III.

P, Q, R and S show the graphs of
$$y = ( \cos x ) ^ { \cos x } , y = ( \sin x ) ^ { \sin x } , y = ( \cos x ) ^ { \sin x } \text { and } y = ( \sin x ) ^ { \cos x }$$
for $0 < x < \frac { \pi } { 2 }$ in some order.
Which row in the following table correctly identifies the graphs?

	$y = ( \cos x ) ^ { \cos x }$	$y = ( \sin x ) ^ { \sin x }$	$y = ( \cos x ) ^ { \sin x }$	$y = ( \sin x ) ^ { \cos x }$
A	P	Q	R	S
B	P	Q	S	R
C	Q	P	R	S
D	Q	P	S	R
E	R	S	P	Q
F	R	S	Q	P
G	S	R	P	Q
H	S	R	Q	P

A polygon has $n$ vertices, where $n \geq 3$. It has the following properties:

• Every vertex of the polygon lies on the circumference of a circle $C$.
• The centre of the circle $C$ is inside the polygon.
• The radii from the centre of the circle $C$ to the vertices of the polygon cut the polygon into $n$ triangles of equal area.

For which values of $n$ are these properties sufficient to deduce that the polygon is regular?
A no values of $n$
B $n = 3$ only
C $n = 3$ and $n = 4$ only
D $\quad n = 3$ and $n \geq 5$ only
E all values of $n$

The functions $f _ { 1 }$ to $f _ { 5 }$ are defined on the real numbers by
$$\begin{aligned} & \mathrm { f } _ { 1 } ( x ) = \cos x \\ & \mathrm { f } _ { 2 } ( x ) = \sin ( \cos x ) \\ & \mathrm { f } _ { 3 } ( x ) = \cos ( \sin ( \cos x ) ) \\ & \mathrm { f } _ { 4 } ( x ) = \sin ( \cos ( \sin ( \cos x ) ) ) \\ & \mathrm { f } _ { 5 } ( x ) = \cos ( \sin ( \cos ( \sin ( \cos x ) ) ) ) \end{aligned}$$
where all numbers are taken to be in radians. These functions have maximum values $m _ { 1 } , m _ { 2 } , m _ { 3 } , m _ { 4 }$ and $m _ { 5 }$, respectively. Which one of the following statements is true?
A $m _ { 1 } , m _ { 2 } , m _ { 3 } , m _ { 4 }$ and $m _ { 5 }$ are all equal to 1
B $0 < m _ { 5 } < m _ { 4 } < m _ { 3 } < m _ { 2 } < m _ { 1 } = 1$
C $\quad m _ { 1 } = m _ { 3 } = m _ { 5 } = 1$ and $0 < m _ { 2 } = m _ { 4 } < 1$
D $m _ { 1 } = m _ { 3 } = m _ { 5 } = 1$ and $0 < m _ { 4 } < m _ { 2 } < 1$
E $m _ { 1 } = m _ { 3 } = 1$ and $0 < m _ { 2 } = m _ { 4 } < 1$ and $0 < m _ { 5 } < 1$ F $m _ { 1 } = m _ { 3 } = 1$ and $0 < m _ { 4 } < m _ { 2 } < 1$ and $0 < m _ { 5 } < 1$