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TMUA Mock Paper 2 (Paper 2)

20 questions20 marks75Updated August 2026

The TMUA Mock Paper 2 (Paper 2) paper in full: all 20 questions, each with its answer. TMUA is the Test of Mathematics for University Admission. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.

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Question 1

1 mark

Exactly one of the following statements is true. Which one?

A.Every integer is a rational number.
B.If $a^{2}=b^{2}$ then $a=b$ .
C.The sum of any two irrational numbers is irrational.
D. $0!=0$ .
E.Every prime number is odd.

Answer: A

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Question 2

1 mark

Consider the statement: 'if a quadrilateral is a square then it has four equal sides.' Which of the following is its converse?

A.If a quadrilateral is a square then it has four equal sides.
B.If a quadrilateral has four equal sides then it is a square.
C.If a quadrilateral does not have four equal sides then it is not a square.
D.If a quadrilateral is not a square then it does not have four equal sides.
E.A quadrilateral is a square if and only if it has four equal sides.

Answer: B

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Question 3

1 mark

The statement '

\forall x\in\mathbb{R},\ \exists y\in\mathbb{R}

such that

xy=1

' is false. Which of the following is its negation?

A. $\forall x\in\mathbb{R},\ \exists y\in\mathbb{R}$ such that $xy\ne 1$
B. $\exists x\in\mathbb{R}$ such that $\forall y\in\mathbb{R},\ xy\ne 1$
C. $\exists x,y\in\mathbb{R}$ such that $xy\ne 1$
D. $\forall x,y\in\mathbb{R},\ xy\ne 1$
E. $\exists x\in\mathbb{R}$ such that $\exists y\in\mathbb{R},\ xy=1$

Answer: B

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Question 4

1 mark

The line

y=mx+c

(with

c\ne 0

) and the circle

x^{2}+y^{2}=a^{2}

(with

a>0

) are given. Consider statement P: 'the line is tangent to the circle', and statement Q: '

c^{2}=a^{2}(1+m^{2})

'. Then Q is:

A.necessary but not sufficient for P.
B.sufficient but not necessary for P.
C.both necessary and sufficient for P.
D.neither necessary nor sufficient for P.

Answer: C

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Question 5

1 mark

To prove that

\sqrt{2}

is irrational, a student writes

\sqrt{2}=\frac{p}{q}

in lowest terms and correctly deduces that

p

and

q

are both even. What makes this a valid proof?

A.It contradicts the assumption that $\frac{p}{q}$ is in lowest terms, so that assumption must be false.
B.It shows that $\sqrt{2}$ is an even number.
C.It proves that $p$ and $q$ do not exist.
D.Even numbers cannot be written as a fraction.
E.The argument is not actually valid.

Answer: A

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Question 6

1 mark

For a real number

x

, consider statement P: '

x^{2}<9

', and statement Q: '

x<3

'. Then P is:

A.necessary but not sufficient for Q.
B.sufficient but not necessary for Q.
C.both necessary and sufficient for Q.
D.neither necessary nor sufficient for Q.

Answer: B

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Question 7

1 mark

Consider the following argument. Line 1:

\frac{1}{4}>\frac{1}{8}

. Line 2:

\left(\frac{1}{2}\right)^{2}>\left(\frac{1}{2}\right)^{3}

. Line 3: Taking

\log_{10}

of both sides,

2\log_{10}\frac{1}{2}>3\log_{10}\frac{1}{2}

. Line 4: Dividing both sides by

\log_{10}\frac{1}{2}

,

2>3

. At which line does the first error occur?

A.Line 1
B.Line 2
C.Line 3
D.Line 4
E.there is no error

Answer: D

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Question 8

1 mark

Consider the claim: 'for every positive integer

n

, the value

n^{2}+n+1

is odd.' Which of the following is a counterexample?

A. $n=2$
B. $n=3$
C. $n=5$
D. $n=4$
E.the statement is always true, so there is no counterexample

Answer: E

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Question 9

1 mark

Which of the following statements are true? I: for every real

x>0

, there is a real

y

with

y^{2}=x

. II: there is a real

y

such that for every real

x>0

,

y^{2}=x

. III: for every real

x

, there is a real

y

with

y^{3}=x

.

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

Answer: F

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Question 10

1 mark

How many ordered pairs

(x,y)

of positive real numbers satisfy both

\log_{2}x+\log_{2}y=5

and

x+y=12

?

A. $2$
B. $0$
C. $1$
D. $3$
E. $4$

Answer: A

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Question 11

1 mark

For a positive integer

n

, the statement '

n^{2}\equiv 1\pmod 8

' is true if and only if:

A. $n$ is even
B. $n$ is prime
C. $4$ divides $n$
D. $n$ is odd
E. $n\equiv 1\pmod 8$

Answer: D

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Question 12

1 mark

A sequence is defined by

u_{1}=2

,

u_{2}=3

and

u_{n+2}=\dfrac{u_{n+1}}{u_{n}}

. Find

u_{2025}

.

A. $2$
B. $3$
C. $\frac{2}{3}$
D. $\frac{3}{2}$
E. $\frac{1}{2}$

Answer: D

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Question 13

1 mark

Three boxes each carry a label. Exactly one box contains a prize, and exactly one of the three labels is true. Box 1 reads 'the prize is in Box 1'; Box 2 reads 'the prize is not in Box 2'; Box 3 reads 'the prize is not in Box 1'. Which box contains the prize?

A.Box 1
B.Box 2
C.Box 3
D.it cannot be determined
E.no arrangement is consistent

Answer: B

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Question 14

1 mark

Integers

a

,

b

and

c

are such that

a+b

,

b+c

and

c+a

are all even. Which of the following must be true? I:

a

,

b

and

c

all have the same parity. II:

abc

is even. III:

a+b+c

is even.

A.I only
B.II only
C.III only
D.I and II
E.I and III
F.II and III
G.all of them
H.none of them

Answer: A

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Question 15

1 mark

What is the minimum value of

\dfrac{1}{2+\cos x}

over all real

x

?

A. $\frac{1}{3}$
B. $1$
C. $\frac{1}{2}$
D. $-1$
E. $\frac{1}{4}$

Answer: A

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Question 16

1 mark

In the expansion of

\left(x^{p}+x^{-q}\right)^{n}

, where

p

,

q

and

n

are positive integers, consider statement P: 'the expansion has a term independent of

x

', and statement Q: '

p+q

divides

np

'. Then Q is:

A.necessary but not sufficient for P.
B.sufficient but not necessary for P.
C.both necessary and sufficient for P.
D.neither necessary nor sufficient for P.

Answer: C

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Question 17

1 mark

Which of the following is the smallest?

A. $\log_{3}5$
B. $\log_{2}3$
C. $\log_{4}9$
D.they are all equal
E. $\log_{2}3$ and $\log_{4}9$ are equal and are the smallest

Answer: A

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Question 18

1 mark

Given that

\displaystyle\int_{0}^{2}f(x)\,dx=7

, find

\displaystyle\int_{0}^{2}f(2-x)\,dx

.

A. $14$
B. $7$
C. $-7$
D. $0$
E. $3.5$

Answer: B

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Question 19

1 mark

How many integers from

1

to

100

inclusive are divisible by neither

2

nor

3

?

A. $34$
B. $17$
C. $50$
D. $33$
E. $67$

Answer: D

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Question 20

1 mark

For which real values of

k

does the equation

x^{4}+kx^{2}+1=0

have at least one real solution?

A. $|k|\ge 2$
B. $k\le -2$
C. $k\ge 2$
D. $k<0$
E. $k\le -2$ or $k\ge 2$

Answer: B

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TMUA Mock Paper 2 (Paper 2): Questions & Worked Solutions | tmua.fyi