TMUA 2020 D513/01

20 questions20 marks75Updated September 2025

The TMUA 2020 D513/01 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

Which of the following is an expression for the first derivative with respect to

x

\frac{x^3 - 5x^2}{2x \sqrt{x}}

A. $\frac{\sqrt{x}}{2}$
B. $\frac{\sqrt{x}}{4}$
C. $\frac{3x-5}{4\sqrt{x}}$
D. $\frac{3\sqrt{x}-5}{4\sqrt{x}}$
E. $\frac{3\sqrt{x} - 10}{3\sqrt{x}}$
F. $\frac{3x^2 - 10x}{3\sqrt{x}}$

Answer: C

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Question 2

1 mark

(2x + 1)

and

(x – 2)

are factors of

2x^3 + px^2 + q

. What is the value of

2p + q

A.-10
B. $\frac{38}{5}$
C. $\frac{22}{3}$
D. $\frac{22}{3}$
E. $\frac{38}{5}$
F.10

Answer: C

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Question 3

1 mark

Find the complete set of values of

x

for which

(x + 4)(x + 3)(1-x) > 0

and $(x + 2)(x - 2) < 0

A. $1 < x < 2$
B. $-2 < x < 1$
C. $-2 < x < 2$
D. $x < -2$ or $x > 1$
E. $x < -4$ or $x > 2$
F. $x < -4$ or $-3 < x < 1$
G. $-4 < x < -2$ or $x > 1$

Answer: B

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Question 4

1 mark

The

1^{st}

2^{nd}

and

3^{rd}

terms of a geometric progression are also the

1^{st}

4^{th}

and

6^{th}

terms, respectively, of an arithmetic progression.

The sum to infinity of the geometric progression is 12.

Find the

1^{st}

term of the geometric progression.

Answer: D

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Question 5

1 mark

The curve

S

has equation

y = px^2 + 6x - q

where

p

and

q

are constants.

S

has a line of symmetry at

x = -\frac{1}{4}

and touches the

x

-axis at exactly one point.

What is the value of

p + 8q

A.6
B.18
C.21
D.25
E.38

Answer: A

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Question 6

1 mark

Find the maximum value of the function

f(x) = \frac{1}{5^{2x} - 4(5^x) + 7}

A. $\frac{1}{7}$
B. $\frac{1}{4}$
C. $\frac{1}{3}$
D.3
E.4
F.7

Answer: C

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Question 7

1 mark

Given that

2^{3x} = 8^{(y+3)}

and

4^{(x+1)} = \frac{16^{(y+1)}}{8^{(y+3)}}

what is the value of

x + y

A.-23
B.-22
C.-15
D.-14
E.-11
F.-10

Answer: A

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Question 8

1 mark

The function

f

is defined for all real

x

f(x) = (p - x)(x + 2)

.

Find the complete set of values of

p

for which the maximum value of

f(x)

is less than 4.

A. $-2 - 4\sqrt{2} < p < -2 + 4\sqrt{2}$
B. $-2 - 2\sqrt{2} < p < -2 + 2\sqrt{2}$
C. $-2\sqrt{5} < p < 2\sqrt{5}$
D. $-6 < p < 2$
E. $-4 < p < 0$
F. $-2 < p < 2$

Answer: D

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Question 9

1 mark

The quadratic expression

x^2 - 14x + 9

factorises as

(x − \alpha)(x – \beta)

, where

\alpha

and

\beta

are positive real numbers.

Which quadratic expression can be factorised as

(x - \sqrt{\alpha})(x - \sqrt{\beta})

A. $x^2 - \sqrt{10}x + 3$
B. $x^2 - \sqrt{14}x + 3$
C. $x^2 - \sqrt{20}x + 3$
D. $x^2 - 178x + 81$
E. $x^2 - 176x + 81$
F. $x^2 + 196x + 81$

Answer: C

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Question 10

1 mark

The following sequence of transformations is applied to the curve

y = 4x^2

1. Translation by

\begin{pmatrix} 3 \ -5 \ \end{pmatrix}

2. Reflection in the

x

-axis
3. Stretch parallel to the

x

-axis with scale factor 2

What is the equation of the resulting curve?

A. $y = -x^2 + 12x – 31$
B. $y = -x^2 + 12x - 41$
C. $y = x^2 + 12x + 31$
D. $y = x^2 + 12x + 41$
E. $y = -16x^2 + 48x - 31$
F. $y = -16x^2 + 48x – 41$
G. $y = 16x^2 - 48x + 31$
H. $y = 16x^2 - 48x + 41$

Answer: A

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Question 11

1 mark

The quadratic function shown passes through

(2,0)

and

(q, 0)

, where

q > 2

What is the value of

q

such that the area of region

R

equals the area of region

S

A. $\sqrt{6}$
B.3
C. $\frac{18}{5}$
D.4
E.6
F. $\frac{33}{5}$

Answer: E

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Question 12

1 mark

How many real solutions are there to the equation

3 \cos x = \sqrt{x}

where

x

is in radians?

A.0
B.1
C.2
D.3
E.4
F.5
G.infinitely many

Answer: D

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Question 13

1 mark

Find the coefficient of

x^2y^4

in the expansion of

(1 + x + y^2)^7

A.6
B.10
C.21
D.35
E.105
F.210

Answer: F

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Question 14

1 mark

The area enclosed between the line

y = mx

and the curve

y = x^3

is 6.

What is the value of

m

A.2
B.4
C. $\sqrt{3}$
D. $\sqrt{6}$
E. $2\sqrt{3}$
F. $2\sqrt{6}$

Answer: E

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Question 15

1 mark

Find the positive difference between the two real values of

x

for which

(\log_2 x)^4 + 12 \left(\log_2 \left(\frac{1}{x}\right)\right)^2 - 2^6 = 0

A.4
B.16
C. $\frac{15}{4}$
D. $\frac{17}{4}$
E. $\frac{255}{16}$
F. $\frac{257}{16}$

Answer: C

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Question 16

1 mark

The circle

C_1

has equation

(x + 2)^2 + (y − 1)^2 = 3

The circle

C_2

has equation

(x - 4)^2 + (y − 1)^2 = 3

The straight line

l

is a tangent to both

C_1

and

C_2

and has positive gradient.
The acute angle between

l

and the

x

-axis is

\theta

Find the value of

\tan \theta

A. $\frac{1}{2}$
B.2
C. $\frac{\sqrt{2}}{2}$
D. $\sqrt{2}$
E. $\frac{\sqrt{6}}{2}$
F. $\frac{\sqrt{6}}{3}$
G. $\frac{\sqrt{3}}{3}$
H. $\sqrt{3}$

Answer: C

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Question 17

1 mark

Find the complete set of values of

m

in terms of

c

such that the graphs of

y = mx + c

and

y = \sqrt{x}

have two points of intersection.

A. $0 < m < \frac{1}{4c}$
B. $0 < m < 4c^2$
C. $m > \frac{1}{4c}$
D. $m < \frac{1}{4c}$
E. $m > 4c^2$
F. $m < 4c^2$

Answer: A

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Question 18

1 mark

Find the number of solutions and the sum of the solutions of the equation

1 - 2 \cos^2 x = | \cos x |

where

0 \le x \le 180^{\circ}

A.Number of solutions = 2
Sum of solutions = $180^{\circ}$
B.Number of solutions = 2
Sum of solutions = $240^{\circ}$
C.Number of solutions = 3
Sum of solutions = $180^{\circ}$
D.Number of solutions = 3
Sum of solutions = $360^{\circ}$
E.Number of solutions = 4
Sum of solutions = $240^{\circ}$
F.Number of solutions = 4
Sum of solutions = $360^{\circ}$

Answer: A

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Question 19

1 mark

Find the lowest positive integer for which

x^2 — 52x – 52

is positive.

A.26
B.27
C.51
D.52
E.53
F.54

Answer: E

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Question 20

1 mark

For how many values of

a

is the equation

(x − a)(x^2 – x + a) = 0

satisfied by exactly two distinct values of

x

A.0
B.1
C.2
D.3
E.4
F.more than 4

Answer: C

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