TMUA Challenge Paper 1 Tmua-p1-challenge-3

20 questions20 marks75Updated September 2026

The TMUA Challenge Paper 1 Tmua-p1-challenge-3 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

Let

k

be a real constant. The equation

|x|\,|x-3|=k

has exactly

n

distinct real solutions. Which of the following is the complete list of possible values of

n

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

Answer: B

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Question 2

1 mark

The circle

C

has centre

(1,-2)

and radius

3

. The point

P=(7,6)

lies outside

C

, and the two tangents from

P

touch

C

A

and

B

. What is the radius of the circle that passes through

P

A

and

B

A. $\tfrac{\sqrt{91}}{2}$
B. $5$
C. $\sqrt{34}$
D. $\sqrt{91}$
E. $10$
F. $\tfrac{13}{2}$

Answer: B

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Question 3

1 mark

Which of the following is the necessary and sufficient condition on the real constant

k

for

kx^{4}-4x^{2}+k\ge 0

to hold for all real

x

A. $k>0$
B. $k\ge 1$
C. $k>2$
D. $k\ge 2$
E. $k\ge 4$
F. $k\ge 2$ or $k\le -2$
G. $-2\le k\le 2$

Answer: D

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Question 4

1 mark

How many solutions does the equation

\sin^{2}x=\sin x\cos x

have in the interval

0^{\circ}\le x\le 1000^{\circ}

A. $6$
B. $10$
C. $11$
D. $12$
E. $13$
F. $14$

Answer: D

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Question 5

1 mark

Let

k

be a real constant. What is the sum of all values of

k

for which the equation

(x-k)\left(x^{2}-4x+k\right)=0

has exactly two distinct real roots?

A. $3$
B. $4$
C. $7$
D. $8$
E. $12$
F. $\text{there are infinitely many such } k$

Answer: C

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Question 6

1 mark

The curve

y=x^{2}-4x

undergoes the following transformations in order: a translation by

3

units in the positive

x

-direction; a reflection in the

y

-axis; a stretch parallel to the

x

-axis with scale factor

2

; a translation by

1

unit in the negative

y

-direction; a reflection in the

x

-axis. The resulting curve is

y=g(x)

. What is the value of

g(2)

A. $-76$
B. $-33$
C. $-31$
D. $-4$
E. $31$
F. $33$

Answer: C

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Question 7

1 mark

In triangle

ABC

, angle

A=30^{\circ}

and

AB=8

. The length of

BC

is a positive integer

a

. For how many values of

a

are there exactly two non-congruent triangles satisfying these conditions?

A. $2$
B. $3$
C. $4$
D. $5$
E. $7$
F. $8$

Answer: B

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Question 8

1 mark

A sequence is defined by

a_{1}=1

a_{2}=2

and

a_{n+2}=a_{n+1}-a_{n}

for

n\ge 1

. Let

S_{n}=a_{1}+a_{2}+\dots+a_{n}

. For how many values of

n

with

1\le n\le 100

S_{n}=3

A. $16$
B. $17$
C. $32$
D. $33$
E. $34$
F. $50$

Answer: E

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Question 9

1 mark

What is the remainder when

x^{2026}+x^{1013}+1

is divided by

x^{2}+x+1

A. $0$
B. $1$
C. $x$
D. $x+1$
E. $-x$
F. $2x+1$
G. $3$

Answer: A

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Question 10

1 mark

The distinct real numbers

a

b

c

form an arithmetic progression in that order, and

a

c

b

form a geometric progression in that order. Given that

a+b+c=18

, what is the value of

c

A. $-24$
B. $-12$
C. $-6$
D. $6$
E. $12$
F. $24$

Answer: B

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Question 11

1 mark

How many real values of

x

satisfy

\left(x^{2}-5x+5\right)^{x^{2}-4}=1\,?

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

Answer: C

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Question 12

1 mark

The two tangents from the point

P=(3,4)

to the circle

x^{2}+y^{2}=5

meet the

y

-axis at the points

A

and

B

. What is the length of

AB

A. $5$
B. $6$
C. $10$
D. $12$
E. $15$
F. $18$

Answer: E

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Question 13

1 mark

The function

f

is defined for all real

x

f(x)=|x+1|-2|x-1|+|x-3|.

What is the value of

\displaystyle\int_{-5}^{5}f(x)\,\mathrm{d}x

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

Answer: C

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Question 14

1 mark

Let

f(x)=x^{2}-2

and let

k

be a real constant. For which value of

k

does the equation

f(f(x))=k

have exactly three distinct real solutions?

A. $-2$
B. $-1$
C. $0$
D. $1$
E. $2$
F. $7$

Answer: E

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Question 15

1 mark

A straight line through the point

(0,1)

with gradient

m

meets the curve

y=x^{2}

at two points. As

m

varies, what is the least possible area enclosed between the line and the curve?

A. $\tfrac{2}{3}$
B. $1$
C. $\tfrac{4}{3}$
D. $2$
E. $\tfrac{8}{3}$
F. $4$

Answer: C

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Question 16

1 mark

For how many real values of

k

do the circles

x^{2}+y^{2}=1

and

(x-3)^{2}+(y-4)^{2}=k^{2}

touch each other?

A. $1$
B. $2$
C. $3$
D. $4$
E. $6$
F. $\text{infinitely many}$

Answer: D

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Question 17

1 mark

Let

k

be a real constant. The equation

x^{4}-4x^{3}+k=0

has exactly

n

distinct real roots. Which of the following is the complete list of possible values of

n

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

Answer: B

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Question 18

1 mark

How many integers

x

satisfy

\sqrt{x+5}>x-1

A. $3$
B. $4$
C. $5$
D. $8$
E. $9$
F. $10$

Answer: E

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Question 19

1 mark

Let

A

be the sum of all real solutions of

2\log_{2}(x-1)=\log_{2}(x+5)

, and let

B

be the sum of all real solutions of

\log_{2}\left((x-1)^{2}\right)=\log_{2}(x+5)

. What is the value of

A-B

A. $-3$
B. $-1$
C. $0$
D. $1$
E. $3$
F. $4$

Answer: D

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Question 20

1 mark

What is the remainder when

3^{2026}+5^{2026}

is divided by

13

A. $0$
B. $2$
C. $4$
D. $8$
E. $9$
F. $11$

Answer: B

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