NDA Previous Year Question Paper 2024-2nd-Mathematics-

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1. Let X be a matrix of order 3 × 3, Y be a matrix of order 2 × 3 and Z be a matrix of order 3 × 2. Which of the following statements are correct?
A. (ZY)X is defined and is a square matrix of order 3.
B. Y(XZ) is defined and is a square matrix of order 2.
C. X(YZ) is not defined.
Select the answer using the code given below.
(a) A and B only
(b) B and C only
(c) A and C only
(d) A, B and C

2. Consider the following statements:
A. The set of all irrational numbers between √12 and √15 is an infinite set.
B. The set of all odd integers less than 1000 is a finite set.
Which of the statements given above is/are correct?
(a) A only
(b) B only
(c) Both A and B
(d) Neither A nor B

3. How many 4-digit numbers are there having all digits as odd?
(a) 625
(b) 400
(c) 196
(d) 120

4. If ω ≠ 1 is a cube root of unity, then what is (1 + ω – ω²)¹⁰⁰ + (1 – ω + ω²)¹⁰⁰ equal to?
(a) 2¹⁰⁰ ω²
(b) 2¹⁰⁰ ω
(c) 2¹⁰⁰
(d) –2¹⁰⁰

5. Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?
(a) Both A and B are null matrices
(b) Either A or B is a null matrix
(c) B is a null matrix if A is a non-singular matrix
(d) Both A and B are singular matrices

6. In the expansion of (1 + x)ᵖ (1 + x)q, if the coefficient of x³ is 35, then what is the value of (p + q)?
(a) 5
(b) 6
(c) 7
(d) 8

7. If p times the pᵗʰ term of an AP is equal to q times the qᵗʰ term (p ≠ q), then what is the (p + q)ᵗʰ term equal to?
(a) 0
(b) p + q
(c) pq
(d) pq(p + q)

8. Let p = ln(x), q = ln(x²) and r = ln(x³), where x > 1. Which of the following statements is/are correct?
A. p, q and r are in AP.
B. p, q and r can never be in GP.
Select the answer using the code given below:
(a) A only
(b) B only
(c) Both A and B
(d) Neither A nor B

9. If Z = (1/3) [i 2i 1 ; 2i 3i 2 ; 3 1 3] = x + iy ; i = √(–1), then what is modulus of Z equal to?
(a) 1
(b) √2
(c) 2
(d) √3

10. What is the value of the sum ∑ₙ₌₁²⁰ (iⁿ⁻¹ + iⁿ + iⁿ⁺¹) where i = √(–1)?
(a) –2i
(b) 0
(c) 1
(d) 2i

11. Let x > 1, y > 1, z > 1 be in GP. Then 1/(1 + ln x), 1/(1 + ln y), 1/(1 + ln z) are
(a) in AP
(b) in GP
(c) in HP
(d) neither in AP nor in GP nor in HP

12. If ω = –1/2 + i √3 / 2, then what is
| 1 + ω 1 + ω² ω + ω² |
| 1 ω ω² |
| 1/ω –1/ω² 1 | equal to?
(a) 0
(b) ω
(c) ω²
(d) 1 – ω²

13. If the sum of the first n terms of a series is n(2n + 1), then what is the nᵗʰ term?
(a) 4n – 1
(b) 4n
(c) 4n + 1
(d) 4n + 3

14. In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd positions?
(a) 3
(b) 6
(c) 9
(d) 12

15. The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?
(a) 240
(b) 720
(c) 1440
(d) 1620

16. If n is a root of the equation x² + p x + m = 0 and m is a root of the equation x² + p x + n = 0, where m ≠ n, then what is the value of p + m + n?
(a) –1
(b) 0
(c) 1
(d) 2

17. In how many ways can a student choose (n – 2) courses out of n courses if 2 courses are compulsory (n > 4)?
(a) (n – 3)(n – 4)
(b) (n – 1)(n – 2)
(c) (n – 3)(n – 4)/2
(d) (n – 2)(n – 3)

18. If Dₙ = | n² 20 30 |
| n² 40 50 |
| n² 60 70 |, then what is the value of ∑ Dₙ from n=1 to 4?
(a) –10000
(b) –10
(c) 10
(d) 10000

19. Consider the following in respect of the matrices
P = | 0 c –b |
| –c 0 a |
| b –a 0 |
and
Q = | a² ab ac |
| ab b² bc |
| ac bc c² |
A. PQ is a null matrix.
B. QP is an identity matrix of order 3.
C. PQ = QP
Which of the above is/are correct?
(a) A only
(b) B only
(c) A and C
(d) B and C

20. If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
(a) –1
(b) 0
(c) 1
(d) 3

21. If 4 sin⁻¹ x + cos⁻¹ x = π, then what is sin⁻¹ x + 4 cos⁻¹ x equal to?
(a) π/2
(b) π
(c) 3π/2
(d) 2π

22. What is cot²(sec⁻¹ 2) + tan²(cosec⁻¹ 3) equal to?
(a) 11/12
(b) 11/24
(c) 7/24
(d) 1

23. In a triangle ABC, a / cos A = b / cos B = c / cos C. What is the area of the triangle if a = 6 cm?
(a) 9√3 square cm
(b) 12 square cm
(c) 18√3 square cm
(d) 24 square cm

24. The roots of the equation 7x² – 6x + 1 = 0 are tan α and tan β, where 2α and 2β are the angles of a triangle. Which one of the following is correct?
(a) The triangle is equilateral
(b) The triangle is isosceles but not right-angled
(c) The triangle is right-angled
(d) The triangle is right-angled isosceles

25. In a triangle ABC, ∠A = 75° and ∠B = 45°. What is 2a – b equal to?
(a) c
(b) √2 c
(c) 2c
(d) 2√2 c

26. What is the number of solutions of the equation cot 2x – cot 3x = 1 for 0 < x < π?
(a) Only one
(b) Only two
(c) Only five
(d) More than five

27. What is the general solution of cos¹⁰⁰ x – sin¹⁰⁰ x = 1?
(a) nπ
(b) (2n + 1)π
(c) 2nπ
(d) (2n + 1)π/2
where n is an integer.

28. In a triangle ABC, tan A + tan B + tan C = k. What is the value of cot A cot B cot C?
(a) 0.5 k
(b) 1/k
(c) 3/k
(d) 1/k³

29. What is sin 12° sin 48° equal to?
(a) (√5 – 1)/4
(b) (√5 + 1)/4
(c) (√5 – 1)/8
(d) (√5 + 1)/8

30. What is (cos 17° – sin 17°) / (cos 17° + sin 17°) equal to?
(a) tan 34°
(b) cot 34°
(c) tan 62°
(d) cot 62°

31. Consider the following numbers:
A. tan 22.5°
B. cot 22.5°
C. tan 22.5° – cot 22.5°
How many of the above are irrational numbers?
(a) None
(b) Only one
(c) Only two
(d) All three

32. If x / cos θ = y / cos(2π/3 – θ) = z / cos(2π/3 + θ), then what is x + y + z equal to?
(a) –1
(b) 0
(c) 1
(d) 3

33. If p tan(θ – 30°) = q tan(θ + 120°), then what is (p + q)/(p – q) equal to?
(a) sin 2θ
(b) cos 2θ
(c) 2 sin 2θ
(d) 2 cos 2θ

34. Let P and Q be two non-void relations on a set A. Which of the following statements are correct?
A. P and Q are reflexive ⇒ P ∩ Q is reflexive.
B. P and Q are symmetric ⇒ P ∪ Q is symmetric.
C. P and Q are transitive ⇒ P ∩ Q is transitive.
Select the answer using the code given below.
(a) A and B only
(b) B and C only
(c) A and C only
(d) A, B and C

35. If A and B are two non-empty sets having 10 elements in common, then how many elements do A × B and B × A have in common?
(a) 10
(b) 20
(c) 40
(d) 100

36. What is the remainder when 7ⁿ – 6n is divided by 36 for n = 100?
(a) 0
(b) 1
(c) 2
(d) 6

37. What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is between lines as well as circle and lines)?
(a) 6
(b) 10
(c) 14
(d) 16

38. In an AP, the ratio of the sum of the first p terms to the sum of the first q terms is p² : q². Which one of the following is correct?
(a) The first term is equal to the common difference
(b) The first term is equal to twice the common difference
(c) The common difference is equal to twice the first term
(d) The first term is equal to square of the common difference

39. What is the number of real roots of the equation (x – 1)² + (x – 3)² + (x – 5)² = 0?
(a) None
(b) Only one
(c) Only two
(d) Three

40. In a class of 240 students, 180 passed in English, 130 passed in Hindi and 150 passed in Sanskrit. Further, 60 passed in only one subject, 110 passed in only two subjects and 10 passed in none of the subjects. How many passed in all three subjects?
(a) 60
(b) 55
(c) 40
(d) 35

41. Let Z₁ and Z₂ be any two complex numbers such that Z₁² + Z₂² + Z₁ Z₂ = 0. What is the value of |Z₁ / Z₂|?
(a) 1
(b) 2
(c) 3
(d) 4

42. What is the value of 1/2 + Re(Z₁ / Z₂)?
(a) –1
(b) 0
(c) 1
(d) 2

43. The product of 5 consecutive terms of an AP is 229635. The first, second and fifth terms are in GP. What is the common difference?
(a) 3
(b) 4
(c) 5
(d) 6

44. What is the sum of all five terms?
(a) 60
(b) 65
(c) 75
(d) 80

45. Let (8 + 3√7)²⁰ = U + V and (8 – 3√7)²⁰ = W, where U is an integer and 0 < V < 1. What is V + W equal to?
(a) 8
(b) 4
(c) 2
(d) 1

46. What is the value of (U + V) W?
(a) 1/2
(b) 1
(c) 3/2
(d) 2

47. The roots of the quadratic equation a²(b² – c²)x² + b²(c² – a²)x + c²(a² – b²) = 0 are equal (a² ≠ b² ≠ c²). Which one of the following statements is correct?
(a) a², b², c² are in AP.
(b) a², b², c² are in GP.
(c) a², b², c² are in HP.
(d) a², b², c² are neither in AP nor in GP nor in HP.

48. Which one of the following is a root of the equation?
(a) b²(c² – a²) / a²(c² – b²)
(b) b²(c² – a²) / a²(b² – c²)
(c) b²(c² – a²) / 2a²(c² – b²)
(d) b²(c² – a²) / 2a²(b² – c²)

49. What is A (adj A) equal to?
(Options with matrices)

50. What is A⁻¹ equal to?
(Options with matrices)

51. What is the value of λ if (2î + 3ĵ + 2k̂) × (î + λĵ + 3k̂) is a null vector?
(a) 3/2
(b) 3
(c) 9/2
(d) 27/2

52. For what value of the angle between the vectors a⃗ and b⃗ is the quantity |a⃗ × b⃗| + 3 |a⃗ · b⃗| maximum?
(a) 30°
(b) 45°
(c) 60°
(d) 90°

53. Let θ be the angle between two unit vectors a⃗ and b⃗. If a⃗ + 2b⃗ is perpendicular to 5a⃗ – 4b⃗, then what is cos θ equal to?
(a) 0
(b) 1/2
(c) 1/√2
(d) √3 / 2

54. Let ABCDEF be a regular hexagon. If AD = m BC and CF = n AB, then what is mn equal to?
(a) –4
(b) –2
(c) 2
(d) 4

55. The vectors a⃗, b⃗ and c⃗ are of the same length. If taken pairwise, they form equal angles. If a⃗ = î + ĵ and b⃗ = ĵ + k̂, then what can c⃗ be equal to?
(a) î – k̂
(b) –î + k̂
Select the correct answer using the code given below.
(a) A only
(b) B only
(c) Both A and B
(d) Neither A nor B

56. The diagonals of a quadrilateral ABCD are along the lines x + 2y = 1 and 4x + 2y = 3. The quadrilateral ABCD may be a
(a) rectangle
(b) cyclic quadrilateral
(c) parallelogram
(d) square

57. The foci of the ellipse 4x² + 9y² = 1 are at Q and R. If P(x, y) is any point on the ellipse, then what is PQ + PR equal to?
(a) 2
(b) 1
(c) 1/2
(d) 1/√2

58. If P(2, 4), Q(8, 12), R(10, 14) and S(x, y) are vertices of a parallelogram, then what is (x + y) equal to?
(a) 8
(b) 10
(c) 12
(d) 14

59. The equation of a circle is (x – 4)² + (y – 6)² + 8 = 0. Which of the following statements are correct?
A. The end points of a diameter of the circle are at (1, 4) and (7, 8).
B. The end points of a diameter of the circle are at (1, 4) and (3, 2).
C. The end points of a diameter of the circle are at (2, 4) and (4, 2).
Select the answer using the code given below.
(a) A and B only
(b) B and C only
(c) A and C only
(d) A, B and C

60. Consider the points P(4k, 4k) and Q(4k, 4) lying on the parabola y² = 16x. If the vertex is A, then what is ∠PAQ equal to?
(a) 60°
(b) 90°
(c) 120°
(d) 135°

61. Triangle ABC is inscribed in the circle x² + y² = 100. B and C have coordinates (6, 8) and (–8, 6) respectively. What is ∠BAC equal to?
(a) 30°
(b) 45°
(c) 60°
(d) 90°

62. What are the coordinates of A?
(a) (8, –6)
(b) (–6, 8)
(c) (6√2, 6√2)
(d) (–6√2, –6√2)

63. ABCD is an isosceles trapezium and AD is parallel to BC. If A(–3, 0), B(–1, 2), C(1, 2) and D(x, y), then what is the value of x + y?
(a) –2
(b) –1
(c) 0
(d) 1

64. What are the coordinates of the centroid of the triangle formed by the points (1, 2), (3, 4) and (5, 6)?
(Options)

65. What is the diameter of the sphere x² + y² + z² – 4x – 6y – 8z + 15 = 0?
(a) 4
(b) 6
(c) 8
(d) 10

66. The centre of the sphere lies on the plane
(a) 2x + 3y + 4z – 3 = 0
(b) 3x + 4y + 5z – 3 = 0
(c) 4x + 5y + 6z – 15 = 0
(d) x + y + z – 3 = 0

67. Which of the following are the direction ratios of the line of intersection of the planes x + y + z + 1 = 0 and 2x – y – z + 8 = 0?
(a) ⟨1, 1, –2⟩
(b) ⟨0, –1, 2⟩
(c) ⟨2, –1, –1⟩
(d) ⟨1, –2, 1⟩

68. If ⟨l, m, n⟩ are direction cosines of the line of intersection, then what is the value of l + m + n?
(a) 0
(b) 1
(c) √2
(d) √3

69. What are the direction ratios of the line?
(a) ⟨2, 1, –1⟩
(b) ⟨0, –1, 2⟩
(c) ⟨1, –2, 1⟩
(d) ⟨–1, 2, –1⟩

70. What is the point of intersection of the line and the plane?
(a) (4, –3, 3)
(b) (4, –3, –3)
(c) (–4, 3, 3)
(d) (4, 3, –3)

71. Let f(x) = x² + 2x + 1 and g(x) = x² – 2x + 1. If h(x) = f(x) + g(x) and k(x) = f(x) – g(x), then what is the value of k?
(Options)

72. What is the domain of the function f(x) = logₓ (x² + 2x + 11)?
(a) (0, 1) ∪ (1, ∞)
(b) (1, ∞)
(c) (0, ∞)
(d) ℝ – {0, 1}

73. What is lim (x→0) (sin x / x) equal to?
(a) Limit exists and is equal to 1
(b) Limit exists and is equal to 0
(c) Limit exists and is equal to –1
(d) Limit does not exist

74. What is the remainder when x³ + 3x² + 3x + 1 is divided by x + 1?
(Options)

75. What is √(a² + b² + c² + 2ab + 2bc + 2ca) equal to?
(a) |a + b + c|
(b) √(a² + b² + c²)
(c) √(a² + b²) – c
(d) √(a² + b²) + c

76. If f(2x) = 4x² + 1, then for how many real values of x will f(2x) be the GM of f(x) and f(4x)?
(a) Four
(b) Two
(c) One
(d) None

77. If f(x) = [x]² – 3[x] + 2 = 0, where [x] is the greatest integer function, then what is the sum of all integer solutions?
(a) 3
(b) 4
(c) 5
(d) 6

78. If f(x) = xⁿ – 1 and g(x) = xⁿ + 1, then what is f′(x) / g′(x) equal to when n is even?
(Options)

79. What is the number of real solutions of the equation |x|² – 3|x| + 2 = 0?
(a) 1
(b) 2
(c) 3
(d) 4

80. Which of the following statements is/are correct?
(A) f(x) is continuous at x = 0
(B) f(x) is differentiable at x = 0
Select the correct answer using the code given below.
(a) A only
(b) B only
(c) Both A and B
(d) Neither A nor B

81. Which one of the following is f(x)?
(a) cos x
(b) cos² x
(c) cos³ x
(d) cos⁴ x

82. Which one of the following is g(x)?
(a) |cos x|
(b) cos |x|
(c) |cos x|²
(d) cos² |x|

83. What is f(0.999) + f(1.001) equal to?
(a) –1
(b) 0
(c) 1
(d) 2

84. Consider the following statements:
A. f is continuous at x = 0.
B. f is differentiable at x = 0.
Which of the statements given above is/are correct?
(a) A only
(b) B only
(c) Both A and B
(d) Neither A nor B

85. What is the greatest value of f(θ)?
(a) 2√2
(b) 3√2
(c) 4√2
(d) 5√2

86. What is the least value of f(θ)?
(a) √2
(b) 2√2
(c) –√2
(d) –2√2

87. What is the value of k?
(a) 1
(b) 2
(c) 3
(d) 4

88. What is the area of the parabola bounded by the axes?
(a) 1 square unit
(b) 2 square units
(c) 4 square units
(d) 8 square units

89. What are the order and degree respectively of the differential equation?
(a) 1 and 1
(b) 1 and 2
(c) 2 and 1
(d) 2 and 2

90. What is the solution of the differential equation?
(a) y = 2eˣ + 3e⁻ˣ
(b) y = 2eˣ – 3e⁻ˣ
(c) 2y + x = eˣ
(d) xy – y + x = 0

91. What is f′(–4) equal to?
(a) –1
(b) 0
(c) 1
(d) 2

92. What is ∫ f(x) dx equal to?
(a) 2
(b) 3
(c) 4
(d) 5

93. Consider the following statements:
A. f(x) is an odd function.
B. g(x) is an odd function.
Select the correct answer using the code given below.
(a) A only
(b) B only
(c) Both A and B
(d) Neither A nor B

94. What is ∫₀¹ g(x) dx equal to?
(a) –1
(b) 0
(c) 1
(d) 2

95. What is M′(0) equal to?
(a) –2
(b) –1
(c) 0
(d) 1

96. What is g′(0) equal to?
(a) –4
(b) –2
(c) 0
(d) 2

97. What is ∫ f(x) dx equal to?
(a) sin x
(b) cos x
(c) –cos x
(d) –sin x

98. What is ∫ g(x) dx equal to?
(a) sin x
(b) cos x
(c) x sin x
(d) x cos x

99. What is U(x) – V(x) equal to?
(a) 0
(b) 1
(c) 2
(d) 3

100. What is U(x) V(x) equal to?
(a) 1
(b) x
(c) x²
(d) eˣ

101. If bₓᵧ and bᵧₓ are regression coefficients of lines of regression of y on x and x on y respectively, then what is the value of bₓᵧ + 7 bᵧₓ?
(a) –2
(b) –1
(c) 1
(d) 2

102. The mean of n observations 1, 4, 9, 16, …, n² is 130. What is the value of n?
(a) 9
(b) 10
(c) 11
(d) 12

103. Three distinct natural numbers are chosen at random from 1 to 10. What is the probability that they are consecutive?
(a) 1/15
(b) 1/20
(c) 1/30
(d) 1/40

104. If A, B, C are three mutually exclusive and exhaustive events such that P(A) = 2P(B) = 3P(C), then what is P(A) equal to?
(a) 2/9
(b) 1/3
(c) 6/11
(d) 6/13

105. A die has two faces with number 4, three faces with number 5 and one face with number 6. If the die is rolled once, then what is the probability of getting 4 or 5?
(a) 1/2
(b) 2/3
(c) 5/6
(d) 1

106. A bag contains 2 black, 4 yellow and 6 white balls. Three balls are drawn in succession with replacement. What is the probability that all three are of the same colour?
(a) 1/36
(b) 1/27
(c) 1/18
(d) 1/9

107. A can hit a target 5 times in 6 shots, B can hit 4 times in 5 shots and C can hit 3 times in 4 shots. What is the probability that A and C may hit but B may miss?
(a) 1/12
(b) 1/16
(c) 1/20
(d) 1/24

108. The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants occupy odd positions?
(a) 1/35
(b) 2/35
(c) 3/35
(d) 4/35

109. If the mean of a frequency distribution is 100 and the coefficient of variation is 45%, then what is the value of standard deviation?
(a) 45
(b) 50
(c) 55
(d) 60

110. The variance of 20 observations is 5. If each observation is multiplied by 2, then what is the new variance?
(a) 5
(b) 10
(c) 20
(d) 40

111. Two cards are drawn at random from a pack of 52 cards. What is the probability that both are aces?
(a) 1/221
(b) 1/221 × 2
(c) 2/221
(d) 1/13

112. A problem in Mathematics is given to three students A, B and C. Their probabilities of solving the problem are 1/2, 1/3 and 1/4 respectively. What is the probability that the problem is solved?
(a) 1/4
(b) 1/2
(c) 3/4
(d) 11/12

113. The probability that a student is not a swimmer is 1/5. What is the probability that out of 5 students, exactly 4 are swimmers?
(a) (4/5)⁴ × (1/5)
(b) ⁵C₄ × (4/5)⁴ × (1/5)
(c) ⁵C₄ × (1/5)⁴ × (4/5)
(d) (1/5)⁴ × (4/5)

114. If A and B are two independent events such that P(A) = 0.3 and P(B) = 0.4, then what is P(A ∪ B) equal to?
(a) 0.12
(b) 0.58
(c) 0.7
(d) 0.82

115. The mean of a binomial distribution is 20 and standard deviation is 4. What is the value of n?
(a) 25
(b) 50
(c) 75
(d) 100

116. In a Poisson distribution, if P(X = 2) = P(X = 3), then what is the value of the mean?
(a) 2
(b) 3
(c) 4
(d) 5

117. The regression equations are 3x + 2y = 26 and 6x + y = 31. What is the value of the correlation coefficient?
(a) –1/√2
(b) 1/√2
(c) –√2
(d) √2

118. If the two regression coefficients are 0.8 and 0.2, then what is the value of the coefficient of correlation?
(a) 0.16
(b) 0.4
(c) 0.5
(d) 1

119. The mean of 100 observations is 50 and standard deviation is 10. What is the sum of the squares of all observations?
(a) 250000
(b) 260000
(c) 270000
(d) 280000

120. If the variance of the data 2, 4, 5, 6, 8, 17 is 23.2, then what is the variance of the data 4, 8, 10, 12, 16, 34?
(a) 23.2
(b) 46.4
(c) 92.8
(d) 185.6

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