TMUA Mock Paper 1

20 questions20 marks75Updated August 2026

The TMUA Mock Paper 1 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.

Questions and answers are free. Full step-by-step worked solutions unlock with a free account. Start practising.

Question 1

1 mark

The first three terms of a geometric sequence are

\sqrt{2}

2

2\sqrt{2}

. What is the seventh term?

A. $8$
B. $8\sqrt{2}$
C. $16$
D. $16\sqrt{2}$
E. $32$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 2

1 mark

Evaluate

\displaystyle\int_{1}^{4}\frac{3x-2}{\sqrt{x}}\,dx

A. $6$
B. $8$
C. $10$
D. $12$
E. $14$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 3

1 mark

The line

y=2x+c

does not intersect the curve

y=x^{2}-4x+7

. Find the set of values of

c

A. $c>-2$
B. $c<2$
C. $c>2$
D. $c<-2$
E. $-2<c<2$

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 4

1 mark

Find the sum of all real solutions of

x-5\sqrt{x}+6=0

A. $5$
B. $13$
C. $6$
D. $25$
E. $90$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 5

1 mark

Find

\displaystyle\sum_{n=0}^{\infty}\frac{\cos(n\pi)}{3^{n}}

A. $\frac{3}{2}$
B. $\frac{2}{3}$
C. $\frac{3}{4}$
D. $\frac{1}{2}$
E. $\frac{9}{4}$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 6

1 mark

The circle

C

has equation

x^{2}+y^{2}-6x+8y+21=0

. Find the length of the tangent to

C

drawn from the origin.

A. $\sqrt{13}$
B. $\sqrt{5}$
C. $21$
D. $\sqrt{29}$
E. $\sqrt{21}$

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 7

1 mark

Evaluate

\displaystyle\prod_{n=2}^{99}\left(1-\frac{1}{n^{2}}\right)

A. $\frac{50}{99}$
B. $\frac{1}{99}$
C. $\frac{100}{99}$
D. $\frac{1}{2}$
E. $\frac{25}{99}$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 8

1 mark

Find the area of the region enclosed between the graphs of

y=|x-1|

and

y=3-|x-1|

A. $9$
B. $2.25$
C. $4.5$
D. $3$
E. $6$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 9

1 mark

Find the maximum value of

f(x,y)=-2x^{2}-y^{2}+8x+6y-5

A. $8$
B. $12$
C. $9$
D. $3$
E. $20$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 10

1 mark

Find the coefficient of

x^{5}

in the expansion of

(1+x)^{3}(1+2x)^{4}

A. $168$
B. $144$
C. $96$
D. $208$
E. $120$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 11

1 mark

For how many integer values of

k

does the equation

x^{3}-3x^{2}=k

have three distinct real solutions?

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

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 12

1 mark

A quadratic curve has a minimum point at

(2,-3)

and passes through

(0,5)

. What is the coefficient of

x^{2}

A. $2$
B. $\frac{1}{2}$
C. $4$
D. $8$
E. $-2$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 13

1 mark

Given that

\log_{a}b=3

and

\log_{b}c=4

, find

\log_{a}c

A. $7$
B. $\frac{4}{3}$
C. $12$
D. $64$
E. $81$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 14

1 mark

How many solutions does

2\sin^{2}x=\sin x+1

have in the interval

0\le x<2\pi

A. $4$
B. $2$
C. $5$
D. $3$
E. $1$

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 15

1 mark

Let

F(n)=\displaystyle\int_{0}^{n}(2x-1)\,dx

for positive integers

n

. What is the largest

n

for which

F(n)<90

A. $9$
B. $10$
C. $8$
D. $11$
E. $45$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 16

1 mark

How many real solutions does

9^{x}-10\cdot 3^{x}+9=0

have?

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

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 17

1 mark

The parabola

y=x^{2}

is rotated

180^{\circ}

about the origin and then translated by the vector

\binom{2}{0}

. What is the equation of the resulting curve?

A. $y=-x^{2}+4x-4$
B. $y=-x^{2}-4x-4$
C. $y=x^{2}-4x+4$
D. $y=-x^{2}+4x+4$
E. $y=-(x+2)^{2}$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 18

1 mark

Evaluate

\cos^{2}1^{\circ}+\cos^{2}2^{\circ}+\cos^{2}3^{\circ}+\cdots+\cos^{2}89^{\circ}

A. $45$
B. $44.5$
C. $44$
D. $89$
E. $88.5$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 19

1 mark

The region between the curves

y=x

and

y=x^{3}

for

0\le x\le 1

is divided by the vertical line

x=a

into two parts of equal area. Which equation does

a

satisfy?

A. $2a^{4}-4a^{2}+1=0$
B. $2a^{4}-4a^{2}-1=0$
C. $a^{4}-2a^{2}+1=0$
D. $2a^{4}-4a^{2}+2=0$
E. $4a^{4}-2a^{2}+1=0$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 20

1 mark

The line

L

is a tangent to the curve

y=x^{4}-4x^{3}+8x

at two distinct points. Find the

y

-intercept of

L

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

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →