NDA Previous Year Question Paper 2023 Mathematics
English and General Abilities Solved Question Paper
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HINTS & SOLUTIONS
MATHEMATICS
1. (d)
Since z = x + iy
Now (x + iy)(x – iy) = |x + iy + x – iy|
⇒ x² – (iy)² = |2x|
⇒ x² + y² = 2x
⇒ x² – 2x + 1 + y² = 1
⇒ (x – 1)² + y² = 1
So locus of z is a circle.
2. (c)
1! + 3! + 5! + 7! + ⋯ + 199!
= 1 + 6 + 5×24 + 7×6×5×24 + ⋯ + 199!
= 1 + 6 + 24k + 7
= 24 + 24k = 24(1 + k) + 7
Remainder = 7
3. (c)
Let √(12 + 5i) = x + iy
⇒ 12 + 5i = (x + iy)²
⇒ x² – y² = 12 and 2xy = 5
Solving, x = 5/√2 , y = 1/√2
So √(12 + 5i) = 5/√2 + i/√2
Similarly √(12 – 5i) = 5/√2 – i/√2
∴ √(12 + 5i) + √(12 – 5i) = 5√2
4. (a)
I + AAᵀ =
[[1 0 0] [1] [[2 2 3]
[0 1 0] + [2] [1 2 3] = [2 5 6]
[0 0 1] [3] [3 6 10]]
|I + AAᵀ| = 2(50 – 36) – 2(20 – 18) + 3(12 – 15) = 15
5. (a)
Given |BC| = 2|A|
|2 A⁻¹ BC| = 2³ |A⁻¹| |BC| = 8 × (1/|A|) × 2|A| = 16
6. (c)
tₙ = (2n + 5)/7
t₁ = 7/7 = 1
t₁₄₀ = (2×140 + 5)/7 = 285/7
S₁₄₀ = (140/2) × (t₁ + t₁₄₀) = 70 × (7/7 + 285/7) = 2920
7. (a)
A is skew-symmetric of order 3 (odd) ⇒ |A| = 0
|4A⁴| – |3A³| + |2A²| – |A| + |–I|
= 4³ |A|⁴ – 3³ |A|³ + 2² |A|² – |A| + (–1)³ |I|
= 0 – 0 + 0 – 0 – 1 = –1
8. (a)
A is skew-symmetric of order 3 ⇒ |A| = 0
A² is symmetric and |A²| = |A|² = 0
Hence the correct statement is (a).
9. (b)
A = diag(2, 3, 4) is non-zero diagonal
⇒ Aⁿ is also non-zero diagonal for any positive integer n
and Aⁿ is symmetric.
So statements 2 and 3 are correct.
10. (c)
a+b, 2b, b+c are in H.P.
⇒ 1/(a+b), 1/(2b), 1/(b+c) are in A.P.
⇒ b² = ac
⇒ a, b, c are in G.P.
11. (c)
tₙ = a rⁿ⁻¹
(t₁ t₃ ⋯ t₂₁)^(1/11) = t₁₁
12. (b)
√(–√(–1)) = (1 – i)/√2
13. (c)
Maximum points of intersection of 10 circles = ¹⁰C₂ × 2 = 90
14. (b)
n(S) = 2n + 1
Number of subsets with at most n elements = 4096
⇒ 2ⁿ = 2¹² ⇒ n = 6
15. (b)
Putting x = 0 in the determinant gives e = 0
16. (b)
Determinant of a 3×3 matrix with entries ±1 is always even.
17. (d)
|A| = 5
|adj(adj A)| = |A|⁽ⁿ⁻¹⁾² = 5⁴ = 625
18. (a)
A = I ⇒ 23A³ – 19A² – 4A = 0 (null matrix)
19. (c)
|A| = 3
|C| = |adj A| = |A|² = 9
|C²| = 81
20. (d)
|Aₖ| = 2k – 1
Sum from k=1 to 100 = 1 + 3 + 5 + ⋯ + 199 = 10000
21. (c)
Both statements are correct. A = {0,1,2,3}
22. (b)
Domain of R = {1,4,7,10,13}
Range of R = {1,5,9,13,17}
23. (a)
Only statement 1 is correct (f is a function, g is not).
25. (d)
f(2) + f(3) + f(4) + f(5) + f(6) = 31/64
26. (c)
2 cot(½ cos⁻¹(√5/3)) = 3 + √5
27. (a)
p⁻² + q⁻² = 1
28. (a)
1 + sin²(cos⁻¹(3/√17)) = 25/17
29. (c)
8 sin²(θ + π/4) = 1
30. (c)
cos(α + 2β) = 1/√2
31. (b)
Number of real roots = 2
32. (b)
Sum of all roots = 12
33. (b)
Roots of equation-I: –1, ω, ω²
34. (d)
ω is a root of equation-II
35. (c)
Number of common roots = 2
36. (b)
Roots are real and unequal
37. (c)
Roots are 1 and –c/(a+b)
38. (d)
T₁ + 2T₂ + ⋯ + nTₙ = n 2ⁿ⁻¹
39. (d)
1 – T₁ + 2T₂ – 3T₃ + ⋯ = 1
40. (b)
T₁ + T₂ + ⋯ + Tₙ = 2ⁿ – 1
41. (b)
g(x) = √(1+x) – √4 [blocked] + 1
42. (c)
g(15) = 3
43. (b)
f(0.5) = 2/3
44. (c)
f′(0.5) = 2
45. (a)
Function is decreasing on [–28/3, 0]
46. (c)
Local minimum at x = 0
47. (b)
Maximum value of y = 3
48. (a)
Maximum value of xy = 9/4
49. (b)
Range = [π, π+1]
50. (b)
Period = π
51. (a)
Directrix: y = –1/8
52. (b)
Length of latus rectum = 1/2
53. (b)
lim = (1/2) ln(ab)
54. (a)
lim = √(ab)
55. (b)
Domain = [–2, 2]
56. (c)
Greatest value = √8
57. (b)
lim (x→0⁺) h(x) = –1
58. (a)
lim (x→0⁻) h(x) = –2
59. (d)
a = 3
60. (b)
From continuity at x = 3:
–1 + a = a – b = 1 + b
⇒ b = 1, a = 3
61. (b)
∫ (sin⁴x + cos⁴x) / (1 + 3ˣ) dx from –2π to 2π
= ∫ (sin⁴x + cos⁴x) dx from 0 to 2π (even function property)
= (3π)/4
62. (c)
I = ∫₀^{2π} (sin⁴x + cos⁴x)/(1 + 3ˣ) dx = 3π/4
63. (a)
For differentiability at x = 1: a + b = –1/3
64. (b)
lim (x→0) f(x) = –2/3
65. (a)
f(x) = |ln |x||
f(0.5) = –2
66. (d)
dy/dx = –(3/x²) cos(ln((2x–3)/x))
67. (d)
∫₀^{8π} |sin x| dx = 16
68. (a)
Area = 2/3
69. (d)
Order = 3, Degree = 2
70. (d)
Differential equation of y² = 4a(x – b) is
y (d²y/dx²) + (dy/dx)² = 0
71. (a)
a₁ + a₅ – a₁₀ – a₁₅ – a₂₀ – a₂₅ + a₃₀ + a₃₄ = 0
72. (d)
∑ aₙ from n=1 to 34 = 1275
73. (c)
p + q = 0
74. (a)
pq = –1/16
75. (a)
pq = 1
76. (c)
1/p = 0 has only two values of x
77. (b)
sin 3x + sin 3y = 0
78. (a)
cos³x + cos³y = 3√3 / 8
79. (c)
a + b + √2 c = 3b
80. (c)
a² : b² : c² = 2 : 2 + √3 : 2
81. (a)
Directrix: x + 1 = 0 (since b = –1)
82. (d)
The points are collinear
83. (d)
Point (4, 7) lies on the pair of lines
84. (b)
Mid-point of the chord = (3/2, 1/2)
85. (c)
All three statements are correct
(k=19 → ellipse, k=12 → hyperbola, k=20 → circle)
86. (b)
Only two statements are correct
(eccentricity = sec θ, distance between foci = 2)
87. (b)
Only two statements are correct
(The circle touches both axes, centre lies on x + y = a)
88. (b)
k = –1, 2
89. (b)
PR : RQ = 1 : 2
90. (c)
Sum of intercepts = 8√6
91. (b)
Length of projection = 2/√17
92. (a)
|a⃗| = 3
93. (c)
0 ≤ θ ≤ π/2
94. (c)
β = –40
95. (d)
Neither 1 nor 2
(Number of unit vectors perpendicular to both is two; angle is π/3)
96. (d)
Points are (13/7, –11/7, 19/7) and (1/7, –3/7, –5/7)
97. (b)
2x – y – 2z + 5 = 0 is parallel to the given line
98. (d)
Angle between the lines = 90°
99. (d)
x² + y² + z² – 2x – 6y – 8z = 0
100. (b)
Sum of other two coordinates = –13
101. (d)
n = 100
102. (a)
∑ [(3xᵢ – 4)/5]² = 0
103. (d)
Standard deviation = 4
104. (c)
Probability = 19/108
105. (d)
77 P(A) = 30
106. (d)
Probability = 1/36
107. (c)
Probability = 5/18
108. (c)
Mean = 5.5
109. (c)
Probability = 0.2048
110. (b)
Probability = 0.19
111. (a)
Correlation coefficient = –r
112. (c)
Probability = 1/32
113. (c)
270/H = 17/14
114. (b)
Probability = 2/n
115. (c)
p = 3/4
116. (b)
Probability that all three boys sit together = 1/4
117. (b)
Probability that boys and girls sit alternatively = 1/10
118. (d)
Probability that no two girls sit together = 1/5
119. (a)
Probability that P and Q take the two end positions = 1/15
120. (d)
Probability that Q and U sit together = 1/3
GENERAL KNOWLEDGE Answers (selected)
146. (b) Only statement 3 is correct (Kavach has SIL-4).
147. (c) Ukraine
148. (b) Australia
149. (b) Antardrishti
150. (d) Chola Empire
NDA Previous Year question paper solved 2023
