Question 1

1 mark

A laboratory culture dish has a surface area of

5.0 \times 10^{-3} \text{ m}^2

. A population of bacteria is grown such that it forms a single layer covering

60\%

of the total surface area of the dish. Each individual bacterium is modeled as a square with a side length of

2.0 \times 10^{-6} \text{ m}

.

How many bacteria are in the dish, expressed in standard form?

A. $7.5 \times 10^8$
B. $1.25 \times 10^9$
C. $1.5 \times 10^9$
D. $7.5 \times 10^9$
E. $1.5 \times 10^3$

Answer: A

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Question 2

1 mark

A material is composed of three components:

X, Y,

and

Z

. Initially, component

X

makes up

30\%

of the total mass, and component

Y

makes up

\frac{2}{5}

of the total mass. The remainder of the mass is component

Z

.

The material then undergoes two sequential changes:

1. First,

25\%

of the mass of component

X

currently in the material is removed and replaced with an equal mass of component

Z

.
2. Second, an additional mass of component

Y

is added to the material such that the total mass of the material increases by

20\%

.

What fraction of the final material's mass is component

Z

?

A. $\frac{5}{16}$
B. $\frac{3}{8}$
C. $\frac{1}{4}$
D. $\frac{19}{48}$
E. $\frac{1}{3}$

Answer: A

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Question 3

1 mark

Simplify the following expression as far as possible:

\frac{\sqrt{14 - 6\sqrt{5}}}{\sqrt{5} - 2} + \frac{4}{\sqrt{5} + 1}

A. $2\sqrt{5}$
B. $2$
C. $2\sqrt{5} + 2$
D. $2\sqrt{5} - 2$
E. $0$

Answer: A

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Question 4

1 mark

The initial velocity

u

of a particle is measured to be

10\text{ m s}^{-1}

, correct to the nearest

2\text{ m s}^{-1}

. The final velocity

v

of the particle is measured to be

30\text{ m s}^{-1}

, correct to the nearest

2\text{ m s}^{-1}

. The time

t

taken for this change in velocity is measured to be

5.0\text{ s}

, correct to the nearest

0.5\text{ s}

. Assuming the acceleration

a

is constant, which one of the following expressions gives the difference between the maximum possible value of

a

and the minimum possible value of

a

?

A. $\frac{160}{133}\text{ m s}^{-2}$
B. $\frac{160}{399}\text{ m s}^{-2}$
C. $\frac{4}{5}\text{ m s}^{-2}$
D. $\frac{320}{399}\text{ m s}^{-2}$
E. $\frac{800}{399}\text{ m s}^{-2}$

Answer: A

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Question 5

1 mark

The length

L

of a rectangle is measured as

12\,\text{cm}

, correct to the nearest centimetre. The width

W

of the same rectangle is

5\,\text{cm}

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

A

is the area of the rectangle, which of the following is the correct error interval for

A

?

A. $57.5\,\text{cm}^2 \le A < 75.0\,\text{cm}^2$
B. $57.5\,\text{cm}^2 < A < 75.0\,\text{cm}^2$
C. $51.75\,\text{cm}^2 \le A < 68.75\,\text{cm}^2$
D. $51.75\,\text{cm}^2 < A < 68.75\,\text{cm}^2$
E. $60.0\,\text{cm}^2 \le A < 78.0\,\text{cm}^2$

Answer: A

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Question 6

1 mark

A nature reserve has an actual area of

4.5\,\text{km}^2

. On a map with a scale of

1 : 50,000

, what is the area of the reserve in square centimetres

(\text{cm}^2)

?

A.1.125
B.2.25
C.9.0
D.18.0
E.22.5

Answer: D

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Question 7

1 mark

A tank is being filled by two pipes,

P

and

Q

, each operating at a constant rate. Pipe

P

can fill the entire tank alone in

15

minutes. When both pipes

P

and

Q

work together, they can fill the entire tank in

6

minutes. Express the volume of water delivered by Pipe

P

in

10

minutes as a fraction of the volume of water delivered by Pipe

Q

in

8

minutes.

A. $\frac{2}{3}$
B. $\frac{5}{6}$
C. $\frac{6}{5}$
D. $\frac{3}{2}$
E. $\frac{5}{4}$

Answer: B

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Question 8

1 mark

A chemical mixture is composed of three substances: P, Q, and R. The ratio of the mass of P to the mass of Q is

4:3

, and the ratio of the mass of Q to the mass of R is

5:2

. The initial total mass of the mixture is

1.64\,\text{kg}

. A quantity of substance R is added to the mixture such that the ratio of the mass of P to the mass of R becomes

5:3

. What mass of substance R, in grams, was added?

A. $120$
B. $200$
C. $240$
D. $360$
E. $480$

Answer: C

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Question 9

1 mark

Liquid

P

is a mixture of salt and water in the ratio

1 : 3

by volume. Liquid

Q

is also a mixture of salt and water. When liquid

P

and liquid

Q

are mixed in the ratio

2 : 1

by volume, the concentration of salt by volume in the resulting mixture is

C_1

. When they are mixed in the ratio

1 : 2

by volume, the concentration of salt by volume in the resulting mixture is

C_2

. If

C_1 : C_2 = 7 : 8

, what is the ratio of salt to water in liquid

Q

in its simplest form?

A. $3 : 5$
B. $5 : 3$
C. $3 : 8$
D. $1 : 1$
E. $1 : 5$

Answer: A

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Question 10

1 mark

A fruit punch is a mixture of orange juice, apple juice, and grape juice.

The ratio of the volume of orange juice to the volume of apple juice is

3:2

.

The ratio of the volume of apple juice to the volume of grape juice is

4:5

.

What fraction of the total volume of the fruit punch is apple juice?

A. $\frac{4}{15}$
B. $\frac{1}{3}$
C. $\frac{2}{5}$
D. $\frac{4}{9}$
E. $\frac{3}{5}$

Answer: A

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Question 11

1 mark

A scientist creates a mixture

M

by combining two reagents,

R_1

and

R_2

. Reagent

R_1

is a mixture of compounds

X

and

Y

in the ratio

2:3

by mass. Reagent

R_2

is a mixture of compounds

X

and

Z

in the ratio

5:2

by mass. The two reagents are mixed in a proportion such that compound

X

makes up exactly

\frac{1}{2}

of the total mass of the final mixture

M

. What fraction of the total mass of mixture

M

is compound

Y

?

A. $\frac{1}{11}$
B. $\frac{3}{11}$
C. $\frac{7}{22}$
D. $\frac{9}{22}$
E. $\frac{15}{22}$

Answer: D

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Question 12

1 mark

A saline solution contains

15\%

salt by mass. After

40~\text{g}

of pure salt is added to the solution and completely dissolved, the new solution contains

25\%

salt by mass. What was the mass, in

\text{g}

, of the original saline solution?

A.200
B.300
C.340
D.400
E.440

Answer: B

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Question 13

1 mark

Which one of the following expressions is equivalent to

\frac{(25x^4 y^{-2})^{1/2}}{(5x^{-1} y)^2}

for all positive real numbers

x

and

y

?

A. $\frac{x^4}{5y^3}$
B. $\frac{1}{5y^3}$
C. $\frac{x^4}{y^3}$
D. $\frac{5x^4}{y^3}$
E. $\frac{x^4}{5y}$

Answer: A

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Question 14

1 mark

A metallic alloy is composed of three metals:

X

,

Y

, and

Z

.
The masses of the three metals in the alloy are in the ratio

3 : 2 : 1

respectively.
The density of metal

X

is

\rho_x

, the density of metal

Y

is

\rho_y

, and the density of metal

Z

is

\rho_z

.
Which of the following expressions represents the volume of metal

X

as a fraction of the total volume of the alloy?

A. $\frac{3\rho_y \rho_z}{3\rho_y \rho_z + 2\rho_x \rho_z + \rho_x \rho_y}$
B. $\frac{3\rho_x}{3\rho_x + 2\rho_y + \rho_z}$
C. $\frac{2\rho_x \rho_z}{3\rho_y \rho_z + 2\rho_x \rho_z + \rho_x \rho_y}$
D. $\frac{\rho_x \rho_y}{3\rho_y \rho_z + 2\rho_x \rho_z + \rho_x \rho_y}$
E. $\frac{\rho_y \rho_z}{\rho_y \rho_z + \rho_x \rho_z + \rho_x \rho_y}$

Answer: A

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Question 15

1 mark

Which one of the following is a simplification of the expression

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

?

A. $24x + 35$
B. $24x + 37$
C. $37$
D. $12x + 37$
E. $8x^2 + 24x + 35$

Answer: B

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Question 16

1 mark

Which one of the following is a simplification of

\frac{2x^2 + 7x - 15}{4x^2 - 9} - \frac{x - 2}{2x + 3}

?

A. $1$
B. $\frac{3}{2x + 3}$
C. $\frac{7}{2x + 3}$
D. $\frac{7}{2x - 3}$
E. $\frac{x + 7}{2x + 3}$

Answer: C

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Question 17

1 mark

Which one of the following is a simplification of

2 - \frac{x^2 - x - 6}{x^2 + 2x} \div \frac{x^2 - 9}{x^2}

A. $\frac{x + 6}{x + 3}$
B. $\frac{x}{x + 3}$
C. $\frac{3x + 6}{x + 3}$
D. $\frac{x - 6}{x + 3}$
E. $\frac{x - 3}{x}$

Answer: A

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Question 18

1 mark

Let

a, b,

and

c

be real constants. Consider the following equation in

x

:

a(x + 1)^2 + b(x - 1)^2 = c(x^2 + 1) + 4x

Consider these three conditions:
1.

a = 1, b = -1

and

c = 0

2.

a + b = c

3.

a - b = 2

Which of these conditions is/are **necessary** for the equation to be an **identity** in

x

?

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

Answer: E

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Question 19

1 mark

The points

P(-1, 6)

and

Q(5, -2)

are the endpoints of a diagonal of a square. What is the area of the square?

A.25
B.40
C.50
D.80
E.100

Answer: C

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Question 20

1 mark

A straight line

L_1

with a positive gradient

m

passes through the point

(4, 3)

. A second line

L_2

is perpendicular to

L_1

and also passes through the point

(4, 3)

. Both lines intersect the

y

-axis,

L_1

at point

P

and

L_2

at point

Q

. If the area of the triangle with vertices at

P

,

Q

and the intersection point

(4, 3)

is 20 square units, what is the sum of all possible values of

m

?

A.2.0
B.2.5
C.3.0
D.4.0
E.5.0

Answer: B

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Question 21

1 mark

The curve with equation

y = x^2 - 6x + 5

intersects the

x

-axis at two points,

P

and

Q

. The vertex (turning point) of the curve is the point

V

. What is the area of the triangle

PQV

?

A.4
B.8
C.10
D.12
E.16

Answer: B

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Question 22

1 mark

The graph of the reciprocal function

y = \frac{1}{x}

is translated by the vector

\begin{pmatrix} -1 \ 2 \end{pmatrix}

to produce the curve

C

. At how many distinct points does

C

intersect the graph of the quadratic function

y = x^2 + 2x

?

A.0
B.1
C.2
D.3
E.4

Answer: D

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Question 23

1 mark

A car travels a total distance of

120\,\text{km}

. The journey is split into two equal stages of

60\,\text{km}

each. In the first stage, the car travels at a constant speed of

40\,\text{km\,h}^{-1}

. In the second stage, it travels at a constant speed of

v\,\text{km\,h}^{-1}

. The average speed for the whole journey is denoted by

V_{avg}

. Which of the following correctly describes the graph of

V_{avg}

against

v

for

v > 0

?

A.A straight line passing through the points $(0, 20)$ and $(40, 40)$ .
B.A curve starting at the origin, increasing with a decreasing gradient, and approaching a horizontal asymptote at $V_{avg} = 40\,\text{km\,h}^{-1}$ .
C.A curve starting at the origin, increasing with a decreasing gradient, and approaching a horizontal asymptote at $V_{avg} = 80\,\text{km\,h}^{-1}$ .
D.A curve starting at the origin, increasing with a decreasing gradient, and approaching a horizontal asymptote at $V_{avg} = 120\,\text{km\,h}^{-1}$ .
E.A curve starting at the origin, increasing with an increasing gradient, such that $V_{avg}$ is proportional to $v^2$ .

Answer: C

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Question 24

1 mark

The velocity

v

(in

\text{m s}^{-1}

) of a particle at time

t

(in

\text{s}

) is proportional to the square of the time for

0 \le t \le 6

. At

t = 6\,\text{s}

, the acceleration of the particle is

4\,\text{m s}^{-2}

.

An estimate for the total distance travelled by the particle over the first

6\,\text{s}

is calculated using the trapezium rule with three strips of equal width.

What is the ratio of this estimated distance to the exact distance travelled?

A. $10/9$
B. $19/18$
C. $37/36$
D. $9/8$
E. $19/27$

Answer: B

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Question 25

1 mark

Consider the simultaneous equations:

y = 2x + c

y = x^2 - 4x + 7

where

c

is a real constant. Suppose these equations have two distinct real solutions for

x

, denoted by

x_1

and

x_2

. If

x_1^2 + x_2^2 = 20

, what is the value of

c

?

A.-11
B.-1
C.1
D.9
E.15

Answer: B

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Question 26

1 mark

A sequence

u_1, u_2, u_3, \dots

is defined by the recurrence relation

u_{n+1} = u_n + 2n - 1

for all integers

n \ge 1

. Given that

u_1 = 5

, what is the value of the term

u_{21}

?

A.366
B.400
C.405
D.441
E.446

Answer: C

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Question 27

1 mark

ABCDEF

is a regular hexagon. A square

ABGH

is constructed outside the hexagon on the side

AB

, and an equilateral triangle

BCI

is constructed outside the hexagon on the side

BC

.

What is the size of the angle

\angle GCI

?

A. $15^\circ$
B. $30^\circ$
C. $45^\circ$
D. $60^\circ$
E. $75^\circ$

Answer: C

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Question 28

1 mark

In the trapezium

ABCD

, the side

AB

is parallel to

CD

. The diagonals

AC

and

BD

intersect at right angles.
The length of

AB

is

4\,\text{cm}

and the length of

CD

is

9\,\text{cm}

.
The length of the diagonal

AC

is

5\,\text{cm}

.
What, in square centimetres, is the area of the trapezium

ABCD

?

A.26
B.30
C.32.5
D.42.25
E.60

Answer: B

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Question 29

1 mark

In triangle

ABC

, the point

D

lies on the side

AB

and the point

E

lies on the side

AC

such that the line segment

DE

is parallel to

BC

. The point

F

lies on the side

BC

such that the line segment

EF

is parallel to

AB

.

Given that the area of triangle

ADE

is

X

and the area of triangle

EFC

is

Y

, what is the area of the parallelogram

BDEF

?

A. $\sqrt{XY}$
B. $2\sqrt{XY}$
C. $X + Y$
D. $\frac{X + Y}{2}$
E. $(\sqrt{X} + \sqrt{Y})^2$

Answer: B

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Question 30

1 mark

Point

P

lies on the sphere with equation

x^2 + y^2 + z^2 - 2x - 2y - 2z = 13

.

Point

Q

lies on the sphere with equation

x^2 + y^2 + z^2 - 6x - 14y - 20z + 154 = 0

.

What is the minimum possible distance between

P

and

Q

?

A.5
B.7
C.9
D.11
E.17

Answer: A

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ESAT Mock Maths 1 Esat-maths1-bank-3

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