The following notations are used in the question paper:

R is the set of real numbers,

C is the set of complex numbers,
Z is the set of integers,
� �
n n!
= for all n = 1, 2, 3, . . . and r = 0, 1, . . . , n.
r r!(n − r)!

1. For a real number x,

x3 − 7x + 6 > 0

if and only if

(A) x > 2. (B) −3 < x < 1.

(C) x < −3 or 1 < x < 2. (D) −3 < x < 1 or x > 2.

2. Define a polynomial f (x) by

� �
� 1 x x�
� �
� �
f (x) = �x 1 x�
� �
�x x 1 �

for all x ∈ R, where the right hand side above is a determinant.

Then the roots of f (x) are of the form
(A) α, β ± iγ where α, β, γ ∈ R, γ �= 0 and i is a square root of
−1.
(B) α, α, β where α, β ∈ R are distinct.
(C) α, β, γ where α, β, γ ∈ R are all distinct.
(D) α, α, α for some α ∈ R.

1
3. Let S be the set of those real numbers x for which the identity
∞
�
cosn x = (1 + cos x) cot2 x
n=2

is valid, and the quantities on both sides are finite. Then

(A) S is the empty set.
(B) S = {x ∈ R : x �= nπ for all n ∈ Z}.
(C) S = {x ∈ R : x �= 2nπ for all n ∈ Z}.
(D) S = {x ∈ R : x �= (2n + 1)π for all n ∈ Z}.

4. The number of consecutive zeroes adjacent to the digit in the

unit’s place of 40150 is

(A) 3. (B) 4. (C) 49. (D) 50.

5. Consider a right angled triangle �ABC whose hypotenuse AC

is of length 1. The bisector of ∠ACB intersects AB at D. If
BC is of length x, then what is the length of CD?
�
2x2 1
(A) 1+x
(B) √2+2x
� x x
(C) 1+x
(D) √1−x 2

6. Consider a triangle with vertices (0, 0), (1, 2) and (−4, 2). Let A
be the area of the triangle and B be the area of the circumcircle
B
of the triangle. Then A
equals

π 5π √3 π.
(A) 2
. (B) 4
. (C) 2
(D) 2π.

2
7. Let f, g be continuous functions from [0, ∞) to itself,
� 3x
h(x) = f (t) dt , x > 0 ,
2x

and � h(x)
F (x) = g(t) dt , x > 0 .
0

If F � is the derivative of F , then for x > 0,

(A) F � (x) = g(h(x)).
(B) F � (x) = g(h(x)) [f (3x ) − f (2x )].
(C) F � (x) = g(h(x)) [x3x−1 f (3x ) − x2x−1 f (2x )].
(D) F � (x) = g(h(x)) [3x f (3x ) ln 3 − 2x f (2x ) ln 2].

8. How many numbers formed by rearranging the digits of 234578

are divisible by 55?

(A) 0 (B) 12 (C) 36 (D) 72

9. Let
�� � �
πθ 1 πθ
S= θ sin , cos : θ ∈ R, θ > 0
1+θ θ 1+θ

and � �
1
T = (x, y) : x ∈ R, y ∈ R, xy = .
2
How many elements does S ∩ T have?

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

3
10. The limit
3 � �1
lim n− 2 (n + 1)(n+1) (n + 2)(n+2) . . . (2n)(2n) n2
n→∞

equals
1 3
(A) 0. (B) 1. (C) e− 4 . (D) 4e− 4 .

11. Suppose x and y are positive integers. If 4x + 3y and 2x + 4y

are divided by 7, then the respective remainders are 2 and 5. If
11x + 5y is divided by 7, then the remainder equals

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

12. The value of � � � �

202
�
k 202 kπ
(−1) cos
k=0
k 3
equals
� 202 � � 202 �
(A) sin 3
π . (B) − sin 3
π .
� 202 � � π
�
(C) cos 3 π . (D) cos202 3 .

13. For real numbers a, b, c, d, a� , b� , c� , d� , consider the system of

equations

ax2 + ay 2 + bx + cy + d = 0 ,
a� x2 + a� y 2 + b� x + c� y + d� = 0 .

If S denotes the set of all real solutions (x, y) of the above

system of equations, then the number of elements in S can
never be

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

4
14. The limit
� � � � � �
1 1 1
lim cos(x) + cos − cos(x) cos −1
x→0 x x x

(A) equals 0. (B) equals 12 .

(C) equals 1. (D) does not exist.

15. Let n be a positive integer having 27 divisors including 1 and

n, which are denoted by d1 , . . . , d27 . Then the product of
d1 , d2 , . . . , d27 equals
27
(A) n13 . (B) n14 . (C) n 2 . (D) 27n.

16. Suppose F : R → R is a continuous function which has exactly

one local maximum. Then which of the following is true?
(A) F cannot have a local minimum.
(B) F must have exactly one local minimum.
(C) F must have at least two local minima.
(D) F must have either a global maximum or a local minimum.

17. Suppose z ∈ C is such that the imaginary part of z is non-zero

and z 25 = 1. Then
2023
�
zk
k=0

equals

(A) 0 . (B) 1 . (C) −1 − z 24 . (D) −z 24 .

5
18. Let f : R → R be a twice differentiable one-to-one function. If
f (2) = 2, f (3) = −8 and
� 3
f (x) dx = −3 ,
2

then � 2
f −1 (x) dx
−8

equals

(A) −25. (B) 25. (C) −31. (D) 31.

19. If f : [0, ∞) → R is a continuous function such that

� x
f (x) + ln 2 f (t) dt = 1 , x ≥ 0 ,
0

then for all x ≥ 0,

(A) f (x) = ex ln 2. (B) f (x) = e−x ln 2.

� �x
(C) f (x) = 2x . (D) f (x) = 12 .

20. If [x] denotes the largest integer less than or equal to x, then
� √ �
(9 + 80)20

equals
√ √
(A) (9 + 80)20 − (9 − 80)20 .
√ √
(B) (9 + 80)20 + (9 − 80)20 − 20.
√ √
(C) (9 + 80)20 + (9 − 80)20 − 1.
√
(D) (9 − 80)20 .

6
21. The limit � �2−n
−2n+1 −2n−1
lim 2 +2
n→∞

equals

(A) 1. (B) √1 . (C) 0. (D) 1

.
2 4

22. In the following figure, OAB is a quarter-circle. The unshaded

region is a circle to which OA and CD are tangents.

If CD is of length 10 and is parallel to OA, then the area of the

shaded region in the above figure equals

(A) 25π. (B) 50π. (C) 75π. (D) 100π.

23. Three left brackets and three right brackets have to be arranged
in such a way that if the brackets are serially counted from the
left, then the number of right brackets counted is always less
than or equal to the number of left brackets counted. In how
many ways can this be done?

(A) 3 (B) 4 (C) 5 (D) 6

7
24. The polynomial x10 + x5 + 1 is divisible by

(A) x2 + x + 1. (B) x2 − x + 1.
(C) x2 + 1. (D) x5 − 1.

25. Suppose a, b, c ∈ R and

f (x) = ax2 + bx + c , x ∈ R .

If 0 ≤ f (x) ≤ (x − 1)2 for all x, and f (3) = 2, then

(A) a = 12 , b = −1, c = 12 . (B) a = 13 , b = − 13 , c = 0.

(C) a = 23 , b = − 53 , c = 1. (D) a = 34 , b = −2, c = 54 .

26. As in the following figure, the straight line OA lies in the

second quadrant of the (x, y)-plane and makes an angle θ with
the negative half of the x-axis, where 0 < θ < π2 .

The line segment CD of length 1 slides on the (x, y)-plane in

such a way that C is always on OA and D on the positive side
of the x-axis. The locus of the mid-point of CD is
(A) x2 + 4xy cot θ + y 2 (1 + 4 cot2 θ) = 14 .
1
(B) x2 + y 2 = 4
+ cot2 θ.
(C) x2 + 4xy cot θ + y 2 = 14 .
(D) x2 + y 2 (1 + 4 cot2 θ) = 14 .

8
27. Suppose that f (x) = ax3 + bx2 + cx + d where a, b, c, d are real
numbers with a �= 0. The equation f (x) = 0 has exactly two
distinct real solutions. If f � (x) is the derivative of f (x), then
which of the following is a possible graph of f � (x)?

(A) (B)

(C) (D)

28. Consider the function f : C → C defined by

f (a + ib) = ea (cos b + i sin b) , a, b ∈ R ,
where i is a square root of −1. Then
(A) f is one-to-one and onto.
(B) f is one-to-one but not onto.
(C) f is onto but not one-to-one.
(D) f is neither one-to-one nor onto.

9
29. Suppose f : Z → Z is a non-decreasing function. Consider the
following two cases:

Case 1. f (0) = 2 , f (10) = 8 ,

Case 2. f (0) = −2 , f (10) = 12 .

In which of the above cases it is necessarily true that there exists

an n with f (n) = n?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.

30. How many functions f : {1, 2, . . . , 10} → {1, . . . , 2000}, which

satisfy
f (i + 1) − f (i) ≥ 20 , for all 1 ≤ i ≤ 9 ,

are there?
� � � �
1829 1830
(A) 10! (B) 11!
10 11
� � � �
1829 1830
(C) (D)
10 11

10