Question 1

1 mark

Which of the following functions

f(x)

, defined for all real numbers

x

, is a one-to-one mapping?

A. $f(x) = |x + 2|$
B. $f(x) = x^2 - 2x$
C. $f(x) = x^3 + x$
D. $f(x) = x^4 - 1$
E. $f(x) = \sin x$

Answer: C

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Question 2

1 mark

The sequence

x_k

is defined by

x_1 = 5

and the recurrence relation

x_{n+1} = 1 - \frac{1}{x_n}

for

n \ge 1

. What is the value of the sum

\sum_{k=1}^{61} x_k

?

A.111.00
B.112.85
C.116.00
D.116.55
E.338.55

Answer: C

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Question 3

1 mark

The sequence

a_n

is defined by the formula

a_n = 1 + \frac{1}{n^2 + 2n}

for

n \ge 1

. What is the value of the product

a_1 \times a_2 \times \dots imes a_{48}

?

A.0.98
B.1.92
C.1.96
D.1.98
E.2.00

Answer: C

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Question 4

1 mark

An arithmetic series has first term

a

and common difference

d

. Let

S_n

denote the sum of the first

n

terms of the series. If

S_{10} = 5S_5

, which of the following expresses the relationship between

a

and

d

?

A. $d = -3a$
B. $d = 3a$
C. $a = -3d$
D. $d = -a$
E. $a = 3d$

Answer: A

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Question 5

1 mark

The line

L_1

passes through the points

A(-2, 5)

and

B(4, 2)

. A second line

L_2

is perpendicular to

L_1

and passes through the midpoint of the line segment

AB

. Which of the following is the equation of

L_2

?

A.4x - 2y + 3 = 0
B.4x - 2y - 1 = 0
C.x + 2y - 8 = 0
D.4x + 2y - 11 = 0
E.2x - 4y + 13 = 0

Answer: A

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Question 6

1 mark

The lines with equations

3x - y + 4 = 0

and

x + 2y - 1 = 0

intersect at the point

P

. A third line

L

passes through

P

and is parallel to the line

y = 5x - 7

. What is the equation of the line

L

?

A.5x - y + 6 = 0
B.5x - y - 4 = 0
C.x + 5y - 4 = 0
D.5x + y + 4 = 0
E.3x - y + 4 = 0

Answer: A

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Question 7

1 mark

A triangle

ABC

has side lengths

AB = 12

and

BC = 8

. The angle

\angle BAC = \theta

is such that two non-congruent triangles are possible for this specific value of

\theta

. If the difference between the areas of these two possible triangles is

16\sqrt{3}

, what is the value of

\sin \theta

?

A. $\frac{1}{6}$
B. $\frac{1}{3}$
C. $\frac{1}{2}$
D. $\frac{2}{3}$
E. $\frac{3}{4}$

Answer: B

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Question 8

1 mark

A chord of length

L

in a circle of radius

r

divides the circle into two segments. The area of the minor segment is

A_{\text{seg}}

and the area of the triangle formed by the chord and the centre of the circle is

A_{\text{tri}}

. Given that

3\sqrt{3}(A_{\text{seg}} + A_{\text{tri}}) = 4\pi A_{\text{tri}}

, which one of the following is the value of

L

?

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

Answer: D

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Question 9

1 mark

A sequence of right-angled triangles

T_1, T_2, T_3, \dots

is constructed such that for each

n \geq 1

:
- The hypotenuse of

T_1

has length

1

.
- Each triangle

T_n

contains an angle of

30^\circ

.
- The hypotenuse of

T_{n+1}

is the side of

T_n

adjacent to the

30^\circ

angle.
Find the sum of the areas of all triangles in the infinite sequence.

A. $\frac{\sqrt{3}}{4}$
B. $\sqrt{3}$
C. $\frac{\sqrt{3}}{2}$
D. $\frac{\sqrt{3}}{6}$
E. $\frac{1}{2}$

Answer: C

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Question 10

1 mark

Find the complete set of values of

x

in the interval

0 \le x \le \pi

for which the inequality

\cos(2x) < \cos(x)

is satisfied.

A. $0 < x < \frac{\pi}{3}$
B. $0 < x < \frac{2\pi}{3}$
C. $\frac{\pi}{3} < x < \frac{2\pi}{3}$
D. $\frac{2\pi}{3} < x < \pi$
E. $\frac{\pi}{3} < x < \pi$

Answer: B

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Question 11

1 mark

How many solutions does the equation

\sin(2x) = \sin\left(\frac{x}{2}\right)

have in the interval

0 \le x \le 2\pi

?

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

Answer: D

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Question 12

1 mark

An angle

\theta

satisfies the conditions

\tan \theta = \frac{3}{4}

and

\cos \theta < 0

. What is the value of

\sin \theta

?

A. $-0.8$
B. $-0.6$
C. $0.6$
D. $0.8$
E. $0.75$

Answer: B

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Question 13

1 mark

Find the sum of all possible values of

x

in the range

0 \le x \le 2\pi

which satisfy the equation

4 \sin^4 x + \cos^2 x = 1

Give your answer in radians.

A. $2\pi$
B. $3\pi$
C. $4\pi$
D. $5\pi$
E. $7\pi$

Answer: E

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Question 14

1 mark

Find the number of real solutions of the equation

\tan x + \cos x = \sec x

in the range

0 \le x \le 2\pi

.

A.1
B.2
C.3
D.4
E.5

Answer: C

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Question 15

1 mark

The graph of

y = 2^x

is reflected in the

y

-axis and then translated by

c

units in the positive

y

-direction to produce the graph of

y = g(x)

. The point

(-2, 13)

lies on the graph of

g

. What is the

y

-intercept of the graph of

g

?

A.8
B.9
C.10
D.11
E.13

Answer: C

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Question 16

1 mark

The curves

C_1

and

C_2

have equations

y = 2^x + 13

and

y = 2^{x+2} + 1

respectively. What is the

x

-coordinate of the point of intersection of

C_1

and

C_2

?

A.1
B.2
C.3
D.4
E.6

Answer: B

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Question 17

1 mark

Given that

b > 1

and

x

and

y

are positive real numbers such that:

\log_b(x^2 y) = 8

\log_b\left(\frac{x}{y^2}\right) = -1

what is the value of

\log_b(xy^2)

?

A.2
B.3
C.5
D.7
E.8

Answer: D

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Question 18

1 mark

The function

f

is defined for

x > 0

by

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

, where

a

and

b

are non-zero constants. The tangent to the graph

y = f(x)

at

x = 2

has the equation

y = 2x - 4

. What is the value of

f''(1)

?

A. $-\frac{14}{3}$
B. $-\frac{7}{3}$
C. $0$
D. $\frac{10}{3}$
E. $\frac{14}{3}$

Answer: A

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Question 19

1 mark

A curve has the equation

y = x^3 + ax^2 + bx + 5

, where

a

and

b

are constants. The gradient of the curve at

x = 1

is

1

. The rate of change of this gradient with respect to

x

is zero at

x = 2

. What is the minimum possible value of the gradient of this curve?

A. $-12$
B. $-6$
C. $-2$
D. $1$
E. $10$

Answer: C

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Question 20

1 mark

The curve

C

has equation

y = x^2 - \frac{16}{3}x\sqrt{x} + 10x

for

x > 0

.

What is the minimum possible value of the gradient of

C

?

A.2
B.4
C.0
D.10
E.-8

Answer: A

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Question 21

1 mark

The curve

C

has the equation

y = 3x^2 - 12x + 7

. The line

L

is normal to the curve

C

at the point where

x = 1

.

The

x

-coordinate of the point where the line

L

cuts the x-axis is

A. $-11$
B. $1$
C. $11$
D. $13$
E. $15$

Answer: D

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Question 22

1 mark

The function

f(x) = x^3 + bx^2 + 12x + 5

, where

b

is a real constant, is defined for all real

x

.

The complete set of values of

b

for which

f'(x) > 0

for all real

x

is

A. $b < -6$ or $b > 6$
B. $b > -6$
C. $-6 < b < 6$
D. $-6 \leq b \leq 6$
E. $-12 < b < 12$

Answer: C

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Question 23

1 mark

What is the area of the finite region bounded by the curve

y = \frac{(x-1)^2}{\sqrt{x}}

, the

x

-axis, and the lines

x = 1

and

x = 4

?

A. $\frac{76}{15}$
B. $\frac{16}{15}$
C. $\frac{92}{15}$
D. $\frac{46}{15}$
E. $\frac{106}{15}$

Answer: A

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Question 24

1 mark

The gradient of a curve is given by

\frac{dy}{dx} = \sqrt{x} \left( x + \frac{1}{x^2} \right)

for

x > 0

. Given that the curve passes through the point

(4, 10)

, find the

y

-coordinate of the point on the curve where

x = 1

.

A. $-\frac{17}{5}$
B. $-\frac{7}{5}$
C. $\frac{1}{5}$
D. $-\frac{9}{5}$
E. $\frac{3}{5}$

Answer: A

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Question 25

1 mark

A continuous function

f

is defined for all

t > 0

and satisfies the equation:

\int_4^{x^2} f(t) \, dt = x^3 - 8

for all

x > 0

. What is the value of

f(9)

?

A.1.5
B.3.0
C.4.5
D.13.5
E.27.0

Answer: C

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Question 26

1 mark

A continuous function

f(x)

satisfies the following conditions:

\int_0^4 f(x) \, dx = 10

\int_2^4 f(x) \, dx = 6

What is the value of

\int_0^1 [f(2x) + 3] \, dx

?

A.2
B.5
C.6
D.7
E.8

Answer: B

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Question 27

1 mark

A student uses the trapezium rule with

n

equal strips (

n > 1

) to estimate the area under each of the following three curves between

x = 0

and

x = 1

:

(1)

y = \ln(x + 1)

(2)

y = \ln\left(\frac{2}{x + 1}\right)

(3)

y = \ln(2 - x)

For which of these curves does the rule produce an underestimate?

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 28

1 mark

The trapezium rule with

n

equal strips is used to estimate the integral:

\int_{0}^{2} (2x - x^2) \, dx

Which of the following is the correct expression for the estimate produced?

A. $\frac{4n^2 - 3}{3n^2}$
B. $\frac{4(n^2 - 1)}{3n^2}$
C. $\frac{4(n^2 + 1)}{3n^2}$
D. $\frac{4n^2 - 2}{3n^2}$
E. $\frac{4(n - 1)}{3n}$

Answer: B

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Question 29

1 mark

The function

f(x) = \ln(x + a) + b

, where

a

and

b

are real constants, passes through the origin

(0, 0)

and the point

(e^2 - 1, 2)

. What is the value of

f(e - 1)

?

A. $0$
B. $1$
C. $2$
D. $e$
E. $e - 1$

Answer: B

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Question 30

1 mark

How many distinct real solutions are there to the equation

|(x - 1)^2 (x - 4)| = 2

?

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

Answer: D

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ESAT Mock Maths 2 Esat-maths2-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