ESAT Mock Maths 2 Esat-maths2-challenge-3

10 questions10 marks40Updated August 2026

The ESAT Mock Maths 2 Esat-maths2-challenge-3 paper in full: all 10 questions, each with its answer. ESAT is the Engineering and Science Admissions Test. Sit it cold under exam timing, mark it, then work back through anything you missed using the solutions below.

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Question 1

1 mark

Consider the graph of

y = \tan x

for

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

, together with the line

y = x

. Which of the following statements is/are true?
1. For

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

, the graph of

y=\tan x

lies entirely above the line

y = x

.
2. The graphs of

y = \tan x

and

y = x

intersect at exactly one point in

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

.
3. For

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

, the graph of

y = \tan x

lies entirely below the line

y = x

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

Answer: F

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Question 2

1 mark

The graph of

y = \cos(2x) + \cos(3x)

has period

P

. A student claims

P = 2\pi

because each of

\cos(2x)

and

\cos(3x)

individually repeats within

2\pi

. Which one of the following is correct?

A.The student is correct that $2\pi$ is a period, but the shortest period is actually $\frac{2\pi}{3}$ .
B.The student is correct that $2\pi$ is a period, but the shortest period is actually $\pi$ .
C.The student is wrong: $4\pi$ is a period but $2\pi$ is not.
D.The student is correct that $2\pi$ is a period, but the shortest period is actually $\frac{\pi}{2}$ .
E.The student is correct: $P = 2\pi$ , and this is also the shortest period.
F.The student is wrong: the function is not periodic at all.

Answer: E

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Question 3

1 mark

Let

f(x) = \sin^4 x + \cos^4 x

for real

x

. Which one of the following correctly gives the period of

f

and the difference between its maximum and minimum values?

A.period $2\pi$ ; difference $1$
B.period $2\pi$ ; difference $\frac{1}{2}$
C.period $\frac{\pi}{2}$ ; difference $\frac{1}{2}$
D.period $\frac{\pi}{2}$ ; difference $1$
E.period $\pi$ ; difference $\frac{1}{2}$
F.period $\pi$ ; difference $1$

Answer: C

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Question 4

1 mark

The real numbers

p

and

q

satisfy

p > q > 1

and

\log_2 p + \log_2 q = 6

. Let

S = \log_2 p - \log_2 q

. Which of the following gives the complete range of possible values of

S

A. $S \geq 0$
B. $0 < S < 6$
C. $S > 6$
D. $S > 0$
E. $0 < S < 3$
F. $0 < S \leq 6$
G. $1 < S < 6$
H. $0 \leq S < 6$

Answer: B

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Question 5

1 mark

Consider the three statements below, where

x

and

y

are positive real numbers with

x \neq 1

and

y \neq 1

: Statement 1:

\log_2 x + \log_2 y = \log_2(x+y)

is true for some choice of

x, y

with

x, y > 0

. Statement 2: If

\log_a(x) = \log_a(y)

for some base

a>0

a \ne 1

, then

x = y

. Statement 3:

2\log_2(x) = \log_2(x^2)

holds for all real

x \neq 0

. Which of these statements is/are true?

A.None of them
B.1 and 3 only
C.2 only
D.3 only
E.1, 2 and 3
F.2 and 3 only
G.1 and 2 only
H.1 only

Answer: G

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Question 6

1 mark

For real numbers

x

, let

f(x) = x^2 + bx + c

where

b

and

c

are real constants. It is given that

f(x) \ge 0

for all real

x

, and that

f(1) = 0

. What is the complete set of possible values of

c

A. $c \ge 1$
B. $0 \le c \le 1$
C. $c \le 1$
D. $c = 1$
E. $c > 0$
F. $c \ge 0$
G.all real $c$

Answer: D

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Question 7

1 mark

A quadratic function

f(x) = ax^2 + bx + c

(with

a \ne 0

) satisfies

f(0) = f(4)

and has a discriminant equal to

16a^2

. Given that the graph of

f

touches the x-axis at exactly one point when

a=1

, find the value of

c

when

a = 1

A. $8$
B. $4$
C. $2$
D. $-4$
E. $-2$
F. $0$

Answer: B

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Question 8

1 mark

For a real constant

k

, consider the pair of inequalities

x^2 - 5x + 6 \leq 0 \quad \text{and} \quad x^2 - (k+5)x + 5k \leq 0

It is given that every

x

satisfying the first inequality also satisfies the second. What is the complete range of possible values of

k

A. $k \leq 3$
B. $k \geq 2$
C. $k \leq 2$
D.No such $k$ exists
E. $k \leq 2$ or $k \geq 3$
F. $k \geq 3$
G. $2 \leq k \leq 3$
H. $k = 5$

Answer: C

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Question 9

1 mark

Let

a

and

b

be real numbers with

a \neq 0

. The quadratic inequality

ax^2 + bx + 1 < 0

has no real solutions for

x

. Which one of the following must be true?

A. $a > 0$ and $b^2 \geq 4a$
B. $a \geq 0$ and $b^2 \leq 4a$
C. $a > 0$ and $b^2 < 4a$
D. $a > 0$ or $b^2 \leq 4a$
E. $a < 0$ and $b^2 \leq 4a$
F. $a > 0$ and $b^2 \leq 4a$
G.There are no such $a, b$
H. $b^2 \leq 4a$ regardless of the sign of $a$

Answer: F

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Question 10

1 mark

A real number

x

satisfies both of the following:

|x - 3| < 2 \qquad \text{and} \qquad x^2 - 7x + c < 0

for some real constant

c

. It is given that every

x

satisfying the first inequality also satisfies the second, and that this fails to be true if

c

is decreased by any amount, however small. What is the value of

c

A. $18$
B. $0$
C. $6$
D. $10$
E. $4$
F. $-4$
G. $5$
H. $12$

Answer: D

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