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

The equation

x^3 - 2x^2 + 3x - 4 = 0

has roots

\alpha

\beta

and

\gamma

. Find

\alpha^3+\beta^3+\gamma^3

A. $2$
B. $-2$
C. $0$
D. $8$
E. $20$

Answer: A

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Question 2

1 mark

Evaluate

\displaystyle\int_{0}^{\pi/2}\frac{\sin^{3}x}{\sin^{3}x+\cos^{3}x}\,\mathrm{d}x

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

Answer: B

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Question 3

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 4

1 mark

The curves

y = x^2

and

y = \ln(kx)

, where

k>0

, touch each other. Find

k

A. $\sqrt{e}$
B. $2\sqrt{e}$
C. $\tfrac{e}{2}$
D. $\sqrt{2e}$
E. $e\sqrt{2}$

Answer: D

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Question 5

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 6

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 7

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 8

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 9

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 10

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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