Sets, Relations & Functions PYQs Set 4 (PDF)

Set 4 of 25 · Questions 91-120 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 4 (Q91-120)
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91. $If y = tan^{- 1} (\frac{5 cos x - 12 sin x}{12 cos x + 5 sin x}), then \frac{dy}{dx} is equal to$ (1) $1$ (2) $-1$ (3) $-2$ (4) $\frac{1}{2}$ 92. $A and B are subsets of R. Every element x of A is mapped to an element of B by the rule y (x) = (\frac{5 x}{( x - 3 ) ( x + 3 )}, if x \neq = - 1 - 1, if x = - 1) then A is equal to$ (1) $R / {- 3, + 3, - 0}$ (2) $R / {- 3, + 3}$ (3) $R / {- 3, 3, 0, - 1}$ (4) $R$ 93. $If y = e^{log_{e} (1 + x + x^{2} + \dots)}, then \frac{dy}{dx} =$ (1) $\frac{- 1}{( 1 - x ) ^{2}}$ (2) $\frac{1}{( 1 + x ) ^{2}}$ (3) $\frac{1}{( 1 - x ) ^{2}}$ (4) $\frac{- 1}{( 1 + x ) ^{2}}$ 94. $A function f (x) is given by f (x) = \frac{5 ^{x}}{5 ^{x} + 5}, then the sum of the serics f (\frac{1}{20}) + f (\frac{2}{20}) + f (\frac{3}{20}) + \dots \dots + f (\frac{39}{20}) is equal to:$ (1) $\frac{19}{2}$ (2) $\frac{49}{2}$ (3) $\frac{39}{2}$ (4) $\frac{29}{2}$ 95. $If f (x) = 8 x^{3}, g (x) = x^{1/3}, then f_{0} g (x) is$ (1) $8^{3} x$ (2) $8 x^{3}$ (3) $8 x$ (4) $(8 x)^{1/3}$ 96. $If f (x) = sin (cosh (\frac{x ^{2} + 1}{x ^{2} + 2})) then f^{’} (1) =$ (1) $\frac{2}{9} sinh (\frac{2}{3}) cos (cosh (\frac{2}{3}))$ (2) $sinh (\frac{2}{3}) cos (cosh (\frac{2}{3}))$ (3) $\frac{2}{9} cos (cosh (\frac{2}{3}))$ (4) $\frac{2}{9} cosh (\frac{2}{3}) cos (sinh (\frac{2}{3}))$ 97. $The number of functions f : (1, 2, 3, 4) \to {a ϵ Z : \| a \| \leq 8} satisfying f (n) + \frac{1}{n} f (n + 1) = 1, \forall n ϵ 1, 2, 3 is$ (1) $3$ (2) $4$ (3) $1$ (4) $2$ 98. $If y = tan^{- 1} (\frac{sin x + cos x}{cos x - sin x}), then \frac{dy}{dx} is equal to$ (1) $0$ (2) $\frac{1}{2}$ (3) $\frac{π}{4}$ (4) $1$ 99. $lim_{x \to 0} \frac{27 ^{x} - 9 ^{x} - 3 ^{x} + 1}{5 - 4 + cos x} =$ (1) $5 (log 3)^{2}$ (2) $85 (log 3)^{2}$ (3) $165 log 3$ (4) $85 log 3$ 100. $If f (x) = log_{e} (\frac{1 - x}{1 + x}), (x) < 1, then f (\frac{2 x}{1 + x ^{2}}) is equal to :$ (1) $2 f (x)$ (2) $2 f (x^{2})$ (3) $(f (x))^{2}$ (4) $- 2 f (x)$ 101. $Let f (x) = x^{2} . x \in R . for any A \subseteq R, define g (A) = (x \in R : f (x) \in A) . If S = (0, 4), then which one of the following statements is not true ?$ (1) $g (f (S)) \neq = S$ (2) $f (g (S)) = S$ (3) $g (f (S)) = g (S)$ (4) $f (g (S)) \neq = f (S)$ 102. $Let A = 0, 1, 2, 3, 4, 5 . Let R be a relation on A defined by (x, y) \in R if and only if max x, y \in 3, 4 . Then among the statements (S_{1}) : The number of elements in R is 18, and (S_{2}) : The relation R is symmetric but neither reflexive nor transitive$ (1) $both are false$ (2) $only (S_{1}) is true$ (3) $only (S_{2}) is true$ (4) $both are true$ 103. $If f (x) = 1 + nx + \frac{n ( n - 1 )}{2} x^{2} + \frac{n ( n - 1 ) ( n - 2 )}{6} x^{3} + \dots + x^{n}, then f^{’ ’} (1) =$ (1) $n (n - 1) 2^{n - 2}$ (2) $n (n - 1) 2^{n - 1}$ (3) $n (n - 1) 2^{n}$ (4) $(n - 1) 2^{n - 1}$ 104. $Let the domain of the function f (x) = log_{3} log_{5} (7 - log_{2} (x^{2} - 10 x + 85)) + sin^{- 1} ((\frac{3 x - 7}{17 - x})) be (α, β] . Then α + β is equal to:$ (1) $9$ (2) $10$ (3) $12$ (4) $8$ 105. $If f (x) = (tan^{- 1} x, when \| x \| \leq 1 \frac{1}{2} (\| x \| - 1) when \| x \| > 1), then the domain of \frac{d}{dx} f (x) is$ (1) $R - {- 1, 1}$ (2) $R - (- 1, 1)$ (3) $R - [- 1, 1]$ (4) $R - {- 1}$ 106. $If x = 32 cos^{3} θ and y = 4 tan^{2} θ then (\frac{dy}{dx})_{θ = \frac{π}{4}} =$ (1) $\frac{32 2}{9}$ (2) $\frac{16}{9}$ (3) $- \frac{16}{9}$ (4) $- \frac{32}{9}$ 107. $If f (x) = (sin x if x \leq 0 x^{2} + a^{2} if 0 < x < 1 bx + 2 if 1 \leq x \leq 20 if x > 2) is continuous on IR, then a + b + ab =$ (1) $-2$ (2) $0$ (3) $2$ (4) $-1$ 108. $If y = \frac{sin h ^{- 1} x}{1 + x ^{2}}, then (1 + x^{2}) y_{2} + 3 x y_{1} + y =$ (1) $2$ (2) $1$ (3) $-1$ (4) $0$ 109. $The domain of f (x) = \frac{log _{(x + 1)} ( x - 2 )}{e ^{2 log_{e} x} - ( 2 x + 3 )}, x ϵ R$ (1) $R - (1 - 3)$ (2) $(2, \infty) - {3}$ (3) $(- 1, \infty) - (3)$ (4) $R - (3)$ 110. $If f (x) = \frac{4 x + 3}{6 x - 4}, x \neq = \frac{2}{3} and (f \circ f) (x) = g (x), where g : R - (\frac{2}{3}) \to R - (\frac{2}{3}), then (gogog) (4) is equal to$ (1) $- \frac{19}{20}$ (2) $\frac{19}{20}$ (3) $-4$ (4) $4$ 111. $If y = a sin x + b cos x, then y^{2} + (\frac{dy}{dx})^{2} is a$ (1) $function of x and y$ (2) $function of x$ (3) $constant$ (4) $function of y$ 112. $If f : R \to R is defined by f (x) = x^{3} then f^{- 1} (8) =$ (1) ${2, - 2}$ (2) ${2, 2}$ (3) ${2}$ (4) $({2, 2 ω, 2 ω}^{2})$ 113. $If f : R - 2 \to R is a function defined by f (x) = \frac{x ^{2} - 4}{x - 2}, then its range is$ (1) $R$ (2) $R - {2}$ (3) $R - {4}$ (4) $R - {- 2, 2}$ 114. $Let f : R \to R be defined by f (x) = 3 x^{2} - 5 and g : R \to R by g (x) = \frac{x}{x ^{2} + 1} then gof is$ (1) $\frac{3 x ^{2}}{x ^{4} + 2 x ^{2} - 4}$ (2) $\frac{3 x ^{2} - 5}{9 x ^{4} - 30 x ^{2} + 26}$ (3) $\frac{3 x ^{2}}{9 x ^{4} + 30 x ^{2} - 2}$ (4) $\frac{3 x ^{2} - 5}{9 x ^{4} - 6 x ^{2} + 26}$ 115. $f : R \to R defined by f (x) = ⎩ ⎨ ⎧ 2 x, x^{2}, 3 x, x > 3 1 < x \leq 3 x \leq 1 then f (- 2) + f (3) + f (4) is$ (1) $14$ (2) $9$ (3) $5$ (4) $11$ 116. $If lim_{x \to 0} \frac{\| x \|}{x ^{4} + 4 x ^{2} + 5} = k, lim_{x \to 0} x^{4} sin (\frac{1}{3 x}) = l, Then k + l =$ (1) $0$ (2) $l$ (3) $-1$ (4) $5$ 117. $The domain of the function f : R \to R defined by f (x) = x^{2} - 7 x + 12 is$ (1) $(- \infty, 3] \cup [4, \infty)$ (2) $(- \infty, 3] \cap [4, \infty)$ (3) $(- \infty, 3] \cup (4, \infty)$ (4) $(3, 4)$ 118. $Let the function f : [0, 1] \to R be defined by f (x) = \frac{4 ^{x}}{4 ^{x} + 2} Then the value of f (\frac{1}{40}) + f (\frac{2}{40}) + f (\frac{3}{40}) + \dots . + f (\frac{39}{40}) - f (\frac{1}{2}) is____. (JEE Advanced 2020)$ 119. $Let f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x ϵ R - (\frac{- 3}{2}) For n \geq 2, define f^{n} (x) = f^{1} f^{n - 1} (x) . If f^{5} (x) = \frac{ax + b}{bx + a}, gcd (a, b) = 1 then a + b is equal to _____$ 120. $If A = {x \in R : \|x - 2\| > 1},B = {x \in R : x^{2} - 3 > 1}, C = {x \in R : \|x - 4\| \geq 2} and Z is the set of all integers, then the number of subsets of the set(A \cap B \cap C)^{C} \cap Z is .$