NDA Previous Year Question Paper 2023-II-mathematics  |

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NDA/NA SOLVED PAPER 2023-II
MATHEMATICS (Clean Plain Text)

1. If z conjugate(z) = |z + conjugate(z)|, where z = x + iy, i = √(-1) then the locus of z is a pair of :
(a) Straight lines
(b) rectangular hyperbolas
(c) parabolas
(d) circles

2. If 1! + 3! + 5! + 7! + ... + 199! is divided by 24, what is the remainder?
(a) 3
(b) 6
(c) 7
(d) 9

3. What is the value of √(12 + 5i) + √(12 – 5i), where i = √(-1)?
(a) 24
(b) 25
(c) 5√2
(d) 5(√2 – 1)

4. If A = [1 2 3]ᵀ (column matrix), then what is value of det(I + A Aᵀ), where I is the 3×3 identity matrix?
(a) 15
(b) 6
(c) 0
(d) –1

5. If A, B and C are square matrices of order 3 and det(BC) = 2 det(A), then what is the value of det(2 A⁻¹ BC)?
(a) 16
(b) 8
(c) 4
(d) 2

6. If the nᵗʰ term of a sequence is (2n + 5)/7, then what is the sum of its first 140 terms?
(a) 2840
(b) 2780
(c) 2920
(d) 5700

7. Let A be a skew-symmetric matrix of order 3. What is the value of
det(4A⁴) – det(3A³) + det(2A²) – det(A) + det(–I)
Where I is the identity matrix of order 3?
(a) –1
(b) 0
(c) 1
(d) 2

8. If A = [[0, 3, 4], [–3, 0, 5], [–4, –5, 0]], then which one of the following statements is correct?
(a) A² is symmetric matrix with det(A²) = 0
(b) A² is symmetric matrix with det(A²) ≠ 0
(c) A² is skew-symmetric matrix with det(A²) = 0
(d) A² is skew-symmetric matrix with det(A²) ≠ 0

9. If A = [[2, 0, 0], [0, 3, 0], [0, 0, 4]], then which of the following statements are correct?

1. Aⁿ will always be singular for any positive integer n.
2. Aⁿ will always be a diagonal matrix for any positive integer n.
3. Aⁿ will always be a symmetric matrix for any positive integer n.
   Select the correct answer using the code given below:
   (a) 1 and 2 only
   (b) 2 and 3 only
   (c) 1 and 3 only
   (d) 1, 2 and 3

10. If (a + b), 2b, (b + c) are in HP, then which one of the following is correct?
(a) a, b and c are in AP
(b) a – b, b – c and c – a are in AP
(c) a, b and c are in GP
(d) a – b, b – c and c – a are in GP

11. Let t₁, t₂, t₃, ... be in GP. What is (t₁ t₃ ... t₂₁)^(1/11) equal to?
(a) t₁₀
(b) t₁₀²
(c) t₁₁
(d) t₁₁²

12. Which one of the following is a square root of –√(–1)?
(a) 1 + i
(b) (1 – i)/√2
(c) (1 + i)/√2
(d) (1 – i)/√2

13. What is the maximum number of points of intersection of 10 circles?
(a) 45
(b) 60
(c) 90
(d) 120

14. A set S contains (2n + 1) elements. There are 4096 subsets of S which contain at most n elements. What is n equal to?
(a) 5
(b) 6
(c) 7
(d) 8

15. If |x² – 3x x–1 x+3|
|x+1 –2x x–4|
|x–3 x+4 3x | = ax⁴ + bx³ + cx² + dx + e, then what is the value of e?
(a) –1
(b) 0
(c) 1
(d) 2

16. If all elements of a third order determinant are equal to 1 or –1, then the value of the determinant is:
(a) 0 only
(b) an even number but not necessarily 0
(c) an odd number
(d) 0, 1 or –1

17. If A = [[2, –1, 0], [–1, 3, 0], [1, 0, 1]], then what is the value of det[adj(adj A)]?
(a) 5
(b) 25
(c) 125
(d) 625

18. If A = Identity matrix of order 3, then what is 23A³ – 19A² – 4A equal to?
(a) Null matrix of order 3
(b) Identity matrix of order 3
(c) [[2,0,0],[0,2,0],[0,0,2]]
(d) [[7,0,0],[0,7,0],[0,0,7]]

19. The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C²?
(a) 3
(b) 9
(c) 81
(d) 729

20. If Aₖ = [[k–1, k], [k–2, k+1]], then what is det(A₁) + det(A₂) + ... + det(A₁₀₀) equal to?
(a) 100
(b) 1000
(c) 9900
(d) 10000

21. The Cartesian product A × A has 16 elements among which are (0, 2) and (1, 3). Which of the following statements is/are correct?

1. It is possible to determine set A
2. A × A contains the element (3, 2).
   Select the correct answer using the code given below:
   (a) 1 only
   (b) 2 only
   (c) Both 1 and 2
   (d) Neither 1 nor 2

22. Let A = {1, 2, 3, ..., 20}. Define a relation R from A to A by R = {(x, y) : 4x – 3y = 1}, where x, y ∈ A. Which of the following statements is/are correct?

1. The domain of R is {1, 4, 7, 10, 13, 16}.
2. The range of R is {1, 5, 9, 13, 17}.
3. The range of R is equal to codomain of R.
   Select the correct answer using the code given below:
   (a) 1 only
   (b) 2 only
   (c) 1 and 2
   (d) 2 and 3

23. Consider the following statements:

1. The relation f defined by f(x) = x³ (0 ≤ x ≤ 2) and 4x (2 ≤ x ≤ 8) is a function
2. The relation g defined by g(x) = x² (0 ≤ x ≤ 4) and 3x (4 ≤ x ≤ 8) is a function.
   Which of the statements given above is/are correct?
   (a) 1 only
   (b) 2 only
   (c) Both 1 and 2
   (d) Neither 1 nor 2

24. Consider the following statements:

1. A = (A ∪ B) ∪ (A – B)
2. A ∪ (B – A) = (A ∪ B)
3. B = (A ∪ B) – (A – B)
   Which of the statements given above are correct?
   (a) 1 and 2 only
   (b) 2 and 3 only
   (c) 1 and 3 only
   (d) 1, 2 and 3

25. A function satisfies f(x – y) = f(x)/f(y), where f(y) ≠ 0. If f(1) = 0.5, then what is f(2) + f(3) + f(4) + f(5) + f(6) equal to?
(a) 15/32
(b) 17/32
(c) 29/64
(d) 31/64

26. What is 2 cot(½ cos⁻¹ (√5 / 3)) equal to?
(a) –1
(b) 1
(c) 3 + √5
(d) 3 – √5

27. If sec⁻¹ p – cosec⁻¹ q = 0, where p > 0, q > 0; then what is the value of p⁻² + q⁻²?
(a) 1
(b) 2
(c) 1/2
(d) 1/(2√2)

28. What is 1 + sin²(cos⁻¹ (3/√17)) equal to?
(a) 25/17
(b) 8/17
(c) 9/17
(d) 47/17

29. If tan(π cos θ) = cot(π sin θ), 0 < θ < π/2; then what is the value of 8 sin²(θ + π/4)?
(a) 16
(b) 2
(c) 1
(d) 1/2

30. If tan α = 1/7, sin β = 1/√10; 0 < α, β < π/2 then what is the value of cos(α + 2β)?
(a) 1/2
(b) –1/√2
(c) 1/√2
(d) 1

31. What is the number of real roots of the equation (1 – x)⁴ + (5 – x)⁴ = 82?
(a) 0
(b) 2
(c) 4
(d) 8

32. What is the sum of all the roots of the equation?
(a) 24
(b) 12
(c) 10
(d) 6

33. What are the roots of equation-I: z³ + 2z² + 2z + 1 = 0?
(a) 1, ω, ω²
(b) –1, ω, ω²
(c) 1, –ω, ω²
(d) –1, –ω, –ω²

34. Which one of the following is a root of equation-II: z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0?
(a) –1
(b) –ω
(c) –ω²
(d) ω

35. What is the number of common roots of equation-I and equation-II?
(a) 0
(b) 1
(c) 2
(d) 3

36. If k = c/2 (c ≠ 0), then the roots of the equation (a+b)x² – (a+b+c)x + k = 0 are:
(a) Real and equal
(b) Real and unequal
(c) Real iff a > c
(d) Complex but not real

37. If k = c, then the roots of the equation are:
(a) (a+c)/(a+b) and b/(a+b)
(b) (a+c)/(a+b) and –b/(a+b)
(c) 1 and –c/(a+b)
(d) –1 and –c/(a+b)

38. What is T₁ + 2T₂ + 3T₃ + ... + nTₙ equal to?
(a) 0
(b) 1
(c) 2ⁿ
(d) n 2ⁿ⁻¹

39. What is 1 – T₁ + 2T₂ – 3T₃ + ... + (–1)ⁿ nTₙ equal to?
(a) 0
(b) –2ⁿ⁻¹
(c) n 2ⁿ⁻¹
(d) 1

40. What is T₁ + T₂ + T₃ + ... + Tₙ equal to?
(a) 2ⁿ
(b) 2ⁿ – 1
(c) 2ⁿ⁻¹
(d) 2ⁿ + 1

41. Which one of the following is a possible expression for g(x)?
(a) √(x+1) – √(x+1)
(b) √(x+1) – √(x+1) + 1
(c) √(x+1) + √(x+1)
(d) x + 1 – √(x+1) + 1

42. What is g(15) equal to?
(a) 1
(b) 2
(c) 3
(d) 4

43. What is f(0.5) equal to?
(a) 1/2
(b) 2/3
(c) 1
(d) 2

44. If f is differentiable, then what is f'(0.5) equal to?
(a) 1/4
(b) 2/3
(c) 2
(d) 4

45. The function is decreasing on:
(a) [–28/3, 0]
(b) [0, 28/3]
(c) [0, 50/3]
(d) [0, 56/3]

46. The function attains local minimum value at:
(a) x = –28/3
(b) x = –1
(c) x = 0
(d) x = 28/3

47. What is the maximum value of y?
(a) 3/2
(b) 3
(c) 4
(d) 6

48. What is the maximum value of xy?
(a) 9/4
(b) 3/2
(c) 4/9
(d) 2/3

49. What is the range of the function f(x) = π + sin²x?
(a) [0, 1]
(b) [π, π+1]
(c) [π–1, π+1]
(d) [π–1, π–1]

50. What is the period of the function?
(a) 2π
(b) π
(c) π/2
(d) The function is non-periodic

51. What is the directrix of the parabola?
(a) y = –1/8
(b) y = 1/8
(c) x = –1/8
(d) x = 1/8

52. What is the length of latus rectum of the parabola?
(a) 1
(b) 1/2
(c) 1/4
(d) 1/8

53. What is lim (x→1) [f(x) – 1]/g(x) equal to?
(a) ln(ab)/4
(b) ln(ab)/2
(c) ln(ab)
(d) 2 ln(ab)

54. What is lim (x→1) f(x)^g(x) equal to?
(a) √(ab)
(b) ab
(c) 2ab
(d) √(ab)/2

55. What is the domain of the function f(x) = √(2 – x) + √(2 + x)?
(a) (–2, 2)
(b) [–2, 2]
(c) R – (–2, 2)
(d) R – [–2, 2]

56. What is the greatest value of the function?
(a) √3
(b) √6
(c) √8
(d) 4

57. What is lim (x→0⁺) h(x) equal to?
(a) –2
(b) –1
(c) 0
(d) 1

58. What is lim (x→0⁻) h(x) equal to?
(a) –2
(b) –1
(c) 0
(d) 2

66. If f'(x) = cos(ln x) and y = f((2x – 3)/x), then what is dy/dx equal to?
(a) cos(ln((2x–3)/x))
(b) –(3/x²) sin(ln((2x–3)/x))
(c) (3/x²) cos(ln((2x–3)/x))
(d) –(3/x²) cos(ln((2x–3)/x))

67. What is ∫ from 0 to 8π of |sin x| dx equal to?
(a) 2
(b) 4
(c) 8
(d) 16

68. What is the area between the curve f(x) = x|x| and x-axis for x ∈ [–1, 1]?
(a) 2/3
(b) 1/2
(c) 1/4
(d) 1/3

69. What are the order and the degree respectively of the differential equation x² (d³y/dx³)² + (dy/dx)⁴ + sin x = 0?
(a) 3, 4
(b) 1, 4
(c) 2, 2
(d) 3, 2

70. What is the differential equation of all parabolas of the type y² = 4a(x – b)?
(a) d²y/dx² + (dy/dx)² = 0
(b) d²y/dx² + x² (dy/dx)² = 0
(c) y² d²y/dx² + (dy/dx)² = 0
(d) y d²y/dx² + (dy/dx)² = 0

71. What is a₁ + a₅ – a₁₀ – a₁₅ – a₂₀ – a₂₅ + a₃₀ + a₃₄ equal to?
(a) 0
(b) 25
(c) 125
(d) 250

72. What is ∑ from n=1 to 34 of aₙ equal to?
(a) 900
(b) 1025
(c) 1200
(d) 1275

73. What is the value of p + q?
(a) –1/2
(b) –1/4
(c) 0
(d) 1/2

74. What is the value of pq?
(a) –1/16
(b) –1/4
(c) 1/4
(d) 1/16

75. What is pq equal to?
(a) 1
(b) 2
(c) 8/3
(d) –8/3

76. For how many values of x does 1/p become zero?
(a) No value
(b) Only one value
(c) Only two values
(d) Only three values

77. What is a value of sin 3x + sin 3y?
(a) –1
(b) 0
(c) 1
(d) 3

78. What is a value of cos³x + cos³y?
(a) 3√3 / 8
(b) 3√6 / 8
(c) 3√6 / 4
(d) 1

79. What is the value of a + b + √2 c equal to?
(a) 3a
(b) 2b
(c) 3b
(d) 2c

80. What is the ratio of a² : b² : c²?
(a) 2 : 2 + √3 : 3
(b) 2 : 2 – √3 : 2
(c) 2 : 2 + √3 : 2
(d) 2 : 2 – √3 : 3

81. What is the equation of directrix of parabola y² = 4bx, where b < 0 and b² + b – 2 = 0?
(a) x + 1 = 0
(b) x – 2 = 0
(c) x – 1 = 0
(d) x + 2 = 0

82. The points (–a, –b), (0, 0), (a, b) and (a², ab) are:
(a) lying on the same circle
(b) vertices of a square
(c) vertices of a parallelogram that is not a square
(d) collinear

83. Given that 16p² + 49q² – 4r² – 56pq = 0, which one of the following is a point on a pair of straight lines (px + qy + r)(px + qy – r) = 0?
(a) (2/2, 7/2)
(b) (2/2, –7/2)
(c) (4, –7)
(d) (4, 7)

84. If 3x + y – 5 = 0 is the equation of a chord of the circle x² + y² – 25 = 0, then what are the coordinates of the midpoint of the chord?
(a) (3/4, 1/4)
(b) (3/2, 1/2)
(c) (3/4, –1/4)
(d) (3/2, –1/2)

85. Consider the following in respect of the equation x²/(24–k) + y²/(k–16) = 2.

1. The equation represents an ellipse if k = 19.
2. The equation represents a hyperbola if k = 12.
3. The equation represents a circle if k = 20.
   How many of the statements given above are correct?
   (a) Only one
   (b) Only two
   (c) All three
   (d) None

86. Consider the following statements in respect of hyperbola x²/cos²θ – y²/sin²θ = 1:

1. The two foci are independent of θ.
2. The eccentricity is sec θ.
3. The distance between the two foci is 2 units.
   How many of the statements given above are correct?
   (a) Only one
   (b) Only two
   (c) All three
   (d) None

87. Consider the following in respect of the circle 4x² + 4y² – 4ax – 4ay + a² = 0:

1. The circle touches both the axes.
2. The diameter of the circle is 2a.
3. The centre of the circle lies on the line x + y = a.
   How many of the statements given above are correct?
   (a) Only one
   (b) Only two
   (c) All three
   (d) None

88. For what value of k is the line (k–3)x – (5–k²)y + k² – 7k + 6 = 0 parallel to the line x + y = 1?
(a) –1, 1
(b) –1, 2
(c) 1, –2
(d) 2, –2

89. The line x + y = 4 cuts the line joining P(–1, 1) and Q(5, 7) at R. What is PR : RQ equal to?
(a) 1 : 1
(b) 1 : 2
(c) 2 : 1
(d) 1 : 3

90. What is the sum of the intercepts of the line whose perpendicular distance from origin is 4 units and the angle which the normal makes with positive direction of x-axis is 15°?
(a) 8
(b) 4√6
(c) 8√6
(d) 16

91. What is the length of projection of the vector î + 2ĵ + 3k̂ on the vector 2î + 3ĵ – 2k̂?
(a) 1/√17
(b) 2/√17
(c) 3/√17
(d) 2/√14

92. If (a⃗ × b⃗)² + (a⃗ · b⃗)² = 144 and |b⃗| = 4, then what is the value of |a⃗|?
(a) 3
(b) 4
(c) 5
(d) 6

93. If θ is the angle between vectors a⃗ and b⃗ such that a⃗ · b⃗ ≥ 0, then which one of the following is correct?
(a) 0 ≤ θ ≤ π
(b) π/2 ≤ θ ≤ π
(c) 0 ≤ θ ≤ π/2
(d) 0 < θ < π/2

94. The vectors 60î + 3ĵ, 40î – 8ĵ & βî – 52ĵ are collinear if
(a) β = 20
(b) β = 40
(c) β = –40
(d) β = 26

95. Consider the following in respect of the vectors a⃗ = (0,1,1) and b⃗ = (1,0,1):

1. The number of unit vectors perpendicular to both a⃗ and b⃗ is only one.
2. The angle between the vectors is π/3.
   Which of the statement given above is/are correct?
   (a) 1 only
   (b) 2 only
   (c) both 1 and 2
   (d) Neither 1 nor 2

96. If L is the line with direction ratios <3, –2, 6> and passing through (1, –1, 1), then what are the coordinates of the points on L whose distance from (1, –1, 1) is 2 units?
(a) (11/7, 13/7, 19/7) and (1/7, 3/7, 5/7)
(b) (19/7, –11/7, 13/7) and (–1/7, 3/7, –5/7)
(c) (13/7, 11/7, 19/7) and (–1/7, –3/7, 5/7)
(d) (13/7, –11/7, 19/7) and (1/7, –3/7, –5/7)

97. Which one of the planes is parallel to the line (x–2)/3 = (y–3)/4 = (z–4)/5?
(a) 2x + 2y + z – 1 = 0
(b) 2x – y – 2z + 5 = 0
(c) 2x + 2y – 2z + 1 = 0
(d) x – 2y + z – 1 = 0

98. What is the angle between the lines 2x = 3y = –z and 6x = –y = –4z?
(a) 0°
(b) 30°
(c) 60°
(d) 90°

99. What is the equation of the sphere concentric with the sphere x² + y² + z² – 2x – 6y – 8z – 5 = 0 and which passes through the origin?
(a) x² + y² + z² – 2x – 8z = 0
(b) x² + y² + z² – 2x – 6y = 0
(c) x² + y² + z² – 6y – 8z = 0
(d) x² + y² + z² – 2x – 6y – 8z = 0

100. A point P lies on the line joining A(1, 2, 3) and B(2, 10, 1). If z-coordinate of P is 7, what is the sum of other two coordinates?
(a) –15
(b) –13
(c) –11
(d) –9

101. The sum of deviations of n numbers from 10 and 20 are p and q respectively. If (p – q)² = 10000, then what is the value of n?
(a) 10
(b) 20
(c) 50
(d) 100

102. If x̄ = 20 is the mean of 10 observations x₁ to x₁₀, then what is the value of ∑ [(3xᵢ – 4)/5]² from i=1 to 10?
(a) 0
(b) 12
(c) 112
(d) 1012

103. If the mean and the sum of squares of 10 observations are 40 and 16160 respectively, then what is the standard deviation?
(a) 16
(b) 6
(c) 5
(d) 4

104. Three dice are thrown. What is the probability of getting a sum which is a perfect square?
(a) 17/108
(b) 5/108
(c) 19/108
(d) 23/108

105. A, B, C and D are mutually exclusive and exhaustive events. If 2P(A) = 3P(B) = 4P(C) = 5P(D), then what is 77P(A) equal to?
(a) 12
(b) 15
(c) 20
(d) 30

106. Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
(a) 1/81
(b) 1/72
(c) 1/18
(d) 1/36

107. Two dice are thrown. What is the probability that difference of number on them is 2 or 3?
(a) 7/36
(b) 7/18
(c) 5/18
(d) 11/36

108. What is the mean of the numbers 1, 2, 3, ..., 10 with frequencies ⁹C₀, ⁹C₁, ..., ⁹C₉ respectively?
(a) 1.1 × 2⁸
(b) 1.2 × 7⁴
(c) 5.5
(d) 0.55

109. The probability that a person recovers from a disease is 0.8. What is the probability that exactly 2 persons out of 5 will recover from the disease?
(a) 0.00512
(b) 0.02048
(c) 0.2048
(d) 0.0512

110. Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
(a) 0.10
(b) 0.19
(c) 0.45
(d) 0.95

111. If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (–3y + 3)?
(a) –r
(b) r
(c) √(3r)
(d) –√(3r)

112. A fair coin is tossed 6 times. What is the probability of getting a result in the 6th toss which is different from those obtained in the first five tosses?
(a) 7/16
(b) 1/16
(c) 1/32
(d) 1/64

113. If H is the Harmonic Mean of three numbers ¹⁰C₄, ¹⁰C₅ and ¹⁰C₆, then what is the value of 270/H?
(a) 1
(b) 14/17
(c) 17/14
(d) 1/31

114. In a class, there are n students including the students P and Q. What is the probability that P and Q sit together if seats are assigned randomly?
(a) 1/n
(b) 2/n
(c) 4/n
(d) 1/(2n)

115. In a Binomial distribution B(n, p), n = 6 and 9P(X=4) = P(X=2). What is p equal to?
(a) 1/4
(b) 1/2
(c) 3/4
(d) 4/5

116. What is the probability that all three boys sit together?
(a) 1/5
(b) 1/4
(c) 1/3
(d) 1/12

117. What is the probability that boys and girls sit alternatively?
(a) 4/5
(b) 1/10
(c) 5/6
(d) 1/7

118. What is the probability that no two girls sit together?
(a) 2/5
(b) 3/5
(c) 1/18
(d) 1/5

119. What is the probability that P and Q take the two end positions?
(a) 1/15
(b) 7/15
(c) 14/15
(d) 11/45

120. What is the probability that Q and U sit together?
(a) 2/5
(b) 1/4
(c) 5/6
(d) 1/3

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