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TMUA Mock Paper 1 (Paper 2)

20 questions20 marks75Updated August 2026

The TMUA Mock Paper 1 (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

An arithmetic sequence has first term

5

and common difference

3

. What is the sum of its first

20

terms?

A. $670$
B. $640$
C. $1340$
D. $335$
E. $610$

Answer: A

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Question 2

1 mark

Find the equation of the normal to the curve

y=x^{2}-3x

at the point where

x=2

.

A. $y=x-4$
B. $y=-x+4$
C. $y=-x$
D. $y=x$
E. $y=-x-4$

Answer: C

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Question 3

1 mark

A fair six-sided die is rolled three times. What is the probability that the three numbers obtained are strictly increasing?

A. $\frac{1}{6}$
B. $\frac{5}{54}$
C. $\frac{5}{108}$
D. $\frac{1}{36}$
E. $\frac{7}{27}$

Answer: B

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Question 4

1 mark

An open-topped box is made from a square sheet of card of side

12

cm by cutting a square of side

x

cm from each corner and folding up the sides. Which value of

x

gives the maximum volume?

A. $3$
B. $4$
C. $2$
D. $6$
E. $1$

Answer: C

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Question 5

1 mark

How many positive integers are divisors of

720

?

A. $30$
B. $24$
C. $15$
D. $20$
E. $36$

Answer: A

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Question 6

1 mark

Two circles have equations

x^{2}+y^{2}=4

and

(x-6)^{2}+(y-8)^{2}=9

. Find the shortest distance between them.

A. $10$
B. $15$
C. $3$
D. $5$
E. $2$

Answer: D

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Question 7

1 mark

Find the term independent of

x

in the expansion of

\left(2x-\dfrac{1}{x^{2}}\right)^{6}

.

A. $-240$
B. $60$
C. $240$
D. $160$
E. $15$

Answer: C

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Question 8

1 mark

A sequence is defined by

u_{1}=1

and

u_{n+1}=\dfrac{6}{u_{n}+1}

. Given that it converges, find its limit.

A. $-3$
B. $2$
C. $3$
D. $6$
E. $1$

Answer: B

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Question 9

1 mark

What is the maximum value of

f(x)=3\sin x+4\cos x

?

A. $7$
B. $5$
C. $25$
D. $\sqrt{7}$
E. $12$

Answer: B

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Question 10

1 mark

The line

y=kx-2

is a tangent to the curve

y=x^{2}-3x+2

for two values of

k

. Find the sum of those two values.

A. $-7$
B. $6$
C. $8$
D. $-6$
E. $-8$

Answer: D

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Question 11

1 mark

The graph of

y=\sqrt{x}

is transformed, in order, first by a translation of

3

in the positive

x

-direction and then by a stretch of scale factor

2

parallel to the

y

-axis. What is the equation of the resulting graph?

A. $y=2\sqrt{x}-3$
B. $y=\sqrt{2x-3}$
C. $y=2\sqrt{x+3}$
D. $y=2\sqrt{x-3}$
E. $y=\sqrt{2(x-3)}$

Answer: D

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Question 12

1 mark

The cubic

f(x)=x^{3}+ax^{2}+bx-6

has both

(x-1)

and

(x-2)

as factors. What is its third root?

A. $-3$
B. $6$
C. $1$
D. $-6$
E. $3$

Answer: E

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Question 13

1 mark

Which of the following is the largest?

A. $2^{1/2}$
B. $3^{1/3}$
C. $6^{1/6}$
D.they are all equal
E.none is larger than the others

Answer: B

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Question 14

1 mark

Given that

\sin\theta+\cos\theta=\dfrac{1}{2}

, find the value of

\sin\theta\cos\theta

.

A. $\frac{3}{8}$
B. $-\frac{1}{8}$
C. $-\frac{3}{8}$
D. $\frac{1}{4}$
E. $-\frac{3}{4}$

Answer: C

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Question 15

1 mark

Evaluate

\displaystyle\int_{0}^{3}\lfloor x\rfloor\,dx

, where

\lfloor x\rfloor

denotes the greatest integer not exceeding

x

.

A. $4.5$
B. $6$
C. $3$
D. $2$
E. $9$

Answer: C

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Question 16

1 mark

Solve

\log_{2}x+\log_{2}(x-2)=3

for

x>2

.

A. $-2$
B. $4$
C. $8$
D. $2$
E. $6$

Answer: B

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Question 17

1 mark

The sum of the first

n

terms of a sequence is

S_{n}=n^{2}+2n

. Find the tenth term of the sequence.

A. $21$
B. $120$
C. $19$
D. $23$
E. $39$

Answer: A

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Question 18

1 mark

The points

(1,2)

and

(5,8)

are the endpoints of a diameter of a circle. Find the radius of the circle.

A. $\sqrt{52}$
B. $\sqrt{26}$
C. $\sqrt{13}$
D. $13$
E. $2\sqrt{13}$

Answer: C

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Question 19

1 mark

In how many ways can

8

identical balls be placed into

3

distinct boxes so that no box is empty?

A. $28$
B. $21$
C. $15$
D. $45$
E. $36$

Answer: B

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Question 20

1 mark

How many ordered pairs of positive integers

(x,y)

satisfy

\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{6}

?

A. $9$
B. $5$
C. $8$
D. $10$
E. $18$

Answer: A

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