$P Q R S$ is a quadrilateral, labelled anticlockwise.
Which one of the following is a necessary but not sufficient condition for $P Q R S$ to be a parallelogram?

A multiple-choice test question offered the following four options relating to a certain statement:
A The statement is true if and only if $x > 1$
B The statement is true if $x > 1$
C The statement is true if and only if $x > 2$
D The statement is true if $x > 2$
Given that exactly one of these options was correct, which one was it?

A student makes the following claim: For all integers $n$, the expression $4 \left( \frac { 9 n + 1 } { 2 } - \frac { 3 n - 1 } { 2 } \right)$ is divisible by 3 . Here is the student's argument:
$$\begin{aligned} 4 \left( \frac { 9 n + 1 } { 2 } - \frac { 3 n - 1 } { 2 } \right) & = 2 \left( 2 \left( \frac { 9 n + 1 } { 2 } - \frac { 3 n - 1 } { 2 } \right) \right) \\ & = 2 ( 9 n + 1 - 3 n - 1 ) \\ & = 2 ( 6 n ) \\ & = 12 n \\ & = 3 ( 4 n ) \end{aligned}$$
which is always a multiple of 3 .
So the expression $4 \left( \frac { 9 n + 1 } { 2 } - \frac { 3 n - 1 } { 2 } \right)$ is always divisible by 3 .
Which one of the following is true?
A The argument is correct.
B The argument is incorrect, and the first error occurs on line (I).
C The argument is incorrect, and the first error occurs on line (II).
D The argument is incorrect, and the first error occurs on line (III).
E The argument is incorrect, and the first error occurs on line (IV). F The argument is incorrect, and the first error occurs on line (V). G The argument is incorrect, and the first error occurs on line (VI).

Consider the following statement: Every positive integer $N$ that is greater than 6 can be written as the sum of two non-prime integers that are greater than 1 .
Which of the following is/are counterexample(s) to this statement? I $\quad N = 5$ II $\quad N = 7$ III $N = 9$
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

Consider the following conditions on a parallelogram $P Q R S$, labelled anticlockwise:
I length of $P Q =$ length of $Q R$ II The diagonal $P R$ intersects the diagonal $Q S$ at right angles III $\angle P Q R = \angle Q R S$ Which of these conditions is/are individually sufficient for the parallelogram $P Q R S$ to be a square?

A student is asked to prove whether the following statement (*) is true or false: (*) For all real numbers $a$ and $b , | a + b | < | a | + | b |$
The student's proof is as follows:
Statement (*) is false. A counterexample is $a = 3 , b = 4$, as $| 3 + 4 | = 7$ and $| 3 | + | 4 | = 7$, but $7 < 7$ is false.
Which of the following best describes the student's proof?
A The statement ( $*$ ) is true, and the student's proof is not correct.
B The statement (*) is false, but the student's proof is not correct: the counterexample is not valid.
C The statement (*) is false, but the student's proof is not correct: the student needs to give all the values of $a$ and $b$ where $| a + b | < | a | + | b |$ is false.
D The statement (*) is false, but the student's proof is not correct: the student should have instead stated that for all real numbers $a$ and $b , | a + b | \leq | a | + | b |$.
E The statement (*) is false, and the student's proof is fully correct.

Nine people are sitting in the squares of a 3 by 3 grid，one in each square，as shown． Two people are called neighbours if they are sitting in squares that share a side． （People in diagonally adjacent squares，which only have a point in common，are not called neighbours．）
Each of the nine people in the grid is either a truth－teller who always tells the truth， or a liar who always lies．
Every person in the grid says：＇My neighbours are all liars＇． Given only this information，what are the smallest number and the largest number of people who could be telling the truth？

$x$ is a real number and f is a function. Given that exactly one of the following statements is true, which one is it?
A $x \geq 0$ only if $\mathrm { f } ( x ) < 0$
B $x < 0$ if $\mathrm { f } ( x ) \geq 0$
C $\quad x \geq 0$ only if $\mathrm { f } ( x ) \geq 0$
D $\mathrm { f } ( x ) < 0$ if $x < 0$
E $\quad \mathrm { f } ( x ) \geq 0$ only if $x \geq 0$ F $\quad \mathrm { f } ( x ) \geq 0$ if and only if $x < 0$

Consider the following claim about positive integers $a , b$ and $c$ : if $a$ is a factor of $b c$, then $a$ is a factor of $b$ or $a$ is a factor of $c$
Which of the following provide(s) a counterexample to this claim? I $a = 5 , b = 10 , c = 20$ II $\quad a = 8 , b = 4 , c = 4$ III $a = 6 , b = 7 , c = 12$
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 function f is such that
$$\mathrm { f } ( m n ) = \begin{cases} \mathrm { f } ( m ) \mathrm { f } ( n ) & \text { if } m n \text { is a multiple of } 3 \\ m n & \text { if } m n \text { is not a multiple of } 3 \end{cases}$$
for all positive integers $m$ and $n$. Given that $\mathrm { f } ( 9 ) + \mathrm { f } ( 16 ) - \mathrm { f } ( 24 ) = 0$, what is the value of $\mathrm { f } ( 3 )$ ? A $\frac { 8 } { 3 }$ B $2 \sqrt { 2 }$ C 3 D $\frac { 16 } { 5 }$ E $3 \sqrt { 2 }$ F 4

On which line is the first error in the following argument?
A $\sin ^ { 2 } x + \cos ^ { 2 } x = 1$ for all values of $x$.
B Therefore $\cos x = \sqrt { 1 - \sin ^ { 2 } x }$ for all values of $x$.
C Hence $1 + \cos x = 1 + \sqrt { 1 - \sin ^ { 2 } x }$ for all values of $x$.
D Thus $( 1 + \cos x ) ^ { 2 } = \left( 1 + \sqrt { 1 - \sin ^ { 2 } x } \right) ^ { 2 }$ for all values of $x$.
E Substituting $x = \pi$ gives $0 = 4$.

Consider the following two statements about the polynomial $\mathrm { f } ( x )$ : $P : \quad \mathrm { f } ( x ) = 0$ for exactly three real values of $x$ $Q : \quad \mathrm { f } ^ { \prime } ( x ) = 0$ for exactly two real values of $x$ Which one of the following is correct?
A $P$ is necessary but not sufficient for $Q$.
B $P$ is sufficient but not necessary for $Q$.
C $P$ is necessary and sufficient for $Q$.
D $P$ is not necessary and not sufficient for $Q$.

Consider the following statement about the polynomial $\mathrm { p } ( x )$, where $a$ and $b$ are real numbers with $a < b$ : (*) There exists a number $c$ with $a < c < b$ such that $\mathrm { p } ^ { \prime } ( c ) = 0$.
Which one of the following is true?
A The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary and sufficient for ( $*$ )
B The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary but not sufficient for (*)
C The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is sufficient but not necessary for ( $*$ )
D The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is not necessary and not sufficient for ( $*$ )

Consider the following statements about a polynomial $\mathrm { f } ( x )$ : I $\mathrm { f } ( x ) = p x ^ { 3 } + q x ^ { 2 } + r x + s$, where $p \neq 0$. II There is a real number $t$ for which $\mathrm { f } ^ { \prime } ( t ) = 0$. III There are real numbers $u$ and $v$ for which $\mathrm { f } ( u ) \mathrm { f } ( v ) < 0$. Which of these statements is/are sufficient for the equation $\mathrm { f } ( x ) = 0$ to have a real solution?

	Statement I is sufficient	Statement II is sufficient	Statement III is sufficient
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

A student attempts to solve the following problem, where $a$ and $b$ are non-zero real numbers:
Show that if $a ^ { 2 } - 4 b ^ { 3 } \geq 0$ then there exist real numbers $x$ and $y$ such that $a = x y ( x + y )$ and $b = x y$.
Consider the following attempt:
$$\begin{aligned} & ( x - y ) ^ { 2 } \geq 0 \\ & \text { so } \quad x ^ { 2 } + y ^ { 2 } - 2 x y \geq 0 \\ & \text { so } \quad ( x + y ) ^ { 2 } - 4 x y \geq 0 \\ & \text { so } \quad x ^ { 2 } y ^ { 2 } ( x + y ) ^ { 2 } - 4 x ^ { 3 } y ^ { 3 } \geq 0 \\ & \text { so } \quad a ^ { 2 } - 4 b ^ { 3 } \geq 0 \end{aligned}$$
Which of the following best describes this attempt?
A It is completely correct.
B It is incorrect, but it would be correct if written in the reverse order.
C It is incorrect, but the student has correctly proved the converse.
D It is incorrect because there is an error in line (II). $\mathbf { E }$ It is incorrect because there is an error in line (III). F It is incorrect because there is an error in line (IV).

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

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

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.

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

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.

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 $n$

A student attempts to answer the following question. What is the largest number of consecutive odd integers that are all prime? The student's attempt is as follows: I There are two consecutive odd integers that are prime (for example: 17, 19). II Any three consecutive odd integers can be written in the form $n - 2 , n , n + 2$ for some $n$. III If $n$ is one more than a multiple of 3 , then $n + 2$ is a multiple of 3 . IV If $n$ is two more than a multiple of 3 , then $n - 2$ is a multiple of 3 . V The only other possibility is that $n$ is a multiple of 3 . VI In each case, one of the integers is a multiple of 3 , so not prime. VII Therefore the largest number of consecutive odd integers that are all prime is two.
Which of the following best describes this attempt?
A It is completely correct. B It is incorrect, and the first error is on line I. C It is incorrect, and the first error is on line II. D It is incorrect, and the first error is on line III. E It is incorrect, and the first error is on line IV. F It is incorrect, and the first error is on line V. G It is incorrect, and the first error is on line VI. H It is incorrect, and the first error is on line VII.

A student draws a triangle that is acute-angled or obtuse-angled but not right-angled. The student counts the number of straight lines that divide the triangle into two triangles, at least one of which is right-angled.
Which of the following statements is/are true? I The student can draw a triangle for which there is exactly 1 such straight line. II The student can draw a triangle for which there are exactly 2 such straight lines. III The student can draw a triangle for which there are exactly 3 such straight lines.
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

Consider the following statement about a pentagon P: (*) If at least one of the interior angles in P is $108 ^ { \circ }$, then all the interior angles in P form an arithmetic sequence.
Which of the following is/are true? I The statement (*) II The contrapositive of (*) III The converse of (*)
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