MAT 2022 MCQb

10 questions40 marks60 minutesUpdated July 2026

The MAT 2022 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

Whenever I toss a particular coin, it lands on heads with probability

\cos^2 \alpha

for some fixed real number

\alpha

(and the outcome is independent of other tosses). I toss the coin three times. The probability that the coin lands on heads two or more times is equal to

A.1 + 3 $\sin$ ^4 $\alpha$ - 2 $\sin$ ^6 $\alpha$ ,
B.1 - 3 $\sin$ ^4 $\alpha$ - 2 $\sin$ ^6 $\alpha$ ,
C.1 + 3 $\sin$ ^4 $\alpha$ + 2 $\sin$ ^6 $\alpha$ ,
D.1 - 3 $\sin$ ^4 $\alpha$ + 2 $\sin$ ^6 $\alpha$ ,
E.1 + 8 $\sin$ ^6 $\alpha$ .

Answer: D

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Question 2

4 marks

Which of the following graphs is a sketch of

e^{-x/2} - e^{-x}

for

x > 0

A.[Graph (a)]
B.[Graph (b)]
C.[Graph (c)]
D.[Graph (d)]
E.[Graph (e)]

Answer: B

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Question 3

4 marks

For precisely which non-zero real values of

x

is it true that

x^2 - 3x + 2 < \frac{x - 1}{x} \quad ?

A. $x < 1 - \sqrt{2}$ or $x > 1 + \sqrt{2}$ ,
B. $1 - \sqrt{2} < x < 0$ or $1 < x < 1 + \sqrt{2}$ ,
C. $1 < x < 1 + \sqrt{2}$ ,
D. $1 - \sqrt{2} < x < 1 + \sqrt{2}$ ,
E. $1 - \sqrt{2} < x < 0$ .

Answer: A

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Question 4

4 marks

Consider the two inequalities

1 \le x^2 + y^2 \le 4 \text{ and } x^2 \ge 3y^2.

The **total** area of all regions of the

(x, y)

-plane where **both** inequalities hold is

A. $\pi \sqrt{3}$ ,
B. $\pi$ ,
C. $2\pi$ ,
D. $\frac{\pi}{2}$ ,
E. $\frac{\pi}{6}$ .

Answer: B

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Question 5

4 marks

The points

(0, 1)

and

(p, q)

are on opposite ends of the diameter of circle

C

. The

x

-axis is a tangent to the circle

C

if and only if

A. $p = 1 + q$ ,
B. $pq = 1$ ,
C. $p^2 = 4q$ ,
D. $p^2 + (q - 1)^2 = 1$ ,
E. $p + q = 1$ .

Answer: C

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Question 6

4 marks

The series

1 + (1 + x - x^2) + (1 + x - x^2)^2 + (1 + x - x^2)^3 + \dots

converges to

\frac{1}{x(x - 1)}

for precisely which real values of

x

A.If and only if $-1 < x < 1$ ,
B.If and only if we have both $x \neq 0$ and $x \neq 1$ ,
C.If and only if either $-1 < x < 0$ or $1 < x < 2$ ,
D.If and only if either $-2 < x < -1$ or $0 < x < 1$ ,
E.For all real $x$ .

Answer: C

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Question 7

4 marks

Given that

y = f(x)

is a solution to

\frac{dy}{dx} = y^{1/4}

, it follows that one of the following functions is a solution to

\frac{dy}{dx} = 2y^{1/4}

. Which one?

A. $y = (2)^{-4} f(x)$ ,
B. $y = (2)^3 f(x)$ ,
C. $y = (2)^{4/3} f(x)$ ,
D. $y = (2)^{-3} f(x)$ ,
E. $y = (2)^4 f(x)$ .

Answer: C

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Question 8

4 marks

Suppose that a function

f(n)

on the positive integers is defined such that

f(1) = 1

and then for

n \ge 1

f(2n) = f(n) \quad \text{and} \quad f(2n + 1) = f(n) + f(n + 1).

How many values of

n

are there such that

f(n) = 3

and also

n

is a multiple of 35?

A.0,
B.1,
C.2,
D.3,
E.Infinitely many.

Answer: E

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Question 9

4 marks

The number of positive solutions

x

to the equation

\log_2 x = \log_2 (x + a) + b,

where

a

and

b

are non-zero real numbers, is

A.zero if $ab < 1$ , or one if $ab > 1$ ,
B.one if $ab < 1$ , or two if $ab > 1$ ,
C.one if $ab < 0$ , or zero if $ab > 0$ ,
D.zero if $ab < 0$ , or one if $ab > 0$ ,
E.one if $ab < 1$ , or zero if $ab > 1$ .

Answer: C

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Question 10

4 marks

Given that

\int_1^2 \frac{x^2}{1 + x^4} \, dx = A

where

A

is some positive real number (which you should not attempt to determine), it follows that the value of

\int_1^2 \frac{x^{-2}}{1 + x^4} \, dx

is equal to

A.1 - A,
B.-A,
C. $\frac{1}{A}$ ,
D.A - 1,
E. $\frac{1}{2} - A$ .

Answer: A

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