ESAT Mock Maths 1 ESAT-MATHS1-MOCK-1: Questions & Worked Solutions

Question 1

1 mark

A rectangular water tank has a base area of

1.5\,\text{m}^2

. The tank is filled with water to a depth of

40\,\text{cm}

. The density of water is

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

.

What is the total mass of the water in the tank in kilograms?

A. $6\,\text{kg}$
B. $60\,\text{kg}$
C. $600\,\text{kg}$
D. $6000\,\text{kg}$
E. $60,000\,\text{kg}$

Answer: C

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Question 2

1 mark

A cylindrical water tank has a constant cross-sectional area of

0.5\,\text{m}^2

and a total height of

2.0\,\text{m}

. The tank is initially empty and is being filled by a pipe at a constant rate of

2.5\,\text{litres per second}

. The density of water is

1000\,\text{kg m}^{-3}

.

How many minutes will it take for the tank to reach

75\%

of its maximum capacity?

(

1\,\text{m}^3 = 1000\,\text{litres}

)

A.2.5
B.5.0
C.6.7
D.7.5
E.300

Answer: B

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Question 3

1 mark

Let

P

and

Q

be defined by the following expressions:

P = \left( -\frac{2}{3} \right)^2 - \frac{5}{6}

Q = \frac{1}{2} \div \left( -\frac{3}{4} \right) + \frac{1}{3}

Consider the following statements:
I.

P < Q

II.

P + Q > -\frac{2}{3}

III.

\frac{P}{Q} > 1

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

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Question 4

1 mark

Evaluate the following expression:

X = \left[ \left( 1 - \frac{1}{1 - \frac{1}{3}} \right)^{-1} + \left( 1 + \frac{1}{1 + \frac{1}{3}} \right)^{-1} \right]^{-1}

A.-1.4
B.-0.7
C.0.7
D.1.4
E.7.0

Answer: B

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Question 5

1 mark

Let

f(n) = \frac{n^3 - 1}{n^3 + 1}

for integers

n > 1

.

What is the value of the product

P = f(2) \times f(3) \times f(4) \times f(5)

?

A. $\frac{28}{45}$
B. $\frac{31}{45}$
C. $\frac{2}{3}$
D. $\frac{14}{15}$
E. $\frac{124}{135}$

Answer: B

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Question 6

1 mark

Given that

x

is a positive real number satisfying the equation:

\sqrt{x + \sqrt{x + \sqrt{x + \dots}}} = 3\sqrt{x - \sqrt{x - \sqrt{x - \dots}}}

Where both sides represent infinite nested radicals, what is the value of

x

?

A.0.25
B.0.50
C.0.75
D.1.50
E.2.00

Answer: C

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Question 7

1 mark

A security code consists of 4 digits chosen from the set

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

. The digits in the code must be in non-decreasing order (from left to right).

How many such codes contain at least one repeated digit?

A.15
B.105
C.111
D.126
E.1281

Answer: C

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Question 8

1 mark

How many 4-digit positive integers (from 1000 to 9999 inclusive) have the property that the product of their four digits is exactly 24?

A.44
B.52
C.60
D.64
E.72

Answer: D

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Question 9

1 mark

A square grid consists of 16 dots arranged in 4 rows and 4 columns. How many different squares can be formed such that all four vertices of the square are dots in the grid?

A.14
B.18
C.20
D.30
E.50

Answer: C

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Question 10

1 mark

For a positive real number

a

, the following relationship holds:

\sqrt[3]{a^2} = \sqrt{\frac{1}{a^k}}

What is the value of

k

?

A. $-\frac{4}{3}$
B. $-\frac{3}{4}$
C. $\frac{3}{4}$
D. $\frac{4}{3}$
E. $-\frac{1}{3}$

Answer: A

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Question 11

1 mark

Which one of the following statements is NOT true for all real values of

x

?

A. $\sqrt{x^2} = |x|$
B. $(\sqrt[3]{x})^3 = x$
C.If $x$ is a square root of $49$ , then $x = \sqrt{49}$
D. $\sqrt[3]{-x} = -\sqrt[3]{x}$
E. $\sqrt{x^4} = x^2$

Answer: C

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Question 12

1 mark

For

x > 0

, the expression

\frac{(x^{\frac{3}{4}} + x^{-\frac{1}{4}})^2}{x^{\frac{1}{2}} + x^{-\frac{1}{2}}}

is equivalent to:

A. $x + 1$
B. $x - 1$
C. $x^{\frac{1}{2}} + 1$
D. $x^2 + 1$
E. $x^{\frac{3}{2}} + x^{\frac{1}{2}}$

Answer: A

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Question 13

1 mark

Two real numbers

x

and

y

are defined by

x = \sqrt[3]{4\sqrt{2}}

and

y = \sqrt{2\sqrt[3]{4}}

. The product

xy

can be expressed in the form

2^k

for some constant

k

. What is the value of

k

?

A. $\frac{5}{6}$
B. $\frac{5}{3}$
C. $\frac{7}{3}$
D. $\frac{25}{36}$
E. $1$

Answer: B

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Question 14

1 mark

Two spherical planets,

X

and

Y

, have surface areas

S_X

and

S_Y

respectively such that

S_X = 1.6 \times 10^7 S_Y

. The average density of planet

X

is

\rho_X = 1.25 \times 10^3 \text{ kg m}^{-3}

and the average density of planet

Y

is

\rho_Y = 1.0 \times 10^7 \text{ kg m}^{-3}

. What is the ratio of the mass of planet

X

to the mass of planet

Y

?

You may use the following formulae for a sphere of radius

r

:

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

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

A.8.0 $\times$ 10^6
B.5.0 $\times$ 10^{-1}
C.1.6 $\times$ 10^7
D.8.0 $\times$ 10^3
E.6.4 $\times$ 10^{10}

Answer: A

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Question 15

1 mark

A laser pulse has a total energy of

E = 9.9 \times 10^{-4} \text{ J}

. Each photon in the pulse has an energy

E_{ph}

given by the formula

E_{ph} = \frac{hc}{\lambda}

.

Take the following values:
- Planck's constant

h = 6.6 \times 10^{-34} \text{ J s}

- Speed of light

c = 3.0 \times 10^8 \text{ m s}^{-1}

- Wavelength

\lambda = 6.0 \times 10^{-7} \text{ m}

How many photons are contained in this laser pulse?

A.3.0 $\times$ 10^{15}
B.3.0 $\times$ 10^{16}
C.5.0 $\times$ 10^{14}
D.3.3 $\times$ 10^{15}
E.3.0 $\times$ 10^{14}

Answer: A

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Question 16

1 mark

A rectangular plot of land has an area of

1.0 \times 10^{13} \text{ m}^2

and a perimeter of

2.2 \times 10^7 \text{ m}

. What is the positive difference between the length and the width of the rectangle, expressed in standard form?

A.9.0 $\times$ 10^6 $\text{ m}$
B.1.0 $\times$ 10^6 $\text{ m}$
C.1.1 $\times$ 10^7 $\text{ m}$
D.1.0 $\times$ 10^7 $\text{ m}$
E.8.1 $\times$ 10^6 $\text{ m}$

Answer: A

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Question 17

1 mark

Let

x = 0.2\dot{5}

and

y = 0.\dot{2}\dot{5}

. What is the value of

x - y

expressed as a fraction in its simplest form?

A. $\frac{1}{330}$
B. $\frac{1}{33}$
C. $\frac{1}{110}$
D. $\frac{5}{198}$
E. $\frac{1}{90}$

Answer: A

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Question 18

1 mark

A chemical process involves mixing three solutions of acid:

X

,

Y

, and

Z

.

Solution

X

contains

40\%

acid by volume.

Solution

Y

contains

\frac{1}{4}

acid by volume.

Solutions

X

and

Y

are mixed in the ratio

2 : 3

by volume to create a base mixture

M

.

A final mixture is then formed by combining mixture

M

with solution

Z

in the ratio

5 : 2

by volume.

If the final mixture is

25\%

acid by volume, what is the acid concentration of solution

Z

as a percentage?

A. $10\%$
B. $12.5\%$
C. $15\%$
D. $17.5\%$
E. $20\%$

Answer: A

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Question 19

1 mark

The wholesale cost of a machine is first increased by

25\%

to determine its retail price.

During a promotion, the retail price is discounted to a 'special price' which is

10\%

less than the original wholesale cost.

The special price is subsequently increased by

33\frac{1}{3}\%

to give a final price.

What is the ratio of the final price to the retail price?

A. $24 : 25$
B. $25 : 24$
C. $4 : 5$
D. $1 : 1$
E. $5 : 6$

Answer: A

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Question 20

1 mark

Simplify the following expression as far as possible:

\frac{10}{\sqrt{5}} + \frac{1}{\sqrt{5} - 2}

A. $2\sqrt{5} + 2$
B. $3\sqrt{5} - 2$
C. $3\sqrt{5} + 2$
D. $11\sqrt{5} + 2$
E. $\sqrt{5} + 2$

Answer: C

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Question 21

1 mark

A car travels a distance of

120\text{ km}

, measured correct to the nearest

10\text{ km}

. The time taken for the journey is

2.0\text{ hours}

, measured correct to the nearest

0.1\text{ hours}

. Which one of the following expressions gives the difference between the maximum possible average speed and the minimum possible average speed for this journey?

A. $\frac{12800}{1599}\text{ km/h}$
B. $\frac{12200}{1599}\text{ km/h}$
C. $\frac{12800}{1600}\text{ km/h}$
D. $5\text{ km/h}$
E. $\frac{12000}{1599}\text{ km/h}$

Answer: A

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Question 22

1 mark

A rectangular block has dimensions

20\text{ cm}

by

30\text{ cm}

by

40\text{ cm}

, each measured correct to the nearest

2\text{ cm}

. What is the difference between the maximum possible volume and the minimum possible volume of the block?

A. $5202\text{ cm}^3$
B. $5200\text{ cm}^3$
C. $10416\text{ cm}^3$
D. $2601\text{ cm}^3$
E. $4800\text{ cm}^3$

Answer: A

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Question 23

1 mark

A cylindrical copper pipe has an outer diameter of

100\,\text{mm}

and an inner diameter of

90\,\text{mm}

. The pipe is

2\,\text{m}

long. Given that the density of copper is

8960\,\text{kg}\,\text{m}^{-3}

, which of the following is the best estimate for the mass of the pipe?

A. $3\,\text{kg}$
B. $9\,\text{kg}$
C. $27\,\text{kg}$
D. $54\,\text{kg}$
E. $110\,\text{kg}$

Answer: C

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Question 24

1 mark

A rectangular nature reserve has an area of

8\,\text{cm}^2

on a map with a scale of

1 : 25,000

. What is the actual area of the reserve in square kilometres

(\text{km}^2)

?

A.0.2
B.0.5
C.2.0
D.5.0
E.20.0

Answer: B

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Question 25

1 mark

The distance between two cities

X

and

Y

is measured on two different maps. On Map A, with a scale of

1 : n

, the distance is

12\,\text{cm}

. On Map B, with a scale of

1 : (n + 5000)

, the distance is

10\,\text{cm}

. What is the actual distance between the two cities in kilometres?

A.0.3
B.2.5
C.3.0
D.6.0
E.30.0

Answer: C

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Question 26

1 mark

A construction company has two teams, Alpha and Beta. Team Alpha can complete a specific project in

10

days. When Team Alpha and Team Beta work together, they can complete the same project in

4

days. Assuming both teams work at constant daily rates, express the daily work rate of Team Beta as a fraction of the daily work rate of Team Alpha.

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

Answer: C

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Question 27

1 mark

An alloy consists of three metals: copper, zinc, and nickel. The ratio of the mass of copper to the mass of zinc is

3:2

. The ratio of the mass of zinc to the mass of nickel is

4:3

. What fraction of the total mass of the alloy is copper?

A. $\frac{6}{13}$
B. $\frac{3}{5}$
C. $\frac{3}{13}$
D. $\frac{4}{7}$
E. $\frac{1}{4}$

Answer: A

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