Sets, Relations & Functions PYQs Set 5 (PDF)

Set 5 of 25 · Questions 121-150 of 734

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 5 (Q121-150)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
121. $Let R be a relation defined on the set 1, 2, 3, 4 \times 1, 2, 3, 4 by R = ((a, b), (c, d)) : 2a+3b=3c+4d . Then the number of elements in R is$ (1) $18$ (2) $6$ (3) $15$ (4) $12$ 122. $If lim_{x \to 2} \frac{1 + 1 + 4 log _{2} x}{2 + ( 2 x + sin ^{2} x + 2 cos x ) ( 2 x - 4 )} = m, then m (m - 1) =$ (1) $0$ (2) $log_{2} e$ (3) $1$ (4) $\frac{1 + 3}{2}$ 123. $lim_{x \to \infty} x sin (\frac{2}{x}) is equal to$ (1) $2$ (2) $\frac{1}{2}$ (3) $\infty$ (4) $0$ 124. $lim x \to 0 \frac{x ^{4} + x ^{3} + x ^{2}}{sin ^{- 1} ( \frac{x}{1 + x ^{2}} ) tan ^{- 1} x} =$ (1) $\frac{1}{2}$ (2) $0$ (3) $1$ (4) $\frac{- 1}{2}$ 125. $The domain of the function f (x) = \frac{1}{log _{10} ( 1 - x )} + x + 2 is$ (1) $[- 2, 0) \cap (0, 1)$ (2) $[- 2, 1)$ (3) $[- 2, 0)$ (4) $[- 2, 0) \cup (0, 1)$ 126. $Let f (x) = sin (\frac{π}{6} sin (\frac{π}{2} sinx)) for all x \in R and g (x) = \frac{π}{2} sinx for all x \in R . Let (fog) (x) denote f (g (x)) and (gof)(x) denote g (f (x)) . Then which of the following is (are) true? (JEE Advanced 2015)$ (1) $Range of f is [- \frac{1}{2}, \frac{1}{2}]$ (2) $Range of fog is [- \frac{1}{2}, \frac{1}{2}]$ (3) $lim x \to 0 \frac{f ( x )}{g ( x )} = \frac{π}{6}$ (4) $There is an x \in R such that (gof) (x) = 1$ 127. $Let R be a relation defined on N as a R b is 2 a + 3 b is a multiple of 5, a, b ϵ N . Then R is$ (1) $Not reflective$ (2) $Transitive but not symmetric$ (3) $symmetric but not transitive$ (4) $an equivalence relation$ 128. $lim_{x \to 1} \frac{log x}{( 1 - x )} is equal to$ (1) $1$ (2) $-1$ (3) $0$ (4) $\frac{- 1}{2}$ 129. $The function f (x) = [x], where [x] denotes greatest integer function is continuous at$ (1) $-2$ (2) $1.5$ (3) $4$ (4) $1$ 130. $If y = tan^{- 1} (\frac{a cos x - b sin x}{b cos x + a sin x}), then \frac{dy}{dx} is equal to$ (1) $0$ (2) $\frac{a}{b}$ (3) $-1$ (4) $2$ 131. $Let f be a function defined by f (xy) = \frac{f ( x )}{y} for all positive real numbers x and y . If f (30) = 20, then f (40) =$ (1) $10$ (2) $15$ (3) $25$ (4) $17$ 132. $If y = e^{x x x \dots \infty}, x > 1 then \frac{d ^{2} y}{d x ^{2}} at x = lo g_{e} 3 is$ (1) $3$ (2) $0$ (3) $5$ (4) $1$ 133. $If a > 0, n \in R, then lim_{x \to a} x^{n} =$ (1) $n a^{n}$ (2) $(n - 1) a^{n}$ (3) $n a^{n - 1}$ (4) $a^{n}$ 134. $If f is differentiable, f (x + y) = f (x) f (y) for all x, y \in R, f (3) = 3, f^{’} (0) = 11, then f^{’} (3) =$ (1) $\frac{3}{11}$ (2) $\frac{11}{3}$ (3) $8$ (4) $33$ 135. $The function f : (- \infty, \infty) \to (- \infty, 1), defined by f (x) = \frac{2 ^{x} - 2 ^{- x}}{2 ^{x} + 2 ^{- x}} is:$ (1) $Neither one-one nor onto$ (2) $Onto but not one-one$ (3) $Both one-one and onto$ (4) $One-one but not onto$ 136. $If f (x) = 1 + nx + \frac{n ( n - 1 )}{2} x^{2} + \frac{n ( n - 1 ) ( n - 2 )}{6} x^{3} + \dots .. + x^{n} then f^{’ ’} (1) =$ (1) $n (n - 1) 2^{n}$ (2) $(n - 1) 2^{n - 1}$ (3) $2^{n - 1}$ (4) $n (n - 1) 2^{n - 2}$ 137. $lim x \to 2 ((x^{2} - 4 x + 4) cos (\frac{2}{x - 2}) + \frac{x ^{2} - 4}{x ^{3} - 2 x - 4}) =$ (1) $0$ (2) $\infty$ (3) $1$ (4) $\frac{2}{5}$ 138. $If the range of the function f (x) = - 3 x - 3 is {3, - 6, - 9, - 18}, then which of the following elements is not in the domain of f ?$ (1) $-1$ (2) $-2$ (3) $1$ (4) $2$ 139. $Let S = 1, 2, 3, \dots, 10 . Suppose M is the set of all the subsets of S, then the relation R = {(A, B) : A \cap B \neq = φ; A, B \in M} is :$ (1) $symmetric and reflexive only$ (2) $reflexive only$ (3) $symmetric and transitive only$ (4) $symmetric only$ 140. $If f (x) = \frac{g ( x ) + g ( - x )}{2} + \frac{2}{{ h ( x ) + h ( - x ) } ^{- 1}} where g and h are differentiable functions, then f^{’} (0) =$ (1) $\frac{3}{2}$ (2) $0$ (3) $1$ (4) $1/2$ 141. $Let * be a binary operation defined on R by a * b = \frac{a + b}{4} \forall a, b \in R then the operation * is$ (1) $Commutative and Associative$ (2) $Commutative but not Associative$ (3) $Associative but not Commutative$ (4) $Neither Associative nor Commutative$ 142. $If f (x^{5}) = 5 x^{3}, then f^{’} (x) =$ (1) $\frac{3}{x}$ (2) $5 x$ (3) $\frac{3}{5 x ^{2}}$ (4) $\frac{3}{5 x}$ 143. $Define f (x) = (x^{2} + bx + c, x < 1 x, x \geq 1) . If f (x) is differentiable at x = 1, then (b - c) is equal to$ (1) $-2$ (2) $0$ (3) $1$ (4) $2$ 144. $Given, sin x = \sum_{n = 1}^{\infty} (- 1)^{n - 1} \frac{x ^{2 n - 1}}{( 2 n - 1 ) !} . If the function f (x) given by f (x) = \frac{cos ( sin x ) - cos x}{x ^{4}} (x \neq = 0) and f (0) = k, is continuous at x = 0, then k =$ (1) $\frac{1}{6}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $0$ 145. $Which of the following is not correct for relation R on the set of real numbers?$ (1) $(x, y) \in R \Leftrightarrow 0 < \|x\| - \|y\| \leq 1 is neither transitive nor symmetric.$ (2) $(x, y) \in R \Leftrightarrow 0 < \|x - y\| \leq 1 is symmetric and transitive.$ (3) $(x, y) \in R \Leftrightarrow \|x\| - \|y\| \leq 1 is reflexive but not symmetric.$ (4) $(x, y) \in R \Leftrightarrow \|x - y\| \leq 1 is reflexive and symmetric.$ 146. $Consider the sets A = ((x, y) \in R \times R : x^{2} + y^{2} = 25), B = ((x, y) \in R \times R : x^{2} + 9 y^{2} = 144), C = ((x, y) \in Z \times Z : x^{2} + y^{2} \leq 4) and D = A \cap B . The total number of one-one functions from the set D to the set C is:$ (1) $15120$ (2) $18290$ (3) $17160$ (4) $19320$ 147. $Let A = 1, 2, 3, 5, 8, 9 . Then the number of possible functions f : A \to A such that f (m . n) = f (m) . f (n) for every m, n ϵ A with m . n ϵ A is equal to ______$ 148. $Let S = 1, 2, 3, 4, 5, 6 . Then the number of one-one functions f : S \to P (S), where P (S) denote the power se of S, such that f (n) \subset f (m) where n < m is ________$ 149. $For Some a, b, cϵ N, let f (x) = ax - 3 and g (x) = x^{b} + c, x ϵ R . If (fog)^{- 1} (x) = (\frac{x - 7}{2})^{\frac{1}{3}} then (fog) (ac) + (gof) (b) is equal to _________$ 150. $If a function f satisfies f (m + n) = f (m) + f (n) for all m, n \in N and f (1) = 1, then the largest natural number λ such that \sum_{k = 1}^{2022} f (λ + k) \leq (2022)^{2} is equal to_______$