ESAT Mock Maths 2 Esat-maths2-tmm4-1

2 questions2 marks40Updated August 2026

The ESAT Mock Maths 2 Esat-maths2-tmm4-1 paper in full: all 2 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

The function

f(x) = \sin x + \cos(kx)

, where

k

is a positive integer, is periodic. Let

T

denote the smallest positive period of

f

. Which of the following statements about

T

is true?

A. $T = \frac{2\pi}{k}$ for every positive integer $k$
B. $T$ depends on $k$ in a way that cannot be determined without calculus
C. $T = 2\pi$ if $k$ is odd, and $T = \pi$ if $k$ is even
D. $T = 2\pi$ whenever $k$ is even, and $T$ is not $2\pi$ but some other multiple of $\pi$ when $k$ is odd and $k \neq 1$
E. $T = 2\pi$ for every positive integer $k$
F. $T = 2\pi$ if $k$ is even, and $T = \frac{2\pi}{k}$ if $k$ is odd

Answer: E

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Question 2

1 mark

Let

f(x) = \cos^2 x - \sin x

for

0 \le x \le 2\pi

. Consider the two statements below.
1:

f

has exactly one line of symmetry in the interval

0 \le x \le 2\pi

.
2: The maximum value of

f

is greater than the value of

f

at every point where

f

is decreasing.
Which of the following is correct?

A.Both statements are false only because of behaviour at $x = 2\pi$
B.Both statements are true
C.Statement 1 is true and statement 2 is false only because of a boundary point at $x=0$
D.Statement 2 only is true
E.Statement 1 only is true
F.Neither statement is true

Answer: D

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