Sets, Relations & Functions PYQs Set 13 (PDF)

Set 13 of 25 · Questions 361-390 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 13 (Q361-390)
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361. $lim x \to 0 \frac{8}{sin ^{8} x} (1 - cos (\frac{x ^{2}}{2}) - cos (\frac{x ^{2}}{4}) + cos (\frac{x ^{2}}{2}) cos (\frac{x ^{2}}{4})) =$ (1) $\frac{1}{16}$ (2) $\frac{1}{32}$ (3) $\frac{1}{64}$ (4) $\frac{1}{8}$ 362. $For k > 0, \sum_{x = 0}^{\infty} \frac{k ^{x}}{x !} lim_{n \to \infty} \frac{n !}{( n - x ) !} (1 - \frac{k}{n})^{n - x} (\frac{1}{n})^{x} =$ (1) $0$ (2) $k$ (3) $x$ (4) $1$ 363. $Let R_{1} and R_{2} be two relations defined on R by a R_{1} b \Leftrightarrow ab \geq 0 and a R_{2} b \Leftrightarrow a \geq b, then$ (1) $R_{1} is an equivalence relation but not R_{2}$ (2) $R_{2} is an equivalence relation but not R_{1}$ (3) $both R_{1} and R_{2} are equivalence relations$ (4) $neither R_{1} nor R_{2} is an equivalence relation$ 364. $For any two real numbers, an operation * defined by a * b = 1 + ab is$ (1) $neither commutative nor associative$ (2) $commutative but not associative$ (3) $both commutative and associative$ (4) $associative but not commutative.$ 365. $If x + y - x - y = c, then \frac{d ^{2} y}{d x ^{2}} =$ (1) $\frac{1}{y} (\frac{dy}{dx})^{2}$ (2) $\frac{- c ^{4}}{4 y ^{3}}$ (3) $y (\frac{dy}{dx})^{2}$ (4) $\frac{- c ^{2}}{4 y ^{3}}$ 366. $Find the function g (t) if f (t) = 3 t - 2 and (gof)^{- 1} (t) = t - 2 .$ (1) $g (t) = \frac{( t - 8 )}{3}$ (2) $g (t) = \frac{( t + 8 )}{3}$ (3) $g (t) = \frac{( 8 - t )}{3}$ (4) $g (t) = 3 t - 8$ 367. $Let D be the domain of a twice differentiable function f . For all x \in D, f^{’ ’} (x) + f (x) = 0 and f (x) = \int g (x) dx + constant. If h (x) = f (x)^{2} + g (x)^{2} and h (0) = 5, then h (2015) - h (2014) is equal to$ (1) $5$ (2) $3$ (3) $0$ (4) $1$ 368. $If X = {4^{n} - 3 n - 1 : n \in N} and Y = {9 (n - 1) : n \in N}, where N is the set of natural numbers, then X \cup Y is equal to$ (1) $X$ (2) $Y$ (3) $N$ (4) $Y - X$ 369. $If y = f (x^{2} + 2) and f^{’} (3) = 5, then \frac{dy}{dx} at x = 1 is$ (1) $25$ (2) $10$ (3) $5$ (4) $15$ 370. $In P (X), the power set of a non-empty set X, a binary operation * is defined by A * B = A \cup B \forall A, B \in P (X) . Under *, a TRUE statement is$ (1) $inverse law is not satisfied$ (2) $identity law is not satisfied$ (3) $associative law is not satisfied$ (4) $commutative law is not satisfied.$ 371. $Let f, g : R \to R be two functions defined as f (x) = \| x \| + x and g (x) = \| x \| - x \forall x \in R . Then (f o g) (x) for x < 0 is$ (1) $0$ (2) $4 x$ (3) $- 4 x$ (4) $2 x$ 372. $If x = \frac{1 - y}{1 + y}, then (x + 1) \frac{d ^{2} y}{d x ^{2}} + (\frac{3 y + 1}{y}) \frac{dy}{dx} equals$ (1) $- 2 y$ (2) $0$ (3) $- y$ (4) $y$ 373. $If R = ((x, y); x, y \in Z, x^{2} + 3 y^{2} \leq 8) is a relation on the set of integers Z, then the domain of R^{- 1} is:$ (1) ${0, 1}$ (2) $(- 2, - 1, 1, 2)$ (3) $(- 1, 0, 1)$ (4) $(- 2, - 1, 0, 1, 2)$ 374. $Let the relation R be defined in N by aRb if 3 a + 2 b = 27 then R is$ (1) ${(1, 12), (3, 9), (5, 6), (7, 3), (9, 0)}$ (2) ${(1, 12), (3, 9), (5, 6), (7, 3)}$ (3) ${(2, 1), (9, 3), (6, 5), (3, 7)}$ (4) $((0, \frac{27}{2}), (1, 12), (3, 9), (5, 6), (7, 3))$ 375. $f : R \to R and g : [0, \infty) \to R are defined by f (x) = x^{2} and g (x) = x . Which one of the following is not true?$ (1) $(f og) (2) = 2$ (2) $(g o f) (4) = 4$ (3) $(g o f) (- 2) = 2$ (4) $(f o g) (- 4) = 4$ 376. $If f (x) = x^{3} - x and g (x) = sin 2 x, then f (g (\frac{π}{12})) =$ (1) $0$ (2) $1$ (3) $- \frac{3}{8}$ (4) $2$ 377. $The derivative of tan^{- 1} (\frac{x}{1 + 1 - x ^{2}}) with respect to sec^{- 1} (\frac{1}{2 x ^{2} - 1}) is$ (1) $\frac{1}{2}$ (2) $\frac{1}{4}$ (3) $\frac{- 1}{4}$ (4) $\frac{- 1}{2}$ 378. $If 2^{x} + 2^{y} = 2^{x + y}, then \frac{dy}{dx} =$ (1) $1 - 2^{y}$ (2) $2^{y} - 1$ (3) $1 + 2^{y}$ (4) $1 + 2^{- y}$ 379. $Let A = {1, 3,4,6,9} and B = {2,4, 5, 8,10}. Let # be a relation defined on A x B such that R ={(a_{1}, b_{1}), (a_{2}, b_{2})^{:} a_{1} \leq b_{2} and b_{1} \leq a_{2}} - Then the number of elements in the set R is$ (1) $160$ (2) $52$ (3) $26$ (4) $180$ 380. $If f : R \to R is defined by f (x) = \| x \|, then$ (1) $f^{- 1} (x) = - x$ (2) $f^{- 1} (x) =1\|x\|$ (3) $the function f^{- 1} (x) does not exist$ (4) $f^{- 1} (x) =1x$ 381. $If f : R \to R is defined by f (x) = (\| [x - 5] \|, for x < 5 [\| x - 5 \|], for x \geq 5) Then, (fof) (- \frac{7}{2}) = (here, [x] is the greatest integer not exceeding x)$ (1) $(f o f) (- \frac{11}{2})$ (2) $(f o f) (- \frac{9}{2})$ (3) $(f o f) (3)$ (4) $(f o f) (\frac{9}{2})$ 382. $Let [x] denote the greatest integer less than or equal to x and f (x) = 2 x - [2 x] . If lim_{x \to 2^{-}} f (x) = l_{1} and lim_{x \to 2^{+}} f (x) = l_{2} then l_{1} + l_{2} =$ (1) $1$ (2) $2$ (3) $0$ (4) $4$ 383. $If \frac{y}{x} cos^{4} α + \frac{x}{y} sin^{4} α = 2 sin^{2} α \cdot cos^{2} α, then \frac{dy}{dx} =$ (1) $sin^{3} αcos α$ (2) $sin^{2} α cos^{2} α$ (3) $\frac{sin ^{2} α}{cos ^{2} α}$ (4) $sin α cos^{3} α$ 384. $If y = log (log x), then \frac{d ^{2} y}{d x ^{2}} is equal to$ (1) $\frac{( 1 + log x )}{x ^{2} log x}$ (2) $\frac{- ( 1 + log x )}{( x log x ) ^{2}}$ (3) $\frac{( 1 + log x )}{( x log x ) ^{2}}$ (4) $\frac{- ( 1 + log x )}{x ^{2} log x}$ 385. $Let f : R \to R be defined as f (x) = (2 sin (- \frac{πx}{2}), if x < - 1 (a x^{2} + x + b), if - 1 \leq x \leq 1 sin (πx) if x > 1) If f (x) is continuous on R, then a + b equals :$ (1) $3$ (2) $- 1$ (3) $- 3$ (4) $1$ 386. $If f (1) = 1, f^{’} (1) = 3 then the derivative of f (f (f (x))) + (f (x))^{2} at x = 1 is$ (1) $10$ (2) $35$ (3) $33$ (4) $12$ 387. $Let f :N \to N be a function such that f(m + n) = f(m) + f(n) for every m, n \inN . If f(6) = 18, then f(2) \cdotf(3) is equal to :$ (1) $6$ (2) $54$ (3) $18$ (4) $36$ 388. $Let S={1, 2, 3,4}. Then the number of elements in the set {f: S x S \to S : f is onto and f(a, b) = f(b, a) \geq a \forall_{(} a, b_{)} \in S \times S} is$ 389. $Consider the function f : R \to R defined by f (x) = \frac{2 x}{1 + 9 x ^{2}} . If the composition of f, (f \circ f \circ f \circ \dots \circ f) 10 times (x) = \frac{2 ^{10} x}{1 + 9 α x ^{2}}, then the value of 3 α + 1 is equal to______$ 390. $The number of solutions of the equation, l o g_{(x + 1)} (2 x^{2} + 7x + 5) + l o g_{(2 x + 5)} (x + 1)^{2} - 4 = 0 x > 0 is$