Sets, Relations & Functions PYQs Set 12 (PDF)

Set 12 of 25 · Questions 331-360 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 12 (Q331-360)
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331. $If x^{2} + xy + y^{2} = k, then \frac{d ^{2} y}{d x ^{2}} is equal to$ (1) $\frac{- 6 k}{( x + 2 y ) ^{3}}$ (2) $\frac{- 6 k}{( x + 2 y ) ^{2}}$ (3) $\frac{x ^{2} + xy + y ^{2}}{( 2 x + y ) ^{2}}$ (4) $0$ 332. $Let A and B two sets containing 2 elements and 4 elements respectively. The number of subsets of A \times B having 3 or more elements is$ (1) $256$ (2) $220$ (3) $219$ (4) $211$ 333. $If \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, then \frac{d ^{2} y}{d x ^{2}} =$ (1) $- \frac{b ^{4}}{a ^{2} y ^{3}}$ (2) $\frac{b ^{2}}{a y ^{2}}$ (3) $\frac{- b ^{3}}{a ^{2} y ^{3}}$ (4) $\frac{b ^{3}}{a ^{2} y ^{2}}$ 334. $lim x \to 1 (1 - x) tan (\frac{π}{2} x) =$ (1) $\frac{π}{2}$ (2) $\frac{2}{π}$ (3) $1$ (4) $0$ 335. $lim_{y \to 0} \frac{3 + y ^{3} - 3}{y ^{3}} =$ (1) $\frac{1}{2 3}$ (2) $23$ (3) $\frac{1}{3 2}$ (4) $32$ 336. $If R is the smallest equivalence relation on the set 1, 2, 3, 4 such that (1, 2), (1, 3) \subset R, then the number of elements in R is$ (1) $10$ (2) $12$ (3) $8$ (4) $15$ 337. $If f : R \to R is defined as f (x) = x^{2} - 2 x - 3 then f is$ (1) $one-one but not onto$ (2) $onto but not one-one$ (3) $neither one-one nor onto$ (4) $a bijection$ 338. $If x = e^{θ} sin θ, y = e^{θ} cos θ where θ is a parameter, then \frac{dy}{dx} at (1, 1) is equal to$ (1) $0$ (2) $- \frac{1}{2}$ (3) $\frac{1}{2}$ (4) $- \frac{1}{4}$ 339. $lim n \to \infty 2 (\frac{( 2 + 2 ) ^{n} + ( 2 - 2 ) ^{n}}{( 2 + 2 ) ^{n} - ( 2 - 2 ) ^{n}}) =$ (1) $2 - 2$ (2) $2-3$ (3) $1$ (4) $2$ 340. $If y = x + y + x + y + \dots \infty, then \frac{dy}{dx} =$ (1) $\frac{y ^{3} - x}{2 y ^{2} - 2 xy + 1}$ (2) $\frac{x + y ^{3}}{2 y ^{2} - x}$ (3) $\frac{y + x}{y ^{2} - 2 x}$ (4) $\frac{y ^{2} - x}{2 y ^{3} - 2 xy - 1}$ 341. $Let f : R \to R be a function such that f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} + 1} . Then$ (1) $f(x) is many - one in (- \infty, - 1)$ (2) $f(x) is many-one in (1, \infty)$ (3) $f(x) one-one in [1, \infty) but not in -\infty, \infty$ (4) $f (x) is one- one in (- \infty, \infty)$ 342. $If the domain of the function f (x) = \frac{[ x ]}{1 + x ^{2}}, where [x] is greatest integer \leq x is (2, 6), then its range is$ (1) $(\frac{5}{26}, \frac{2}{5}) - (\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53})$ (2) $(\frac{5}{26}, \frac{2}{5})$ (3) $(\frac{5}{37}, \frac{2}{5}) - (\frac{9}{29}, \frac{27}{109} \frac{18}{89}, \frac{9}{53})$ (4) $(\frac{5}{37}, \frac{2}{5})$ 343. $The number of non-empty equivalence relations on the set 1, 2, 3 is:$ (1) $6$ (2) $5$ (3) $7$ (4) $4$ 344. $The inverse of 2010 in the group Q^{+} of all positive rationals under the binary operation * defined by a * b = \frac{ab}{2010}, \forall a, b \in Q^{+}, is$ (1) $2011$ (2) $2009$ (3) $2010$ (4) $1$ 345. $If a function f defined by f (x) = (\frac{1 - 2 sin x}{π - 4 x}, if x \neq = \frac{π}{4} k, if x = \frac{π}{4}) is continuous at x = \frac{π}{4}, then k =$ (1) $\frac{1}{4}$ (2) $1$ (3) $\frac{- 1}{4}$ (4) $2$ 346. $If f (x) = 2 x^{2}, find \frac{f ( 3.8 ) - f ( 4 )}{3.8 - 4} .$ (1) $156$ (2) $0.156$ (3) $1.56$ (4) $15.6$ 347. $If x^{x} = y^{y}, then \frac{dy}{dx} is$ (1) $- \frac{y}{x}$ (2) $- \frac{x}{y}$ (3) $1 + log (\frac{x}{y})$ (4) $\frac{1 + log x}{1 + log y}$ 348. $Let f : X \to Y be a function and A_{y} = (\frac{f ^{- 1} ( y )}{y \in Y}) . Then A_{i} \cap A_{j} = φ (i \neq = j) \forall i, j \in Y and \cup y \in Y A_{y} = X, if$ (1) $f is onto function only$ (2) $f is one-one function only$ (3) $f is any function$ (4) $X and Y are finite sets only$ 349. $If \frac{d}{dx} (Alog (\frac{1 - x ^{3} + B}{1 - x ^{3} + 1})) = \frac{1}{x 1 - x ^{3}}, then AB =$ (1) $\frac{1}{3}$ (2) $\frac{- 1}{3}$ (3) $\frac{- 2}{3}$ (4) $\frac{2}{3}$ 350. $If e^{x} = y + y^{2} - 1, then \frac{dy}{dx} =$ (1) $sinh x$ (2) $cosh x$ (3) $tanh x$ (4) $- sinh x$ 351. $If the operation \oplus is defined by a \oplus b = a^{2} + b^{2} for all real numbers ' a' and ' b, then (2 \oplus 3) \oplus 4 =$ (1) $182$ (2) $185$ (3) $181$ (4) $184$ 352. $If the function is f (x) = \frac{1}{x + 2}, then the point of discontinuity of the composite function y = f (f (x)) is$ (1) $\frac{2}{5}$ (2) $\frac{- 5}{2}$ (3) $\frac{1}{2}$ (4) $\frac{5}{2}$ 353. $\frac{d}{d x} lo g x + x^{2} + a^{2} =$ (1) $x^{2} + a^{2}$ (2) $\frac{1}{x ^{2} + a ^{2}}$ (3) $\frac{1}{2 x ^{2} + a ^{2}}$ (4) $\frac{1}{2 ( x + x ^{2} + a ^{2} )}$ 354. $The relation R = {(x, y) : x, y \in Z and x + y is even} is:$ (1) $reflexive and symmetric but not transitive$ (2) $an equivalence relation$ (3) $symmetric and transitive but not reflexive$ (4) $reflexive and transitive but not symmetric$ 355. $Let f (x) = a^{x} (a > 0) be written as f (x) = f_{1} (x) + f_{2} (x), where f_{1} (x) is an even function and f_{2} (x) is an odd function. Then f_{1} (x + y) + f_{1} (x - y) equals:$ (1) $2 f_{1} (x) f_{1} (y)$ (2) $2 f_{1} (x + y) f_{1} (x - y)$ (3) $2 f_{1} (x) f_{2} (y)$ (4) $2 f_{1} (x + y) f_{2} (x - y)$ 356. $If f (x) is a function such that f^{’ ’} (x) + f (x) = 0 and g (x) = [f (x)]^{2} + ((f^{’} (x))^{2}) and g (3) = 8, then g (8) =$ (1) $0$ (2) $3$ (3) $5$ (4) $8$ 357. $lim x \to 0 \frac{( 2 ^{x} - 1 ) ( 1 + sin x ) ^{\frac{2}{sin x}}}{log ( 1 + 2 x )} =$ (1) $e^{2} log 4$ (2) $e log 2$ (3) $e^{2} log 2$ (4) $e^{2} log 2$ 358. $Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8 x^{2} - 2x, and g(f(x)) = 4 x^{2} + 6x + 1, then the value of f(2) + g(2) is_____.$ 359. $Let A = (a, b, c) and B = (1, 2, 3, 4) . Then the number of elements in the set C = (f : A \to B \| 2 \in f (A) and f is not one - one) is$ 360. $Let A = {1, 2, 3,4,..........10} and B ={0,1, 2,3,4}. The number of elements in the relation R = ((a, b) \in A \times A : 2 (a - b)^{2} + 3 (a - b) \in B) is$