Question 1

1 mark

The first term of a convergent geometric series is

12

. The sum of the first two terms of the series is

15

.

What is the sum to infinity of this series?

A. $15$
B. $16$
C. $18$
D. $48$
E. $60$

Answer: B

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Question 2

1 mark

The first three terms in the expansion of

(1 + ax)^n

in ascending powers of

x

are

1

,

20x

, and

180x^2

, where

a

is a constant and

n

is a positive integer. What is the coefficient of

x^3

in this expansion?

A.120
B.240
C.480
D.840
E.960

Answer: E

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Question 3

1 mark

The coefficient of

x^2

in the expansion of

(k + x^2)^5

in ascending powers of

x

is

80

, where

k

is a positive constant. What is the coefficient of

x^3

in the expansion of

(1 + kx)^6

?

A.20
B.40
C.80
D.160
E.320

Answer: D

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Question 4

1 mark

A line

L_1

passes through the midpoint of the line segment joining the points

A(1, 8)

and

B(5, 2)

.

L_1

is perpendicular to the line with equation

2x - 3y + 5 = 0

. The line

L_1

intersects a second line

L_2

at a point on the

x

-axis. If the equation of

L_2

is

y = mx + 1

, what is the value of

m

?

A. $-\frac{3}{19}$
B. $\frac{2}{9}$
C. $-\frac{1}{5}$
D. $-\frac{19}{3}$
E. $-\frac{3}{2}$

Answer: A

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Question 5

1 mark

The line

L

has the equation

(k+2)x + (1-k)y = k+5

, where

k

is a real constant. It can be shown that all such lines pass through a fixed point

P

. Find the equation of the straight line that is parallel to the line segment

PQ

and passes through the

y

-intercept of the line

2x - 3y + 6 = 0

, where

Q

is the point

(1, -2)

.

A. $3x - y + 2 = 0$
B. $3x + y - 2 = 0$
C. $x + 3y - 6 = 0$
D. $3x - y - 2 = 0$
E. $x - 3y + 6 = 0$

Answer: A

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Question 6

1 mark

A circle

C

is tangent to the

y

-axis at the point

(0, 4)

and passes through the point

(2, 0)

. A horizontal line is drawn with equation

y = 7

. What is the length of the chord formed by the intersection of the circle

C

and this line?

A.4
B.6
C.8
D. $2\sqrt{21}$
E.10

Answer: C

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Question 7

1 mark

A circle passes through the origin

O(0, 0)

and the points

A(6, 0)

and

B(0, 8)

. A point

P

is chosen on the circle such that the area of the quadrilateral

OAPB

is maximized. What is the value of this maximum area?

A.24
B.44
C.48
D.49
E.50

Answer: D

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Question 8

1 mark

In triangle

PQR

, the side

PQ = 5\text{ cm}

and the angle

\angle PQR = 60^{\circ}

. The area of the triangle is

5\sqrt{3}\text{ cm}^2

. Find the length of the side

PR

, in cm.

A. $\sqrt{21}$
B. $\sqrt{41}$
C. $\sqrt{61}$
D. $4$
E. $\sqrt{19}$

Answer: A

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Question 9

1 mark

A triangle

ABC

has side lengths

AB = 10\text{ cm}

and

BC = 5\sqrt{2}\text{ cm}

. The angle

\angle BAC = 30^{\circ}

. Which of the following describes all possible values for the angle

\angle ACB

?

A. $45^{\circ}$ only
B. $135^{\circ}$ only
C. $45^{\circ}$ or $135^{\circ}$
D. $60^{\circ}$ or $120^{\circ}$
E. $30^{\circ}$ or $150^{\circ}$

Answer: C

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Question 10

1 mark

A planar region is bounded by two concentric circular arcs of radii

r

and

3r

, and two straight line segments. The arcs subtend an angle of

heta

radians at their common centre. The area of the region is

A

and its perimeter is

P

. Given that

P^2 = 18A

, which one of the following is a possible value for

heta

?

A.0.5
B.1.0
C.1.5
D.2.5
E.3.0

Answer: A

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Question 11

1 mark

What is the value of the following expression?

\frac{\tan^2 60^\circ + 4 \sin^2 45^\circ}{6 \cos 60^\circ}

A. $1$
B. $\frac{4}{3}$
C. $\frac{5}{3}$
D. $2$
E. $\frac{7}{3}$

Answer: C

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Question 12

1 mark

How many solutions does the equation

\sin(4x) = \sin(2x)

have in the interval

0 \leq x \leq \pi

?

A.2
B.3
C.4
D.5
E.6

Answer: D

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Question 13

1 mark

What is the sum of all values of

x

in the range

0 \le x \le 2\pi

that satisfy the equation

2\cos^2 x = 3\sin x + 3

?

A. $\frac{3\pi}{2}$
B. $\frac{5\pi}{2}$
C. $3\pi$
D. $\frac{7\pi}{2}$
E. $\frac{9\pi}{2}$

Answer: E

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Question 14

1 mark

For

0 < x < \frac{\pi}{2}

, let

x

be the solution to the equation

\frac{1}{\cos^2 x} + \tan^2 x = 5

. What is the value of

\sin^2 x

?

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

Answer: C

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Question 15

1 mark

The graph of

y = 2^x

is translated by 2 units in the negative

x

-direction and 3 units in the positive

y

-direction to produce the graph of

y = f(x)

. The graph of

y = f(x)

intersects the graph of

y = 11 - 2^{x+1}

at the point

P(p, q)

. What is the value of

2^p

?

A. $\frac{2}{3}$
B. $\frac{4}{3}$
C.2
D. $\frac{7}{3}$
E.4

Answer: B

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Question 16

1 mark

What is the complete set of real values of

x

that satisfy the equation

2\log_3 x - \log_3\left(x - \frac{2}{3}\right) = 1

?

A. $x = 1$ or $x = 2$
B. $x = 1$ only
C. $x = 2$ only
D. $x = 3$ or $x = -1$
E. $x = 1$ or $x = -2$

Answer: A

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Question 17

1 mark

A particle moves in a straight line such that its displacement

s

(in metres) at time

t > 0

(in seconds) is given by

s = t^3 + \frac{k}{t}

, where

k

is a constant. The velocity

v

and acceleration

a

of the particle are defined as

v = \frac{ds}{dt}

and

a = \frac{d^2s}{dt^2}

. Given that the acceleration at

t = 2\,\text{s}

is

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

, what is the velocity of the particle at

t = 1\,\text{s}

?

A. $-5\,\text{m\,s}^{-1}$
B. $-1\,\text{m\,s}^{-1}$
C. $3\,\text{m\,s}^{-1}$
D. $4\,\text{m\,s}^{-1}$
E. $7\,\text{m\,s}^{-1}$

Answer: B

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Question 18

1 mark

The function

f

is defined for

x > 0

by

f(x) = \frac{x^2 + c}{x}

, where

c

is a constant. The gradient of the tangent to the graph

y = f(x)

at the point where

x = 2

is

\frac{3}{4}

. What is the value of

f''(4)

?

A. $\frac{1}{64}$
B. $\frac{1}{32}$
C. $\frac{1}{8}$
D. $\frac{1}{4}$
E. $1$

Answer: B

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Question 19

1 mark

A curve

C

has the equation

y = \frac{(x+a)^2}{\sqrt{x}}

where

a

is a real constant and

x > 0

.
The gradient of the tangent to

C

at the point where

x=1

is denoted by

G

.
What is the maximum possible value of

G

as

a

varies?

A.0.5
B.1
C.1.5
D.2
E.2.5

Answer: D

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Question 20

1 mark

The function

f(x) = \frac{1}{3}x^3 + kx^2 + (k+6)x + 5

, where

k

is a real constant, is strictly increasing for all real values of

x

.

Given that a function is strictly increasing if

f'(x) > 0

for all

x

, what is the complete set of possible values for

k

?

A. $k < -2$ or $k > 3$
B. $-2 < k < 3$
C. $k < -3$ or $k > 2$
D. $-3 < k < 2$
E. $-1 < k < 6$

Answer: B

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Question 21

1 mark

A line is drawn normal to the curve

y = \frac{8}{x}

at the point on the curve where

x = 2

.

This line intersects the

x

-axis at point

P

and the

y

-axis at point

Q

.

What is the area of the triangle

OPQ

, where

O

is the origin

(0,0)

?

A. $4$
B. $8$
C. $9$
D. $16$
E. $18$

Answer: C

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Question 22

1 mark

For a positive constant

k

, the area of the region strictly enclosed between the curve

y = x^2 - kx

and the

x

-axis is

36

square units. What is the total area of the region bounded by the curve

y = x^2 - kx

, the

x

-axis, and the vertical lines

x = 0

and

x = 2k

?

A.72
B.144
C.180
D.216
E.252

Answer: D

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Question 23

1 mark

What is the total area enclosed between the curve

y = 1 - \frac{9}{x^2}

, the

x

-axis, and the lines

x = 1

and

x = 9

?

A.0
B.4
C.8
D.10
E.16

Answer: C

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Question 24

1 mark

The function

f

is continuous for all

x

and satisfies the equation:

\int_{2}^{x} f(t) \, dt = x^4 - 8x^2 + 16

What is the value of

f(3)

?

A.25
B.60
C.76
D.84
E.108

Answer: B

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Question 25

1 mark

Let

f(x)

be a continuous function. Given that:

\int_0^2 (f(x) + f(2-x)) \, dx = 20

\int_0^1 (f(x) - f(2-x)) \, dx = 6

What is the value of

\int_1^2 f(x) \, dx

?

A.2
B.4
C.7
D.8
E.10

Answer: A

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Question 26

1 mark

The continuous function

f(x)

is defined for all real

x

and satisfies the relation

f(x) + f(x+2) = 3

. Given that

\int_0^2 f(x) \, dx = 5

, evaluate

\int_0^6 f(x) \, dx

.

A.3
B.7
C.9
D.11
E.15

Answer: D

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Question 27

1 mark

The function

y = f(x)

is such that its second derivative

f''(x)

is positive for all

x

in the interval

a \leq x \leq b

. The trapezium rule with

n

equal strips is used to estimate the integral

I = \int_a^b f(x) \, dx

.

Consider the following three statements:
I. The estimate is always greater than the exact value of

I

.
II. If the same rule is used to estimate

\int_a^b (f(x) - 10) \, dx

, the result will be an underestimate.
III. If the same rule is used to estimate

\int_a^b -2f(x) \, dx

, the result will be an underestimate.

Which of these statements is/are necessarily true?

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

Answer: D

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Question 28

1 mark

A curve

y = f(x)

has a gradient given by

\frac{dy}{dx} = \frac{3x^2 - 1}{\sqrt{x}}

for

x > 0

. Given that the curve passes through the point

(1, 4)

, what is the

y

-coordinate of the point on the curve where

x = 4

?

A. $\frac{196}{5}$
B. $\frac{176}{5}$
C. $\frac{156}{5}$
D. $\frac{118}{5}$
E. $\frac{108}{5}$

Answer: A

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Question 29

1 mark

A function

f(x)

satisfies the differential equation

\frac{dy}{dx} = |2x - 4|

for all real

x

. The function is continuous everywhere and

f(0) = 5

. Find the value of

f(3)

.

A.2
B.8
C.9
D.10
E.13

Answer: D

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Question 30

1 mark

How many distinct real solutions are there to the equation

||x| - 4| = \frac{1}{2}x + 2

?

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

Answer: D

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ESAT Mock Maths 2 Esat-maths2-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