TMUA 2021 D513/01

20 questions20 marks75Updated July 2025

The TMUA 2021 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

Two circles have the same radius.
The centre of one circle is

(-2, 1)

.
The centre of the other circle is

(3, -2)

.
The circles intersect at two distinct points.
What is the equation of the straight line through the two points at which the circles intersect?

A. $3x - 5y = 4$
B. $3x + 5y = -1$
C. $5x - 3y = -4$
D. $5x - 3y = -1$
E. $5x - 3y = 1$
F. $5x - 3y = 4$
G. $5x + 3y = 1$

Answer: F

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Question 2

1 mark

The curve

y = x^3 - 6x + 3

has turning points at

x = \alpha

and

x = \beta

, where

\beta > \alpha

.
Find

\int_{\alpha}^{\beta} x^3 - 6x + 3 \,dx

A. $-8\sqrt{2}$
B. $-10$
C. $-10+6\sqrt{2}$
D. $0$
E. $12 - 8\sqrt{2}$
F. $6\sqrt{2}$
G. $12$

Answer: F

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Question 3

1 mark

An arithmetic progression and a convergent geometric progression each have first term

\frac{1}{2}

.
The sum of the second terms of the two progressions is

\frac{1}{2}

.
The sum of the third terms of the two progressions is

\frac{1}{8}

.
What is the sum to infinity of the geometric progression?

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

Answer: G

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Question 4

1 mark

Find the minimum value of the function

2^{2x} - 2^{x+3} + 4

A. $-16$
B. $-12$
C. $-8$
D. $0$
E. $4$
F. $20$

Answer: B

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Question 5

1 mark

The function

f

is such that

for all positive integers

m

and

n

Given that

f(9) + f(16) - f(24) = 0

, what is the value of

f(3)

A. $\frac{8}{3}$
B. $2\sqrt{2}$
C. $3$
D. $\frac{16}{5}$
E. $3\sqrt{2}$
F. $4$

Answer: F

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Question 6

1 mark

The function

f

is given by

f(x) = \frac{\cos x + 3}{7+5 \cos x - \sin^2 x}

.
Find the positive difference between the maximum and the minimum values of

f(x)

A. $0$
B. $\frac{1}{3}$
C. $\frac{1}{2}$
D. $\frac{2}{3}$
E. $1$
F. $2$

Answer: D

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Question 7

1 mark

The function

f

is such that

f(0) = 0

, and

xf(x) > 0

for all non-zero values of

x

.
It is given that

\int_{-2}^{2} f(x)\,dx = 4

and

\int_{-2}^{2} |f(x)|\, dx = 8

Evaluate

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

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

Answer: G

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Question 8

1 mark

The line

y = 2x + 3

meets the curve

y = x^2 + bx + c

at exactly one point.
The line

y = 4x - 2

also meets the curve

y = x^2 + bx + c

at exactly one point.
What is the value of

b-c

A. $-9$
B. $-5.5$
C. $-1$
D. $5$
E. $6$
F. $14$

Answer: A

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Question 9

1 mark

Find the area enclosed by the graph of

|x| + |y| = 1

A. $\frac{1}{2}$
B. $1$
C. $2$
D. $4$
E. $\frac{1}{2}\sqrt{2}$
F. $\sqrt{2}$
G. $2\sqrt{2}$

Answer: C

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Question 10

1 mark

Use the trapezium rule with 3 strips to estimate

\int_{1/2}^{2} 2 \log_{10} x \,dx

A. $\log_{10} \frac{\sqrt{6}}{2}$
B. $\log_{10} \frac{3}{2}$
C. $\log_{10} \frac{9}{4}$
D. $\log_{10} 3$
E. $\log_{10} \frac{81}{16}$
F. $\log_{10} \frac{\sqrt{23}}{2}$

Answer: B

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Question 11

1 mark

The function

f

is given by

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

Find the fraction of the interval

0 < x < 1

for which

f(x)

is decreasing.

A. $\frac{2}{15}$
B. $\frac{1}{5}$
C. $\frac{1}{3}$
D. $\frac{1}{2}$
E. $\frac{2}{3}$
F. $\frac{4}{5}$
G. $\frac{13}{15}$

Answer: A

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Question 12

1 mark

The minimum value of the function

x^4 - p^2x^2

-9

p

is a real number.
Find the minimum value of the function

x^2 - px + 6

A. $-3$
B. $6 - \frac{3\sqrt{2}}{2}$
C. $\frac{3}{2}$
D. $3$
E. $\frac{9}{2}$
F. $6 + \frac{3\sqrt{2}}{2}$

Answer: E

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Question 13

1 mark

The function

f

is such that, for every integer

n

\int_{n}^{n+1} f(x)\,dx = n + 1

Evaluate

\sum_{r=1}^{8} (\int_{0}^{r} f(x)\,dx)

A. $36$
B. $84$
C. $120$
D. $165$
E. $204$
F. $288$

Answer: C

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Question 14

1 mark

This question uses radians.
Find the number of distinct values of

x

that satisfy the equation

(x + 1)(3 - x) = 2(1 - \cos(\pi x))

A. $2$
B. $3$
C. $4$
D. $5$
E. $6$
F. $7$

Answer: B

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Question 15

1 mark

The diagram shows the graph of

y = f(x)

The graph consists of alternating straight-line segments of gradient

1

and

-1

and continues in this way for all values of

x

.
The function

g

is defined as

g(x) = \sum_{r=1}^{10} f (2^{r-1}x)

Find the value of

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

A. $\frac{1023}{1024}$
B. $\frac{1023}{512}$
C. $5$
D. $10$
E. $\frac{55}{2}$
F. $55$

Answer: C

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Question 16

1 mark

Consider the expansion of

(a + bx)^n

.
The third term, in ascending powers of

x

, is

105x^2

.
The fourth term, in ascending powers of

x

, is

210x^3

.
The fourth term, in descending powers of

x

, is

210x^3

.
Find the value of

\frac{a^2}{b}

A. $\frac{1}{4}$
B. $\frac{4}{9}$
C. $\frac{25}{36}$
D. $\frac{5}{6}$
E. $1$

Answer: B

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Question 17

1 mark

Which of the following sketches shows the graph of

\sin(x^2 + y^2) = \frac{1}{2}

where

x^2 + y^2 < 8\pi

A.Graph A
B.Graph B
C.Graph C
D.Graph D
E.Graph E

Answer: A

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Question 18

1 mark

The curve with equation

x = y^2 - 6y + 11

is rotated

90^{\circ}

clockwise about the point

P

to give the curve

C

P

has

x

-coordinate

-2

and

y

-coordinate

3

What is the equation of

C

A. $y = -x^2 - 4x - 3$
B. $y = -x^2 - 4x - 5$
C. $y = -x^2 - 6x - 7$
D. $y = -x^2 - 6x - 11$
E. $y = x^2 - 4x + 5$
F. $y = x^2 + 4x + 3$
G. $y = x^2 - 6x + 11$
H. $y = x^2 + 6x + 7$

Answer: B

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Question 19

1 mark

The equation

\sin^2 (4^{\cos \theta} \times 60^{\circ}) = \frac{3}{4}

has exactly three solutions in the range

0^{\circ} \le \theta < x^{\circ}

.
What is the range of all possible values of

x

A. $90 ≤ x < 120$
B. $90 ≤ x < 270$
C. $120 ≤ x < 240$
D. $270 ≤ x < 300$
E. $300 ≤ x < 360$
F. $450 ≤ x < 630$

Answer: B

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Question 20

1 mark

Find the length of the curve with equation

2\log_{10} (x - y) = \log_{10} (2 - 2x) + \log_{10} (y + 5)

A. $5$
B. $10$
C. $15$
D. $3\pi$
E. $9\pi$
F. $12\pi$

Answer: D

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