TMUA Mock Paper 2

20 questions20 marks75Updated August 2026

The TMUA Mock 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. $\sqrt{144}=\pm 12$ .
B.For every real number $x$ , $\sqrt{x^{2}}=x$ .
C. $0.\overline{9}=1$ .
D. $3^{0}=0$ .
E.The product of any two irrational numbers is irrational.

Answer: C

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Question 2

1 mark

Let

n

be a positive integer. Consider statement P: '

n

is prime', and statement Q: '

n

is of the form

6k\pm 1

for some integer

k

'. 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: D

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Question 3

1 mark

Consider the claim: 'For every integer

n\ge 2

, the value

n^{2}-n+11

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

A. $n=5$
B. $n=10$
C. $n=11$
D. $n=2$
E.there is no counterexample

Answer: C

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Question 4

1 mark

Let

x

and

y

be real numbers. Consider statement P: '

x+y>0

', and statement Q: '

x^{3}+y^{3}>0

'. Then Q is:

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

Answer: C

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Question 5

1 mark

f

is a function defined for all real numbers, with

f(1)=3

and

f(4)=-2

. Consider the statements: I:

f(x)=0

has a solution with

1<x<4

; II:

f(x)=0

has an odd number of solutions with

1<x<4

; III:

f

takes the value

1

somewhere in

1<x<4

. Which of these must be true?

A.none of them
B.I only
C.I and II only
D.I and III only
E.I, II and III

Answer: A

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Question 6

1 mark

f

is a differentiable function. Consider statement P: '

x=2

is a turning point of

f

', and statement Q: '

f'(2)=0

'. 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: A

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Question 7

1 mark

The following argument claims to show that every real number equals

0

. Line 1: Let

a=b

. Line 2: Then

a^{2}=ab

. Line 3: So

a^{2}-b^{2}=ab-b^{2}

. Line 4: Factorising,

(a-b)(a+b)=b(a-b)

. Line 5: Dividing by

a-b

a+b=b

. Line 6: Hence

a=0

. At which line does the first error occur?

A.Line 2
B.Line 3
C.Line 4
D.Line 5
E.Line 6

Answer: D

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Question 8

1 mark

Consider the statement: 'If a function

g

is defined on

[0,1]

, then

\int_{0}^{1}g(x)\,dx

equals the area enclosed between the graph of

g

and the

x

-axis over

[0,1]

.' Which of the following is a counterexample?

A. $g(x)=x^{2}$
B. $g(x)=x-1$
C. $g(x)=\frac{1}{x}$
D. $g(x)=1$
E.there is no counterexample

Answer: B

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Question 9

1 mark

Which of the following statements are true? I: There exists a real

y

such that for all real

x

x^{2}\ge y

. II: For all real

x

, there exists a real

y

such that

y>x^{2}

. III: There exists a real

y

such that for all real

x

y>x^{2}

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: E

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Question 10

1 mark

How many real solutions with

x>0

does the equation

(\ln x)^{2}-3\ln x+1=0

have?

A. $0$
B. $1$
C. $2$
D. $3$
E.infinitely many

Answer: C

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Question 11

1 mark

For a positive integer

n

, let

R_{n}

be the repunit consisting of

n

ones (so

R_{1}=1

R_{2}=11

R_{3}=111

, and so on). The statement '

3

divides

R_{n}

' is true if and only if:

A. $n$ is even
B. $3$ divides $n$
C. $9$ divides $n$
D. $n$ is odd
E. $n>3$

Answer: B

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Question 12

1 mark

What is the maximum value of

f(x)=2x^{3}-9x^{2}+12x+1

on the closed interval

0\le x\le 3

A. $6$
B. $10$
C. $5$
D. $1$
E. $12$

Answer: B

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Question 13

1 mark

S(x)

is a property of a real number

x

, and exactly one of the following four claims about

S

is true. Which one? A:

S(x)

holds whenever

x^{2}<4

. B:

S(x)

holds precisely when

x^{2}<4

. C:

S(x)

holds whenever

x^{2}<9

. D:

S(x)

holds precisely when

x^{2}<9

Answer: A

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Question 14

1 mark

b

is a real number, and

x=b

is a root of the equation

bx^{2}-(b^{2}+1)x+b=0

. Which of the following describes the complete set of possible values of

b

A. $b=0$ or $b=1$
B. $b=\pm 1$
C.all real numbers
D. $b\ne 0$
E.there is no such $b$

Answer: C

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Question 15

1 mark

What is the maximum value of

x+2y

subject to

x^{2}+y^{2}\le 5

A. $5$
B. $\sqrt{5}$
C. $2\sqrt{5}$
D. $3$
E. $10$

Answer: A

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Question 16

1 mark

Consider the statement about an integer

n

: 'if

n^{2}

is even then

n

is even'. Which of the following is its contrapositive?

A.If $n$ is even then $n^{2}$ is even.
B.If $n$ is odd then $n^{2}$ is odd.
C.If $n^{2}$ is odd then $n$ is odd.
D.If $n$ is odd then $n^{2}$ is even.
E.If $n^{2}$ is even then $n$ is odd.

Answer: B

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Question 17

1 mark

Which of the following statements are true? I: no integer of the form

4k+3

is the sum of two perfect squares. II: the product of two integers, each of the form

4k+1

, is again of the form

4k+1

. III: every prime number is of the form

4k+1

4k+3

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: E

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Question 18

1 mark

f

is a continuous function and

a<b

. Consider statement P: '

f(a)

and

f(b)

have opposite signs', and statement Q: '

f

has a root in the open interval

(a,b)

'. 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 19

1 mark

For which of these functions is it true that

f'(x)>0

for all real

x

? I:

f(x)=x^{3}

. II:

f(x)=e^{x}

. III:

f(x)=x+\sin x

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

Answer: C

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Question 20

1 mark

A sequence of integers satisfies

u_{n+1}=3u_{n}-4

, with

u_{1}

an integer. Which of the following statements are true? I:

u_{n}-2

is a multiple of

3^{n-1}

for every

n

. II: if

u_{1}

is even then every term is even. III:

u_{10}

is divisible by

3

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: E

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