Sets, Relations & Functions PYQs Set 3 (PDF)

Set 3 of 25 · Questions 61-90 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 3 (Q61-90)
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61. $If f : A \to B is an onto function such that f (x) = \| x \| - x + \frac{1}{\| x \| - x}, then A and B are respectively.$ (1) $(- \infty, \infty), (0, \infty)$ (2) $(- \infty, 0), [2, \infty)$ (3) $(0, \infty), (2, \infty)$ (4) $(- \infty, 0], (0, \infty)$ 62. $Let f (x) = sin 2 x + cos 2 x and g (x) = x^{2} - 1, then g (f (x)) is invertible in the domain$ (1) $x \in (\frac{- π}{2}, \frac{π}{2})$ (2) $x \in (\frac{- π}{4}, \frac{π}{4})$ (3) $x \in (0, \frac{π}{4})$ (4) $x \in (\frac{- π}{8}, \frac{π}{8})$ 63. $If u = sin (\frac{x}{y}), x = e^{t} and y = t^{2}, then t^{6} (\frac{du}{dt})^{2} \div e^{2 t} (t - 2)^{2} =$ (1) $2 u$ (2) $u^{2}$ (3) $1 - u^{2}$ (4) $cos u$ 64. $If the function f (x) satisfies lim_{x \to 1} \frac{f ( x ) - 2}{x ^{2} - 1} = π, then lim_{x \to 1} f (x) =$ (1) $1$ (2) $2$ (3) $0$ (4) $3$ 65. $Let A be the set of all functions f : Z \to Z and R be a relation on A such that R = {(f, g) : f (0) = g (1) and f (1) = g (0)} . Then R is:$ (1) $Symmetric and transitive but not reflective$ (2) $Symmetric but neither reflective nor transitive$ (3) $Transitive but neither reflexive nor symmetric$ (4) $Reflexive but neither symmetric nor transitive$ 66. $lim n \to \infty \frac{A + e ^{nx}}{x + A e ^{nx}} =$ (1) $\frac{A}{x}, when x < 0$ (2) $1 , when x > 0$ (3) $0 , when \forall x \in R$ (4) $A, when x = 0$ 67. $If U is the universal set with 100 elements; A and B are two sets such that n (A) = 50, n (B) = 60, n (A \cap B) = 20 then n (A^{’} \cap B^{’}) =$ (1) $40$ (2) $90$ (3) $20$ (4) $10$ 68. $If y = (1 + x^{2}) tan^{- 1} x - x then \frac{dy}{dx} is$ (1) $2 x tan^{- 1} x$ (2) $x^{2} tan^{- 1} x$ (3) $\frac{tan ^{- 1} x}{x}$ (4) $x tan^{- 1} x$ 69. $If y = cosec^{- 1} (x) and \frac{dy}{dx} = \frac{- 1}{\| x \| x ^{2} - 1}, then$ (1) $y \in (- \frac{π}{2}, 0)$ (2) $y \in (- \frac{π}{2}, 2 π)$ (3) $y \in (- \frac{π}{2}, 0) \cup (0, \frac{π}{2})$ (4) $y \in R$ 70. $If f (x), defined below, is continuous at x = 4, then f (x) = (\frac{x - 4}{\| x - 4 \|} + a, x < 4 a + b, x = 4 \frac{x - 4}{\| x - 4 \|} + b, x > 4)$ (1) $a = 0 and b = 0$ (2) $a = 1 and b = 1$ (3) $a = - 1 and b=1$ (4) $a=1 and b = - 1$ 71. $If y = 2 x^{n + 1} + \frac{3}{x ^{n}}, then x^{2} \frac{d ^{2} y}{d x ^{2}} is$ (1) $n (n + 1) y$ (2) $x \frac{dy}{dx} + y$ (3) $y$ (4) $6 n (n + 1) y$ 72. $Let a function f : (0, \infty) \to [0, \infty) be defined by f (x) = 1 - \frac{1}{x} . Then f is :$ (1) $not injective but it is surjective$ (2) $injective only$ (3) $neither injective nor surjective$ (4) $both injective as well as surjective$ 73. $On the set of all natural numbers N, which one of the following * is a binary operation?$ (1) $a * b = a + 3 b$ (2) $a * b = 3 a - 4 b$ (3) $a * b = ab$ (4) $a * b = \frac{a - b}{a + b}$ 74. $If af (x) + bf (\frac{1}{x}) = x + 1, and \frac{d}{dx} (x^{2} \cdot f (x)) = 2 x^{2} + 2 x + \frac{1}{3}, then a - b =$ (1) $2$ (2) $3$ (3) $0$ (4) $1$ 75. $Let f (x) = x - \frac{1}{x}, then f^{’} (- 1) is$ (1) $0$ (2) $2$ (3) $1$ (4) $-2$ 76. $The range of the function f (x) = sin [x], - \frac{π}{4} < x < \frac{π}{4} where [x] denotes the greatest integer \leq x, is$ (1) ${0}$ (2) ${0, - 1}$ (3) ${0, \pm sin 1}$ (4) ${0, - sin 1}$ 77. $If f (x) = \frac{1 - x + 9 x ^{2} + 10 x + 1}{2 x}, then lim_{x \to - 1^{-}} f (x) =$ (1) $1$ (2) $-1$ (3) $0$ (4) $- 1/5$ 78. $If f (x) = \| x - 1 \| + \| x - 2 \| + \| x - 3 \|, 2 < x < 3, then f is$ (1) $an onto function but not one-one$ (2) $one-one function but not onto$ (3) $a bijection$ (4) $neither one-one nor onto$ 79. $If f (x) = \| cos x - sin x \|, then f^{’} (π /6) is equal to$ (1) $- \frac{1}{2} (1 + 3)$ (2) $\frac{1}{2} (1 + 3)$ (3) $- \frac{1}{2} (1 - 3)$ (4) $\frac{1}{2} (1 - 3)$ 80. $Let A = 2, 3, 5, 7, 9 . Let R be the relation on A defined by xRy if and only if 2 x \leq 3 y . Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to:$ (1) $21$ (2) $27$ (3) $23$ (4) $25$ 81. $If f : R \to R is defined by f (x) = \frac{x}{x ^{2} + 1}, find f (f (2)) .$ (1) $\frac{10}{29}$ (2) $29$ (3) $\frac{1}{29}$ (4) $\frac{29}{10}$ 82. $Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g(1) is :$ (1) $\frac{1}{10}$ (2) $\frac{1}{15}$ (3) $\frac{1}{5}$ (4) $\frac{1}{30}$ 83. $The domain of the function f (x) = \frac{1}{\| x \| - x} is$ (1) $R$ (2) $(- \infty, 0)$ (3) $(0, \infty)$ (4) $(- \infty, 1)$ 84. $Let f : [2, \infty) \to R be the function defined by f (x) = x^{2} - 4 x + 5, then the range of f is$ (1) $[1, \infty)$ (2) $[5, \infty)$ (3) $(1, \infty)$ (4) $(- \infty, \infty)$ 85. $Let X = ((ab c d) : a; b, c, d \in R), Define f : X \to R by f (A) = det (A), \forall A \in X . Then f is$ (1) $one one but not onto$ (2) $onto but not one-one$ (3) $one-one and onto$ (4) $neither one-one nor onto$ 86. $Let [t] be the greatest integer less than or equal to t . Let A be the set of all prime factors of 2310 and f : A \to Z be the function f (x) = (log_{2} (x^{2} + (\frac{x ^{3}}{5}))) . The number of one-to-one functions from A to the range of f is$ (1) $25$ (2) $24$ (3) $20$ (4) $120$ 87. $Let A ={0, 3,4,6, 7, 8, 9,10} and R be the relation defined on A such that R{(x, y) A x A : x – y is odd positive integer or x - y = 2}. The minimum number of elements that must be added to the relation R, so that it is a symmetric relation, is equal to__________$ 88. $Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m ∙ n is \dots$ 89. $Let A = 2, 3, 6, 7 and B = 4, 5, 6, 8 . Let R be a relation defined on A \times B by (a_{1}, b_{1}) R (a_{2}, b_{2}) if and only if a_{1} + a_{2} = b_{1} + b_{2} . Then the number of elements in R is_______$ 90. $Let A = 1, 2, 3, \dots 7 and let P (1) denote the power set of A . If the number of functions f : A \to P (A) such that a \in f (a), \forall a \in A is m^{n}, m and n \in N and m is least, then m + n is equal to________$