NDA Mathematics 13 April 2025 | NDA Year Wise Previous Years Questions

Mathematics

1. The sum of the first k terms of a series S is 3k2+5k. Which one of the following is correct? 2. The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If r≠1 is the common ratio, then what i 3. If one root of the equation x2−kx+k=0 exceeds the other by $2\sqrt{3}$ then which one of the following is a value of k? 4.  If $x+\frac{5}{y}=4$, and then $y+\frac{5}{x}= -4$  what is (x + y) equal to? 5. If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference? 6. If p, 1, q are in AP and p, 2, q are in GP, then which of the following statements is/are correct? I. p, 4, q are in HP 7.  If x=(1111)2, y=(1001)2 and z=(110)2, then what is x3−y3−z3−3xyz equal to? 8. If $\Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} $  and 9. . Consider the following statements in respect of the determinant  $ \Delta = \begin{vmatrix} k(k+2) & 2k+1 &a 10. If $ \begin{vmatrix} 2 & 3+i & -1 \\ 3-i & 0 & i -1 \\ -1 &-1 -i & 1 \end{vmatrix} = A + iB 11. If A2+B2+C2=0, then what is the value of the following? $\begin{vmatrix} 1& cosC& cosB\\ cosC&1&cosA \\ 12. If ω is a non-real cube root of unity, then what is a root of the following equation?$ \begin{vmatrix} x+1 & \omega 13. What is $ \left( \frac{\sqrt{3}+i}{\sqrt{3}-i} \right)^3 $ equal to? 14.  If x2− x +1 = 0, then what is $(x-\frac{1}{x})^2+(x-\frac{1}{x})^4+(x-\frac{1}{x})^8$ 15. How many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that a 16. If $z\ne0$ is a complex number, then what is amp(z)+amp(zˉ) equal to? 17. How many sides are there in a polygon that has 20 diagonals? 18. In how many ways can the letters of the word DELHI be arranged, keeping the positions of vowels and consonants unchanged 19. What is the number of positive integer solutions of x+y+z=5? 20. What is the number of rational terms in the expansion of $(\sqrt{3}+5^\frac{1}{4})^{12}$ 21. If the sum of binomial coefficients in the expansion of (x+y)n is 256, then the greatest binomial coefficient occurs in 22. If $k<(\sqrt{2}+1)^3<k+2,$ where k is a natural number, the value of k? 23. If  $\begin{bmatrix} x & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 24. If  $A = \begin{bmatrix} y & z & x \\ z & x & y \\ x & y & z \end{bmatrix} $ where x,y,z 25. Consider the following in respect of a non-singular matrix M: I. ∣M2∣=∣M∣2 II. ∣M∣=∣M−1∣ III. ∣M∣=∣MT∣ How many of t 26. If $f(\theta) = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$&nbsp 27. If $A=\begin{bmatrix}1&2&2 \\ 2&1&2\\2&2&1\end{bmatrix}$ then what is A2−4A equal to? 28. If the number of selections of r as well as (n+r) things from 5n different things are equal, then what is the value of r 29. What is the number of selections of at most 3 things from 6 different things? 30. If $A=\begin{bmatrix}x&y&z\\y&z&x\\z&x&y\end{bmatrix}$ where x,y,z are integers, is an ort 31. What is (p+q+r) equal to? 32. What is (pq+qr+rp) equal to? 33. What is (p2+q2+r2) equal to? 34. What is the minimum value of p? 35. What is the maximum value of p? 36. Consider the following statements: I. The triangle is obtuse-angled triangle. II. The sum of acute angles of the trian 37. What is ∠B equal to? 38. What is the area of the triangle? 39. What is PN equal to? 40. What is MN equal to? 41. What is (p/q) equal to? 42. What is (p+q) equal to? 43. What is tan2α equal to? 44. What is tanα equal to? 45. What is 2sin2α+cos2α equal to? 46. One of the angles of the triangle is 47. Consider the following statements: I. The triangle is right-angled. II. One of the sides of the triangle is 3 times th 48. A man at M, standing 100 m away from the base (P) of a chimney of height 50 m, observes the angle of elevation of the hi 49. If k is a root of x2−4x+1=0, then what is tan−1k+tan−1$\frac{1}{k}$ equal to? 50. If tan−1k+tan−1 $\frac{1}{2}$=$\frac{π}{2} $, then what is the value of k? 51. Under what condition will the lines m2x+ny−1=0 and n2x−my+2=0 be perpendicular? 52. If p and q are real numbers between 0 and 1 such that the points (p,1), (1,q) and (0,0) form an equilateral triangle, th 53. The vertices of a triangle are A(1, 1),B(0, 0) and C(2, 0). The angular bisectors of the triangle meet at P. What are th 54. Let A(3, -1) and B(1, 1) be the end points of line segment AB. Let P be the middle point of the line segment AB. Let Q b 55. ABC is an equilateral triangle and AD is the altitude on BC. If the coordinates of A are (1,2) and that of D are (−2,6), 56. What is the equation of the circle whose diameter is 10 cm and the equations of two of its diameters are x+y=0 and x−y=0 57. A square is inscribed in a circle x 2 + y 2 + 2x + 2y + 1 = 0 and its sides are parallel to coordinate axes. Which one o 58. A tangent to the parabola y2 = 4x is inclined at an angle 45° deg with the positive direction of x-axis. What is the poi 59. . What is the distance between the two foci of the hyperbola 25x2 - 75y 2= 225 ? 60. If any point on an ellipse is (3sin$\alpha$, 5cos$\alpha$), then what is the eccentricity of the ellipse? 61. If a line in 3 dimensions makes angles α, β and γ with the positive directions of the coordinat 62. A(1,2,−1), B(2,5,−2) and C(4,4,−3) are three vertices of a rectangle. What is the area of the rectangle? 63. ABC is a triangle right-angled at B. If A(k,1,−1), B(2k,0,2) and C(2+2k,k,1) are the vertices of the triangle, then what 64. If a line  $\frac{x+1}{p} = \frac{y-1}{q} = \frac{z-2}{r}$  where p=2q=3r, makes an angle θ with the 65. What is the equation of the plane passing through the point (1,1,1) and perpendicular to the line whose direction ratios 66. A line makes angles α, β and γ with the positive directions of the coordinate axes. If $\vec{a}=(\sin^2 \alpha)\hat{i} + 67. Consider the following statements with respect to a vector d = (a × b) × c: I. d is coplanar with a and b. I 68. The position vectors of three points A, B and C are a" style="display:block;position:absolute;width:100%;height:inherit; 69. The position vectors of three points A, B and C respectively, where  $\vec{a} ,\vec{b} $ and $\vec{c 70. Let $\vec{a},\vec{b} ,(\vec{a}\times\vec{b})$ be unit vectors. What is $\vec{a}.\vec{b}$ equal to? 71. What is $(\frac{dy}{dx})^2$ equal to? 72. What  $\left[ \frac{x^2+4}{y^2+4} \frac{dy}{dx} \left( x^2+4 \frac{d^2y}{dx^2} - 16y \right) \right] $ &n 73. What is ∠A equal to if the area of the triangle is maximum? 74. What is the maximum area of the triangle? 75. The derivative of y with respect to x 76. If p+q=10, then what is $\frac{dy}{dx}$ equal to? 77. What is the nature of the curve? 78. . What is the area bounded by the curve, the x-axis and the line x = 4? 79. What is  $\lim_{x \to 0} f'(x)$ equal to? 80. Consider the following statements: I. The function is continuous at x=−1. II. The function is differentiable at x=1. 81. What is the range of the function? 82. What is ∫ydx equal to?  where c is the constant of integration. 83. What $\lim_{x \to 0} {f(x) g(x)}$ is  equal to? 84. What is $\lim_{x \to 0} \frac{f(x)}{g(x)}$ equal to? 85. What is the domain of the function (f(x))? 86. What is the area bounded by the curve f(x) and y = 3? 87. If f(x) = px + q then what is the value of (p + q) ? 88. Consider the following statements: I. f is one-one function. II. f is onto function if the codomain is the set of natura 89. What is $\lim_{x \to 1} \{f \circ f(x)\}$ equal to? 90. What is the area bounded by the function f(x) and the x-axis? 91. What is y equal to? 92. What is  $\frac{dy}{dx}$ euqal to ? 93. What is  $\lim_{x \to 0} \frac{\sqrt{f(x)} - 3}{\sqrt{f(x)+7} - 4}$ equal to? 94. Consider the following statements: I. f(x) is an increasing function. II. f(x) has local maximum at x = 0 Which of the s 95. What is f(16) equal to? 96. What is f(1)f(4) equal to? 97. What is f(0) equal to? 98. What is f(20)+f(-20) equal to? 99. What  $\int_{\sqrt{2}}^{\sqrt{3}} f(x) dx$ equal to? 100. $\int_{\sqrt{2}}^{2} f(x) dx$  is equal to ? 101. What is the total number of students whose height is less than or equal to 165 cm? 102. What is the height of the class? 103. The height which occurs most frequently in the class is 104. The most appropriate graphical representation of the given frequency distribution is 105. Which one of the following is correct? 106. Which one of the following statements is correct? 107. What is the value of k? 108. What is the value of P * (X = 3) ^ 0 109. What is the probability that the committee includes exactly 3 gentlemen? 110. What is the probability that the committee includes at least 2 ladies? 111. What is the probability that the bonus scheme will be introduced? 112. If the bonus scheme has been introduced, then what is the probability that the manager appointed was B? 113. The arithmetic mean of 100 observations is 50. If 5 is subtracted from each observation and then divided by 20, then wha 114. The standard deviation of 100 observations is 10. If 5 is added to each observation and then divided by 20, then what wi 115. If P(A)=1/3, P(B)=1/2 and P(A∩B)=1/4, then what is the value of P(A|BC)? 116. If P(A)=1/3, P(B)=1/2 and P(A∩B)=1/4, then what is the value of P(AC∩BC)? 117. If two fair dice are tossed, then what is the probability that the sum of the numbers on the faces of the dice is strict 118. The probability of a man hitting a target is 1/5. If the man fires 7 times, then what is the probability that he hits th 119. Let X be a random variable following binomial distribution whose mean and variance are 200 and 160 respectively. What is 120. What is the arithmetic mean of 82,92,102,...,152?

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NDA Mathematics 13 April 2025

MCQ (Single Correct Answer)

+2.5

-0.833

Consider the following for the two (02) items that follow:
The slope of the tangent to the curve y = f(x) at (x, f(x) ) is 4 for every real number x and the curve passes through the origin.

. What is the area bounded by the curve, the x-axis and the line x = 4?

8 square units

16 square units

32 square units

64 square units

NDA Mathematics 13 April 2025

MCQ (Single Correct Answer)

+2.5

-0.833

Consider the following for the two (02) items that follow:

$\text{Let } f(x)= \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \ge 1 \end{cases} \\$

What is $\lim_{x \to 0} f'(x)$ equal to?

Limit does not exist

NDA Mathematics 13 April 2025

MCQ (Single Correct Answer)

+2.5

-0.833

Consider the following for the two (02) items that follow:

$\text{Let } f(x)= \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \ge 1 \end{cases} \\$

Consider the following statements:

I. The function is continuous at $x = - 1$ .

II. The function is differentiable at $x = 1$ .

Which of the statements given above is/are correct?

I only

II only

Both I and II

Neither I nor II

NDA Mathematics 13 April 2025

MCQ (Single Correct Answer)

+2.5

-0.833

Consider the following for the two (02) items that follow:

Let the function y = (1 - cos x)^-1 where$x \ne 2n\pi$ and n is an integer

What is the range of the function?

[0,∞)

[0.5,∞)

[1,∞)

(−∞,0.5]

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