Mathematics
1. If z z̅ = |z + z̅ |, where z = x + iy, i = $\sqrt{-1}$, then the locus of z is a pair of: 2. If 1! + 3! + 5! + 7! + ... + 199! is divided by 24, what is the remainder? 3. What is the value of $\sqrt{12+5 i}+\sqrt{12-5 i}$ where $i=\sqrt{-1}$ ? 4. If $A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$, then what is the value of det(I + AA'), where I is the 5. If A, B and C are square matrices of order 3 and det(BC) = 2 det(A), then what is the value of det(2A-1BC)? 6. If the nth term of a sequence is $\frac{2 n+5}{7}$, then what is the sum of its first 140 terms? 7. Let A be a skew-symmetric matrix of order 3.
What is the value of det(4A4) - det(3A3) + det(2A2) - det(A) + det(-I 8. If $A=\left[\begin{array}{rrr} 0 & 3 & 4 \\ -3 & 0 & 5 \\ -4 & -5 & 0 \end{array}\right]$, then 9. If $A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]$, the 10. If (a + b), 2b, (b + c) are in HP, then which one of the following is correct? 11. Let t1, t2, t3 ... be in GP. What is $\rm \left(t_1 t_3 \ldots t_{21}\right)^{\frac{1}{11}}$ equal to ? 12. Which one of the following is a square root of $-\sqrt{-1} $? 13. What is the maximum number of points of intersection of 10 circles? 14. A set S contains (2n + 1) elements. There are 4096 subsets of S which contain at most n elements. What is n equal to? 15. If $\left|\begin{array}{ccc} x^2+3 x & x-1 & x+3 \\ x+1 & -2 x & x-4 \\ x-3 & x+4 & 3 x \en 16. If all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is: 17. If $A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 3 & 0 \\ 1 & 0 & 1 \end{array}\right]$, then w 18. If $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$, then 19. The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, t 20. If $A_k=\left[\begin{array}{cc} k-1 & k \\ k-2 & k+1 \end{array}\right] $, then what is det(A1) + det( 21. The Cartesian product A × A has 16 elements among which are (0, 2) and (1, 3). Which of the following sta 22. Let A = {1, 2, 3, ..., 20}. Define a relation R from A to A by R = {(x, y) : 4x - 3y = 1}, where x, y ∈ A. Whi 23. Consider the following statements:
1. The relation f defined by $f(x)= \begin{cases}x^3, & 0 \leq x 24. Consider the following statements
1. A = (A ∪ B) ∪ (A - B),
2. A ∪ (B - 25. A function satisfies $f(x-y)=\frac{f(x)}{f(y)}$, where f(y) ≠ 0. If f(1) = 0.5, then what is f(2) + f(3) + f(4 26. What is 2 cot $\left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)$ equal to ? 27. If sec-1 p - cosec-1q = 0, where p > 0, q > 0; then what is the value of p-2 + q-2 ? 28. What is $1+\sin ^2\left(\cos ^{-1}\left(\frac{3}{\sqrt{17}}\right)\right)$ equal to ? 29. If tan (π cos θ) = cot (π sin θ), $0<\theta<\frac{\pi}{2}$; then what is the value of $8 30. If $\tan \alpha=\frac{1}{7}$, $\sin \beta=\frac{1}{\sqrt{10}}$; $0<\alpha, \beta<\frac{\pi}{2}$, then what is the 31. What is the number of real roots of the equation? 32. What is the sum of all the roots of the equation? 33. What are the roots of equation-I ? 34. Which one of the following is a root of equation-II? 35. What is the number of common roots of equation-I and equation-II? 36. If $\rm {k}=\frac{{c}}{2},({c} \neq 0)$, then the roots of the equation are : 37. If k = c, then the roots of the equation are: 38. What is T1 + 2T2 + 3T3 + ... + nTn equal to ? 39. What is 1 - T1 + 2T2 - 3T3 + ... + (-1)nnTn equal to ? 40. What is T1 + T2 + T3 + ... + Tn equal to ? 41. Which one of the following is a possible expression for g(x)? 42. What is g(15) equal to ? 43. What is f(0.5) equal to ? 44. If f is differentiable, then what is f'(0.5) equal to? 45. The function is decreasing on : 46. The function attains local minimum value at : 47. What is the maximum value of y? 48. What is the maximum value of xy ? 49. What is the range of the function? 50. What is the period of the function? 51. What is the directrix of the parabola ? 52. What is the length of latus rectum of the parabola? 53. What is $\displaystyle \lim _{x \rightarrow 1} \frac{f(x)-1}{g(x)}$ equal to? 54. What is $\displaystyle \lim _{x \rightarrow 1} f(x)^{\frac{1}{g(x)}}$ equal to? 55. What is the domain of the function? 56. What is the greatest value of the function? 57. What is $\displaystyle \lim _{x \rightarrow 0+}$ h(x) equal to ? 58. What is $\displaystyle \lim _{x \rightarrow 0-}$ h(x) equal to ? 59. What is the value of a ? 60. What is the value of b? 61. What is $\displaystyle \int_0^\pi\left(\sin ^4 x+\cos ^4 x\right) d x$ equal to? 62. What is I equal to? 63. If the function f(x) is differentiable at x = 1, then what is the value of (a + b)? 64. What is $\displaystyle \lim _{x \rightarrow 0} $ f(x) equal to ? 65. If f(x) = |ln|x|| where 0 < x < 1, then what is f'(0.5) equal to ? 66. If f'(x) = cos (In x) and $y=f\left(\frac{2 x-3}{x}\right)$, then what is $\frac{d y}{d x}$ equal to ? 67. What is $\displaystyle \rm \int_0^{8 \pi}|\sin x| d x$ equal to? 68. What is the area between the curve f(x) = x |x| and x-axis for x = [-1, 1]? 69. What are the order and the degree respectively of the differential equation $\rm x^2\left(\frac{d^3 y}{d x^3}\ 70. What is the differential equation of all parabolas of the type y2 = 4a (x - b)? 71. What is a1 + a5 - a10 - a15 - a20 - a25 + a30 + a34 equal to ? 72. What is $\displaystyle \sum_{n=1}^{34} a_n$ equal to ? 73. What is the value of p + q? 74. What is the value of pq? 75. What is pq equal to ? 76. For how many values of x does $\frac{1}{p}$ become zero? 77. What is a value of sin 3x + sin 3y? 78. What is a value of cos3 x + cos3 y? 79. What is the value of a + b + √2 c equal to ? 80. What is the ratio of a2 ∶ b2 ∶ c2 ? 81. What is the equation of directrix of parabola y2 = 4bx, where b < 0 and b2 + b - 2 = 0? 82. The points (-a, -b), (0, 0), (a, b) and (a2, ab) are: 83. Given that 16p2 + 49q2 - 4r2 - 56pq = 0. Which one of the following is a point on a pair of str 84. If 3x + y - 5 = 0 is the equation of a chord of the circle x2 + y2 - 25 = 0, then what are the coordinate 85. Consider the following in respect of the equation $\frac{x^2}{24-k}+\frac{y^2}{k-16}=2$.
1. The equation repr 86. Consider the following statements in respect of hyperbola $\frac{x^2}{\cos ^2 \theta}-\frac{y^2}{\sin ^2 \thet 87. Consider the following in respect of the circle 4x2 + 4y2 - 4ax - 4ay + a2 = 0:
1. The circle touch 88. For what values of k is the line (k - 3)x - (5 - k2) y + k2 - 7k + 6 = 0 parallel to the line x 89. The line x + y = 4 cuts the line joining P(-1, 1) and Q(5, 7) at R. What is PR ∶ RQ equal to ? 90. What is the sum of the intercepts of the line whose perpendicular distance from origin is 4 units and the angle which th 91. What is the length of projection of the vector $\rm \hat{i}+2 \hat{j}+3 \hat{k}$ on the vector $\rm2 \hat{i}+3 92. If $\rm (\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=144$ and $\rm|\vec{b}|=4 $, then what is the 93. If θ is the angle between vectors $\vec{a}$ and $ \vec{b}$ such that $\vec{a} \cdot \vec{b} \geq 0$, then whic 94. The vectors $\rm 60 \hat{i}+3 \hat{j}, 40 \hat{i}-8 \hat{j}$ and $\rm \beta \hat{i}-52 \hat{j}$ are collinear 95. Consider the following in respect of the vectors $\rm \vec{a}=(0,1,1)$ and $\rm \vec{b}=(1,0,1) $ :
1.&n 96. If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are t 97. Which one of the planes is parallel to the line $\rm \frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ ? 98. What is the angle between the lines 2x = 3y = -z and 6x = -y = -4z? 99. What is the equation of the sphere concentric with the sphere x2 + y2 100. A point P lies on the line joining A(1, 2, 3) and B(2, 10, 1). If z-coordinate of P is 7, what is the sum of o 101. The sum of deviations of n numbers from 10 and 20 are p and q respectively. If (p - q)2 = 10000, then what is the v 102. If X̅ = 20 is the mean of 10 observations x1, x2, ... x10; then what is the value of $\displaystyle \sum_{i=1} 103. If the mean and the sum of squares of 10 observations are 40 and 16160 respectively, then what is the standard deviation 104. Three dice are thrown. What is the probability of getting a sum which is a perfect square? 105. A, B, C and D are mutually exclusive and exhaustive events.
If 2P(A) = 3P(B) = 4P(C) = 5P(D), then what is 77 106. Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in 107. Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ? 108. What is the mean of the numbers 1, 2, 3, ... 10 with frequencies 9C0, 9C1, 9C2 ..., 9 109. The probability that a person recovers from a disease is 0.8. What is the probability that exactly 2 persons out of 110. Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The ch 111. If r is the coefficient of correlation between x and y, then what is the correlation coefficient bet 112. A fair coin is tossed 6 times. What is the probability of getting a result in the 6th toss which is diffe 113. If H is the Harmonic Mean of three numbers 10C4, 10C5, and 10C6, then what is the value of $\fr 114. In a class, there are n students including the students P and Q. What is the probability that P and Q sit together 115. In a Binomial distribution B(n, p), n = 6 and 9P(X = 4) = P(X = 2). What is p equal to ? 116. What is the probability that all three boys sit together? 117. What is the probability that boys and girls sit alternatively? 118. What is the probability that no two girls sit together? 119. What is the probability that P and Q take the two end positions? 120. What is the probability that Q and U sit together?
1
NDA Mathematics 3 September 2023
MCQ (Single Correct Answer)
+2.5
-0.83
Consider the following for the next items that follow:
Let $f(x)=\left\{\begin{array}{cc} a x(x+1)+b, & x<1 \\ x-1, & 1 \leq x \leq 2 \end{array}\right.$
What is $\displaystyle \lim _{x \rightarrow 0} $ f(x) equal to ?
A
$- \frac{1}{3} $
B
$-\frac{2}{3}$
C
0
D
1
2
NDA Mathematics 3 September 2023
MCQ (Single Correct Answer)
+2.5
-0.83
If f(x) = |ln|x|| where 0 < x < 1, then what is f'(0.5) equal to ?
A
-2
B
-1
C
0
D
2
3
NDA Mathematics 3 September 2023
MCQ (Single Correct Answer)
+2.5
-0.83
If f'(x) = cos (In x) and $y=f\left(\frac{2 x-3}{x}\right)$, then what is $\frac{d y}{d x}$ equal to ?
A
$ \cos \left(\ln \left(\frac{2 \mathrm{x}-3}{\mathrm{x}}\right)\right)$
B
$-\frac{3}{\mathrm{x}^2} \sin \left(\ln \left(\frac{2 \mathrm{x}-3}{\mathrm{x}}\right)\right)$
C
$\frac{3}{\mathrm{x}^2} \cos \left(\ln \left(\frac{2 \mathrm{x}-3}{\mathrm{x}}\right)\right)$
D
$-\frac{3}{\mathrm{x}^2} \cos \left(\ln \left(\frac{2 \mathrm{x}-3}{\mathrm{x}}\right)\right)$
4
NDA Mathematics 3 September 2023
MCQ (Single Correct Answer)
+2.5
-0.83
What is $\displaystyle \rm \int_0^{8 \pi}|\sin x| d x$ equal to?
A
2
B
4
C
8
D
16
Paper analysis
Total Questions
Mathematics
120
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