Mathematics
1. If x2 + x + 1 = 0, then what is the value of x199 + x200 + x201? 2. If x, y, z are in GP, then which of the following is/are correct?
1. ln(3x), ln(3y), ln(3z) are in AP
2. xyz + ln(x), 3. If log102, log10(2x - 1), log10(2x + 3) are in AP, then what is x equal to? 4. Let S = {2, 3, 4, 5, 6, 7, 9}. How many different 3-digit numbers (with all digits different) from S can be made which a 5. If p = (1111 ... up to n digits), then what is the value of 9p2 + p? 6. The quadratic equation 3x2 - (k2 + 5k)x + 3k2 - 5k = 0 has real roots of equal magnitude and opposite sign. Which one of 7. If an = n(n!), then what is a1 + a2 + a3 +...+ a10 equal to? 8. If p and q are the non-zero roots of the equation x2 + px + q = 0, then how many possible values can q have? 9. If $\rm \Delta = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}$ then 10. $\frac{1}{b+c}, \frac{1}{c+a},\frac{1}{a+b}$ are in HP, then which of the following is/are correct?
1. a, b, c are 11. $\rm A=\begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}$ where a ∈ ℕ, then is A100 - A50 - 2A25 equal to?
12. If $\rm \begin{vmatrix} a & -b & a - b - c\\ -a & b & -a + b - c\\ -a & -b & -a - b + c \en 13. What is $\rm \sum\limits_{n=1}^{8n+7} i^n$ equal to, where i = √-1? 14. If z = x + iy, where i = √-1, then what does the equations zz̅ + ∣z∣2 + 4(z + z̅) - 48 = 0 represent 15. Which one of the following is a square root of $\rm 2a+2\sqrt{a^2 + b^2}$, where a, b ∈ ℝ? 16. If sinθ and cosθ are the roots of the equation ax2 + bx + c = 0, then which one of the following is correct? 17. If C(n, 4), C(n, 5) and C(n, 6) are in AP, then what is the value of n? 18. How many 4 - letter words (with or without meaning) containing two vowels can be constructed using only the letters (wit 19. Suppose 20 distinct points are placed randomly on a circle. Which of the following statements is/are correct?
1. The nu 20. How many terms are there in the expansion of $\rm \left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)^{21}$ where a 21. For what values of k is the system of equations 2k2x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent? 22. The inverse of a matrix A is given by $\rm \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatri 23. What is the period of the function f(x) = ln(2 + sin2x)? 24. If sin(A + B) = 1 and 2 sin(A - B) = 1, where 0 < A, B < $\frac{\pi}{2}$, then what is tan A ∶ tan B equ 25. Consider a regular polygon with 10 sides. What is the number of triangles that can be formed by joining the vertices whi 26. Consider all the real roots of the equation x4 - 10x2 + 9 = 0. What is the sum of the absolute values of the roots? 27. Consider the expansion of (1 + x)n. Let p, q, r and s be the coefficients of first, second, nth and (n + 1)th terms resp 28. Let sin-1x + sin-1y + sin-1z = $\frac{3\pi}{2}$ for 0 ≤ x, y z ≤ 1. What is the value of x1000 + y10 29. Let sin x + sin y = cos x + cos y for all x, y ∈ ℝ. What is $\rm tan \left(\frac{x}{2}+\frac{y}{2}\right)$ equ 30. Let $A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$ and (mI + nA)2 = A where m, n are positive rea 31. What is the value of following?
$\rm cot \left[sin^{-1} \frac{3}{5}+cot^{-1}\frac{3}{2} \right]$
32. Let 4sin2 x = 3, where 0 ≤ x ≤ π. What is tan3x equal to? 33. Let p, q and 3 be respectively the first, third and fifth terms of an AP. Let d be the common difference. If the product 34. Consider the following statements in respect of the roots of the equation x3 - 8 = 0 :
1. The roots are non-collinear.
35. Let the equation sec x.cosec x = p have a solution, where p is a positive real number. What should be the smallest value 36. For what value of θ, where 0 < θ < $\frac{\pi}{2}$, does sin θ + sin θ cos θ maximum value? 37. Consider the following statements in respect of sets:
1. The union over the intersection of sets is distributive.
2. T 38. Consider three sets X, Y and Z having 6, 5 and 4 elements respectively. All these 15 elements are distinct. Let S = (X - 39. Consider the following statements in respect of relations and functions:
1. All relations are functions but all functio 40. If log10 2 log2 10 + log10(10x) = 2, then what is the value of x? 41. Let ABC be a triangle. If cos2A + cos2B + cos2C = -1 then which one of the following is correct? 42. What is the value of the following determinant?
$\begin{vmatrix} \cos \rm C & \tan \rm A & 0\\ \sin \rm B & 43. Suppose set A consists of first 250 natural numbers that are multiple of 3 and set B consists of first 200 even natural 44. Let Sk denote the sum of first k terms of an AP. What is $\rm \frac{S_{30}}{S_{20}-S_{10}}$ equal to? 45. If the roots of the equation 4x2 - (5k + 1)x + 5k = 0 differ by unity, then which one of the following is a possible val 46. Consider the digits 3, 5, 7, 9. What is the number of 5-digit numbers formed by these digits in which each of these four 47. How many distinct matrices exist with all four entries taken from (1, 2)? 48. If i = √-1, then how many values does i-2n have for different n ∈ ℤ? 49. If $x = \frac{a}{b-c}$, $y = \frac{b}{c - a}$, $z = \frac{c}{a - b}$ then what is the value of the f 50. Consider the following in respect of the matrix $\rm A = \begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 51. What are the coordinates of the center of the circle? 52. If r is the radius of the circle, then which one of the following is correct? 53. Consider the following statements:
1. The third vertex has at least one irrational coordinate.
2. The area is irration 54. The difference of coordinates of the third vertex is 55. What is the equation of the diagonal BD? 56. What is the area of the parallelogram? 57. What is the equation of the altitude through B on AC? 58. What are the coordinates of circumcentre of the triangle? 59. What is the maximum number of parabolas that can be drawn through these two points as end points of latus rectum? 60. Consider the following statements in respect of such parabolas:
1. One of the parabolas passes through the origin (0, 0 61. The locus of a point P(x, y, z) which moves in such a way that z = 7 is a 62. Consider the following statements:
1. A-line in space can have infinitely many direction ratios.
2. It is possible for 63. The xy-plane divides the line segment joining the points (-1, 3, 4) and (2, -5, 6) 64. The number of spheres of radius r touching the coordinate axes is 65. ABCDEFGH is a cuboid with base ABCD. Let A(0, 0, 0), B(12, 0, 0), C(12, 6, 0) and G(12, 6, 4) be the vertices. If α 66. Let $\rm \vec{a}, \vec{b}$ and $\rm\vec{c}$ be unit vectors such that $\rm\vec{a} \times \vec{b 67. If $\rm\vec{a}+3\vec{b} = 3\hat{i}- \hat{j}$ and $\rm2\vec{a}+\vec{b} = \hat{i}- 2\hat{j}$, then what is 68. If $\rm(\vec {a} + \vec{b})$ is perpendicular to $\rm\vec {a}$ and magnitude of $\rm\vec {b}$&n 69. Let $\rm\vec {a}, \vec{b}$ and $ \rm \vec {c}$ be three vectors such that $\rm\vec {a}, \ 70. If the position vectors of A and B are (√2 - 1)î - ĵ and î + (√2 + 1)ĵ respectively, then what is the magn 71. If y = (1 + x)(1 + x2)(1 + x4)(1 + x8)(1 + x16) then what is $\frac{dy}{dx}$ at x = 0 equal to? 72. If y = cos x ⋅ cos 4x ⋅ cos 8x, then what is $\rm \frac{1}{y}\frac{dy}{dx}$ at $\rm x = \frac{\ 73. Let f(x) be a polynomial function such that f ∘ f(x) = x4. What is f'(1) equal to ? 74. What is $\rm \displaystyle\lim_{n \rightarrow \infty} \frac{a^n+b^n}{a^n-b^n}$ where a > b > 1, equal to 75. Let $\rm f(x) = \left\{\begin{matrix} 1+\frac{x}{2k}, & 0 < x < 2\\\ kx, & 2 \le x < 4 \end {matri 76. Consider the following statements in respect of f(x) = |x| - 1
1. f(x) is continuous at x = 1.
2. f(x) is differentiab 77. If $f(x) = \frac{[x]}{|x|}$, x ≠ 0, where [⋅] denotes the greatest integer function, then what is the right-ha 78. Consider the following statements in respect of the function $\rm f(x) = sin \left(\frac{1}{x^2}\right)$, x ≠ 79. What is the range of the function f(x) = 1 - sinx defined on entire real line? 80. What is the slope of the tangent of y = cos-1 (cos x) at x = $-\frac{\pi}{4}$? 81. What is the integral of f(x) = 1 + x2 + x4 with respect to x2? 82. Consider the following statements in respect of the function f(x) = x2 + 1 in the interval [1, 2]:
1. The maximum value 83. If f(x) satisfies f(1) = f(4), then what is $\rm \int^4_1f'(x) dx$ equal to? 84. What is $\rm \int^\frac{\pi}{2}_0 e^{ln(cos x)} dx$ equal to? 85. If $\rm \int \sqrt{1 - sin 2x} \space dx$ = A sinx + B cosx + C, where 0 < x < $\frac{\pi}{4}$, the 86. What is the order of the differential equation of all ellipses whose axes are along the coordinate axes? 87. What is the degree of the differential equation of all circles touching both the coordinate axes in the first quadrant? 88. What is the differential equation of $\rm y = A- \frac{B}{x}$? 89. What is $\rm \int^\pi _0 ln\left(tan\frac{x}{2}\right) dx$ equal to? 90. Where does the tangent to the curve y = ex at the point (0, 1) meet x-axis? 91. Consider the following statements in respect of the function f(x) = x + $\rm \frac{1}{x}$:
1. The local maximum va 92. What is the maximum area of a rectangle that can be inscribed in a circle of radius 2 units? 93. What is $\rm \int \frac{dx}{x(x^2 + 1)}$ equal to? 94. What is the derivative of $\rm e^{e^x}$ with respect to ex? 95. What is the condition that f(x) = x3 + x2 + kx has no local extremum? 96. If f(x) = 2x, then what is $\int^{10}_2\frac{f'(x)}{f(x)}dx$ equal to? 97. If $\rm \int^0_{-2} f(x)dx=k$, then $\rm \int^0_{-2}|f(x)|dx$ is 98. If the function f(x) = x2 - kx is monotonically increasing in the interval (1, ∞), then which one of the following is co 99. What is the area bounded by y = [x], where [⋅] is the greatest integer function, the x-axis and the lines x = -1.5 and x 100. The tangent to the curve x2 = y at (1, 1) makes an angle θ with the positive direction of x-axis. Which one of the 101. Consider the following relations for two events E and F :
1. P(E ∩ F) ≥ P(E) + P(F) - 1
2. P(E ∪ F) = P 102. If P(A|B) < P(A), then which one of the following is correct? 103. When the measure of central tendency is available in the form of mean, which one of the following is the most reliable a 104. A problem is given to three students A, B and C, whose probabilities of solving the problem independently are $\rm 105. In a cricket match, a batsman hits a six 8 times out of 60 balls he plays. What is the probability that on a ball played 106. Consider the following statements:
1. The regression line of y on x is $\rm y = \frac{3}{4}x+2$
2. The regression 107. Consider the following statements:
1. The coefficient of correlations r is $\rm \frac{3}{4}$.
2. The means of x a 108. What is the median? 109. What is the mode? 110. What is the mean of natural numbers in the interval [15, 64]? 111. For the set of numbers x, x, x + 2, x + 3, x + 10 where x is a natural number, which of the following is/are correct?
1 112. The mean of 10 observations is 5.5. If each observation is multiplied by 4 and subtracted from 44, then what is the new 113. If g is the geometric mean of 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, then which one of the following is correct? 114. If the harmonic mean of 60 and x is 48, then what is the value of x? 115. What is the mean deviation of first 10 even natural numbers? 116. If $\rm \displaystyle \sum_{i = 1}^{10}x_i = 110$ and $\rm \displaystyle \sum_{i = 1}^{10}x_i^2 = 1540$&n 117. 3-digit numbers are formed using the digits 1, 3, 7 without repetition of digits. A number is randomly selected. What is 118. What is the probability that the roots of the equation x2 + x + n = 0 are real, where n ∈ ℕ and n < 4? 119. If A and B are two events such that P(not A) = $\rm \frac{7}{10}$, P(not B) = $\rm \frac{3}{10}$ and P(A| 120. Seven white balls and three black balls are randomly placed in a row. What is the probability that no two black balls ar
1
NDA Mathematics 14 November 2021
MCQ (Single Correct Answer)
+2.5
-0.83
If $f(x) = \frac{[x]}{|x|}$, x ≠ 0, where [⋅] denotes the greatest integer function, then what is the right-hand limit of f(x) at x = 1?
A
-1
B
0
C
1
D
Right-hand limit of f(x) at x = 1 does not exist.
2
NDA Mathematics 14 November 2021
MCQ (Single Correct Answer)
+2.5
-0.83
Consider the following statements in respect of the function $\rm f(x) = sin \left(\frac{1}{x^2}\right)$, x ≠ 0:
1. It is continuous at x = 0, if f(0) = 0.
2. It is continuous at $x = \frac{2}{\sqrt{\pi}}$.
Which of the above statements is/are correct?
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
3
NDA Mathematics 14 November 2021
MCQ (Single Correct Answer)
+2.5
-0.83
What is the range of the function f(x) = 1 - sinx defined on entire real line?
A
(0, 2)
B
[0, 2]
C
(-1, 1)
D
[-1, 1]
4
NDA Mathematics 14 November 2021
MCQ (Single Correct Answer)
+2.5
-0.83
What is the slope of the tangent of y = cos-1 (cos x) at x = $-\frac{\pi}{4}$?
A
-1
B
0
C
1
D
2
Paper analysis
Total Questions
Mathematics
120
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