Sets, Relations & Functions PYQs Set 11 (PDF)

Set 11 of 25 · Questions 301-330 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 11 (Q301-330)
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301. $If f (x) = x (x \geq 0) and g (x) = 1 + x^{2}, then (f \circ g)^{’} (1) = .$ (1) $1$ (2) $\frac{1}{2}$ (3) $2$ (4) $\frac{1}{2}$ 302. $Let a set A = A_{1} U A_{2} U ... U A_{k i} where A_{i} \cap A_{j} = φ for i \neq = j 1 \leq i, j \leq k . Define the relation R from A to A by R = {(x, y) : y \in A_{i} if and only if x \in A_{i}, 1 \leq i \leq k). Then, R is:$ (1) $reflexive, symmetric but not transitive$ (2) $reflexive, transitive but not symmetric$ (3) $reflexive but not symmetric and transitive$ (4) $an equivalence relation$ 303. $If y = tan^{- 1} (\frac{3 x - x ^{3}}{1 - 3 x ^{2}}) + tan^{- 1} (\frac{4 x - 4 x ^{3}}{1 - 6 x ^{2} + x ^{4}}) then \frac{dy}{dx} is equal to$ (1) $\frac{2}{1 + x ^{2}}$ (2) $\frac{4}{1 + x ^{2}}$ (3) $\frac{6}{1 + x ^{2}}$ (4) $\frac{7}{1 + x ^{2}}$ 304. $If x = a cos^{3} θ and y = a sin^{3} θ, then \frac{dy}{dx} =$ (1) $3 \frac{x}{y}$ (2) $3 \frac{y}{x}$ (3) $- 3 \frac{y}{x}$ (4) $- 3 \frac{x}{y}$ 305. $If A = (a, b) : 4a=5b; a, b \in 1, 2, 3, \dots 30, then the number of such ordered pairs (a, b) is$ (1) $4$ (2) $6$ (3) $8$ (4) $10$ 306. $Let [x] denote the greatest integer not more than x . If A and B are the domains of the functions f (x) = \frac{x - [ x ]}{\| x \| - x} and g (x) = \frac{x - [ x ]}{\| x \| + x} respectively, then$ (1) $A \cup B = R$ (2) $A\capB=ϕ$ (3) $A - B = (- \infty, 0)$ (4) $B - A = (0, \infty)$ 307. $Let f : R \to R defined by f (x) = 5 x^{4} + 2 . Then$ (1) $f is one-one but not onto$ (2) $f is onto but not one-one$ (3) $f is both one-one and onto$ (4) $f is neither one-one nor onto$ 308. $Let f (x) = log_{e} (sin x), (0 < x < π) and g (x) = sin^{- 1} (e^{- x}), (x \geq 0) . If α is a positive real number such that a = (fog)^{’} (α) and b = (fog) (α), then :$ (1) $a α^{2} + bα + a = 0$ (2) $a α^{2} - bα - a = 1$ (3) $a α^{2} - bα - a = 0$ (4) $a α^{2} + bα - a = - 2 a^{2}$ 309. $A manager decides to distribute Rs. 20000 between two employees X and Y . He knows X deserves more than Y, but does not know how much more. So, he decides to arbitrarily break ₹ 20000 into two parts and gives X the bigger part. Then, the chance that X gets twice as much as Y or more is$ (1) $2/5$ (2) $1/2$ (3) $1/3$ (4) $2/3$ 310. $Let f : (- \frac{π}{2}, \frac{π}{2}) \to R be given by f (x) = (log (secx + tanx))^{3} . Then (JEE Advanced 2014)$ (1) $f (x) is an odd function$ (2) $f (x) is a one-one function$ (3) $f (x) is an onto function$ (4) $f (x) is an even function$ 311. $Given, the function f (x) = \frac{a ^{x} + a ^{- x}}{2} (a > 2), then f (x + y) + f (x - y) is equal to$ (1) $f (x) - f (y)$ (2) $f (y)$ (3) $2 f (x) f (y)$ (4) $f (x) f (y)$ 312. $If y = cos^{2} \frac{3 x}{2} - sin^{2} \frac{3 x}{2}, then \frac{d ^{2} y}{d x ^{2}} is$ (1) $- 3 1 - y^{2}$ (2) $9 y$ (3) $- 9 y$ (4) $3 1 - y^{2}$ 313. $If x = 4 cos^{3} θ and y = 3 sin^{2} θ, then \frac{d ^{2} y}{d x ^{2}} at θ = \frac{π}{4}, is$ (1) $\frac{1}{3}$ (2) $\frac{1}{6}$ (3) $\frac{- 1}{6}$ (4) $\frac{- 1}{3}$ 314. $Assertion: For x < 0, \frac{d ^{2}}{d x ^{2}} (lo g ∣ x ∣) = \frac{1}{∣ x ∣ ^{2}} . Reason: For x < 0, ∣ x ∣ = - x . The correct option among the following is$ (1) $(A) is true, (R) is true and (R) is the correct explanation to (A) .$ (2) $(A) is true, (R) is true but (R) is not a correct explanation to (A).$ (3) $(A) is true, (R) is false.$ (4) $(A) is false, (R) is true.$ 315. $If y = tan^{- 1} x^{2} - 1, then the ratio \frac{d ^{2} y}{d x ^{2}} : \frac{dy}{dx} =$ (1) $\frac{1 - 2 x ^{2}}{x ( x ^{2} - 1 )}$ (2) $\frac{x ( x ^{2} - 1 )}{1 + 2 x ^{2}}$ (3) $\frac{x ( x ^{2} + 1 )}{1 - 2 x ^{2}}$ (4) $\frac{1 + 2 x ^{2}}{x ( x ^{2} + 1 )}$ 316. $If x^{y} = e^{x - y} then \frac{dy}{dx} is equal to$ (1) $\frac{log x}{log ( x - y )}$ (2) $\frac{e ^{x}}{x ^{x - y}}$ (3) $\frac{log x}{( 1 + log x ) ^{2}}$ (4) $\frac{1}{y} - \frac{1}{x - y}$ 317. $If function f (x) = (x - \frac{\| x \|}{x}, x < 0 x + \frac{\| x \|}{x}, x > 01, x = 0), then$ (1) $limx\to0- f (x) does not exist$ (2) $limx\to0+ f (x) does not exist$ (3) $f (x) is continuous at x = 0$ (4) $lim_{x \to 0^{-}} f (x) \neq = lim_{x \to 0^{+}} f (x)$ 318. $If f : R \to R is a function defined by f (x) = [x - 1] cos (\frac{2 x - 1}{2}) π, where [.] denotes the greatest integer function, then f is :$ (1) $discontinuous only at x = 1$ (2) $discontinuous at all integral values of x except at x = 1$ (3) $continuous only at x = 1$ (4) $continuous for every real x$ 319. $f (x) = ⎩ ⎨ ⎧ 4, x^{2} - 1, 4, - \infty < x < - 5 - 5 \leq x \leq 5 5 \leq x < \infty If k is the number of points where f (x) is not differentiable then k - 2 =$ (1) $2$ (2) $1$ (3) $0$ (4) $3$ 320. $Let f : R \to R be defined by f (x) = ⎩ ⎨ ⎧ 2 x, x^{2}, 3 x, x > 3 1 < x \leq 3 x \leq 1 Then f (- 1) + f (2) + f (4) is$ (1) $9$ (2) $14$ (3) $5$ (4) $10$ 321. $If y = (tan^{- 1} x)^{2}, then (x^{2} + 1)^{2} y_{2} + 2 x (x^{2} + 1) y_{1} is equal to$ (1) $4$ (2) $0$ (3) $2$ (4) $1$ 322. $If f (x + y) = f (x) f (y) and \sum f (x) = 2, x, y \in N where N is the set of all natural numbers, then the value of \frac{f ( 4 )}{f ( 2 )} is:$ (1) $\frac{1}{9}$ (2) $\frac{4}{9}$ (3) $\frac{1}{3}$ (4) $\frac{2}{3}$ 323. $lim x \to 0 \frac{a ^{x} - 1}{sin ( x )} =$ (1) $log (a)$ (2) $12 log (a)$ (3) $0$ (4) $1$ 324. $Given below are two statements: Statement I: The function f : R \to R defined by f (x) = \frac{x}{1 + \| x \|} is one-one. Statement II: The function f : R \to R defined by f (x) = \frac{x ^{2} + 4 x - 30}{x ^{2} - 8 x + 18} is many-one. In the light of the above statements, choose the correct answer from the options given below:$ (1) $Both Statement I and Statement II are true$ (2) $Statement I is true but Statement II is false$ (3) $Statement I is false but Statement II is true$ (4) $Both Statement I and Statement II are false$ 325. $The function f defined on (- \frac{1}{3}, \frac{1}{3}) by f (x) = {\frac{1}{x} lo g (\frac{1 + 3 x}{1 - 2 x}), k, x \neq = 0 x = 0 is continuous at x = 0, then k is$ (1) $6$ (2) $1$ (3) $5$ (4) $-5$ 326. $If A = a, b, c, B = b, c, d and C = a, d, c, then (A - B) \times (B \cap C) =$ (1) ${(a, c), (a, d)}$ (2) ${(a, b), (c, d)}$ (3) ${(c, a), (d, a)}$ (4) ${(a, c), (a, d), (b, d)}$ 327. $Let [x] denotes the greatest integer less than or equal to x . Then f (x) = \frac{1 + sin ( [ cos x ] )}{cos ( [ sin x ] )} is$ (1) $continuous on (0, \frac{π}{2})$ (2) $continuous on (0, π)$ (3) $discontinuous on (π, \frac{3 π}{2})$ (4) $continuous on (π, 2 π)$ 328. $If f (x) = ax + b, where a and b are integers, f (- 1) = - 5 and f (3) = 3 then a and b are respectively$ (1) $0,2$ (2) $- 3, - 1$ (3) $2,3$ (4) $2, - 3$ 329. $If a > 0, b > 0 then lim_{n \to \infty} (\frac{a + b ^{1/ n} - 1}{a})^{n} =$ (1) $a^{b}$ (2) $b^{a}$ (3) $b^{1/ a}$ (4) $a^{1/ b}$ 330. $Let A = (x, y) : 2x+3y=23, x, y \in N and B = x : (x, y) \in A . Then the number of one-one functions from A to B is equal to_______$