NDA Mathematics 1st September 2024 | NDA Year Wise Previous Years Questions

Mathematics

1. Let X be a matrix of order 3 x 3, Y be a matrix of order 2 x 3 and Z be a matrix of order 3 × 2. Which of the following 2. Consider the following statements: I. The set of all irrational numbers between $\sqrt{12}$ and $\sqrt{15}$ i 3. How many 4-digit numbers are there having all digits as odd?  4. If ω ≠ 1 is a cube root of unity, then what is (1 + ω - ω2)100 + (1 - ω + ω2)100 equa 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?&nb 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)? 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 8. Let p = ln(x), q = ln(x3) and r = ln(x5), where x > 1. Which of the following statements is/are correct?  9. If $\rm z=\frac{1}{3}\begin{vmatrix}i&2i&1\\\ 2i&3i&2\\\ 3&1&3\end{vmatrix}=x+iy;i=\sqrt{-1 10. What is the value of the sum $\rm \sum\limits_{n=1}^{20}(i^{n-1}+i^n+i^{n+1})$ where i = √-1 ? 11. Let x >1, y >1, z >1 be in GP. Then $\rm \frac{1}{1+ln x}, \frac{1}{1+lny}, \frac{1}{1+\ln z}$ are 12. If $\rm \omega=-\frac{1}{2}+i\frac{\sqrt3}{2}$ then what is $\rm \begin{vmatrix}1+\omega &1+\omega^2& 13. If the sum of the first n terms of a series is n(2n+1), then what is the nth term? 14. In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd 15. The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many 16. If n is a root of the equation x2 + px + m = 0 and m is a root of the equation x2 + px + n = 0, where m&n 17. In how many ways can a student choose (n - 2) courses out of n courses if 2 courses are compulsory (n > 4)?  18. If $\rm D_n=\begin{vmatrix}n&20&30\\\ n^2&40&50\\\ n^3&60&70\end{vmatrix}$ then what i 19. Consider the following in respect of the matrices  $\rm P=\begin{bmatrix}0&c&-b\\\ -c&0&a 20. If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?  21. If 4 sin-1 x + cos-1 x = π, then what is sin-1 x + 4 cos-1 x equal to? 22. What is cot2(sec-1 2) + tan2 (cosec-1 3) equal to ? 23. In a triangle ABC $\rm \frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}$ What is the area of the triangle if 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 25. In a triangle ABC, ∠A = 75° and ∠B = 45°. What is 2a - b equal to? 26. What is the number of solutions of the equation cot2x cot3x = 1 for 0 < x 27. What is the general solution of cos100 x - sin100 x = 1? 28. In a triangle ABC tan A + tan B + tan C = k  What is the value of cot A cot B cot C?  29. What is sin12° sin48° equal to?  30. What is $\rm \frac{\cos 17^\circ-\sin 17^\circ}{\cos 17^\circ+\sin ^\circ}$ equal to 31. Consider the following numbers :  I. tan 22.5°  II. cot 22.5°  III. tan 22.5° - cot 22.5°  How m 32. If $\rm \frac{x}{\cos \theta}=\frac{y}{\cos \left(\frac{2\pi}{3}-\theta\right)}=\frac{z}{\cos\left(\frac{2\pi}{3}+\ 33. If p tan (θ - 30°) = q tan (θ + 120°), then what is (p + q) / (p - q) equal to? 34. Let P and Q be two non-void relations on a set A. Which of the following statements are correct? I. P and Q are reflexi 35. If A and B are two non-empty sets having 10 elements in common, then how many elements do A x B and B x A have in common 36. What is the remainder when 7n - 6n is divided by 36 for n = 100? 37. What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is betwe 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 followi 39. What is the number of real roots of the equation (x - 1)2 + (x - 3)2 + (x - 5)2 = 0? 40. In a class of 240 students, 180 passed in English, 130 passed in Hindi and 150 passed in Sanskrit. Further, 60 passed in 41. what is the value of $\rm \left|\frac{Z_1}{Z_2}\right|$, 42. what is the value of $\rm \frac{1}{2}+Re\left(\frac{Z_1}{Z_2}\right)?$ 43. What is the common difference? 44. What is the sum of all five terms?  45. What is V + W equal to?  46. What is the value of (U + V)W?  47. Which one of the following statements is correct?  48. Which one of the following is a root of the equation?  49. What is A(adj A) equal to?   50. What is A-1 equal to?   51. What is 3α + 2β equal to if (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) is a null vector? 52. For what value of the angle between the vectors $\rm \vec a\ and \ \vec b$ is the quantity $\rm |\vec a\t 53. Let θ be the angle between two unit vectors $\rm \vec a\ and\ \vec b. \ if\ \vec a+2\vec b$ is perpendicu 54. Let ABCDEF be a regular hexagon. If $\rm \vec{AD}=m \vec {BC}\ and\ \vec {CF}=n\vec {AB}$ then what is mn equa 55. The vectors $\rm \vec a, \vec b\ and\ \vec c$ are of the same length. If taken pairwise they form equal a 56. The diagonals of a quadrilateral ABCD are along the lines x - 2y = 1 and 4x + 2y = 3. The quadrilateral ABCD&n 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 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?   59. The equation of a circle is  (x2 - 4x + 3) + (y2 - 6y + 8) = 0 Which of the following statements are co 60. Consider the points P(4k, 4k) and Q(4k, -4k) lying on the parabola y2 = 4kx. If the vertex is A, then what is ∠PAQ equal 61. What is BAC equal to? 62. What are the coordinates of A?  63. What are the coordinates of vertex D? 64. What is the point of intersection of the diagonals of the trapezium? 65. What is the diameter of the sphere?  66. The centre of the sphere lies on the plane  67. Which of the following are the direction ratios of S?  68. If 〈l, m, n〉 are direction cosines of S, then what is the value. of 43 (I2 - m2 - n2)?   69. What are the direction ratios of the line?   70. What is the point of intersection of L and P?  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 f o g(x) = g o f(x), then what is the value of k?   73. What is the minimum value of the function f(x) = log10(x2 + 2x + 11)?  74. Which one of the following is correct regarding $\rm \lim\limits_{x\rightarrow 3}\frac{|x-3|}{x-3}$? 75. What is the maximum value of a cos x + b sin x + 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)?   77. lf f(x) = [x]2 - 30[x] + 221 = 0, where [x] is the greatest integer function, then what is the sum of all 78. f f(x) = 9x - 8√x such that g(x) = f(x) - 1, then which one of the following is correct? 79. What is $\rm \lim\limits_{x\rightarrow \frac{\pi}{2}}(\sec \theta-\tan \theta)$ equal to?  80. Let f(x) f(y) = f(xy) for all real x, y. If f(2) = 4, then what is the value of f(1/2)? 81. Which one of the following is f(x)?  82. Which one of the following is g(x)?  83. What is f(0.999) + f(1.001) equal to 84. Consider the following statements :  I. f(x) is continuous at x = 0.  Il. f(x) is continuous at x = 1  85. What is the greatest value of f(x)?  86. What is the least value of f(x)?  87. What is the value of k?  88. What is the area of the parabola bounded by the latus rectum?  89. What are the order and degree respectively of the differential equation?  90. What is the solution of the differential equation?  91. What is $\rm \int_0^2f(x)dx$ equal to 92. What is $\rm \int_h^3f(x)dx$ equal to 93. Consider the following statements : I. f(t) is an odd function. Il. g(t) is an odd function. Which of the statements 94. What is $\rm \int_{-\pi}^\pi g(t)dt$ equal to 95. What is h’(0) equal to?  96. What is g’(0) equal to?  97. What is $\rm \int_0^{\pi/2}\frac{d(x)}{g(x)}$ equal to 98. What is I equal to? 99. What is |U2(x) - V2(x)| equal to? 100. What is U(x) V(x) equal to?  101. Let x - 3y + 4 = 0 and 2x - 7y + 8 = 0 be two lines of regression computed from some bivariate data. If byx and bxy are 102. The mean of n observations 1, 4, 9, 16 ...., n2 is 130. What is the value of n? 103. Three distinct natural numbers chosen at random from 1 to 10. What is the probability that they are consecutive? 104. A, B, C are three mutually exclusive and exhaustive events associated with a random experiment. If 3P(B) = 4P(A) and 3P( 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 106. A box contains 2 black, 4 yellow and 6 white balls. Three balls are drawn in succession with replacement. What is the pr 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 probab 108. The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants and vowel 109. A natural number x is chosen at random from the first 100 natural numbers. What is the probability that x2 + x > 50? 110. What is the mean deviation of the first 10 natural numbers? 111. Let $\rm \Sigma_{i=1}^9x_i^2=885$ If M is the mean and σ is the standard deviation of  x1, x2, 112. The mean of the series x1, x2...xn is x̅ If xn is replaced by k, then what is the new mean? 113. A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is 114. Urn A contains 2 white and 2 black balls while urn B contains 3 white and 2 black balls. One ball is transferred from ur 115. For two events A and  B, P(A) = P(A|B) = 0.25 and P(BIA) = 0.5. Which of the following are correct? I. A and B are 116. Two perfect dice are thrown. What C is the probability that the sum of the numbers on the faces is neither 9 nor 10? 117. The occurrence of a disease in an industry is such that the workers have 20% chance of suffering from it. What is the pr 118. Three perfect dice are rolled. Under the condition that no two show the same face, what is the probability that one of t 119. Three perfect dice D1, D2 and D3 and are rolled. Let x, represent the numbers on D1, D2 and D3 respective 120. In a binomial distribution, if the mean is 6 and the standard deviation is √2, then what are the values of the parameter

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NDA Mathematics 1st September 2024

MCQ (Single Correct Answer)

+2.5

-0.833

Consider the following in respect of the matrices $\rm P=\begin{bmatrix}0&c&-b\\\ -c&0&a\\\ b&-a&0\end{bmatrix}\ and\ \rm Q=\begin{bmatrix}a^2&ab&ac\\\ ab&b^2&bc\\\ ac&bc&c^2\end{bmatrix}$

I. PQ is a null matrix.

II. QP is an identity matrix of order 3.

III. PQ = QP

Which of the above is/are correct?

I only

II only

I and III

II and III

NDA Mathematics 1st September 2024

MCQ (Single Correct Answer)

+2.5

-0.833

If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?

-1

NDA Mathematics 1st September 2024

MCQ (Single Correct Answer)

+2.5

-0.833

If 4 sin^-1x + cos^-1 x = π, then what is sin^-1 x + 4 cos^-1 x equal to?

π/2

3π/2

2π

NDA Mathematics 1st September 2024

MCQ (Single Correct Answer)

+2.5

-0.833

What is cot²(sec^-1 2) + tan² (cosec^-1 3) equal to ?

11/12

11/24

7/24

1/24

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