1.

The sum of all the solutions of 2 + log2 (x − 2) = log(x−2) 8 in the

interval (2, ∞) is

35 49 55
(A) 8
. (B) 5. (C) 8
. (D) 8
.

2. The value of
1 1 1
1+ + + ··· +
1+2 1+2+3 1 + 2 + 3 + · · · 2021
is

2021 2021 2021 2021

(A) 1010
. (B) 1011
. (C) 1012
. (D) 1013
.

3. The number of ways one can express 22 33 55 77 as a product of two

numbers a and b, where gcd(a, b) = 1, and 1 < a < b, is

(A) 5. (B) 6. (C) 7. (D) 8.

4. Let f : R → R be a continuous function such that

1
f (x + 1) = f (x) for all x ∈ R ,
2
Rn
and let an = f (x) dx for all integers n ≥ 1. Then:
0
R1
(A) limn→∞ an exists and equals 0 f (x) dx.
(B) limn→∞ an does not exist.
R1
(C) limn→∞ an exists if and only if | 0 f (x) dx| < 1.
R1
(D) limn→∞ an exists and equals 2 0 f (x) dx.

5. Let a, b, c, d > 0, be any real numbers. Then the maximum possible

2 2
value of cx + dy, over all points on the ellipse xa2 + yb2 = 1, must be
√ √
(A) a2 c2 + b2 d2 . (B) a2 b2 + c2 d2 .
q q
a2 c2 +b2 d2 a2 b2 +c2 d2
(C) a2 +b2
. (D) c2 +d2
.

6. Let f (x) = sin x + αx, x ∈ R, where α is a fixed real number. The

function f is one-to-one if and only if
(A) α > 1 or α < −1. (B) α ≥ 1 or α ≤ −1.
(C) α ≥ 1 or α < −1. (D) α > 1 or α ≤ −1.

1
 7. The volume of the region S = {(x, y, z) : |x| + 2|y| + 3|z| ≤ 6} is

(A) 36. (B) 48. (C) 72. (D) 6.

d2 f (x)
8. Let f : R → R be a twice differentiable function such that dx2
is
positive for all x ∈ R, and suppose f (0) = 1, f (1) = 4. Which of the
following is not a possible value of f (2)?

(A) 7. (B) 8. (C) 9. (D) 10.

9. Let
f (x) = e−|x| , x ∈ R ,

and Z 1 x
g(θ) = f dx, θ 6= 0 .
−1 θ
Then,
g(θ)
lim
θ→0 θ

(A) equals 0. (B) equals +∞.

(C) equals 2. (D) does not exist.

10. Consider the curves x2 + y 2 − 4x − 6y − 12 = 0, 9x2 + 4y 2 − 900 = 0

and y 2 − 6y − 6x + 51 = 0. The maximum number of disjoint regions
into which these curves divide the XY -plane (excluding the curves
themselves), is

(A) 4. (B) 5. (C) 6. (D) 7.

11. A box has 13 distinct pairs of socks. Let pr denote the probability of
having at least one matching pair among a bunch of r socks drawn at
random from the box. If r0 is the maximum possible value of r such
that pr < 1, then the value of pr0 is

12 13 213 212
(A) 1 − 26 C
12
. (B) 1 − 26 C
13
. (C) 1 − 26 C
13
. (D) 1 − 26 C
12
.

2
12. Consider the following two subsets of C :
1
: |z| = 2 and B = z1 : |z − 1| = 2 .

A= z

Then

(A) A is a circle, but B is not a circle.

(B) B is a circle, but A is not a circle.
(C) A and B are both circles.
(D) Neither A nor B is a circle.

13. Let a, b, c and d be four non-negative real numbers where a+b+c+d =

1. The number of different ways one can choose these numbers such
that a2 + b2 + c2 + d2 = max{a, b, c, d} is

(A) 1. (B) 5. (C) 11. (D) 15.

14. Suppose f (x) is a twice differentiable function on [a, b] such that

f (a) = 0 = f (b)

and
d2 f (x) df (x)
x2 2
+ 4x + 2f (x) > 0 for all x ∈ (a, b) .
dx dx
Then,

(A) f is negative for all x ∈ (a, b).

(B) f is positive for all x ∈ (a, b).
(C) f (x) = 0 for exactly one x ∈ (a, b).
(D) f (x) = 0 for at least two x ∈ (a, b).

15. The polynomial x4 + 4x + c = 0 has at least one real root if and only
if

(A) c < 2. (B) c ≤ 2. (C) c < 3. (D) c ≤ 3.

3
16. The number of different ways to colour the vertices of a square P QRS
using one or more colours from the set {Red, Blue, Green, Yellow},
such that no two adjacent vertices have the same colour is

(A) 36. (B) 48. (C) 72. (D) 84.

17. Define a = p3 + p2 + p + 11 and b = p2 + 1, where p is any prime

number. Let d = gcd (a, b) . Then the set of possible values of d is

(A) {1, 2, 5}. (B) {2, 5, 10}. (C) {1, 5, 10}. (D) {1, 2, 10}.

18. Consider all 2 × 2 matrices whose entries are distinct and taken from
the set {1, 2, 3, 4}. The sum of determinants of all such matrices is

(A) 24. (B) 10. (C) 12. (D) 0.

19. Let f : R → R be any twice differentiable function such that its second
derivative is continuous and
df (x)
6= 0 for all x 6= 0 .
dx
If
f (x)
lim = π,
x→0 x2

then

(A) for all x 6= 0, f (x) > f (0).

(B) for all x 6= 0, f (x) < f (0).
d2 f (x)
(C) for all x, dx2
> 0.
d2 f (x)
(D) for all x, dx2
< 0.

20. The number of all integer solutions of the equation x2 + y 2 + x − y =

2021 is

(A) 5. (B) 7. (C) 1. (D) 0.

4
21. The number of different values of a for which the equation x3 −x+a =
0 has two identical real roots is

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

22. For a positive integer n, the equation

x2 = n + y 2 , x, y integers,

does not have a solution if and only if

(A) n = 2.
(B) n is a prime number.
(C) n is an odd number.
(D) n is an even number not divisible by 4.

23. For 0 ≤ x < 2π, the number of solutions of the equation

sin2 x + 2 cos2 x + 3 sin x cos x = 0

is

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

24. Let f : R → [0, ∞) be a continuous function such that

f (x + y) = f (x)f (y) ,

for all x, y ∈ R. Suppose that f is differentiable at x = 1 and

df (x) 
= 2.
dx x=1


Then, the value of f (1) loge f (1) is

(A) e. (B) 2. (C) loge 2. (D) 1.

5
25. The expression
10
X
2k tan(2k )
k=0

equals
(A) cot 1 + 211 cot (211 ). (B) cot 1 − 210 cot (210 ).
(C) cot 1 + 210 cot (210 ). (D) cot 1 − 211 cot (211 ).

26. Define f : R → R by

1
(1 − cos x) sin

x
, x 6= 0,
f (x) =
0 , x = 0.

Then,

(A) f is discontinuous.
(B) f is continuous but not differentiable.
(C) f is differentiable and its derivative is discontinuous.
(D) f is differentiable and its derivative is continuous.

27. If the maximum and minimum values of sin6 x + cos6 x, as x takes all
real values, are a and b, respectively, then a − b equals

(A) 21 . (B) 23 . (C) 34 . (D) 1.

28. If two real numbers x and y satisfy (x + 5)2 + (y − 10)2 = 196, then
the minimum possible value of x2 + 2x + y 2 − 4y is
√ √
(A) 271 − 112 5. (B) 14 − 4 5.
√ √
(C) 276 − 112 5. (D) 9 − 4 5.

29. Let us denote the fractional part of a real number x by {x} (note:
{x} = x − [x] where [x] is the integer part of x). Then,
n √ o
lim (3 + 2 2)n
n→∞

(A) equals 0. (B) equals 1.

(C) equals 21 . (D) does not exist.

6
30. Let
p(x) = x3 − 3x2 + 2x, x ∈ R ,
R
 x p(t)dt, x ≥ 0,
0
f0 (x) =
− 0 p(t)dt, x < 0 ,
R
x

f1 (x) = ef0 (x) , f2 (x) = ef1 (x) , ... , fn (x) = efn−1 (x) .
dfn (x)
How many roots does the equation = 0 have in the interval
dx
(−∞, ∞)?

(A) 1. (B) 3. (C) n + 3. (D) 3n.