Sets, Relations & Functions PYQs Set 15 (PDF)

Set 15 of 25 · Questions 421-450 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 15 (Q421-450)
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421. $If lim_{n \to \infty} x^{n} log_{e} x = 0, then log_{x} 12 =$ (1) $Negative$ (2) $Positive$ (3) $Zero$ (4) $any value between -1 and 1$ 422. $Given that a, b and c are real numbers such that b^{2} = 4 ac and a > 0 . The maximal possible set D \subseteq R on which the function f : D \to R given by f (x) = log (a x^{3} + (a + b) x^{2} + (b + c) x + c) is defined, is$ (1) $R - {- \frac{b}{2 a}}$ (2) $R - ({- \frac{b}{2 a}} \cup (- \infty, - 2))$ (3) $R - ({- \frac{b}{2 a}} \cup {x : x \geq 1})$ (4) $R - ({- \frac{b}{2 a}} \cup (- \infty, - 1))$ 423. $If f (x) = x^{2} + α for x \geq 0 = 2 x^{2} + 1 + β for x < 0 is continous at x = 0 and f (\frac{1}{2}) = 2, then α^{2} + β^{2} is$ (1) $3$ (2) $\frac{8}{25}$ (3) $\frac{25}{8}$ (4) $\frac{1}{3}$ 424. $If y = 1 + \frac{1}{x} + \frac{1}{x ^{2}} + \frac{1}{x ^{3}} + \dots to \infty with \| x \| > 1 then \frac{dy}{dx} =$ (1) $\frac{x ^{2}}{y ^{2}}$ (2) $x^{2} y^{2}$ (3) $\frac{y ^{2}}{x ^{2}}$ (4) $\frac{- y ^{2}}{x ^{2}}$ 425. $If y = tan^{- 1} (\frac{ax - b}{bx + a}), then y^{’} is equal to$ (1) $\frac{1}{1 + x ^{2}} + \frac{a ^{2}}{a ^{2} + b ^{2}}$ (2) $\frac{1}{1 + x ^{2}}$ (3) $\frac{1}{1 + ( \frac{ax - b}{bx + a} ) ^{2}}$ (4) $\frac{bx + a}{1 + ( ax - b ) ^{2}}$ 426. $If g (x) = 3 x^{2} + 2 x - 3, f (0) = - 3 and 4 g (f (x)) = 3 x^{2} - 32 x + 72, then f (g (2)) is equal to:$ (1) $- \frac{7}{2}$ (2) $- \frac{25}{6}$ (3) $\frac{7}{2}$ (4) $\frac{25}{6}$ 427. $Let A = 0, 1, 2, \dots, 9 . Let R be a relation on A defined by (x, y) \in R if and only if \| x - y \| is a multiple of 3 . Given below are two statements: Statement I: n (R) = 36 . Statement II: R is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below$ (1) $Statement I is incorrect but Statement II is correct$ (2) $Statement I is correct but Statement II is incorrect$ (3) $Both Statement I and Statement II are incorrect$ (4) $Both Statement I and Statement II are correct$ 428. $Let f (x) be a function such that f (x + y) = f (x) . f (y) for all z, y ϵ N . If f (1) = 3 and \sum f (k) = 3279 then the value of n is$ (1) $6$ (2) $8$ (3) $7$ (4) $9$ 429. $The number of elements in the relation R = ((x, y) : 4 x^{2} + y^{2} < 52, x, y \in Z) is$ (1) $77$ (2) $67$ (3) $86$ (4) $89$ 430. $Let f (x) = x^{3} and g (x) = 3^{x}, then the quadratic equation whose roots are solutions of the equation (f o g) (x) = (gof) (x) (for x \neq = 0) is$ (1) $x^{2} - 6 x + 3 = 0$ (2) $x^{2} - 6 x + 9 = 0$ (3) $x^{2} - x + 3 = 0$ (4) $x^{2} - 3 = 0$ 431. $If f : Z \to Z is defined by f (x) = x^{9} - 11 x^{8} - 2 x^{7} + 22 x^{6} + x^{4} - 12 x^{3} + 11 x^{2} + x - 3, \forall x \in Z, then f (11) is equal to$ (1) $7$ (2) $8$ (3) $6$ (4) $9$ 432. $Let the range of the function f (x) = \frac{1}{2 + sin 3 x + cos 3 x}, x \in R be [a, b] . If α and β are respectively the A.M. and the G.M. of a and b, then \frac{α}{β} is equal to$ (1) $π$ (2) $π$ (3) $2$ (4) $2$ 433. $If x^{2019} \cdot y^{2020} = (x + y)^{4039}, then \frac{dy}{dx} =$ (1) $0$ (2) $\frac{x}{y}$ (3) $\frac{y}{x}$ (4) $1$ 434. $lim_{z \to 1} \frac{z ^{(1/3)} - 1}{z ^{(1/6)} - 1} is equal to$ (1) $-1$ (2) $1$ (3) $2$ (4) $-2$ 435. $Let A = 1, 2, 3, 4 and B = 1, 4, 9, 16 . Then the number of many-one functions f : A \to B such that 1 \in f (A) is equal to:$ (1) $151$ (2) $139$ (3) $163$ (4) $127$ 436. $The inverse function of f (x) = \frac{8 ^{2 x} - 8 ^{- 2 x}}{8 ^{2 x} + 8 ^{- 2 x}}, x \in (- 1, 1) is$ (1) $\frac{1}{4} (log_{8} e) log_{e} (\frac{1 + x}{1 - x})$ (2) $\frac{1}{2} log_{8} (\frac{1 - x}{1 + x})$ (3) $\frac{1}{4} (log_{8} e) log_{e} (\frac{1 - x}{1 + x})$ (4) $\frac{1}{2} log_{8} (\frac{1 + x}{1 - x})$ 437. $Let A = -2, -1, 0, 1, 2, 3, 4 . Let R be a relation on A defined by xRy if and only if 2 x + y ⩽ 2 . Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l + m + n is equal to:$ (1) $35$ (2) $34$ (3) $33$ (4) $32$ 438. $If y = acos (sin 2 x) + bsin (sin 2 x), then y^{’ ’} + (2 tan 2 x) y^{’} is equal to$ (1) $0$ (2) $4 (cos^{2} 2 x)$ (3) $- 4 (cos^{2} 2 x) y$ (4) $- (cos^{2} 2 x) y$ 439. $The function f (x) = \| x - 2 \| + x is$ (1) $differentiable at x = 2 but not at x = 0$ (2) $differentiable at both x = 2 and x = 0$ (3) $continuous at both x = 2 and x = 0$ (4) $continuous at x = 2 but not at x = 0$ 440. $If sin (x + y) + cos (x + y) = log (x + y), then \frac{d ^{2} y}{d x ^{2}} =$ (1) $- y / x$ (2) $0$ (3) $-1$ (4) $1$ 441. $If n (A) = 2 and total number of possible relations from set A to set B is 1024 , then n (B) is$ (1) $20$ (2) $10$ (3) $5$ (4) $512$ 442. $f (x) is a twice differentiable function such that f^{’ ’} (x) = - f (x) and f^{’} (x) = g (x) . If h (x) = (f (x))^{2} + (g (x))^{2} and h (1) = 2, then h (2) =$ (1) $0$ (2) $1$ (3) $2$ (4) $4$ 443. $The right hand and left hand limit of the function f (x) = (\frac{e ^{1/ x} - 1}{e ^{1/ x} + 1}, if x \neq = 00, if x = 0) are respectively$ (1) $1 and -1$ (2) $-1 and -1$ (3) $-1 and 1$ (4) $1 and 1$ 444. $Let f, g : (1, \infty) \to R be defined as f (x) = \frac{2 x + 3}{5 x + 2} and g (x) = \frac{2 - 3 x}{1 - x} . If the range of the function fog: [2, 4] \to R is [α, β], then \frac{1}{β - α} is equal to$ (1) $56$ (2) $2$ (3) $29$ (4) $68$ 445. $If f (x) = e^{x}, h (x) = (fof) (x), then \frac{h ^{’} ( x )}{h ( x )} =$ (1) $h (x)$ (2) $1 h (x)$ (3) $log h (x)$ (4) $- log h (x)$ 446. $If f (x) = (log (sec^{2} x)^{cot^{2} x} for x \neq = 0 K for x = 0) is continuous at x = 0 then K is$ (1) $e^{- 1}$ (2) $1$ (3) $e$ (4) $0$ 447. $The number of elements in the set S = {(x, y, z) : x, y, z \in Z, x + 2 y + 3 z = 42, x, y, z \geq 0} equals$ 448. $Let A = {1, 2, 3, \dots \dots . 100} . Let R be a relation on A defined by (x, y) \in R if and only if 2 x = 3 y . Let R_{1} be a symmetric relation on A such that R \subset R_{1} and the number of elements in R_{1} is n . Then, the minimum value of n is______$ 449. $The minimum number of elements that must be added to the relation R = ((a, b), (b, c), (b, d)) on the set a, b, c, d so that it is an equivalence relation, is _______$ 450. $The number of points, at which the function f (x) = \| 2 x + 1 \| - 3 \| x + 2 \| + (x^{2} + x - 2), x \in R is not differentiable, is$