MAT 2021 MCQb

10 questions40 marks60Updated July 2026

The MAT 2021 MCQb 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: E

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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.A graph showing a curve starting at $(0,0)$ , increasing to a peak near $c=0.7$ , and decreasing to $(1,1)$ .
B.A graph showing a straight line from $(0,0)$ to $(1,1)$ .
C.A graph showing a concave down curve increasing from $(0,0)$ to $(1,1)$ .
D.A graph showing a curve starting at $(0,0)$ , increasing to a peak near $c=0.9$ , and ending at $(1,1)$ .
E.A graph showing a concave up curve increasing from $(0,0)$ to $(1,1)$ .

Answer: D

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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: A

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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. $(-5\sqrt{2}, 5\sqrt{2})$ or $(5\sqrt{2}, -5\sqrt{2})$
B. $(-4, 3)$ or $(4, -3)$
C. $(-5, 5\sqrt{3})$ or $(5, -5\sqrt{3})$
D. $(-8, 6)$ or $(8, -6)$
E. $(-5\sqrt{3}, 5)$ or $(5\sqrt{3}, -5)$

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: D

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Question 8

4 marks

A function

f(x)

satisfies the following equation

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

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

it follows that the value of

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

must be equal to

Answer: C

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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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