Question 1

1 mark

A security code consists of four characters. The first two characters must be letters chosen from the set

\{P, Q, R\}

, and the last two characters must be digits chosen from the set

\{1, 2, 3\}

. Characters may be repeated. How many different such codes contain at least one letter

P

or at least one digit

1

?

A.16
B.25
C.45
D.65
E.81

Answer: D

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Question 2

1 mark

For positive real numbers

a

and

b

, let

k

and

m

be the constants that satisfy the following equations:

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

\sqrt{b \sqrt[3]{b}} = b^m

What is the value of

k + m

?

A.1
B. $\frac{7}{6}$
C. $\frac{3}{2}$
D. $\frac{5}{3}$
E. $\frac{11}{6}$

Answer: C

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Question 3

1 mark

A rectangular plot of land has an area of

1.0 \times 10^9\,\text{m}^2

and a perimeter of

1.3 \times 10^5\,\text{m}

. What is the ratio of the length of the plot to its width, expressed in standard form? (Assume that length

\ge

width.)

A.1.6 $\times$ 10^0
B.2.5 $\times$ 10^0
C.4.0 $\times$ 10^0
D.6.25 $\times$ 10^{-1}
E.1.6 $\times$ 10^1

Answer: A

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Question 4

1 mark

A fuel tank contains a mixture of petrol, ethanol, and a performance additive.

Petrol makes up

75\%

of the total volume of the mixture.

The performance additive makes up

\frac{1}{5}

of the volume that is not petrol.

The remaining

12\,\text{litres}

of the mixture is ethanol.

What is the total volume of the mixture in the tank?

A. $48\,\text{litres}$
B. $60\,\text{litres}$
C. $80\,\text{litres}$
D. $120\,\text{litres}$
E. $240\,\text{litres}$

Answer: B

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Question 5

1 mark

A hemispherical bowl has an internal radius of

\sqrt{30}\,\text{cm}

. The bowl is completely filled with a liquid that has a density of

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

. Given that the volume of a sphere is

V = \frac{4}{3}\pi r^3

, which of the following is the best estimate for the mass of the liquid in the bowl?

A.210\, $\text{g}$
B.360\, $\text{g}$
C.410\, $\text{g}$
D.620\, $\text{g}$
E.820\, $\text{g}$

Answer: C

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Question 6

1 mark

A metalworker has two different alloys of copper and tin. Alloy A contains copper and tin in the ratio

3 : 4

by mass. Alloy B contains copper and tin in the ratio

2 : 1

by mass. A sample is taken from each alloy such that the mass of tin in the sample of Alloy A is three times the mass of tin in the sample of Alloy B. Express the mass of copper in the sample of Alloy A as a fraction of the mass of copper in the sample of Alloy B.

A. $\frac{1}{8}$
B. $\frac{8}{9}$
C. $\frac{9}{8}$
D. $2$
E. $\frac{9}{2}$

Answer: C

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Question 7

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

4:3

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

5:2

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

A. $\frac{6}{41}$
B. $\frac{15}{41}$
C. $\frac{3}{7}$
D. $\frac{20}{41}$
E. $\frac{5}{7}$

Answer: B

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Question 8

1 mark

[diagram not to scale]

In a square

WXYZ

,

M

is a point on the side

WX

such that

WM : MX = 1 : 2

.

N

is a point on the side

XY

such that

XN : NY = k : 1

.

P

is the midpoint of the side

YZ

, meaning

YP : PZ = 1 : 1

.

The area of triangle

MNP

is

\frac{5}{18}

of the area of the square

WXYZ

.

What is the value of

k

?

A. $\frac{1}{2}$
B. $1$
C. $2$
D. $3$
E. $5$

Answer: C

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Question 9

1 mark

A construction mixture is made from sand, cement, and gravel. The ratio of the mass of sand to the mass of cement is

5:2

. The ratio of the mass of cement to the mass of gravel is

3:4

. If

rac{1}{5}

of the mass of the sand in the mixture is removed and replaced by an equal mass of gravel, what fraction of the total mass of the final mixture is cement?

A. $\frac{6}{29}$
B. $\frac{3}{13}$
C. $\frac{3}{16}$
D. $\frac{2}{11}$
E. $\frac{11}{29}$

Answer: A

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Question 10

1 mark

An object moves away from a radiation source such that its distance

r

from the source is directly proportional to the square root of the time

t

for which it has been moving. The intensity

I

of radiation recorded at the object's position is inversely proportional to the square of its distance

r

from the source.

If the time

t

is increased by 300%, what is the percentage decrease in the recorded intensity

I

?

A.25%
B.50%
C.75%
D.87.5%
E.93.75%

Answer: C

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Question 11

1 mark

Two mathematically similar solid prisms,

P

and

Q

, have total surface areas in the ratio

4 : 9

.

The volume of prism

P

is

24 \text{ cm}^3

.

What is the volume, in

\text{cm}^3

, of prism

Q

?

A.36
B.54
C.81
D.121.5
E.162

Answer: C

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Question 12

1 mark

Isotope

X

decays into isotope

Y

, which then decays into other stable products. At the start of a study, a sample contains

100\,\text{g}

of isotope

X

and no isotope

Y

. Each day, the following changes occur: (1) the mass of

X

decreases by

10\%

of the mass of

X

that was present at the start of that day, with all of the lost mass converting into isotope

Y

; and (2) the mass of

Y

decreases by

20\%

of the mass of

Y

that was present at the start of that day. What is the mass of isotope

Y

in the sample, in

\text{g}

, after 3 days?

A.17.0
B.17.36
C.21.7
D.27.1
E.30.0

Answer: C

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Question 13

1 mark

Simplify fully the following expression:

\frac{(2x^2y)^3}{x^2y + x^2y + x^2y + x^2y}

where

x

and

y

are non-zero.

A. $2x^3y^2$
B. $2x^4y^2$
C. $2x^4y^3$
D. $4x^4y^2$
E. $8x^4y^2$

Answer: B

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Question 14

1 mark

Which one of the following is a simplification of the expression

\frac{(4x^4 y^{-2})^{\frac{1}{2}} \times (2xy^3)^{-2}}{x^{-1}y}

where

x

and

y

are positive?

A. $\frac{x}{2y^8}$
B. $\frac{x}{2y^6}$
C. $\frac{1}{2xy^8}$
D. $\frac{x}{y^8}$
E. $\frac{x}{8y^8}$

Answer: A

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Question 15

1 mark

The heat capacity

C

of a particular substance at low temperatures is modeled by the formula:

C = aT + bT^3

where

T

is the absolute temperature in Kelvin, and

a

and

b

are constant coefficients.

An experiment yields the following measurements:
- At

T = 1\,\text{K}

,

C = 4\,\text{J\,K}^{-1}\text{mol}^{-1}

- At

T = 2\,\text{K}

,

C = 14\,\text{J\,K}^{-1}\text{mol}^{-1}

What is the predicted value of

C

at

T = 3\,\text{K}

?

A. $24\,\text{J\,K}^{-1}\text{mol}^{-1}$
B. $30\,\text{J\,K}^{-1}\text{mol}^{-1}$
C. $36\,\text{J\,K}^{-1}\text{mol}^{-1}$
D. $84\,\text{J\,K}^{-1}\text{mol}^{-1}$
E. $108\,\text{J\,K}^{-1}\text{mol}^{-1}$

Answer: C

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Question 16

1 mark

Which one of the following is an equivalent simplification of the expression

(x+1)^2(x+2)^2 - x(x+1)(x+2)(x+3)

?

A. $4$
B. $x^2 + 3x + 2$
C. $x^2 + 3x + 4$
D. $2(x+1)(x+2)$
E. $2x(x+3)$

Answer: D

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Question 17

1 mark

Which one of the following is a simplification of

2 - \frac{3x^2 - 10x + 3}{x^2 - 9} \times \frac{x^2 - x - 12}{3x^2 + 11x - 4}

for all values of

x

for which the expression is defined?

A. $1$
B. $\frac{x - 4}{x + 4}$
C. $\frac{x + 12}{x + 4}$
D. $\frac{3x + 4}{x + 4}$
E. $\frac{x + 12}{x - 4}$

Answer: C

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Question 18

1 mark

Which one of the following is a simplification of

\frac{4x^2 - 16}{x^2 - 2x} - \frac{x+1}{x}

?

A. $3 + \frac{7}{x}$
B. $3 + \frac{9}{x}$
C. $3 - \frac{7}{x}$
D. $3 - \frac{9}{x}$
E. $5 + \frac{9}{x}$

Answer: A

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Question 19

1 mark

The variables

x

and

y

are related by the equation:

y = \frac{1}{1 + \sqrt{\frac{x}{1-x}}}

where

0 < x < 1

and

y \neq 0

. Which one of the following is a rearrangement for

x

in terms of

y

?

A. $x = \frac{(1-y)^2}{2y^2 - 2y + 1}$
B. $x = \frac{1 - y^2}{2y^2 - 2y + 1}$
C. $x = \frac{(1-y)^2}{2y^2 + 2y + 1}$
D. $x = \frac{(1-y)^2}{1 - 2y}$
E. $x = \frac{y^2}{2y^2 - 2y + 1}$

Answer: A

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Question 20

1 mark

Consider the following three equations:

1.

(x + 4)^2 = x^2 + 16

2.

x^2 - 5x + 6 = (x - 2)(x - 3)

3.

4(x + 1) - 3x = x + 4

Which of these equations is/are identities for all real values of

x

?

A.none of them
B.2 only
C.3 only
D.2 and 3 only
E.1, 2 and 3

Answer: D

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Question 21

1 mark

A point

A

has coordinates

(-2, 5)

. Point

B

is the reflection of point

A

in the line

y = 1

. Point

C

is the reflection of point

B

in the line

x = 3

. Which one of the following is an equation of the circle that has the line segment

AC

as a diameter?

A. $x^2 + y^2 - 6x - 2y - 154 = 0$
B. $x^2 + y^2 - 6x - 2y - 31 = 0$
C. $x^2 + y^2 - 6x + 6y - 7 = 0$
D. $x^2 + y^2 + 4x - 2y - 11 = 0$
E. $x^2 + y^2 + 6x + 2y - 31 = 0$

Answer: B

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Question 22

1 mark

A straight line

L

is perpendicular to the line with equation

3x + 4y = 12

. The line

L

passes through the point

(3, 1)

. What is the area of the triangle bounded by the line

L

, the

x

-axis, and the

y

-axis?

A. $\frac{9}{4}$
B. $\frac{27}{8}$
C. $\frac{25}{24}$
D. $\frac{27}{4}$
E. $\frac{75}{8}$

Answer: B

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Question 23

1 mark

The speed

v

(in

\text{m s}^{-1}

) of an object moving in a straight line is modeled by the equation

v = \frac{1}{5}t^2 + 5

for

0 \le t \le 10

, where

t

is the time in seconds.

A student produces two estimates for the motion of the object:
1. An estimate for the total distance travelled between

t = 0

and

t = 10

using the trapezoidal rule with two intervals of equal width.
2. An estimate for the acceleration of the object at

t = 2.5

by calculating the gradient of the chord between

t = 0

and

t = 5

.

What are the two values obtained by the student?

A.distance: $125\,\text{m}$ , acceleration: $1.0\,\text{m s}^{-2}$
B.distance: $125\,\text{m}$ , acceleration: $2.0\,\text{m s}^{-2}$
C.distance: $150\,\text{m}$ , acceleration: $1.0\,\text{m s}^{-2}$
D.distance: $150\,\text{m}$ , acceleration: $2.0\,\text{m s}^{-2}$
E.distance: $75\,\text{m}$ , acceleration: $1.0\,\text{m s}^{-2}$

Answer: A

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Question 24

1 mark

Two real numbers

x

and

y

satisfy the simultaneous equations:

y - x = 3

x^2 + y^2 = 29

What is the sum of all possible values of

x

?

A. $-5$
B. $-3$
C. $0$
D. $3$
E. $5$

Answer: B

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Question 25

1 mark

Consider the equation

\frac{x^2 - k}{2x - 3} = 4

where

x

is a real variable and

k

is a real constant. Find the complete set of values of

k

for which the equation has two distinct real solutions for

x

.

A. $k > -4$
B. $k > -4$ and $k \neq \frac{3}{2}$
C. $k > -4$ and $k \neq \frac{9}{4}$
D. $k > 4$
E. $k > 4$ and $k \neq \frac{9}{4}$

Answer: C

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Question 26

1 mark

A region in the

xy

-plane is defined by the following two inequalities:

y - 2x \le 2

x + y \ge 5

Consider the following three statements:
1.

x \ge 1

2.

y \ge 4

3.

2x + y \ge 6

Which of these statements must be true for every point

(x, y)

in the region?

A.1 only
B.3 only
C.1 and 2 only
D.1 and 3 only
E.1, 2 and 3

Answer: D

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Question 27

1 mark

A sequence of numbers

a_1, a_2, a_3, \dots

is defined by the following rules:

a_1 = 2

a_{n+1} = 1 - \frac{1}{a_n}

for

n \ge 1

What is the value of the product

a_1 \times a_2 \times a_3 \times \dots \times a_{100}

?

A. $-2$
B. $-1$
C. $-\frac{1}{2}$
D. $1$
E. $2$

Answer: A

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Question 28

1 mark

The first four terms of a quadratic sequence

u_n

are:

8, \quad 16, \quad 26, \quad 38

A linear sequence

v_n

has

3^{rd}

term

v_3 = 32

and

5^{th}

term

v_5 = 56

.

The

n^{th}

terms of these two sequences are equal for two different values of

n

.

What is the sum of these two values of

n

?

A.5
B.6
C.7
D.8
E.12

Answer: C

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Question 29

1 mark

A circle has the equation

x^2 + y^2 - 4x + 2y + 1 = 0

.

The circle is subjected to the following three transformations in the order given:

1. An enlargement by a scale factor of

-3

about the point

(1, 2)

.
2. A rotation of

90^\circ

anticlockwise about the origin

(0, 0)

.
3. A translation by the vector

\begin{pmatrix} 5 \ -4 \end{pmatrix}

.

What are the coordinates of the centre of the resulting circle?

A. $(-6, -6)$
B. $(12, 0)$
C. $(16, -2)$
D. $(2, -10)$
E. $(-7, 3)$

Answer: A

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Question 30

1 mark

A straight line has the equation

3x - 4y + 12 = 0

. This line cuts the

x

-axis at point

P

and the

y

-axis at point

Q

. Point

M

is the midpoint of the line segment

PQ

.

What is the distance from the origin

(0, 0)

to

M

?

A. $1.5$
B. $2.0$
C. $2.5$
D. $3.5$
E. $5.0$

Answer: C

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ESAT Mock Maths 1 Esat-maths1-bank-2

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21

Question 22

Question 23

Question 24

Question 25

Question 26

Question 27

Question 28

Question 29

Question 30