Mathematics
1. How many four-digit natural numbers are there such that all of the digits are odd ? 2. What is $\displaystyle\sum_{r=0}^n$2r C(n, r) equal to ? 3. If different permutations of the letters of the word 'MATHEMATICS' are listed as in a dictionary, how many words (with o 4. Consider the following statements :
1. If f is the subset of Z × Z defined by f = {(xy, x − y); x, y ∈ Z}, then f is a 5. Consider the determinant
Δ = $\left|\begin{array}{lll}\text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \te 6. If A = $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & \sin \theta & 7. If X is a matrix of order 3 × 3, Y is a matrix of order 2 × 3 and Z is a matrix of order 3 × 2, then which of the follow 8. For how many quadratic equations, the sum of roots is equal to the product of roots? 9. Consider the following statements :
1. The set of all irrational numbers between $\sqrt{2}$ and $\sqrt{5}$ is 10. Consider the following statements :
1. 2 + 4 + 6 + ........ + 2n = n2 + n
2. The expression n2 + n + 41 alwa 11. Let p, q(p > q) be the roots of the quadratic equation x2 + bx + c = 0 where c > 0. If p2 + q2 − 11 12. What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm ? 13. Let A = {7, 8, 9, 10, 11, 12, 13; 14, 15, 16} and let f ∶ A → N be defined by f(x) = the highest prime factor of x.
How 14. Let R be a relation from N to N defined by R = {(x, y): x, y ∈ N and x2 = y3}. Which of the following are not corre 15. Consider the following :
1. A ∩ B = A ∩ C ⇒ B = C
2. A ∪ B = A ∪ C ⇒ B = C
Which of the above is/are correct ?
16. What is the modulus of z?
17. What is angle θ such that z is purely real ?
where n is an integer
18. What is angle θ such that z is purely imaginary ?
where n is an integer
19. What is the ratio of the first term of A to that of B ? 20. What is the ratio of their 10th terms ? 21. If d is the common difference of A, and D is the common difference of B, then which one of the following is always corre 22. What is the value of q if the coefficients of x3 and x6 are equal ?
23. What is the ratio of the coefficients of middle terms in the expansion (when expanded in ascending powers of x)?
24. Under what condition the coefficients of x2 and x4 are equal ?
25. How many 4-letter words each of two vowels and two consonants with or without meaning, can be formed ?
26. How many 8-letter words with or without meaning, can be formed such that consonants and vowels occupy alternate position 27. How many 8-letter words with or without meaning, can be formed so that all consonants are together?
28. What is the value of a11C11 + a12C12 + a13C13 ?
29. What is the value of $a_{21}C_{11}+a_{22}C_{12}+a_{23}C_{13}$? 30. What is the value of $\left|\begin{array}{lll}\text{a}_{21} & \text{a}_{31} & \text{a}_{11} \\ \text{a}_{23 31. If $\displaystyle\sum_{x=2}^n$f(x) = 2044, then what is the value of n ?
32. What is $\displaystyle\sum_{x=1}^5$f(2x − 1) equal to ?
33. What is $\displaystyle\sum_{x=1}^6$2x f(x) equal to ?
34. How many received medals in exactly two of the three sports ?
35. How many received medals in at least two of three sports ?
36. How many received medals in exactly one of three sports ?
37. What is P equal to ?
38. What is Q equal to ?
39. What is the minimum value of determinant of A ?
40. What is the height of the lamp post ?
41. What is $\frac{\text{AB}}{\sin \text{C}}$ equal to ?
42. What is cos A + cos B + cos C equal to ?
43. At what height is the top of the tower above the ground level ?
44. If θ is the inclination of the tower to the horizontal, then what is cot θ equal to ?
45. What is the length of the tower ?
46. What is the value of cosec$\left(−\frac{73 \pi}{3}\right)$ ? 47. What is the value of $\cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \ 48. What is the value of tan$\left(\frac{3\pi}{8}\right)$ ? 49. What is tan−1 cot(cosec−1 2) equal to ? 50. In a triangle ABC, a = 4, b = 3, c = 2. What is cos 3C equal to ? 51. What is cos 36° − cos 72° equal to ? 52. If sec x = $\frac{25}{24}$ and x lies in the fourth quadrant, then what is the value of tan x + sin x ? 53. What is the value of tan2 165° + cot2 165° ? 54. What is the value of $\sin\left(2\text{n}\pi+\frac{5\pi}{6}\right)\sin\left(2\text{n}\pi−\frac{5\pi}{6}\right)$&nbs 55. If 1 + 2(sin x + cos x)(sin x − cos x) = 0 where 0 < x < 360°, then how many values does x take ? 56. Consider the following statements in respect of the line passing through origin and inclining at an angle of 75° with th 57. If P(3, 4) is the mid-point of a line segment between the axes, then what is the equation of the line? 58. The base AB of an equilateral triangle ABC with side 8 cm lies along the y-axis such that the mid-point of AB is at the 59. The centre of the circle passing through the origin and making positive intercepts 4 and 6 on the coordinate axes, lies 60. The centre of an ellipse is at (0, 0), major axis is on the y-axis. If the ellipse passes through (3, 2) and (1, 6), the 61. An equilateral triangle is inscribed in a parabola x2 = $\sqrt{3}$y where one vertex of the triangle is at the 62. Consider the points A(2, 4, 6), B(−2, −4, −2), C(4, 6, 4), and D(8, 14, 12). Which of the following statements is/are co 63. Consider the equation of a sphere x2 + y2 + z2 − 4x − 6y − 8z − 16 = 0.
Which of the following statement 64. A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane? 65. Consider the following statements :
1. The direction ratios of y-axis can be <0, 4, 0>
2. The direction ratios o 66. PQRS is a parallelogram. If $\overrightarrow{\text{PR}}=\vec{\text{a}}$ and $\overrightarrow{\text{QS}}=\ 67. Let $\vec{\text{a}}$ and $\vec{\text{b}}$ are two unit vectors such that $\vec{\text{a}}+2 \vec 68. Let $\vec{\text{a}}$, $\vec{\text{b}}$ and $\vec{\text{c}}$ be unit vectors lying on the s 69. What are the values of x for which the angle between the vectors 2x2$\hat{\text{i}}$ + 3x$\hat{\text{j}}$ +&nb 70. The position vectors of vertices A, B and C of triangle ABC are respectively $\hat{\text{j}}+\hat{\text{k}}, 3\hat{ 71. Let z = [y] and y = [x] − x, where [.] is the greatest integer function. If x is not an integer but positive, then what 72. If f(x) = 4x + 1 and g(x) = kx + 2 such that fog(x) = gof(x), then what is the value of k ? 73. What is the minimum value of the function f(x) = log10(x2 + 2x + 11) ? 74. What is ∫(xx)2 (1 + ln x)dx equal to ? 75. What is ∫ex{1 + ln x + x ln x}dx equal to ? 76. What is $\int\frac{(\cos x)^{1.5}−(\sin x)^{1.5}}{\sqrt{\sin x⋅\cos x}}$dx equal to ? 77. If y = $\frac{x \sqrt{x^2−16}}{2} − 8 \ln\left|x + \sqrt{x^2−16}\right|$, then what is $\frac{\text{dy}}{\text 78. If y = (xx)x, then which one of the following is correct ? 79. What is the maximum value of 3(sin x − cos x) + 4(cos3 x − sin3 x) ? 80. What is the area of the region (in the first quadrant) bounded by y = $\sqrt{1−\text{x}^2}$, y = x and y = 0 ? 81. What is the area of the region bounded by x − |y| = 0 and x − 2 = 0 ? 82. If f(α) = $\sqrt{\sec^2\alpha−1}$, then what is $\frac{f(\alpha)+f(\beta)}{1−f(\alpha) f(\beta)}$ equal t 83. If f(x) = ln (x + $\sqrt{1+\text{x}^2}$), then which one of the following is correct ? 84. What is $\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1−\cos 4x}}$ equal to ? 85. What is $\displaystyle\lim_{x\rightarrow \frac{\pi}{2}} \frac{4x−2\pi}{\cos x}$ equal to ? 86. If f(x) = $\frac{x^2+x+|x|}{x}$, then what is $\displaystyle\lim_{x \rightarrow 0}$ f(x) equal to ? 87. What is $\displaystyle\lim_{h \rightarrow 0} \frac{\sin^2(x+h)−\sin^2x}{h}$ equal to ? 88. Let f(x) be a function such that f'(x) = g(x) and f''(x) = −f(x). Let h(x) = {f(x)}2 + {g(x)}2. Then consider the follow 89. If y = ln2$\left(\frac{x^2−x+1}{x^2+x+1}\right)$, then what is $\frac{\text{dy}}{\text{dx}}$ at x = 0 equal to 90. If $\frac{d}{d x}\left(\frac{1+x^4+x^8}{1−x^2+x^4}\right)$ = ax + bx3, then which one of the following is corr 91. Under which one of the following conditions does the function f(x) = (p sec x)2 + (q cosec x)2 attain minimum 92. Where does the function f(x) = $\displaystyle\sum_{j=1}^7$(x − j)2 attain its minimum value ? 93. Consider the following statements in respect of the function f(x) = $\left\{\begin{array}{rc}|x|+1, & 0<|x| 94. What is $\displaystyle\int_0^1 \ln \left(\frac{1}{x}−1\right)$dx equal to ? 95. If $\displaystyle\int_0^{\pi/2}$(sin4 x + cos4 x)dx = k, then what is the value of $\displaystyle\in 96. What is $\displaystyle\int_{−\pi/2}^{\pi/2}$(ecos x sin x + esin x cos x)dx equal to ? 97. What is the area of the region enclosed in the first quadrant by x2 + y2 = π2, y = sin x and x = 0 ? 98. Consider the following statements:
1. The degree of the differential equation $\frac{\text{dy}}{\text{dx}} + \cos 99. What is the differential equation of the family of parabolas having a vertex at origin and axis along positive y-axis? 100. What is the solution of the differential equation (dy − dx) + cos x(dy + dx) = 0 ? 101. Let x be the mean of squares of first n natural numbers and y be the square of mean of first n natural numbers. If 102. What is the probability of getting a composite number in the list of natural numbers from 1 to 50? 103. If n > 7, then what is the probability that C(n, 7) is a multiple of 7? 104. Two numbers x and y are chosen at random from a set of the first 10 natural numbers. What is the probability that (x + y 105. A number x is chosen at random from first n natural numbers. What is the probability that the number chosen satisfies x 106. Three fair dice are tossed once. What is the probability that they show different numbers that are in AP? 107. If P(A) = 0.5, P(B) = 0.7 and P(A ∩ B) = 0.3, then what is the value of P(A' ∩ B') + P(A' ∩ B) + P(A ∩ B') ? 108. Five coins are tossed once. What is the probability of getting at most four tails ? 109. Three fair dice are thrown. What is the probability of getting a total greater than or equal to 15 ? 110. The probability that a person hits a target is 0.5. What is the probability of at least one hit in 4 shots ? 111. A box contains 2 white balls, 3 black balls, and 4 red balls. What is the number of ways of drawing 3 balls from the box 112. During war, one ship out of 5 was sunk on an average in making a certain voyage. What is the probability that exactly 3 113. A card is drawn from a pack of 52 cards. A gambler bets that it is either a spade or an ace. The odds against his winnin 114. The coefficient of correlation between ages of husband and wife at the time of marriage for a given set of 100 couples w 115. The completion of a construction job may be delayed due to strike. The probability of strike is 0.6. The probability tha 116. Which one of the following subjects shows highest variability of marks ?
117. What is the coefficient of variation of marks in Mathematics ?
118. What is the median of the distribution ?
119. What is mean deviation about the median ?
120. What is the mean deviation about the mean ?
1
NDA Mathematics 4 September 2022
MCQ (Single Correct Answer)
+2.5
-0.83
What is ∫ex{1 + ln x + x ln x}dx equal to ?
A
x ex ln x + c
B
x2 ex ln x + c
C
x + ex ln x + c
D
xex + ln x + c
2
NDA Mathematics 4 September 2022
MCQ (Single Correct Answer)
+2.5
-0.83
What is $\int\frac{(\cos x)^{1.5}−(\sin x)^{1.5}}{\sqrt{\sin x⋅\cos x}}$dx equal to ?
A
$\sqrt{\sin \text{x}}−\sqrt{\cos \text{x}}$ + c
B
$\sqrt{\sin \text{x}}+\sqrt{\cos \text{x}}$ + c
C
$2\sqrt{\sin \text{x}}+2 \sqrt{\cos \text{x}}$ + c
D
$\frac{1}{2}\sqrt{\sin \text{x}}+\frac{1}{2} \sqrt{\cos \text{x}}$ + c
3
NDA Mathematics 4 September 2022
MCQ (Single Correct Answer)
+2.5
-0.83
If y = $\frac{x \sqrt{x^2−16}}{2} − 8 \ln\left|x + \sqrt{x^2−16}\right|$, then what is $\frac{\text{dy}}{\text{dx}}$ equal to ?
A
$\text{x}\sqrt{\text{x}^2−16}$
B
$\text{x}−\sqrt{\text{x}^2−16}$
C
$\sqrt{\text{x}^2−16}$
D
$4\sqrt{\text{x}^2−16}$
4
NDA Mathematics 4 September 2022
MCQ (Single Correct Answer)
+2.5
-0.83
If y = (xx)x, then which one of the following is correct ?
A
$\frac{\text{dy}}{\text{dx}}$ + xy(1 + 2 ln x) = 0
B
$\frac{\text{dy}}{\text{dx}}$ − xy(1 + 2 ln x) = 0
C
$\frac{\text{dy}}{\text{dx}}$ − 2xy(1 + ln x) = 0
D
$\frac{\text{dy}}{\text{dx}}$ + 2xy(1 + ln x) = 0
Paper analysis
Total Questions
Mathematics
120
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