ESAT Mock Maths 2 ESAT-MATHS2-MOCK-1: Questions & Worked Solutions

Question 1

1 mark

For

x > 0

, let

y

be defined by the expression:

y = \frac{(8x^6)^{\frac{1}{3}} \times (4x^2)^{\frac{3}{2}}}{2x^4}

Which one of the following is equal to

y

?

A. $2x$
B. $4x$
C. $8x$
D. $16x$
E. $32x$

Answer: C

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Question 2

1 mark

For

x > 0

, which one of the following is equivalent to the expression below?

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

A. $x + 4$
B. $x^2 + 4$
C. $x + 4x$
D. $x^2 + 4x$
E. $x^2 + 4x^2$

Answer: D

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Question 3

1 mark

The vertex of the parabola

y = x^2 + bx + c

lies on the line

y = x

. If the quadratic equation

x^2 + bx + c = 0

has two distinct real roots, which one of the following must be true?

A. $b > 0$
B. $b < 0$
C. $c > 0$
D. $c < 0$
E. $b^2 + 4c = 0$

Answer: A

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Question 4

1 mark

The quadratic function

f(x) = (c-1)x^2 + 4x + (c+2)

takes only positive values for all real values of

x

. What is the complete range of possible values for the constant

c

?

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

Answer: B

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Question 5

1 mark

The polynomial

f(x) = x^4 + kx^3 - x^2 + mx + 6

is exactly divisible by

x^2 + 2x - 3

, where

k

and

m

are constants.

What is the remainder when

f(x)

is divided by

(x + 1)

?

A. $-6$
B. $0$
C. $6$
D. $12$
E. $18$

Answer: D

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Question 6

1 mark

Which of the following functions

f(x)

, defined for all real numbers

x

, is a one-to-one mapping?

A. $f(x) = (x - 2)^2$
B. $f(x) = |x + 5|$
C. $f(x) = 1 - x^3$
D. $f(x) = \cos x$
E. $f(x) = x^2 + 4$

Answer: C

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Question 7

1 mark

The sequence

a_n

is defined by the formula

a_n = \frac{n}{(n+1)!}

for

n \ge 1

. Find the value of the sum

\sum_{k=1}^{10} a_k

.

A. $1 - \frac{1}{10!}$
B. $1 - \frac{1}{11!}$
C. $\frac{10}{11!}$
D. $1 + \frac{1}{11!}$
E. $1 - \frac{11}{12!}$

Answer: B

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Question 8

1 mark

A sequence

x_n

is defined by

x_1 = 2.5

and the recurrence relation

x_{n+1} = x_n^2 - 2

for

n \ge 1

. What is the value of

x_6

?

A. $2^{16} + 2^{-16}$
B. $2^{32} + 2^{-32}$
C. $2^{32} - 2^{-32}$
D. $2^{64} + 2^{-64}$
E. $2.5^{32}$

Answer: B

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Question 9

1 mark

The sum of the first

n

positive integers is given by

S_n = \sum_{i=1}^n i

. For a given positive integer

m

, let

n = 2m + 1

. If

S_n = k S_m

, which of the following is an expression for

k

in terms of

m

?

A. $k = \frac{2m+1}{m}$
B. $k = \frac{4m+2}{m}$
C. $k = \frac{4m+4}{m}$
D. $k = \frac{2m+2}{m}$
E. $k = \frac{m+1}{m}$

Answer: B

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Question 10

1 mark

The first three terms in the expansion of

(1 + ax)^n

in ascending powers of

x

are

1

,

24x

, and

264x^2

, where

a

is a constant and

n

is a positive integer. Find the value of

n - a

.

A.8
B.10
C.12
D.14
E.22

Answer: B

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Question 11

1 mark

The line

L

has the equation

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

P

and is perpendicular to the line

2x + 5y + 7 = 0

.

A. $5x - 2y - 13 = 0$
B. $5x - 2y + 17 = 0$
C. $2x + 5y - 11 = 0$
D. $2x - 5y - 1 = 0$
E. $5x + 2y - 17 = 0$

Answer: A

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Question 12

1 mark

A rectangle has two of its sides on the parallel lines

3x - 4y + 6 = 0

and

3x - 4y - 9 = 0

. A third side of the rectangle passes through the point

(1, 2)

. If the area of the rectangle is

15

square units, which of the following could be the equation of the fourth side?

A. $4x + 3y - 35 = 0$
B. $4x + 3y - 15 = 0$
C. $3x - 4y - 10 = 0$
D. $4x + 3y + 25 = 0$
E. $3x + 4y - 25 = 0$

Answer: A

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Question 13

1 mark

The points

P(1, 2)

and

Q(5, 6)

are the endpoints of a diameter of a circle. What is the area of this circle?

A. $4\pi$
B. $8\pi$
C. $16\pi$
D. $32\pi$
E. $64\pi$

Answer: B

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Question 14

1 mark

A circle with centre

(3, -4)

is tangent to the

x

-axis. Which of the following is the equation of this circle?

A. $x^2 + y^2 - 6x + 8y + 9 = 0$
B. $x^2 + y^2 - 6x + 8y + 16 = 0$
C. $x^2 + y^2 + 6x - 8y + 9 = 0$
D. $x^2 + y^2 - 6x + 8y + 25 = 0$
E. $x^2 + y^2 - 6x + 8y = 0$

Answer: A

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Question 15

1 mark

The circle

C

has the equation

x^2 + y^2 - 4x + 6y - 12 = 0

. What is the equation of the tangent to

C

at the point

(5, 1)

?

A. $4x - 3y = 17$
B. $3x - 4y = 11$
C. $3x + 4y = 19$
D. $4x + 3y = 23$
E. $3x + 4y = 25$

Answer: C

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Question 16

1 mark

A circle has the equation

(x - 1)^2 + (y + 2)^2 = 100

. A horizontal chord of this circle lies on the line

y = 6

. What is the length of this chord?

A.12
B.16
C.6
D.8
E.20

Answer: A

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Question 17

1 mark

A pyramid has a square base

PQRS

in the

xy

-plane with vertices at

P(0,0,0)

,

Q(4,0,0)

,

R(4,4,0)

, and

S(0,4,0)

. The vertex of the pyramid is at

V(2,2,6)

. Let

\theta

be the angle

\angle VQP

. Find

\cos \theta

.

0. $\frac{2\sqrt{11}}{11}$
A. $\frac{1}{11}$
B. $\frac{\sqrt{11}}{11}$
C. $\frac{\sqrt{11}}{22}$
E. $\frac{1}{2}$

Answer: B

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Question 18

1 mark

A triangle

ABC

has side length

AB = \sqrt{2}

and angle

\angle ABC = 135^{\circ}

. Given that the area of the triangle is

Z

, which of the following is an expression for the length of side

AC

?

A. $\sqrt{2Z^2 + 2Z + 2}$
B. $\sqrt{4Z^2 + 2}$
C. $\sqrt{4Z^2 - 4Z + 2}$
D. $\sqrt{4Z^2 + 4Z + 2}$
E. $2Z + \sqrt{2}$

Answer: D

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Question 19

1 mark

A sector of a circle is formed using a piece of wire of length

L

. The area of the sector is

A = rac{3L^2}{50}

. Which one of the following is a possible value for the angle of the sector in radians?

A.2
B.2.5
C.3
D.3.5
E.4

Answer: C

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Question 20

1 mark

Two sectors,

S_1

and

S_2

, are defined within the same circle of radius

r

. The angle subtended at the centre by

S_1

is

\alpha

radians and the angle subtended by

S_2

is

\beta

radians. Given that the area of

S_1

is three times the area of

S_2

, and the perimeter of

S_1

is exactly twice the perimeter of

S_2

, what is the value of

\alpha

?

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

Answer: E

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Question 21

1 mark

The graph of

y = 3^x

is reflected in the line

y = 9

to produce the graph of

y = f(x)

. The graph of

y = f(x)

intersects the graph of

y = 3^{x+2} - 9

at the point

P(p, q)

. What is the value of

3^p

?

A.0.9
B.1.8
C.2.7
D.4.5
E.9.0

Answer: C

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Question 22

1 mark

Consider the equation

4^x - (k+1)2^x + k = 0

, where

k

is a real constant. For which set of values of

k

does this equation have exactly one real solution for

x

?

A. $k = 1$ only
B. $k \le 0$ only
C. $k \le 1$
D. $k \le 0$ or $k = 1$
E. $k \le 0$ or $k \ge 1$

Answer: D

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Question 23

1 mark

What is the complete set of real values of

x

that satisfy the equation

2\log_2 x - \log_2(x+4) = 1

?

A. $x = 4$ only
B. $x = 2$ only
C. $x = 4$ or $x = -2$
D. $x = 2$ or $x = -4$
E. $x = 4$ or $x = 2$

Answer: A

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Question 24

1 mark

The sum of the real roots of the equation

3^{2x+1} + 3 = 10 \cdot 3^x

is

A.-1
B.0
C.1
D. $\frac{10}{3}$
E. $\log_3 10 - 1$

Answer: B

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Question 25

1 mark

The function

f

is defined for

x ≠ 0

by

f(x) = \frac{(x - k)^2}{x}

where

k

is a non-zero constant. The tangent to the graph

y = f(x)

at

x = 2

is parallel to the line

4y - 3x = 12

. What is the value of

f''(2)

?

A. $\frac{1}{16}$
B. $\frac{1}{8}$
C. $\frac{1}{4}$
D. $\frac{1}{2}$
E. $1$

Answer: C

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Question 26

1 mark

A continuous function

f(x)

is defined on the interval

1 \le x \le 5

. Let

A = \int_1^5 |f(x)| \,dx

represent the total area enclosed between the curve

y = f(x)

and the

x

-axis, and let

I = \int_1^5 f(x) \,dx

represent the definite integral over the same interval.

Consider the following three statements:
I.

A \ge |I|

II. If

f(x) = 0

for at least one value

x = c

where

1 < c < 5

, then

A > I

.
III. If

f(1) < 0

and

f(5) > 0

, then

A > I

.

Which of the above statements must be true?

A.I only
B.III 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

The function

f

is continuous for all

x

and satisfies the equation:

\int_{2}^{x} f(t) \, dt = x^2 + ax + 2

where

a

is a constant. What is the value of

f(5)

?

A.4
B.7
C.10
D.12
E.13

Answer: B

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