Mathematics
1. If $Δ_1 = \begin{vmatrix} 1 & p & q \\ 1 & q & r \\ 1 & r & p\end{vmatrix} \ \text{and} \ Δ 2. If (a - b) (b - c) (c - a) = 2 and abc = 6, then what is the value of $\begin{vmatrix}a & b & c \\ a^2 & 3. Under which of the following conditions does the determinant $\begin{vmatrix}a & b & c \\ b & c & a 4. Consider the following in respect of the matrices:
A = [m n], B = [-n -m] & $ C = \begin{bmatrix} m \\ -m\end{ 5. If $A = \begin{bmatrix} 2 \sin \theta & \cos \theta & 0 \\ -2\cos \theta & \sin \theta & 0 \\ -1 &a 6. For what value of k is the matrix $\begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 7. Let A be a non-singular matrix and B = adj A. Which of the following statements is/are correct?
1. AB = BA
2. AB is a 8. Consider the following statements in respect of square matrices A and B of same order :
1. If AB is a null matrix, then 9. If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C 10. Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is 11. How many terms are there in the expansion of $\left(1 + \frac{2}{x}\right)^9\left(1-\frac{2}{x}\right)^9 ? $ 12. Consider the following statements in respect of the expansion of (x + y)10
1. Among all the coefficients of the terms, 13. If C(3n, 2n) = C(3n, 2n - 7), then what is the value of C(n, n - 5)? 14. What is the value of
C(51, 21) - C(51, 22) + C(51, 23) - C(51, 24) + C(51, 25) - C(51, 26) + C(51, 27) - C(51, 28) + C( 15. How many odd numbers between 300 and 400 are there in which none of the digits is repeated ? 16. How many permutations are there of the letters of the word 'TIGER' in which the vowels should not occupy the even p 17. Let α and β be the roots of the equation x2 + px + q = 0. If α3 and β3 are the roots of the equation x2 + mx + n = 18. Let α and β be the roots of the equation x2 - ax - bx + ab - c = 0. What is the quadratic equation whose roots are 19. If the roots of the equation x2 - ax - bx - cx + bc + ca = 0 are equal, then which one of the following is correct? 20. Let α and β (α > β) be the roots of the equation x2 - 8x + q = 0. If α2 - β2 = 16, then what is t 21. What is the maximum value of n such that 5n divides (30! + 35!), where n is a natural number? 22. What is the value of 2(2 × 1) + 3(3 × 2 × 1) + 4(4 × 3 × 2 × 1) + 5(5 × 4 × 3 × 23. If A = {{1, 2, 3}}, then how many elements are there in the power set of A? 24. If a, b, c are in GP where a > 0, b > 0, c > 0, then which of the following are correct?
1. a2, b2, c2 ar 25. If $\frac{a+b}{2}, b, \frac{b+c}{2}$ are in HP, then which one of the following is correct? 26. What is value of cot2 15° + tan2 15° ? 27. In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to? 28. If $α + β = \frac{\pi}{4}$ and 2tan α = 1, then what is tan 2β equal to? 29. If tan (45° + θ) = 1 + sin 2θ, where $-\frac{\pi}{4}< θ <\frac{\pi}{4},$ then what is the value of cos 2 30. Let sin 2θ = cos 3θ, where θ is acute angle. What is the value of 1 + 4 sin θ? (given that $\sin 18^\circ =\fr 31. If $\tan θ = - \frac{5}{12},$ then what can be the value of sin θ? 32. What is the value of $\cos^4\frac{7\pi}{8}+\cos^4\frac{5\pi}{8}?$ 33. What is $\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right)$ equal to? 34. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the p 35. The shadow of a tower becomes x metre longer, when the angle of elevation of sun changes from 60° to θ. If the height of 36. If $\tan^{-1} \left(\frac{1}{2}\right)+\tan^{-1} \left(\frac{x}{3}\right)=\frac{\pi}{4},$ where 0 < x < 37. If 3 sin-1x + cos-1x = π, then what is x equal to? 38. If tan α + tan β = 1 - tan α.tan β, where tan α tan β ≠ 1, then which of the following is 39. If (1 + tan θ) (1 + tan 9θ) = 2, then what is the value of tan(10θ)? 40. What is the value of sin 0° + sin 10° + sin 20° + sin 30° + ⋯ + sin 360°? 41. Consider all the subsets of the set A = {1, 2, 3, 4}. How many of them are supersets of the set {4}? 42. Consider the following statements in respect of two non-empty sets A and B :
1. x ∉ (A ∪ B) ⇒ x ∉ A or x 43. Consider the following statements in respect of two non-empty sets A and B :
1. A ∪ B = A ∩ B if A = B
44. Consider the following statements in respect of the relation R in the set IN of natural numbers defined by xRy if x2 - 5 45. Consider the following statements in respect of any relation R on a set A :
1. If R is reflexive, then R-1 is also 46. What is the principal argument of $\frac{1}{1 + i}$ where $i = \sqrt{-1}?$ 47. What is the modulus of $\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?$ 48. Consider the following statements:
1. $\frac{n!}{3!}$ is divisible by 6, where n > 3
2. $\frac{n!}{3 49. In how many ways can a team of 5 players be selected out of 9 players so as to exclude two particular players ? 50. In the expansion of $\left(x + \frac{1}{x}\right)^{2n} $, what is the (n + 1)th term from the end (when arranged in desc 51. If the sum of the first 9 terms of an AP is equal to sum of the first 11 terms, then what is the sum of the first 20 ter 52. If the 5th term of an AP is $\frac{1}{10}$ and its 10th term is $\frac{1}{5},$ then what is the sum of first 5 53. What is (1110011)2 ÷ (10111)2 equal to ? 54. If x3 + y3 = (100010111)2 and x + y = (11111)2, then what is (x - y)2 + xy equal to ? 55. Consider the inequations 5x - 4y + 12 < 0, x + y < 2, x < 0 and y > 0. Which one of the following points lie 56. Consider the following statements in respect of the function y = [x], x ∈ (-1, 1) where [.] is the greatest integer 57. What is the degree of the differential equation? $1+\left(\frac{dy}{dx}\right)^2 =\left(\frac{d^2y}{dx^2}\right)^{\ 58. A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decay 59. What is the domain of the function $f(x) =\sqrt{1 - (x-1)^2} \ ?$ 60. The area of the region bounded by the parabola y2 = 4kx, where k > 0 and its latus rectum is 24 square units. What is 61. What is $\int\limits^{\frac{\pi}{4}}_0 \frac{dx}{(\sin x + \cos x)^2}$ equal to? 62. What is $\int (\sin x)^{-1/2} (\cos x)^{-3/2}dx$ equal to? 63. If $I_1 =\int\frac{e^x dx}{e^x + e^{-x}}$ and $I_2 =\int\frac{dx}{e^{2x} + 1},$then what is I1 + I2 equal 64. What is $\int\limits^{-1}_{-2}\frac{x}{|x|}dx$ equal to? 65. How many extreme values does sin4x + 2x, where $0 < x < \frac{\pi}{2} $ have ? 66. What is the maximum value of the functions $f(x) = \frac{1}{\tan x+\cot x},$ where $0 < x < \frac{\ 67. If $4f(x) - f \left(\frac{1}{x}\right)=\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right),$ then what is f 68. If f(x) = 4x + 3, then what is f o f o f(-1) equal to? 69. If xyyx = 1, then what is $\frac{dy}{dx}$ at (1, 1) equal to? 70. If y = (xx)x, then what is the value of $\frac{dy}{dx}$ at x = 1? 71. Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect 72. If $\frac{dy}{dx} = (\ln 5)y $ with y(0) = ln 5, then what is y(1) equal to? 73. Consider the following in respect of the function f(x) = 10x :
1. Its domain is (-∞, ∞)
2. It is a continuous function 74. What is $\lim\limits_{x \to 0}x^3(\text{cosec} \ x)^2$ equal to ? 75. What is $\lim\limits_{x \to 1}\frac{x^3 - 1}{\sqrt x - 1}$ equal to? 76. In which one of the following intervals is the function $f(x) = \frac{x^3}{3}-\frac{7x^2}{2} + 6x + 5$ decreas 77. If the derivative of the function$f(x) =\frac{m}{x} +2nx + 1$ vanishes at x = 2, then what is the value of m + 8n ? 78. What is the area included in the first quadrant between the curves y = x and y = x3 ? 79. If xy = 4225 where x, y are natural numbers, then what is the minimum value of x + y ? 80. What does the equation $x \frac{dy}{dx}-2y= 0$ represent ? 81. If the points with coordinates (-5, 0), (5p2, 10p) and (5q2, 10q) are collinear, then what is the value of pq where p ≠ 82. What is the equation of the straight line which passes through the point (1, -2) and cuts off equal intercepts from the 83. What is the equation of the circle which touches both the axes in the first quadrant and the line y - 2 = 0? 84. What is the equation of the parabola with focus (-3, 0) and directrix x - 3 = 0 ? 85. What is the distance between the foci of the ellipse x2 + 2y2 = 1 ? 86. Let a, b, c be the lengths of sides BC, CA, AB respectively of a triangle ABC. If p is the perimeter and q is the area o 87. A straight line passes through the point of intersection of x + 2y + 2 = 0 and 2x - 3y - 3 = 0. It cuts equal intercepts 88. Under which one of the following conditions are the lines ax + by + c = 0 and bx + ay + c = 0 parallel (a ≠ 0, b ≠ 0)? 89. What is the equation of the locus of the mid-point of the line segment obtained by cutting the line x + y = p, (where p 90. If the point (x, y) is equidistant from the points (2a, 0) and (0, 3a) where a > 0, then which one of the following i 91. What is the value of k? 92. If p is the perpendicular distance from the centre of the sphere to the plane, then which one of the following is correc 93. What is the equation of the line through the origin and the centre of the sphere ? 94. What are the direction ratios of the normal to the plane ? 95. If p, q and r are the intercepts made by the plane on the coordinate axes respectively, then what is (p + q + r) equal t 96. If $4\hat i + \hat j - 3\hat k$ and $p\hat i + q\hat j - 2\hat k$ are collinear vectors, then what are the possible 97. If $\vec a, \vec b, \vec c$, are the position vectors of the vertices A, B, C respectively of a triangle ABC and G is th 98. Consider the following statements :
1. Dot product over vector addition is distributive
2. Cross product over vector a 99. Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that $\vec{a}\times \vec{b} = \vec{c} $. 100. Let $\vec a$ and $\vec b$ be two unit vectors such that $|\vec a - \vec b|<2.$ If 2θ is the angle between $\vec 101. Two digits out of 1, 2, 3, 4, 5 are chosen at random and multiplied together. What is the probability that the last digi 102. The frequency curve (assuming unimodal) corresponding to the data obtained in an experiment is skewed to the left. What 103. The variance of five positive observations is 3.6. If four of the observations are 2, 2, 4, 5 then what is the remaining 104. What is the arithmetic mean of 50 terms of an AP with first term 4 and common difference 4 ? 105. What is the coefficient of mean deviation of 21, 34, 23, 39, 26, 37, 40, 20, 33, 27 (taken from mean)? 106. What is the mean of the values? 107. What is the algebraic sum of the deviations of the same set of values measured from 99? 108. If the algebraic sum of the deviations of the same set of values measured from y is 180, then what is the value of y ? 109. What is the mean of the marks ? 110. What is the median of the marks? 111. What is the sum of the deviations measured from the median? 112. What is $P (G \cap \overline T)$ equal to? 113. What is $P(G | \overline T) $ equal to? 114. What is $P(\overline T | \overline G)$ equal to? 115. What is the probability that exactly 3 out of 6 workers suffer from a disease? 116. What is the probability that no one out of 6 workers suffers from a disease? 117. What is the probability that at least one out of 6 workers suffer from a disease ? 118. What is the value of p ? 119. What is the value of q ? 120. If the frequency of each class is doubled, then what would be the mean?
1
NDA Mathematics 10 April 2022
MCQ (Single Correct Answer)
+2.5
-0.83
What is $\lim\limits_{x \to 0}x^3(\text{cosec} \ x)^2$ equal to ?
A
0
B
$\frac{1}{2}$
C
1
D
Limits does not exist
2
NDA Mathematics 10 April 2022
MCQ (Single Correct Answer)
+2.5
-0.83
What is $\lim\limits_{x \to 1}\frac{x^3 - 1}{\sqrt x - 1}$ equal to?
A
0
B
3
C
6
D
Limit does not exist
3
NDA Mathematics 10 April 2022
MCQ (Single Correct Answer)
+2.5
-0.83
In which one of the following intervals is the function $f(x) = \frac{x^3}{3}-\frac{7x^2}{2} + 6x + 5$ decreasing?
A
(-∞, 1) only
B
(1, 6)
C
(6, ∞) only
D
(-∞, 1) ∪ (6, ∞)
4
NDA Mathematics 10 April 2022
MCQ (Single Correct Answer)
+2.5
-0.83
If the derivative of the function$f(x) =\frac{m}{x} +2nx + 1$ vanishes at x = 2, then what is the value of m + 8n ?
A
-2
B
0
C
2
D
Cannot be determined due to insufficient data
Paper analysis
Total Questions
Mathematics
120
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