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TMUA Challenge Paper 1 Tmua-p1-challenge-1

20 questions20 marks75Updated August 2026

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

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Question 1

1 mark

The positive real numbers

x

and

y

satisfy

\log_2 x + \log_4 y = 4

and

\log_2 y + \log_4 x = 5

. Find the value of

\log_2(xy)

.

A. $4$
B. $5$
C. $6$
D. $9$
E. $12$

Answer: C

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Question 2

1 mark

A circle passes through the three points

(0,0)

,

(6,0)

and

(0,8)

. Find the length of the tangent drawn from the point

(10,10)

to this circle.

A. $5$
B. $2\sqrt{15}$
C. $\sqrt{85}$
D. $2\sqrt{10}$
E. $\sqrt{50}$

Answer: B

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Question 3

1 mark

Find the coefficient of

x^5

in the expansion of

(1+x)(1-2x)^7

.

A. $-112$
B. $-672$
C. $560$
D. $-560$
E. $112$

Answer: A

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Question 4

1 mark

A geometric series has all terms positive. The sum of its first two terms is

10

and its sum to infinity is

18

. Find the least value of

n

for which the sum of the first

n

terms exceeds

17

.

A. $6$
B. $7$
C. $8$
D. $9$
E. $10$

Answer: C

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Question 5

1 mark

How many solutions does the equation

\tan x = 2\sin x

have in the interval

0^\circ < x < 360^\circ

?

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

Answer: B

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Question 6

1 mark

Find the minimum value of

\dfrac{x^2 + x + 4}{x}

for

x > 0

.

A. $4$
B. $5$
C. $2\sqrt{5}$
D. $6$
E. $3$

Answer: B

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Question 7

1 mark

Find the area of the finite region enclosed between the curves

y = 4 - x^2

and

y = x^2 - 4

.

A. $\tfrac{32}{3}$
B. $\tfrac{64}{3}$
C. $32$
D. $16$
E. $\tfrac{128}{3}$

Answer: B

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Question 8

1 mark

The cubic

x^3 + ax^2 + bx + 6

has

x = 1

and

x = 2

among its roots. Find its third root.

A. $-3$
B. $3$
C. $-6$
D. $6$
E. $-1$

Answer: A

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Question 9

1 mark

Find the complete set of values of

x

satisfying

\dfrac{x-1}{x-2} \ge 2

.

A. $x \le 3$
B. $2 < x \le 3$
C. $x < 2$ or $x \ge 3$
D. $x \ge 3$
E. $2 \le x \le 3$

Answer: B

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Question 10

1 mark

How many four-digit whole numbers have digits that are strictly increasing from left to right (for example,

1358

)?

A. $126$
B. $210$
C. $3024$
D. $84$
E. $252$

Answer: A

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Question 11

1 mark

How many integers

n

with

1 \le n \le 100

are such that

n^2 + n + 1

is divisible by

3

?

A. $33$
B. $34$
C. $66$
D. $67$
E. $50$

Answer: B

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Question 12

1 mark

As

\theta

varies over all real values, the expression

\dfrac{10}{2 + \cos\theta}

takes a maximum value and a minimum value. Find the sum of these two values.

A. $10$
B. $\tfrac{40}{3}$
C. $\tfrac{50}{3}$
D. $\tfrac{20}{3}$
E. $13$

Answer: B

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Question 13

1 mark

Find the largest integer

x

for which

2^x > 3^{x-1}

.

A. $1$
B. $2$
C. $3$
D. $4$
E.there is no largest such integer

Answer: B

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Question 14

1 mark

The points

A(1,1)

and

B(5,3)

lie above the

x

-axis. A path runs from

A

to a point

P

on the

x

-axis and then to

B

. Find the least possible total length

AP + PB

.

A. $4\sqrt{2}$
B. $2\sqrt{5}$
C. $6$
D. $\sqrt{20}$
E. $4\sqrt{3}$

Answer: A

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Question 15

1 mark

Evaluate

\displaystyle\sum_{r=1}^{20}(2r-1)^2

.

A. $10660$
B. $11480$
C. $5350$
D. $13530$
E. $9880$

Answer: A

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Question 16

1 mark

The function

f

is defined for all real

x \ne 3

by

f(x) = \dfrac{2x+1}{x-3}

. Which value can

f(x)

never take?

A. $3$
B. $2$
C. $-\tfrac{1}{3}$
D. $0$
E. $\tfrac{1}{2}$

Answer: B

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Question 17

1 mark

Find the term independent of

x

in the expansion of

\left(2x^2 - \dfrac{1}{x}\right)^6

.

A. $60$
B. $240$
C. $-60$
D. $15$
E. $-240$

Answer: A

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Question 18

1 mark

The curve

y = x^3 + ax^2 + bx

has a point of inflection at

(1,0)

. Find the gradient of the curve at that point.

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

Answer: A

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Question 19

1 mark

How many real solutions does the equation

|x^2 - 4| = x + 2

have?

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

Answer: C

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Question 20

1 mark

The function

f

is defined for

x \ge 3

by

f(x) = x^2 - 6x + 11

. Find

f^{-1}(6)

.

A. $1$
B. $5$
C. $3$
D. $6$
E. $2$

Answer: B

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