ESAT Mock Maths 1 Esat-maths1-bank-1

15 questions15 marks40Updated August 2026

The ESAT Mock Maths 1 Esat-maths1-bank-1 paper in full: all 15 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.

Questions and answers are free. Full step-by-step worked solutions unlock with a free account. Start practising.

Question 1

1 mark

Let

P

Q

and

R

be defined by the following expressions:

P = \left( \frac{2}{5} - 0.75 \right) \div \frac{1}{2}

Q = -1.2 \times \frac{3}{4} + 0.25

R = -1 \frac{1}{8} \div 1.5

Consider the following statements:
I.

P < Q

II.

Q < R

III.

P + R = -1.45

Which of these statements is/are true?

A.I only
B.II only
C.I and II only
D.I and III only
E.I, II and III

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 2

1 mark

How many positive integers

n < 500

have the property that

n

is a multiple of

11

and the highest common factor of

n

and

30

1

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

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 3

1 mark

A sequence of four integers

(x_1, x_2, x_3, x_4)

is chosen such that each

x_i

is an element of the set

\{1, 2, 3, 4, 5, 6, 7\}

. How many such sequences satisfy the conditions

x_1 < x_2

and

x_2 > x_3 > x_4

A. $85$
B. $105$
C. $175$
D. $210$
E. $350$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 4

1 mark

For any real number

n

, the expression

\frac{(2^{n+1})^3 \times 4^{-n}}{\sqrt[3]{8}}

is equivalent to:

A. $2^{n}$
B. $2^{n+2}$
C. $2^{n+3}$
D. $2^{2n+2}$
E. $2^{5n+2}$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 5

1 mark

Two solid spheres,

X

and

Y

, are made of different metals. Sphere

X

has radius

r_X = 3.0 \times 10^{-2}\,\text{m}

and is made of a metal with density

\rho_X = 4.0 \times 10^3\,\text{kg m}^{-3}

. Sphere

Y

has mass

M_Y = 1.152\pi \times 10^1\,\text{kg}

and is made of a metal with density

\rho_Y = 5.0 \times 10^3\,\text{kg m}^{-3}

.

What is the ratio of the surface area of sphere

Y

to the surface area of sphere

X

?

(You may use the following formulae for a sphere of radius

r

\text{Volume} = \frac{4}{3}\pi r^3

;

\text{Surface Area} = 4\pi r^2

)

A. $1.6 \times 10^1$
B. $4.0 \times 10^0$
C. $6.4 \times 10^1$
D. $1.6 \times 10^{-1}$
E. $8.0 \times 10^0$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 6

1 mark

What is the value of

0.\dot{1} + 0.0\dot{1} + 0.00\dot{1}

expressed as a fraction in its simplest form?

A. $\frac{111}{1000}$
B. $\frac{11}{90}$
C. $\frac{37}{300}$
D. $\frac{41}{333}$
E. $\frac{1}{9}$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 7

1 mark

A container holds a mixture of salt and water. Initially,

37.5\%

of the volume of the mixture is salt. Then,

\frac{1}{5}

of the volume of water currently in the container is removed and replaced with an equal volume of pure salt. Subsequently,

10\%

of this new mixture is removed and replaced with an equal volume of pure water. What is the final percentage, by volume, of salt in the mixture?

A. $33.75\%$
B. $40\%$
C. $45\%$
D. $50\%$
E. $55\%$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 8

1 mark

Given that

x = \sqrt{7 + \sqrt{40}}

, what is the value of

x - \frac{3}{x}

A. $\sqrt{5} - \sqrt{2}$
B. $2$
C. $2\sqrt{2}$
D. $\sqrt{5} + \sqrt{2}$
E. $2\sqrt{5}$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 9

1 mark

A large square has side length

x = 10\,\text{cm}

, measured correct to the nearest centimetre. A smaller square with side length

y = 4\,\text{cm}

is removed from the large square. The length

y

was obtained by truncating a measurement to the nearest whole centimetre. If

A

is the area of the remaining shape, which of the following is the correct error interval for

A

A. $75.00\,\text{cm}^2 < A < 105.00\,\text{cm}^2$
B. $70.00\,\text{cm}^2 < A < 98.00\,\text{cm}^2$
C. $65.25\,\text{cm}^2 < A < 94.25\,\text{cm}^2$
D. $65.25\,\text{cm}^2 \le A < 94.25\,\text{cm}^2$
E. $65.25\,\text{cm}^2 < A \le 94.25\,\text{cm}^2$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 10

1 mark

A solid right circular cone has a base radius of

\sqrt{2}\,\text{cm}

and a slant height of

\sqrt{7}\,\text{cm}

. The material of the cone has a density of

4.5\,\text{g}\,\text{cm}^{-3}

. Given that the volume of a cone is

V = \frac{1}{3}\pi r^2 h

, where

h

is the vertical height, which of the following is the best estimate for the mass of the cone?

A. $7\,\text{g}$
B. $21\,\text{g}$
C. $47\,\text{g}$
D. $63\,\text{g}$
E. $94\,\text{g}$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 11

1 mark

A region of land is represented on two maps, Map X and Map Y. Map X has a scale of

1 : n

. On this map, the region has an area of

100\,\text{cm}^2

. Map Y has a scale of

1 : (n + 20,000)

. On this map, the same region has an area of

64\,\text{cm}^2

. A physical model of a building within this region is constructed with a scale of

1 : m

. The actual volume of the building is

8000\,\text{m}^3

, and the volume of the model is

1\,\text{cm}^3

. What is the value of the ratio

n : m

A. $4 : 1$
B. $10 : 1$
C. $40 : 1$
D. $80 : 1$
E. $400 : 1$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 12

1 mark

A chemist has two different solutions of acid in water. Solution P has acid and water in the ratio

2 : 5

by volume. Solution Q has acid and water in the ratio

3 : 1

by volume. A sample is taken from each solution such that both samples contain the same volume of acid. Express the volume of water in the sample from Solution Q as a fraction of the volume of water in the sample from Solution P.

A. $\frac{2}{15}$
B. $\frac{3}{10}$
C. $\frac{5}{6}$
D. $\frac{6}{5}$
E. $\frac{15}{2}$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 13

1 mark

A metal alloy is composed of copper, tin, and zinc. The ratio of the mass of copper to the mass of tin is

3:4

, and the ratio of the mass of tin to the mass of zinc is

3:2

. The initial total mass of the alloy is

1.45\,\text{kg}

. A quantity of copper is added to the alloy such that the ratio of the mass of copper to the mass of zinc becomes

2:1

. What mass of copper, in grams, was added?

A. $150$
B. $250$
C. $350$
D. $450$
E. $800$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 14

1 mark

Liquid

X

is a mixture of component

A

and component

B

in the ratio

2 : 3

by volume. Liquid

Y

is also a mixture of component

A

and component

B

. When liquid

X

and liquid

Y

are mixed in the ratio

2 : 3

by volume, the concentration of component

A

by volume in the resulting mixture is

C_1

. When liquid

X

and liquid

Y

are mixed in the ratio

3 : 2

by volume, the concentration of component

A

by volume in the resulting mixture is

C_2

. If

C_1 : C_2 = 7 : 8

, what is the ratio of the volume of component

A

to the volume of component

B

in liquid

Y

in its simplest form?

A. $1 : 4$
B. $1 : 5$
C. $4 : 1$
D. $2 : 3$
E. $3 : 7$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 15

1 mark

A cleaning solution is made by mixing three chemicals

X

Y

, and

Z

. The ratio of the mass of

X

to the mass of

Y

3:2

. The ratio of the mass of

Y

to the mass of

Z

5:4

. What fraction of the total mass of the solution is chemical

Y

A. $\frac{10}{33}$
B. $\frac{2}{5}$
C. $\frac{5}{9}$
D. $\frac{8}{33}$
E. $\frac{2}{9}$

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →