ESAT Challenge Mathematics 2 Esat-maths2-challenge-2

10 questions10 marks20Updated August 2026

The ESAT Challenge Mathematics 2 Esat-maths2-challenge-2 paper in full: all 10 questions, each with its answer. ESAT is the Engineering and Science Admissions Test. 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

Find the product of all real solutions of

x^{\log_{10} x} = 100x

A. $1$
B. $100$
C. $10$
D. $1000$
E. $10.1$

Answer: C

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Question 2

1 mark

Find the coefficient of

x^3

in the expansion of

\left(1 + x + x^2\right)^{10}

A. $120$
B. $135$
C. $210$
D. $220$
E. $240$

Answer: C

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Question 3

1 mark

A geometric series with all terms positive and common ratio satisfying

|r|<1

has first three terms summing to

26

and multiplying to

216

. Find the sum to infinity.

A. $26$
B. $36$
C. $39$
D. $27$
E. $54$

Answer: D

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Question 4

1 mark

A right circular cone of the greatest possible volume is inscribed in a sphere. What fraction of the sphere's volume does it occupy?

A. $\tfrac13$
B. $\tfrac49$
C. $\tfrac{8}{27}$
D. $\tfrac12$
E. $\tfrac29$

Answer: C

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Question 5

1 mark

Evaluate

\displaystyle\int_{0}^{4}\left|x^2-3x+2\right|\,\mathrm{d}x

A. $\tfrac{14}{3}$
B. $\tfrac{16}{3}$
C. $6$
D. $\tfrac{19}{3}$
E. $\tfrac{17}{3}$

Answer: E

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Question 6

1 mark

How many pairs of distinct positive integers

(a,b)

with

a<b

satisfy

a^{b} = b^{a}

A. $0$
B. $1$
C. $2$
D. $3$
E.infinitely many

Answer: B

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Question 7

1 mark

Given that

\sin\theta+\cos\theta=\tfrac12

for some

\theta

with

0<\theta<\pi

, find

\sin^3\theta+\cos^3\theta

A. $\tfrac{7}{16}$
B. $\tfrac{11}{16}$
C. $\tfrac18$
D. $\tfrac58$
E. $\tfrac{13}{16}$

Answer: B

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Question 8

1 mark

From a point

P

outside a circle of centre

O

and radius

6

, two tangents are drawn, touching the circle at

A

and

B

. The angle between the two tangents,

\angle APB

, is

\dfrac{\pi}{3}

.

What is the exact area of the region bounded by the two tangents

PA

and

PB

and the minor arc

AB

A. $36\sqrt{3} - 12\pi$
B. $36\sqrt{3} - 24\pi$
C. $18\sqrt{3} - 12\pi$
D. $36\sqrt{3} - 6\pi$
E. $36\sqrt{3} - 18\pi$
F. $72\sqrt{3} - 12\pi$

Answer: A

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Question 9

1 mark

The function

f

is defined for all real

x

f(x) = x^3 + kx^2 + 3x + 7,

where

k

is a constant.

For which values of

k

f

strictly increasing for every real value of

x

A. $k < -3$ or $k > 3$
B. $-3 \le k \le 3$
C. $0 < k < 3$
D. $-\sqrt{3} < k < \sqrt{3}$
E. $k > 3$
F. $-6 < k < 6$
G. $-3 < k < 3$

Answer: G

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Question 10

1 mark

In a non-constant arithmetic sequence, the

2

nd,

5

th and

14

th terms, taken in that order, are three consecutive terms of a geometric sequence.

What is the common ratio of that geometric sequence?

A. $\dfrac{1}{3}$
B. $2$
C. $\dfrac{3}{2}$
D. $9$
E. $3$
F. $4$

Answer: E

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