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TMUA Challenge Paper 1 Tmua-p1-challenge-2

20 questions20 marks75Updated August 2026

The TMUA Challenge Paper 1 Tmua-p1-challenge-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

The diagram shows the graph of

y=f(x)

, where

f(x)=(x-1)^2(x+2)

.

Exam diagram

Find the complete set of values of

k

for which the equation

\bigl|f(x)\bigr|=k

has exactly four distinct real solutions.

A. $k=0$
B. $0\le k<4$
C. $0<k\le 4$
D. $k>4$
E. $0<k<2$
F. $0<k<4$
G. $2<k<4$
H.there is no such $k$

Answer: F

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Question 2

1 mark

The diagram shows the two tangents drawn from the origin

O

to the circle of radius

2

with centre

C=(3,4)

, touching it at

P

and

Q

.

Exam diagram

What is the area of the quadrilateral

OPCQ

?

A. $\sqrt{21}$
B. $2\sqrt{5}$
C. $2\sqrt{21}$
D. $4\sqrt{21}$
E. $5$
F. $10$
G. $\tfrac{21}{2}$
H. $4\sqrt{5}$

Answer: C

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Question 3

1 mark

Let

n

be an integer greater than

6

for which

n-1

and

n+1

are both prime. What is the largest integer that must be a factor of

n^{2}\bigl(n^{2}+16\bigr)

?

A. $12$
B. $48$
C. $120$
D. $144$
E. $240$
F. $360$
G. $720$
H. $1440$

Answer: G

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Question 4

1 mark

How many solutions does

\sin 3x=\cos 2x

have for

0\le x\le 2\pi

?

A. $4$
B. $5$
C. $6$
D. $7$
E. $8$
F. $9$
G. $10$
H. $12$

Answer: B

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Question 5

1 mark

Evaluate

\displaystyle\int_{0}^{1}x(1-x)^{7}\,\mathrm{d}x

.

A. $\tfrac{1}{72}$
B. $\tfrac{1}{80}$
C. $\tfrac{1}{64}$
D. $\tfrac{1}{63}$
E. $\tfrac{1}{56}$
F. $\tfrac{1}{54}$
G. $\tfrac19$
H. $\tfrac18$

Answer: A

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Question 6

1 mark

How many distinct tangents to the curve

y=x^3-3x

pass through the point

(0,2)

?

A. $0$
B. $1$
C. $2$
D. $3$
E. $4$
F. $5$
G. $6$
H.infinitely many

Answer: B

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Question 7

1 mark

How many ordered pairs

(a,b)

of non-negative integers satisfy

\sqrt{a}+\sqrt{b}=\sqrt{2009}

?

A. $0$
B. $1$
C. $2$
D. $4$
E. $7$
F. $8$
G. $14$
H.infinitely many

Answer: F

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Question 8

1 mark

The real numbers

x

,

y

and

z

satisfy

x^{3}+y^{3}+z^{3}-3xyz=1

. What is the smallest possible value of

x^{2}+y^{2}+z^{2}

?

A. $\tfrac13$
B. $\tfrac12$
C. $\tfrac23$
D. $1$
E. $\tfrac32$
F. $2$
G. $3$
H.there is no smallest value

Answer: D

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Question 9

1 mark

The line

y=mx+1

is a tangent to the circle

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

. What is the sum of the possible values of

m

?

A. $-\tfrac32$
B. $-1$
C. $-\tfrac34$
D. $-\tfrac12$
E. $0$
F. $\tfrac34$
G. $1$
H. $\tfrac32$

Answer: C

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Question 10

1 mark

Which sketch shows the graph of

y=\dfrac{1}{x^{2}-1}

?

A.
B.
C.
D.
E.
F.
G.
H.

Answer: B

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Question 11

1 mark

Find the set of real values of

x

for which

|x-2|<|2x+1|

.

A. $-3<x<\tfrac13$
B. $x<-\tfrac13$ or $x>3$
C. $\tfrac13<x<3$
D. $x<-3$
E. $x>\tfrac13$
F. $-\tfrac13<x<3$
G. $x<-3$ or $x>-\tfrac13$
H. $x<-3$ or $x>\tfrac13$

Answer: H

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Question 12

1 mark

The trapezium rule with two strips of equal width is used to estimate

\displaystyle\int_{1}^{3}\frac{1}{x}\,\mathrm{d}x

, as shown.

Exam diagram

Which statement is correct?

A.The estimate is $\tfrac{7}{12}$ , and it is an underestimate.
B.The estimate is $\tfrac{7}{12}$ , and it is an overestimate.
C.The estimate is $\tfrac43$ , and it is an overestimate.
D.The estimate is $\tfrac76$ , and it is an overestimate.
E.The estimate is $\tfrac76$ , and it is an underestimate.
F.The estimate is $\tfrac76$ , and it is exact.
G.The estimate is $\tfrac{11}{6}$ , and it is an overestimate.
H.The estimate is $\tfrac43$ , and it is an underestimate.

Answer: D

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Question 13

1 mark

Two circles, of radii

1

and

2

, touch each other and both touch the line

\ell

. A third circle lies in the region between them, touching both circles and

\ell

, as shown. What is its radius?

Exam diagram

A. $\tfrac13$
B. $\tfrac25$
C. $6-4\sqrt2$
D. $3-2\sqrt2$
E. $\tfrac29$
F. $\sqrt2-1$
G. $\tfrac12$
H. $\tfrac23$

Answer: C

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Question 14

1 mark

Find the sum of the real solutions of

4^{x}-3\cdot 2^{\,x+1}+8=0

.

A. $3$
B. $2$
C. $1$
D. $\log_{2}3$
E. $6$
F. $8$
G. $\log_{2}6$
H. $4$

Answer: A

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Question 15

1 mark

Find the total area of the regions enclosed between the curve

y=x^{3}-x

and the line

y=3x

.

A. $4$
B. $\tfrac92$
C. $6$
D. $\tfrac{15}{2}$
E. $8$
F. $12$
G. $16$
H. $\tfrac{32}{3}$

Answer: E

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Question 16

1 mark

A chord of length

6

is drawn in a circle of radius

6

. The minor segment cut off is shaded.

Exam diagram

What is the area of the shaded segment?

A. $6\pi-18$
B. $6\pi-9$
C. $12\pi-9\sqrt3$
D. $9\sqrt3-6\pi$
E. $6\pi-18\sqrt3$
F. $12\pi-18\sqrt3$
G. $3\pi-9\sqrt3$
H. $6\pi-9\sqrt3$

Answer: H

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Question 17

1 mark

How many ordered pairs

(m,n)

of positive integers satisfy

m^{2}-n^{2}=2025

?

A. $3$
B. $4$
C. $5$
D. $6$
E. $7$
F. $8$
G. $12$
H. $15$

Answer: E

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Question 18

1 mark

An angle

\theta

satisfies

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

. Find

\cos^{2}\theta+\cos^{4}\theta

.

A. $1$
B. $0$
C. $\tfrac12$
D. $2$
E. $\tfrac32$
F. $\sqrt5-2$
G. $\tfrac{3-\sqrt5}{2}$
H.it depends on $\theta$

Answer: A

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Question 19

1 mark

Find the complete set of values of

k

for which the equation

x|x|-2x+k=0

has three distinct real roots.

A. $k>0$
B. $0<k<1$
C. $-1<k<0$
D. $-1<k<1$
E. $0\le k\le 1$
F. $k<1$
G. $-1\le k\le 1$
H.there is no such $k$

Answer: D

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Question 20

1 mark

The diagram shows the graph of

y=ax^{2}+bx+c

.

Exam diagram

Which of the following is true?

A. $a>0$ , $b>0$ , $c>0$
B. $a>0$ , $b>0$ , $c<0$
C. $a>0$ , $b<0$ , $c>0$
D. $a>0$ , $b<0$ , $c<0$
E. $a<0$ , $b<0$ , $c>0$
F. $a<0$ , $b<0$ , $c<0$
G. $a<0$ , $b>0$ , $c>0$
H. $a<0$ , $b>0$ , $c<0$

Answer: G

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