Mathematics
1. What is C(n, 1) + C(n, 2) + _ _ _ _ _ + C(n, n) equal to 2. What is the sum of the coefficients of first and last terms in the expansion of (1 + x)2n, where n is a natural number? 3. If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five t 4. Consider the following statements:
1. If each term of a GP is multiplied by same non-zero number, then the resulting se 5. How many 5-digit prime numbers can be formed using the digits 1, 2, 3, 4, 5 if the repetition of digits is not allowed? 6. If f(x + 1) = x2 - 3x + 2, then what is f(x) equal to? 7. If x2, x, -8 are in AP, then which one of the following is correct? 8. The third term of a GP is 3. What is the product of its first five terms? 9. The element in the ith row and the jth column of a determinant of third order is equal to 2(i + j). What is the value of 10. With the numbers 2, 4, 6, 8, all the possible determinants with these four different elements are constructed. What is t 11. What is the radius of the circle 4x2 + 4y2 - 20x + 12y - 15 = 0? 12. A parallelogram has three consecutive vertices (-3, 4), (0, -4) and (5, 2). The fourth vertex is 13. If the lines y + px = 1 and y - qx = 2 are perpendicular, then which one of the following is correct? 14. If A, B and C are in AP, then the straight line Ax + 2By + C = 0 will always pass through a fixed point. The fixed point 15. If the image of the point (-4, 2) by a line mirror is (4, -2), then what is the equation of the line mirror? 16. $\rm tan^{-1}x+cot^{-1}x=\frac{\pi}{2}$ holds, when 17. If $\tan A=\dfrac{1}{7}$, then what is cos 2A equal to? 18. The sides of a triangle are m, n and $\rm \sqrt{m^2+n^2+mn}$. What is the sum of the acute angles of the triangle? 19. What is the area of the triangle ABC with sides a = 10cm and c = 4cm angle B = 30°? 20. Consider the following statements:
1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets.
2. A = {1, 5, 9} and B = {1 21. Consider the following statements:
1. The null set is a subset of every set.
2. Every set is a subset of itself.
3. I 22. Let R be a relation defined as xRy if and only if 2x + 3y = 20, where x, y ∈ N. How many elements of the form (x, y 23. Consider the following statements:
1. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.
24. Consider the following in respect of a complex number z:
1. $\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}$
25. 1. The difference of Z and its conjugate is an imaginary number.
2. The sum of Z and its conjugate is a real number.
26. What is the modulus of the complex number i2n + 1(-i)2n - 1, where n ∈ N and i = √-1? 27. If α and β are the roots of the equation 4x2 + 2x - 1 = 0, then which one of the following is correct? 28. If one root of 5x2 + 26x + k = 0 is reciprocal of the other, then what is the value of k? 29. In how many ways can a team of 5 players be selected from 8 players so as not to include a particular player? 30. What is the coefficient of the middle term in the expansion of (1 + 4x + 4x2)5? 31. If $\rm tan\:x=-\dfrac{3}{4}$ and x is in the second quadrant, then what is the value of sin x ⋅ cos x? 32. What is the value of the following?
$\rm cosec\left(\dfrac{7\pi}{6}\right)sec\left(\dfrac{5\pi}{3}\right)$
33. If the determinant $\left| {\begin{array}{*{20}{c}} x&1&3\\ 0&0&1\\ 1&x&4 \end{array}} \rig 34. What is the value of the following?
tan 31° tan 33° tan 35° _ _ _ _ _ tan 57° tan 59°
35. If
$f(x)=\left|\begin{array}{ccc}1 & 1 & x+1 \\ 2 x & x(x-1) & x(x+1) \\ 3 x(x-1) & 2(x-1)(x-2) & x(x+1)(x-1)\end{ 36. The equation $sin^{-1}x-cos^{-1}x=\frac{\pi}{6}$ has 37. What is the value of the following?
(sin 24° + cos 66°)(sin 24° - cos 66°)
38. A chord subtends an angle 120° at the centre of a unit circle. What is the length of the chord? 39. What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
40. What is $\rm \frac{1+tan^2\theta}{1+cot^2\theta}-\left(\frac{1-tan\theta}{1-cot\theta}\right)^2$equal to?
41. What is the interior angle of a regular octagon of side length 2 cm? 42. If 7 sinθ + 24 cosθ = 25, then what is the value of (sin θ + cos θ)? 43. A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation 44. The shadow of a tower is found to be x meter longer, when the angle of elevation of the sun changes from 60° to 45°. If 45. If 3cosθ = 4sinθ, then what is the value of tan (45° + θ)? 46. The smallest positive integer n for which
$\rm \left(\frac{1-i}{1+i}\right)^{n^2}=1$
where i = √-1, is
47. The value of x, satisfying the equation $log_{cos x} ~sin x = 1$, where $0<x<\dfrac{\pi}{2}$, is 48. If Δ is the value of the determinant
$\left| {\begin{array}{*{20}{c}} {{a_1}}&{{b_1}}&{{c_1}}\\ {{a_2}}&am 49. If C0, C1, C2, _ _ _ _ _, Cn are the coefficients in the expansion of (1 + x)n, then what is the value of C1 + C2 + C3 + 50. If a + b + c = 4 and ab + bc + ca = 0, then what is the value of the following determinant?
$\left| {\begin{array}{*{20 51. The number of integer values of k, for which the equation 2sinx = 2k + 1 has a solution, is 52. If a1, a2, a3, _ _ _ _ _, a9 are in GP, then what is the value of the following determinant?
$\left| {\begin{array}{*{2 53. If the roots of the quadratic equation x2 + 2x + k = 0 are real, then 54. If n = 100!, then what is the value of the following?
$\rm \dfrac{1}{log_2n}+\dfrac{1}{log_3n}+\dfrac{1}{log_4n}+{..... 55. If Z = 1 + i, where i = √-1, then what is the modulus of $\rm z+\frac{2}{z}?$ 56. If A and B are two matrices such that AB is of order n × n, then which one of the following is correct? 57. How many matrices of different orders are possible with elements comprising all prime numbers less than 30? 58. Let $A = \left| {\begin{array}{*{20}{c}} p&q\\ r&s \end{array}} \right|$
where p, q, r and s are any four 59. If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct? 60. What is cot 2x cot 4x - cot 4x cot 6x - cot 6x cot 2x equal to
61. If M is the mean of n observations x1 - k, x2 - k, x3 - k, _ _ _, xn - k, where k is any real number, then what is the m 62. What is the sum of deviations of the variate values 73, 85, 92, 105, 120 from their mean? 63. Let x be the HM and y be the GM of two positive numbers m and n. If 5x = 4y, then which one of the following is correct? 64. If the mean of a frequency distribution is 100 and the coefficient of variation is 45%, then what is the value of the va 65. Let two events A and B be such that P(A) = L and P(B) = M. Which one of the following is correct? 66. For which of the following sets of numbers do the mean, median and mode have the same value? 67. The mean of 12 observations is 75. If two observation are discarded, then the mean of the remaining observations is 65. 68. If k is one of the roots of the equation x(x + 1) + 1 = 0, then what is its other root? 69. The geometric mean of a set of observations is computed as 10. The geometric mean obtained when each observations xi is 70. If $\rm P(A\cup B)=\dfrac{5}{6}, P(A\cap B)=\dfrac{1}{3}\:and\:P(\bar A)=\dfrac{1}{2}$, then which of the following 71. The average of a set of 15 observations is recorded, but later it is found that for one observation, the digit in the te 72. A coin is tossed twice. If E and F denote occurrence of head on first toss and second toss respectively, then what is P( 73. In a binomial distribution, the mean is $\dfrac{2}{3}$ and variance is $\dfrac{5}{9}$. What is the probability that 74. If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x? 75. If A and B are two events such that $\rm P(A)=\dfrac{3}{4} \:and\: P(B)=\dfrac{5}{8}$, then consider the following 76. What is the derivative of ex with respect to xe? 77. If a differentiable function f(x) satisfies $\mathop {\lim }\limits_{x \to - 1} \dfrac{f(x)+1}{x^2-1}=-\dfrac{3}{2} 78. If the function $\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{ 79. Consider the following statements in respect of the function f(x) = sin x:
1. f(x) increases in the interval (0, π 80. What is the domain of the function f(x) = 3x? 81. If the general solution of a differential equation is y2 + 2cy - cx + c2 = 0, where c is an arbitrary constant, then wha 82. What is the degree of the following differential equation?
$\rm x=\sqrt{1+\frac{d^2y}{dx^2}}$
83. Which one of the following differential equations has the general solution y = aex + be-x?
84. What is the solution of the following differential equation?
$\rm \ln\left(\frac{dy}{dx}\right)+y = x$
85. What is $\rm \int e^{\left(2\ln x+\ln x^2\right)}dx$ equal to? 86. Consider the following measures of central tendency for a set of N numbers:
1. Arithmetic mean.
2. Geometric mean.
Wh 87. The numbers of Science, Arts and Commerce graduates working in a company are 30, 70 and 50 respectively. If these figure 88. For a histogram based on a frequency distribution with unequal class intervals, the frequency of a class should be propo 89. The co-efficient of correlation is independent of: 90. The following tables gives the frequency distribution of number of peas per pea pod of 198 pods:
Number of 91. If $\rm\mathop {\lim }\limits_{x \to a} \frac{a^x -x^a}{x^x -a^a}= - 1$, then what is the value of a?
92. A particle starts from origin with a velocity (in m/s) given by the equation $\rm \frac{dx}{dt}=x+1$. The time (in secon 93. What is $\rm \int_0^a \frac{f(a-x)}{f(x)+f(a-x)}\ dx $ equal to?
94. What is $\rm\mathop {\lim }\limits_{x \to 2}\frac{x^3 + x^2}{x^2 + 3x + 2}$ equal to?
95. If $\rm \int_0^a \left[f(x)+f(-x)\right]dx=\int_{-a}^{\ \ a} g(x)\ dx $, then what is g(x) equal to?
96. What is the area bounded by $\rm y=\sqrt{16-x^2}$, y ≥ 0 and the x-axis? 97. The curve y = -x3 + 3x2 + 2x - 27 has the maximum slope at: 98. A 24 cm long wire is bent to form a triangle with one of the angles as 60°. What is the altitude of the triangle having 99. If f(x) = e|x|, then which one of the following is correct? 100. What is $\rm \int \frac{dx}{\sec x+\tan x}$ equal to?
101. What is $\int \dfrac{dx}{sec^2({tan}^{-1}x)}$ equal to?
102. If x + y = 20 and P = xy, then what is the maximum value of P? 103. What is the derivative of sin(ln x) + cos(ln x) with respect to x at x = e?
104. If x = et cost and y = et sint, then what is $\dfrac{dx}{dy}$ at t = 0 equal to? 105. What is the maximum value of sin 2x ⋅ cos 2x? 106. Consider the following statements in respect of the points (p, p - 3), (q + 3, q) and (6, 3):
1. The points lie on a st 107. What is the acute angle between the lines x - 2 = 0 and √3x - y - 2 = 0? 108. The point of intersection of diagonals of a square ABCD is at the origin and one of its vertices is at A(4, 2). What is 109. If any point on a hyperbola is (3tan θ, 2sec θ), then what is the eccentricity of the hyperbola? 110. Consider the following with regard to eccentricity (e) of a conic section:
1. e = 0 for circle
2. e = 1 for parabola
111. What is the angle between the two lines having direction ratios (6, 3, 6) and (3, 3, 0)? 112. If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z, then what is l4 + m4 + n4 equal to? 113. What is the projection of the line segment joining A(1, 7, -5) and B(-3, 4, -2) on y-axis? 114. What is the number of possible values of k for which the line joining the points (k, 1, 3) and (1, -2, k + 1) also passe 115. The foot of the perpendicular drawn from the origin to the plane x + y + z = 3 is 116. A vector $\vec r=a \hat i+b \hat j$ is equally inclined to both x and y axes. If the magnitude of the vector is 2 u 117. Consider the following statements in respect of a vector $\vec c=\vec a+\vec b$, where $|\vec a|=|\vec b|\ne0$:
1.  118. If $\vec a \:and\: \vec b$ are two vectors such that $|\vec a + \vec b|= |\vec a - \vec b|=4,$ then which one 119. If $\vec a, \vec b\:and \: \vec c$ are coplanar, then what is $(2\vec a\times 3\vec b)\cdot4\vec c+(5\vec b\ti 120. Consider the following statements:
1. The cross product of two unit vectors is always a unit vector.
2. The dot produc
1
NDA Mathematics 18 April 2021
MCQ (Single Correct Answer)
+2.5
-0.83
Which one of the following differential equations has the general solution y = aex + be-x?
A
$\rm \frac{d^2y}{dx^2}+y=0$
B
$\rm \frac{d^2y}{dx^2}-y=0$
C
$\rm \frac{d^2y}{dx^2}+y=1$
D
$\rm \frac{dy}{dx}-y=0$
2
NDA Mathematics 18 April 2021
MCQ (Single Correct Answer)
+2.5
-0.83
What is the solution of the following differential equation?
$\rm \ln\left(\frac{dy}{dx}\right)+y = x$
A
ex + ey = c
B
ex + y = c
C
ex - ey = c
D
ex - y = c
3
NDA Mathematics 18 April 2021
MCQ (Single Correct Answer)
+2.5
-0.83
What is $\rm \int e^{\left(2\ln x+\ln x^2\right)}dx$ equal to?
A
$\rm \frac{x^4}{4}+c$
B
$\rm \frac{x^3}{3}+c$
C
$\rm \frac{2x^5}{5}+c$
D
$\rm \frac{x^5}{5}+c$
4
NDA Mathematics 18 April 2021
MCQ (Single Correct Answer)
+2.5
-0.83
Consider the following measures of central tendency for a set of N numbers:
1. Arithmetic mean.
2. Geometric mean.
Which of the above uses/use all the data?
A
1 only.
B
2 only.
C
Both 1 and 2.
D
Neither 1 nor 2.
Paper analysis
Total Questions
Mathematics
120
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