√

2− x+4
1. Suppose f : R → R is a continuous function such that f (x) = sin 2x
for all x 6= 0. Then the value of f (0) is

(A) − 81 (B) 1
8
(C) 0 (D) − 14

2. A person throws a pair of fair dice. If the sum of the numbers on

the dice is a perfect square, then the probability that the number 3
appeared on at least one of the dice is

(A) 1/9 (B) 4/7 (C) 1/18 (D) 7/36

3. Consider the system of linear equations: x+y+z = 5, 2x+2y+3z = 4.

Then
(A) the system is inconsistent
(B) the system has a unique solution
(C) the system has infinitely many solutions
(D) none of the above is true

4. If g 0 (x) = f (x) then

R
x3 f (x2 )dx is given by

1 2
R R
(A) x2 g(x2 ) − xg(x2 )dx + C (B) 2
x g(x2 ) − xg(x2 )dx + C
x2 g(x2 ) − 21
R R
(C) 2x2 g(x2 ) − xg(x2 )dx + C (D) xg(x2 )dx + C

5. If (n C0 +n C1 )(n C1 +n C2 ) · · · (n Cn−1 +n Cn ) = k n C0 n C1 · · · n Cn−1 ,

then k is equal to

(n+1)n nn (n+1)n (n+1)n+1

(A) n!
(B) n!
(C) nn!
(D) n!

6. Let {fn } be a sequence of functions defined as follows:

fn (x) = xn cos(2πnx), x ∈ [−1, 1].

Then limn→∞ fn (x) exists if and only if x belongs to the interval

(A) (−1, 1) (B) [−1, 1) (C) [0, 1] (D) (−1, 1]

1
 7. Let S be a set of n elements. The number of ways in which n distinct
non-empty subsets X1 , ..., Xn of S can be chosen such that X1 ⊆
X2 · · · ⊆ Xn , is

n

n
n


(A) 1 2
··· n
(B) 1 (C) n! (D) 2n

8. Let A be a 4 × 4 matrix such that both A and Adj(A) are non-null.

If det A = 0, then the rank of A is

(A) 1 (B) 2 (C) 3 (D) 4

9. The set of all a satisfying the inequality

Z a
1 3√ 1 
√ x + 1 − √ dx < 4
a 1 2 x

is equal to the interval

(A) (−5, −2) (B) (1, 4) (C) (0, 2) (D) (0, 4)

10. Let C0 be the set of all continuous functions f : [0, 1] → R and C1

be the set of all differentiable functions g : [0, 1] → R such that the
derivative g 0 is continuous. (Here, differentiability at 0 means right
differentiability and differentiability at 1 means left differentiability.)
If T : C1 → C0 is defined by T (g) = g 0 , then

(A) T is one-to-one and onto

(B) T is one-to-one but not onto
(C) T is onto but not one-to-one
(D) T is neither one-to-one nor onto.

11. Suppose a, b, c are in A.P. and a2 , b2 , c2 are in G.P. If a < b < c and
a + b + c = 32 , then the value of a is

1
(A) √
2 2
(B) − 2√1 2 (C) 1
2
− √1
3
(D) 1
2
− √1
2

2
12. The number of distinct even divisors of
5
Y
k!
k=1

is

(A) 24 (B) 32 (C) 64 (D) 72

13. Let D be the triangular region in the xy-plane with vertices at

(0, 0), (0, 1) and (1, 1). Then the value of
Z Z
2
2
dx dy
D 1+x

is

π π
(A) 2
(B) 2
− ln 2 (C) 2 ln 2 (D) ln 2

14. Given a real number α ∈ (0, 1), define a sequence {xn }n≥0 by the
following recurrence relation:

xn+1 = αxn + (1 − α)xn−1 , n ≥ 1.

If limn→∞ xn = ` then the value of ` is

αx0 +x1 (1−α)x0 +x1

(A) 1−α
(B) 2−α
αx0 +x1 (1−α)x1 +x0
(C) 2−α
(D) 2−α

15. A straight line passes through the intersection of the lines given by
3x − 4y + 1 = 0 and 5x + y = 1 and makes equal intercepts of the
same sign on the coordinate axes. The equation of the straight line is

(A) 23x − 23y + 11 = 0 (B) 23x − 23y − 11 = 0

(C) 23x + 23y + 11 = 0 (D) 23x + 23y − 11 = 0

3
16. The series
X 3 · 6 · 9 · · · 3n
xn , x>0
n
7 · 10 · 13 · · · (3n + 4)

(A) converges for 0 < x ≤ 1 and diverges for x > 1

(B) converges for all x > 0
1 1
(C) converges for 0 < x < 2
and diverges for x ≥ 2
1
(D) converges for 2
< x < 1 and diverges for 0 < x ≤ 12 , x ≥ 1.

17. Suppose A and B are two square matrices such that the largest eigen-
value of (AB − BA) is positive. Then the smallest eigen value of
(AB − BA)

(A) must be positive (B) must be negative

(C) must be 0 (D) is none of the above

18. The number of saddle points of the function f (x, y) = 2x4 − x2 + 3y 2

is

(A) 1 (B) 0 (C) 2 (D) none of the above

19. Suppose G is a cyclic group and a, b ∈ G. There does not exist any
x ∈ G such that x2 = a. Also, there does not exist an y ∈ G such
that y 2 = b. Then,

(A) there exists an element g ∈ G such that g 2 = ab.

(B) there exists an element g ∈ G such that g 3 = ab.
(C) the smallest exponent k > 1 such that g k = ab for some g ∈ G is
4.
(D) none of the above is true.

20. The number of real roots of the polynomial x3 − 2x + 7 is

(A) 0 (B) 1 (C) 2 (D) 3

4
21. Suppose that a 3 × 3 matrix A has an eigen value −1. If the matrix
A + I is equal to  
1 0 −2
 0 0 0 
 

0 0 0
then the eigen vectors of A corresponding to the eigenvalue −1 are in
the form,
   
2t 2t
(A)  0  , t ∈ R (B)  s  , s, t ∈ R
   

t t

   
t t
(C)  0  , t ∈ R (D)  s  , s, t ∈ R
   

−2t 2t

22. Consider the function f : R2 → R defined by f (x, y) = x2 (y − 1). For

~u = ( 21 , 12 ) and ~v = (3, 4), the value of the limit

f (~u + t~v ) − f (~u)

lim
t→0 t
is

3 6
(A) 4
(B) 13
(C) − 21 (D) none of the above

23. Suppose φ is a solution of the differential equation y 00 − y 0 − 2y = 0

such that φ(0) = 1 and φ0 (0) = 5. Then

(A) φ(x) → ∞ as |x| → ∞ (B) φ(x) → −∞ as |x| → ∞

(C) φ(x) → −∞ as x → −∞ (D) φ(x) → ∞ as x → −∞

24. A fair die is rolled five times. What is the probability that the largest
number rolled is 5?

(A) 5/6 (B) 1/6 (C) 1 − (1/6)6 (D) (5/6)5 − (2/3)5

5
25. Two rows of n chairs, facing each other, are laid out. The number of
different ways that n couples can sit on these chairs such that each
person sits directly opposite to his/her partner is

(A) n! (B) n!/2 (C) 2n n! (D) 2n!.

26. Consider the function f : C → C defined on the complex plane C by

f (z) = ez . For a real number c > 0, let A = {f (z)| Re z = c} and
B = {f (z)| Im z = c}. Then

(A) both A and B are straight line segments

(B) A is a circle and B is a straight line segment
(C) A is a straight line segment and B is a circle
(D) both A and B are circles

27. Consider two real valued functions f and g given by

x
f (x) = x−1
for x > 1, and g(x) = 7 − x3 for x ∈ R.

Which of the following statements about inverse functions is true?

(A) Neither f −1 nor g −1 exists

(B) f −1 exists, but not g −1
(C) f −1 does not exist, but g −1 does
(D) Both f −1 and g −1 exist.

28. A circle is drawn with centre at (−1, 1) touching x2 +y 2 −4x+6y−3 = 0

externally. Then the circle touches

(A) both the axes (B) only the x-axis

(C) none of the two axes (D) only the y-axis

6
 f (x)
29. Let f (x − y) = f (y)
for all x, y ∈ R and f 0 (0) = p, f 0 (5) = q. Then the
value of f 0 (−5) is

p p2
(A) q (B) −q (C) q
(D) q

30. Let  
a 1 1
A =  b a 1 .
 

1 1 1
The number of elements in the set
n o
2
(a, b) ∈ Z : 0 ≤ a, b ≤ 2021, rank(A) = 2

is

(A) 2021 (B) 2020 (C) 20212 − 1 (D) 2020 × 2021