MAT 2021 MCQ

10 questions40 marks60Updated July 2026

The MAT 2021 MCQ paper in full: all 10 questions, each with its answer. 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

4 marks

Which of the following expressions has the largest value? Note that all angles are given in degrees.

a. $\cos(10^\circ)$
b. $\sin(115^\circ)$
c. $\cos(375^\circ)$
d. $\sin(85^\circ)$
e. $\cos(-20^\circ)$

Answer: D

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Question 2

4 marks

In the expansion of

(x^2+xy+y^2)^n

, where

n

is a positive whole number, the coefficient of

x^3 y^{2n-3}

a. $\binom{n}{3}$
b. $\binom{n}{3} \times \binom{n}{2}$
c. $\binom{n}{3} + 2 \times \binom{n}{2}$
d. $2 \times \binom{n}{2}$
e. $\binom{n}{3} + \binom{n}{2}$

Answer: C

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Question 3

4 marks

Given a real number

c

with

0 < c < 1

, the line

y = c

intersects the circle

x^2 + y^2 = 1

at two points. These two points, together with

(1, 0)

and

(-1, 0)

, form a quadrilateral. Which of the following graphs is a plot of the area of that quadrilateral against

c

a.Graph (a)
b.Graph (b)
c.Graph (c)
d.Graph (d)
e.Graph (e)

Answer: A

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Question 4

4 marks

A particle moves along the

x

-axis. At time

t = 0

the particle starts at

(0, 0)

with initial speed 1, moving towards

x = 1

. When the particle reaches

x = n

for any positive integer

n

, its speed immediately changes to

2^{-n}

but its direction is unchanged. What is the particle's position at time

t = 100

a. $x = \frac{89}{16}$
b. $x = \frac{105}{16}$
c. $x = \frac{3200}{32}$
d. $x = \frac{421}{64}$
e.The particle has escaped to infinity.

Answer: D

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Question 5

4 marks

The polynomial equation

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

has exactly four real solutions

x

if and only if

a. $k > 1$
b. $k > -\frac{5}{4}$
c. $k > \frac{3}{4}$
d. $k < -\frac{5}{4}$ or $k > \frac{3}{4}$
e. $\frac{3}{4} < k < 1$ or $k > 1$

Answer: E

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Question 6

4 marks

The point

A

has coordinates

(3, 4)

. The origin

(0, 0)

and the point

A

both lie on the circumference of a circle

\mathcal{C}

. The diameter of

\mathcal{C}

through

A

also meets

\mathcal{C}

at another point

B

. The distance between

B

and the origin is 10. It follows that the coordinates of

B

could be either

a. $\left(-5\sqrt{2}, 5\sqrt{2}\right)$ or $\left(5\sqrt{2}, -5\sqrt{2}\right)$
b. $(-4, 3)$ or $(4, -3)$
c. $\left(-5, 5\sqrt{3}\right)$ or $\left(5, -5\sqrt{3}\right)$
d. $(-8, 6)$ or $(8, -6)$
e. $\left(-5\sqrt{3}, 5\right)$ or $\left(5\sqrt{3}, -5\right)$

Answer: D

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Question 7

4 marks

Without calculating it directly, which of the following numbers is the square of 123, 456, 789?

a.15,241,578,710,190,521
b.15,241,578,730,190,521
c.15,241,578,750,190,521
d.15,241,578,770,190,521
e.15,241,578,790,190,521

Answer: C

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Question 8

4 marks

A function

f(x)

satisfies the following equation

f(x) + f(y) = \frac{1}{f(xy)}

\nfor any real positive numbers

x

and

y

, and also satisfies

f(x) > 0

for all real positive numbers

x

. It follows that

f(2021)

is

[Hint: try substituting

x = 1

and

y = 1

into the given expression.]

a.1
b.2021
c. $\log_e 2021$
d. $\frac{1}{\sqrt{2}}$
e. $\frac{1}{\log_e 2021}$

Answer: D

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Question 9

4 marks

Given that there are positive real numbers

a, b, c

that satisfy

\int_a^b \log_c \left(\sin^4 x \tan^2 x\right) dx = 1\quad \text{and} \quad \int_a^b \log_c \left(\sin^2 x \cos^2 x\right) dx = 3,

\nit follows that the value of

\int_a^b \log_c \left(\sin^4 x \cos^2 x\right) dx

\nmust be equal to

[Note that

\sin^4 x

means

(\sin x)^4

Answer: A

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Question 10

4 marks

There is a straight line that is normal to the curve

y = x^3 - kx

at two different points if and only if

a. $k \ge \sqrt{3}$
b. $k^2 \ge 3$
c. $k^2 \ge 1$
d. $k \ge 1$
e. $k \ge \sqrt{3}$ or $k \le -1$

Answer: A

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