Sets, Relations & Functions PYQs Set 17 (PDF)

Set 17 of 25 · Questions 481-510 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 17 (Q481-510)
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481. $The domain of the function f (x) = sin^{- 1} (log_{2} (\frac{x ^{2}}{2})) is$ (1) $[- 2, 0) \cup (0, 1)$ (2) $[1, \infty) \cap [- 2, 2]$ (3) $[- 2, - 1] \cup [1, 2]$ (4) $(- \infty, 1] \cap [- 2, 2]$ 482. $If f : R \to R defined as f (x) = \frac{x ^{3} + 2 x ^{2} + x + 2}{x ^{2} + x - 2} (when x \neq = - 2) is continuous at x = - 2, then f (- 2) is equal to$ (1) $5$ (2) $\frac{- 5}{3}$ (3) $2$ (4) $\frac{3}{5}$ 483. $If x^{2} + y^{2} = t + \frac{2}{t} and x^{4} + y^{4} = t^{2} + \frac{4}{t ^{2}}, then x^{3} y \frac{dy}{dx} equals$ (1) $-1$ (2) $-2$ (3) $\frac{y}{x}$ (4) $xy$ 484. $If x + y = tan^{- 1} y and \frac{d ^{2} y}{d x ^{2}} = f (y) \frac{dy}{dx}, then f (y) =$ (1) $\frac{- 2}{y ^{3}}$ (2) $\frac{2}{y ^{3}}$ (3) $\frac{1}{y}$ (4) $\frac{- 1}{y}$ 485. $Consider function f : A \to B and g : B \to C(A,B,C \subseteq R) such that (g o f)^{- 1} exists then:$ (1) $f and g both are one-one$ (2) $f and g both are onto$ (3) $f is one-one and g is onto$ (4) $f is onto and g is one-one$ 486. $If x^{2} + y^{2} = t + \frac{1}{t} and x^{4} + y^{4} = t^{2} + \frac{1}{t ^{2}}, then \frac{dy}{dx} is equal to$ (1) $- \frac{x}{y}$ (2) $- \frac{y}{x}$ (3) $\frac{x ^{2}}{y ^{2}}$ (4) $\frac{y ^{2}}{x ^{2}}$ 487. $For x \in R - (0, 1), let f_{1} (x) = \frac{1}{x}, f_{2} (x) = 1 - x and f_{3} (x) = \frac{1}{1 - x} be three given functions. If a function, J (x) satisfies (f_{2} oJo f_{1}) (x) = f_{3} (x) then J (x) is equal to :$ (1) $f_{3} (x)$ (2) $\frac{1}{x} f_{3} (x)$ (3) $f_{2} (x)$ (4) $f_{1} (x)$ 488. $lim_{n \to \infty} \frac{n ( 2 n + 1 ) ^{2}}{( n + 2 ) ( n ^{2} + 3 n - 1 )} is equal to$ (1) $0$ (2) $4$ (3) $2$ (4) $\infty$ 489. $Define a relation R over a class of n \times n real matrices A and B as " ARB if there exists a non-singular matrix P such that PAP^{- 1} = B^{’ ’} . Then which of the following is true?$ (1) $R is symmetric, transitive but not reflexive,$ (2) $R is reflexive, symmetric but not transitive$ (3) $R is an equivalence relation$ (4) $R is reflexive, transitive but not symmetric$ 490. $Let A = 1, 2, 3, 4, 5 . Let R be a relation on A defined by xRy if and only if 4 x \leq 5 y . Let m be the number of elements in R and n be the minimum number of elements from A \times A that are required to be added to R to make it a symmetric relation. Then m + n is equal to :$ (1) $25$ (2) $24$ (3) $26$ (4) $23$ 491. $If f (x) = 3 x - 2 and g (x) = x^{2}, then f \circ g (x) =$ (1) $3 x^{2} - 2$ (2) $3 x^{2} + 2$ (3) $3 x - 2$ (4) $2 - 3 x^{2}$ 492. $Let f (x) = cos^{- 1} (\frac{1}{13} (2 cos x - 3 sin x)) . Then f^{’} (0.5) =$ (1) $0.5$ (2) $1$ (3) $0$ (4) $-1$ 493. $If f : R \to R be defined by f (x) = ⎩ ⎨ ⎧ 2 x, x^{2}, 3 x, x > 3 1 < x \leq 3 x \leq 1 then f (- 1) + f (2) + f (4) is$ (1) $5$ (2) $9$ (3) $10$ (4) $14$ 494. $Let the set S = 2, 4, 8, 16, \dots, 512 be partitioned into 3 sets A, B, C with equal number of elements such that A \cup B \cup C = S and A \cap B = B \cap C = A \cap C = φ . The maximum number of such possible partitions of S is equal to:$ (1) $1680$ (2) $1640$ (3) $1520$ (4) $1710$ 495. $\frac{d}{d x} tan^{- 1} (\frac{1 + s i n x - 1 - s i n x}{1 + s i n x + 1 - s i n x}) is equal to$ (1) $1$ (2) $- \frac{1}{2}$ (3) $\frac{1}{2}$ (4) $-1$ 496. $Define a relation R on the interval (0, \frac{π}{2}) by xRy if and only if sec^{2} x - tan^{2} y = 1 . Then R is:$ (1) $both reflexive and transitive but not symmetric$ (2) $an equivalence relation$ (3) $reflexive but neither symmetric not transitive$ (4) $both reflexive and symmetric but not transitive$ 497. $Let f : R \to R be defined by f (x) = x^{4}, then$ (1) $f is one-one but not onto$ (2) $f is neither one-one nor onto$ (3) $f is one-one and onto$ (4) $f may be one-one and onto$ 498. $In the group (Z, ), if a b = a + b - n \forall a, b \in Z, where n is a fixed integer, then the inverse of (- n) is$ (1) $n$ (2) $- 3 n$ (3) $- n$ (4) $3 n$ 499. $lim x \to \infty x^{3} (x^{2} + x^{4} + 1 - 2 x) =$ (1) $0$ (2) $1$ (3) $\frac{1}{4 2}$ (4) $\frac{3}{2 2}$ 500. $Let A = (x \in R : x is not a positive integer) . Define a function f : A \to R as f (x) = \frac{2 x}{x - 1}, the f is :$ (1) $not injective$ (2) $neither injective nor surjective$ (3) $surjective but not injective$ (4) $injective but not surjective$ 501. $lim x \to 1 (lim 7 \to \infty y ((ex)^{1/ y} - 1)) =$ (1) $e$ (2) $0$ (3) $1$ (4) $-1$ 502. $If f (x) = (\frac{x ^{2} log ( cos x )}{log ( 1 + x )}, x \neq = 00, x = 0), then at x = 0, f (x) is$ (1) $not continuous$ (2) $continuous but not differentiable$ (3) $differentiable$ (4) $not continuous, but differentiable$ 503. $If 3 y x = 6 (x + y)^{5}, then \frac{dy}{dx} =$ (1) $x / y$ (2) $x - y$ (3) $x + y$ (4) $y / x$ 504. $The differential coefficient of log_{10} x with respect to log_{x} 10 is$ (1) $1$ (2) $- (log_{10} x)^{2}$ (3) $(log_{x} 10)^{2}$ (4) $\frac{x ^{2}}{100}$ 505. $Let R be the set of real numbers. Statement-1 : A = {(x, y) \in R \times R : y - x is an integer} is an equivalence relation on R. Statement-2 : B = {(x, y) \in R \times R : x = αy for some rational number α} is an equivalence relation on R.$ (1) $Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.$ (2) $Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.$ (3) $Statement-1 is true, Statement-2 is false.$ (4) $Statement-1 is false, Statement-2 is true.$ 506. $If y = (1 + x^{2}) tan^{- 1} x - x then \frac{dy}{dx} is$ (1) $2 x tan^{- 1} x$ (2) $x^{2} tan^{- 1} x$ (3) $\frac{tan ^{- 1} x}{x}$ (4) $x tan^{- 1} x$ 507. $Which one of the following functions is a bijection?$ (1) $f : R - Z \to [0, 1] defined by f (x) = x - [x] . (here [x] represents the greatest integer function)$ (2) $f : R \to (- \infty, 2) defined by f (x) = 4 x - x^{2} - 3$ (3) $f:(5,\infty)\toR-{0} defined by f (x) = \frac{1}{x - 5}$ (4) $f : [0, 4] \to [0, 4] defined by f (x) = 16 - x^{2}$ 508. $If y = 2^{log x}, then \frac{dy}{dx} is$ (1) $\frac{2 ^{log x}}{log 2}$ (2) $2^{log x} \cdot log 2$ (3) $\frac{2 ^{log x}}{x}$ (4) $\frac{2 ^{l o g x} \cdot l o g 2}{x}$ 509. $If y = (cos x^{2})^{2}, then \frac{dy}{dx} is equal to$ (1) $- 4 x sin 2 x^{2}$ (2) $- 2 x sin 2 x^{2}$ (3) $- x sin x^{2}$ (4) $- x cos 2 x^{2}$ 510. $For n \geq 2, let S_{n} denote the set of all subsets of 1, 2, \dots, n with no two consecutive numbers. For example 1, 3, 5 \in S_{6}, but {1, 2, 4} \in / S_{6} . Then n (S_{5}) is equal to________.$