1.

If a, b and c are real numbers such that a > b > c > 0 then the
1
value of lim (an + bn − cn ) n is equal to
n→∞

(A) c (B) 2c (C) a (D) 2a

Z 2024
2. The value of the integral (x − ⌊x⌋)2 dx (where ⌊x⌋ is the
2
greatest integer less than or equal to x) is
2024 2023
(A) 674 (B) 1012 (C) (D)
3 3

3. The area bounded by the curves y = x2 and y = 2−x2 is equal to

2 16 8 5
(A) (B) (C) (D)
3 3 3 3

4. The sum of first n terms of a geometric progression (GP ) is p

and the sum of first 2n terms is 3p. Then the sum of first 3n
terms is
(A) 5p (B) 7p (C) 9p (D) 6p

5. The mean and variance of height of male students in a univer-

sity are 150 cm and 25 cm2 respectively. The height of these
students follow Normal distribution. Then the probability that
height of a randomly chosen male student from this university
to lie between 135 cm and 165 cm is
(A) 0.9973 (B) 0.8889 (C) 0.9999 (D) 0.9500

1
6. A point P on the line 3x + 5y = 15, which is equidistant from
both the coordinate axes, lies in

(A) 1st quadrant only (B) 1st or 2nd quadrant

(C) 1st or 3rd quadrant (D) Any quadrant

2
7. The events A and B are such that P (A ∪ B) = and
3
1
P (A ∩ B c ) = . Then P (B) is
3
1 1
(A) 1 (B) (C) (D) 0
9 3

8. Consider the system of equations in three variables x, y and z:

x + y = α
x + z = β
y + z = γ

Which of the following is correct?

(A) For no values of α, β, γ, the system has a solution
(B) The system always has a unique solution
(C) The system may have infinite solutions
(D) The system has a solution if and only if α = β = γ = 0

9. Suppose I2×2 is the Identity matrix. Let B2×2 be any other

matrix. Then which of the following statements is true?
(A) If B2×2 is not invertible, then I2×2 + B2×2 is not invertible
(B) Even if B2×2 is not invertible, still I2×2 + B2×2 may be
invertible
(C) If B2×2 is invertible, then I2×2 + B2×2 is also invertible
(D) None of the above

2
 √
10. Let i = −1 and let k be the smallest positive integer such
√
that the complex number ( 3 + i)k+2 is real. If p = 506k, then
p p
X 1 Y n
the value of + i is
n=1
in n=1

(A) −1 (B) 0 (C) 1 (D) −2

11. If a + b + c = 1 where a, b and c are positive real numbers, then

a2 + b2 + c2 is

(A) Greater than equal to 1 (B) Equal to 1

1 1
(C) Greater than equal to (D) Equal to
3 3

12. There are 30 balls in a box, 10 of which are red and 20

are blue. We pick up two balls, at a time, at random from
the box. What is the probability that both are of the same color?
235 45 190 45
(A) (B) (C) (D)
435 435 435 235

13. The interval in which the function f (x) = x3 − 3x + 1 is

decreasing in x is

(A) (−∞, 1) (B) (1, ∞) (C) (−∞, ∞) (D) (−1, 1)

14. The equation 2x2 + 3y 2 + 4xy + 5x + 6y + 7 = 0 represents a

(A) a circle (B) a parabola

(C) an ellipse (D) a pair of straight lines

3
15. The mean monthly salary paid to all employees in a certain
company was Rupees 500. The mean monthly salaries paid to
male and female employees were Rupees 520 and Rupees 420,
respectively. Then the percentage of male and female employees
in the company are
(A) 70 and 30 (B) 60 and 40
(C) 80 and 20 (D) 90 and 10

16. Consider A3×3 matrix whose first row is 1, 0, 0, the second row
is 0, 0, 1 and the third row is 0, 1, 0. Then

(A) The matrix A is the inverse of matrix A

(B) The matrix A does not have an inverse
(C) The matrix A is the transpose of matrix A
(D) Both (A) and (C) are true

1 8
17. If the coefficient of x7 of the (ax2 + bx ) is equal to the coefficient
1 8
of x−7 of the (ax − bx2
) then
(A) a2 b2 = −1 (B) ab = 2

(C) a2 b = 1 (D) ab2 = 2

18. Consider a straight line (denoted by l) whose x and y intercepts

are 4 and 3 respectively. Then perpendicular distance from the
origin (0, 0) to line l is
4 3 12 20
(A) (B) (C) (D)
3 4 5 3

4
19. The minimum
Z π value of the function f , defined by
f (x) = cos t cos(x − t)dt, for 0 ≤ x ≤ 2π, is equal
0
to
π π
(A) − (B) 0 (C) (D) π
2 2

20. There are 10 members in a club who are eligible to form a

governing committee that comprises of one president, one vice-
president and three other members. The number of ways in
which this can be done is
(A) 720 (B) 5000 (C) 5040 (D) 30240

21. Consider a magnetic compass. We marked a line connecting

the centre of the compass and a point on the perimeter. Let
the random variable X be the angle between the line and the
magnetic needle which points toward the north pole. Then the
value of P (X ≤ x) is
x x x x
(A) (B) 1 − (C) (D) 1 −
2π 2π π π

x2
22. The limit of , as x → ∞, is
x + ex

(A) 0 (B) 1 (C) 0.5 (D) ∞

23. The slope of a function y = f (x) is given by 2x. Suppose the y

intercept of the function is 9. What is the form of f (x)?

(A) x2 + 9 (B) x2 − 9 (C) 2x2 + 9 (D) 2x2 − 9

5
24. Let M = ((mij )) be a 5 × 5 matrix with mi,i+1 = 1 for 1 ≤ i ≤ 4
and m5,1 = 1 and all other elements of the matrix are zero. Then
which of the following statements is false?

(A) |M | = 1
(B) trace(M ) = 0
(C) M −1 = M
(D) M 5 = I

25. The equation of the line through the intersection of the lines
x + y + 1 = 0 and x − y + 1 = 0 and perpendicular to the line
2x + 3y + 4 = 0 is

(A) 3x + 2y + 3 = 0 (B) 3x − 2y + 3 = 0
(C) 3x − 2y − 3 = 0 (D) 3x + 2y − 3 = 0

26. Box B1 contains 3 white and 5 red balls, and box B2 contains
6 white and 4 red balls. One box is selected at random. The
2
box B1 is selected with probability and B2 with probability
3
1
. A ball is then drawn at random from the selected box. The
3
probability that it is a red ball is
33 49 25 41
(A) (B) (C) (D)
60 60 60 60

6
27. Consider a matrix, An×n , where n ≥ 1 and n is a natural number.
A matrix is called a zero matrix if and only if all the elements of
the matrix are 0s. Otherwise, it is called as a non-zero matrix.
Then, which of the following statements is correct?

(A) For all n, all non-zero matrices have rank n

(B) There exists n for which all non-zero matrices have rank n
(C) There exists n for which all non-zero matrices have rank
less than n
(D) None of the above

1 1
28. Suppose E and F are two events with P (E) = and P (F ) = .
5 3
Which of the following statements is correct?

(A) If E and F are independent, then the probability that at

8
least one of them occurs is
15
(B) If E and F are mutually exclusive, then the probability
7
that at least one of them occurs is
15
(C) If E and F are independent, then the probability that E
2
occurs but F does not occur is
15
(D) If E and F are independent, then the probability that
12
neither event occurs is
15

(−1)n−1 (n!)2 n
29. Let an = p and bn = 5 , where n ∈ N .
(n(n + 1)(n + 2)) (2n)!
Then which one of the following statements is correct?
P P∞
(A) Both ∞ n=1 an and n=1 bn converges
P∞ P∞
(B) n=1 an converges and n=1 bn diverges
P∞ P∞
(C) n=1 an diverges and n=1 bn converges
P∞ P∞
(D) Both n=1 an and n=1 bn diverges

7
30. The value of

 
1 1 1
lim √ √ +√ √ + ... + √ √ is
n→∞ n n+1 n n+2 n n+n

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