Sets, Relations & Functions PYQs Set 14 (PDF)

Set 14 of 25 · Questions 391-420 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 14 (Q391-420)
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391. $If \frac{d ^{n} y}{d x ^{n}} = y_{n} and y = e^{x} + e^{- x}, then 4 x y_{2} + 2 y_{1} =$ (1) $- y$ (2) $y$ (3) $2 y$ (4) $- 2 y$ 392. $Let f (x) = Max cos x, sin x, 0 . If the number of points at which f (x) is not differentiable in (0, 2024 π) is 1012 k, then k =$ (1) $3/2$ (2) $6$ (3) $3$ (4) $2$ 393. $lim_{n \to \infty} \frac{3 \cdot 2 ^{n + 1} - 4 \cdot 5 ^{n + 1}}{5 \cdot 2 ^{n} + 7 \cdot 5 ^{n}} =$ (1) $\frac{- 20}{7}$ (2) $0$ (3) $\frac{3}{5}$ (4) $\frac{- 4}{7}$ 394. $Let f (x) and g (x) be twice differentiable functions such that f (x) = x^{2} + g^{’} (1) x + g^{’ ’} (2) and g (x) = f (1) x^{2} + x f^{’} (x) + f^{’ ’} (x) . Then f (x) - g (x) =$ (1) $2 x + 5$ (2) $3 x^{2} + 6 x + 1$ (3) $x^{2} - 6 x + 2$ (4) $x^{2} - 2$ 395. $If A and B are finite sets and A \subset B, then$ (1) $n (A \cup B) = n (B)$ (2) $n (A \cap B) = n (B)$ (3) $n(A\capB)=ϕ$ (4) $n (A \cup B) = n (A)$ 396. $Let A = 2, 3, 6, 8, 9, 11 and B = 1, 4, 5, 10, 15 . Let R be a relation on A \times B defined by (a, b) R (c, d) if and only if 3 ad - 7 bc is an even integer. Then the relation R is$ (1) $an equivalence relation.$ (2) $reflexive and symmetric but not transitive.$ (3) $transitive but not symmetric.$ (4) $reflexive but not symmetric.$ 397. $If f (t) = \frac{t}{2} + \frac{1}{4} log (2 t - 1), then f^{’} (\frac{t + 1}{2 t + 1}) =$ (1) $t$ (2) $1 + t$ (3) $2 t + 1$ (4) $t - 1$ 398. $If f (x) = \| x - 1 \| + \| x - 2 \|, then f^{’} (- 2023) + f^{’} (\frac{2024}{2023}) + f^{’} (2023) =$ (1) $1$ (2) $-1$ (3) $0$ (4) $3$ 399. $If f : R \to R, g : R \to R are defined by f (x) = 5 x - 3, g (x) = x^{2} + 3, then go f^{- 1} (3) is equal to$ (1) $\frac{25}{3}$ (2) $\frac{111}{25}$ (3) $\frac{9}{25}$ (4) $\frac{25}{111}$ 400. $If x = a t^{2} and y = 2 at, then \frac{d ^{2} y}{d x ^{2}} at t = \frac{1}{2} is$ (1) $- \frac{2}{a}$ (2) $\frac{4}{a}$ (3) $\frac{8}{a}$ (4) $\frac{- 4}{a}$ 401. $lim x \to 0 \frac{4 [ sin ( 2022 x ) - sin ( 2020 x ) ]}{x [ cos ( 2022 x ) + 2 cos ( 2021 x ) + cos ( 2020 x ) ]} =$ (1) $1$ (2) $2$ (3) $2020$ (4) $2021$ 402. $Let A \subseteq R, B \subseteq R and f : A \to B be defined by I (X) = x^{2} - 3 x + 2 . If f is a bijection, then$ (1) $A = (- \infty, 0], B = (- \infty, \frac{- 1}{4})$ (2) $A = (- \infty, \frac{3}{2}), B = (\frac{- 1}{4}, \infty)$ (3) $A = (\frac{3}{2}, \infty), B = (- \infty, \frac{- 1}{4})$ (4) $A = (- \infty, \infty), B = (\frac{- 1}{4}, \infty)$ 403. $For a suitably chosen real constant a, let a function, f : R - -a \to R be defined by f (x) = \frac{a - x}{a + x} . Further suppose that for any real number x \neq = - a and f (x) \neq = - a, (fof) (x) = x . Then f (- \frac{1}{2}) is equal to$ (1) $- 3$ (2) $\frac{1}{3}$ (3) $- \frac{1}{3}$ (4) $3$ 404. $Let R be a relation on R, given by R = {(a, b) : 3 a - 3 b + 7 is an irrational number} . Then R is$ (1) $Reflexive but neither symmetric nor transitive$ (2) $Reflexive and symmetric but not symmetric$ (3) $Reflexive and symmetric but not transitive$ (4) $An equivalence relation$ 405. $Assertion (A): f (x) = \| x - a \| + \| x - b \|, is continuous on R Reason (R): \frac{\| x - α \|}{x - α} is continuous at x\inR-{α}.The correct option among the following is$ (1) $(A) is true, (R) is true and (R) is the correct explanation for A$ (2) $(A) is true, (R) is true but (R) is not the correct explanation for A$ (3) $(A) is true but (R) is false$ (4) $(A) is false but (R) is true$ 406. $If y = (1 + x) (1 + x^{2}) (1 + x^{4}), then \frac{dy}{dx} at x = 1 is$ (1) $20$ (2) $28$ (3) $1$ (4) $0$ 407. $If y = sin^{n} x cos nx, then \frac{dy}{dx} is$ (1) $n sin^{n - 1} x cos nx$ (2) $n sin^{n - 1} x cos (n + 1) x$ (3) $n sin^{n - 1} x sin (n + 1) x$ (4) $n sin^{n - 1} x cos (n - 1) x$ 408. $The number of relations, defined on the set a, b, c, d, which are both reflexive and symmetric, is equal to:$ (1) $16$ (2) $1024$ (3) $64$ (4) $256$ 409. $Let [t] denote the greatest integer \leq t Then the equation in x, [x]^{2} + 2 [x + 2] - 7 = 0 has:$ (1) $exactly two solutions.$ (2) $infinitely many solutions.$ (3) $exactly four integral solutions.$ (4) $no integral solution.$ 410. $If the range of the function f (x) = \frac{5 - x}{x ^{2} - 3 x + 2}, x \neq = 1, 2, is (- \infty, α] \cup [β, \infty), then α^{2} + β^{2} is equal to:$ (1) $188$ (2) $192$ (3) $190$ (4) $194$ 411. $If A, B and C are three sets such that A \cap B = A \cap C and A \cup B = A \cup C, then$ (1) $A = C$ (2) $B = C$ (3) $A \cap B = φ$ (4) $A = B$ 412. $Define a relation R on A = 1, 2, 3, 4 as xRy if x divides y . R is$ (1) $reflexive and transitive$ (2) $reflexive and symmetric$ (3) $symmetric and transitive$ (4) $equivalence$ 413. $If f (x) = (1 + cos x, x \leq 0 a - x, 0 < x \leq 2 x^{2} - b^{2}, x > 2) is continuous everywhere, then a^{2} + b^{2} is equal to$ (1) $4$ (2) $8$ (3) $6$ (4) $12$ 414. $If f is defined in [1, 3] by f (x) = x^{3} + b x^{2} + ax, such that f (1) - f (3) = 0 and f^{’} (c) = 0, where c = 2 + \frac{1}{3}, then (a, b) is equal to$ (1) $(- 6, 11)$ (2) $(2 - \frac{1}{3}, 2 + \frac{1}{3})$ (3) $(11, - 6)$ (4) $(6, 11)$ 415. $If the function g (x) is defined by g (x) = \frac{x ^{200}}{200} + \frac{x ^{199}}{199} + \frac{x ^{198}}{198} + \dots .. + \frac{x ^{2}}{2} + x + 5, then g^{’} (0) =$ (1) $200$ (2) $5$ (3) $1$ (4) $100$ 416. $If the domain of the function f (x) = \frac{1}{10 + 3 x - x ^{2}} + \frac{1}{x + \| x \|} is (a, b), then (1 + a)^{2} + b^{2} is equal to:$ (1) $29$ (2) $30$ (3) $25$ (4) $26$ 417. $\frac{d}{dx} ((x^{\frac{5}{2}} - x^{\frac{3}{2}} + 1) (x^{2} - 3 x + 5)) =$ (1) $\frac{9}{2} x^{\frac{7}{2}} - 14 x^{\frac{5}{2}} + 20 x^{\frac{3}{2}} - \frac{15}{2} x^{\frac{1}{2}} + 2 x - 3$ (2) $\frac{9}{2} x^{\frac{7}{2}} - 7 x^{\frac{5}{2}} + 5 x^{\frac{3}{2}} - \frac{3}{2} x^{\frac{1}{2}} + 2 x - 3$ (3) $9 x^{\frac{7}{2}} - 14 x^{\frac{5}{2}} + 20 x^{\frac{3}{2}} - 15 x^{\frac{1}{2}} + 2 x - 3$ (4) $\frac{9}{2} x^{\frac{7}{2}} - \frac{7}{2} x^{\frac{5}{2}} + \frac{5}{2} x^{\frac{3}{2}} - \frac{15}{2} x^{\frac{1}{2}} + 2 x - 3$ 418. $The number of relations on the set A = 1, 2, 3, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is ______.$ 419. $The number of functions f, from the set A =_{x \in N : x^{2} - 10 x + 9 \leq 0} to the set B =_{n^{2} : n \in N} such that f_{(} x_{)} \leq_{(} x - 3_{)}^{2} + 1, for every x \in A, is_________.$ 420. $Let S = (p_{1}, p_{2} \dots, p_{10}) be the set of first ten prime numbers. Let A = S \cup P, where P is the set of all possible products of distinct elements of S . Then the number of all ordered pairs (x, y), x \in S, y \in A, such that x divides y, is .$