Sets, Relations & Functions PYQs Set 20 (PDF)

Set 20 of 25 · Questions 571-600 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 20 (Q571-600)
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571. $Let α, β and γ be three positive real numbers. Let f_{(} x_{)} = α x^{5} + β x^{3} + γ x, x \in R and g: R \to R be such that g(f(x)) = x for all x g R. If a_{1}, a_{2}, a_{3},..., a_{n} be in arithmetic progression with mean zero, then the value of f (g (\frac{1}{n} \sum_{i = 1}^{n} f (a_{i}))) is equal to :$ (1) $0$ (2) $3$ (3) $9$ (4) $27$ 572. $The function f(x) = \|ax - b\| + c \|x\| \forall x \in (-\infty, \infty), where a > 0, b > 0, c > 0, assumes its minimum value only at one point if$ (1) $a \neq b$ (2) $a \neq c$ (3) $b \neq c$ (4) $a = b = c$ 573. $lim_{x \to 0} \frac{( 1- cos 2 x ) ( 3 + cos x )}{x t an 4 x} is equal to$ (1) $4$ (2) $3$ (3) $2$ (4) $\frac{1}{2}$ 574. $Let f: R \to R be a continuous function such that f (3x) - f (x) = x. If f (8) = 7, then f (14) is equal to :$ (1) $4$ (2) $10$ (3) $11$ (4) $16$ 575. $Let P (S) denote the power set of S = (1, 2, 3, \dots, 10) Define the relation R_{1} and R_{2} on P(S) as AR_{1} B if (A \cap B^{’}) \cup (B \cap A^{’}) = \emptyset and A R_{2} B of A \cup B^{’} = B \cup A^{’}, \forall A, B ϵ P (S) Then$ (1) $Both R_{1} and R_{2} are equvalence relations$ (2) $Only R_{1} is an equivalence relation$ (3) $Only R_{2} is an equivalence relation$ (4) $Both R_{1} and R are not equivalence relations.$ 576. $Let R be the set of all real numbers. Statement I : The function f : (- \frac{π}{2}, \frac{π}{2}) \to R defined by f (x) = sec x + tan x is a one-one function: Statement II : The function f : [0, \infty) \to R defined by f (x) = x^{2} is a one-one function. Which of the above statements is(are) true?$ (1) $Statement I is true, but statement II is false$ (2) $Statement II is true, but statement I is false$ (3) $Both Statement I and statement II are true$ (4) $Both Statement I and statement II are false$ 577. $Let f : R \to R and g : R \to R be defined as f (x) = (log_{e} x, x > 0 e^{- x}, x \leq 0) and g (x) = {x, e^{x}, x \geq 0 x < 0 . Then, gof: R \to R is:$ (1) $one-one but not onto$ (2) $neither one-one nor onto$ (3) $onto but not one-one$ (4) $both one-one and onto$ 578. $If the domain of the function f (x) = \frac{c o s ^{- 1} x ^{2} - x + 1}{s i n ^{- 1} ( \frac{2 x - 1}{2} )} is the interval (α, β], then α + β is equal to :$ (1) $\frac{3}{2}$ (2) $2$ (3) $\frac{1}{2}$ (4) $1$ 579. $f(x) = s i n^{- 1} [2 x^{2} - 3] + l o g_{2} (lo g_{\frac{1}{2}} (x^{2} - 5 x + 5)), where [t] is the greatest integer function, is :$ (1) $(- \frac{5}{2}, \frac{5 - 5}{2})$ (2) $(\frac{5 - 5}{2}, \frac{5 + 5}{2})$ (3) $(1, \frac{5 - 5}{2})$ (4) $[1, \frac{5 + 5}{2})$ 580. $If a set A has m - elements and the set B has n - elements, then the number of injections from A to B is$ (1) $^{n} C_{m} if n\geqm$ (2) $^{n} P_{m} if n\geqm$ (3) $0 if n\geqm$ (4) $m.^{n} C_{m} if n\geqm$ 581. $For x \in R, f(x) = \|log2 - sinx\| and g(x) = f(f(x)), then$ (1) $g'(0) = cos(log2)$ (2) $g'(0) = -cos(log2)$ (3) $g is differentiable at x = 0 and g'(0) = -sin(log2)$ (4) $g is not differentiable at x = 0$ 582. $Let A = {(x, y) \in R \times R : \| x + y \| ⩾ 3} and B = {(x, y) \in R \times R : \| x \| + \| y \| \leq 3} . If C = {(x, y) \in A \cap B : x = 0 or y = 0}, then \sum \| x + y \| is:$ (1) $15$ (2) $24$ (3) $18$ (4) $12$ 583. $The function f : [0, 3] \to [1, 29], defined by f(x) = 2x3 - 15x2 + 36x + 1, is$ (1) $one-one and onto$ (2) $onto but not one-one$ (3) $one-one but not onto$ (4) $neither one-one nor onto$ 584. $Let f (x) = \| x \| and g (x) = \| x \| + a, (a > 0) . For 0 \leq x \leq b, {(x, y) / g (x) \leq y \leq f (x)} represents all the points in the interior of$ (1) $a parallelogram$ (2) $a triangle$ (3) $a square$ (4) $a circle$ 585. $If the domain of the function f (x) = sin^{- 1} (\frac{x - 1}{2 x + 3}) is R - (α, β), then 12 αβ is equal to :$ (1) $32$ (2) $40$ (3) $24$ (4) $36$ 586. $f (x) = sin x + cos x \cdot g (x) = x^{2} - 1, then g (f (x)) is invertible if$ (1) $\frac{- π}{4} \leq x \leq \frac{π}{4}$ (2) $\frac{- π}{2} \leq x \leq 0$ (3) $\frac{- π}{2} \leq x \leq π$ (4) $0 \leq x \leq \frac{π}{2}$ 587. $If x, y, z are in A.P. and tan^{- 1} x, tan^{- 1} y and tan^{- 1} z are also in A.P., then$ (1) $x = y = z$ (2) $2x = 3y = 6z$ (3) $6x = 3y = 2z$ (4) $6x = 4y = 3z$ 588. $The domain of the function f (x) = \frac{1}{[ x ] ^{2} - 3 [ x ] - 10} is (where [x] denotes the greatest integer less than or equal to x)$ (1) $(- \infty, - 3] \cup_{(} 5, \infty_{)}$ (2) $(- \infty, - 2) \cup [6, \infty)$ (3) $(- \infty, - 2) \cup [5, \infty)$ (4) $(- \infty, - 3] \cup [6, \infty)$ 589. $Let Z be the set of integers. If A =and B = x \in Z : -3 < 2 x -1 < 9 then the number of subsets of the set A \times B, is -$ (1) $2^{18}$ (2) $2^{12}$ (3) $2^{15}$ (4) $2^{10}$ 590. $The domain of the definition of the function f (x) = \frac{1}{4 - x ^{2}} + log_{10} (x^{3} - x) is :$ (1) $(- 1, 0) \cup (1, 2) \cup (3, \infty)$ (2) $(- 2, - 1) \cup (- 1, 0) \cup (2, \infty)$ (3) $(- 1, 0) \cup (1, 2) \cup (2, \infty)$ (4) $(1, 2) \cup (2, \infty)$ 591. $The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2,3,4,5} satisfies f(a) + 2f(b) - f(c) = f(d) is :$ (1) $\frac{1}{24}$ (2) $\frac{1}{40}$ (3) $\frac{1}{30}$ (4) $\frac{1}{20}$ 592. $Let A = 1, 3, 7, 9, 11 and B = 2, 4, 5, 7, 8, 10, 12 . Then the total number of one-one maps f : A \to B, such that f (1) + f (3) = 14, is :$ (1) $480$ (2) $240$ (3) $120$ (4) $180$ 593. $Let A = {x \in (0, π) - {\frac{π}{2}} : lo g_{(2/ π)} ∣ sin x ∣ + lo g_{(2/ π)} ∣ cos x ∣ = 2} and B = {x \geq 0 : x (x - 4) - 3∣ x - 2∣ + 6 = 0} . Then n (A \cup B) is equal to:$ (1) $4$ (2) $8$ (3) $6$ (4) $2$ 594. $Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of one-one functions from X to Y and β is the number of onto function form Y to X, then the value of \frac{1}{5 !} (β - α) is ______ .$ 595. $Let the domain of the function f(x) = l o g_{4} (l o g_{5} (l o g_{3} (18x - x^{2} - 77))) be (a,b) Then the value of the integral \int_{a}^{b} \frac{s i n ^{3} x}{( s i n ^{3} x + s i n ^{3} ( a + b - x ) )} d x is equal to$ 596. $Consider the function f (x) = \frac{P ( x )}{s i n ( x - 2 )}, x \neq = 2 =7, x=2 Where P(x) is a polynomial such that P”(x) is always a constant and P(3)=9. If f(x) is continuous at x=2, then P(5) is equal to ___$ 597. $Let f (x) = 2^{x} - x^{2}, x \in R . If m and n are respectively the number of points at which the curves y = f (x) and y = f^{’} (x) intersects the x -axis, then the value of m + n is$ 598. $Let f : [0, 4π] \to [0, π] be defined by f(x) = cos-1 (cos x). The number of points x \in [0, 4π] satisfying the equation f(x) = \frac{10 - x}{10} is$ 599. $The number of distinct solutions of the equation, log_{1/2} \| sin x \| = 2 - log_{1/2} \| cos x \| in the interval [0, 2 π], is$ 600. $Let the domain of the function f (x) = cos^{- 1} (\frac{4 x + 5}{3 x - 7}) be [α, β] and the domain of g (x) = log_{2} (2 - 6 log_{27} (2 x + 5)) be (γ, δ) . Then \| 7 (α + β) + 4 (γ + δ) \| is equal to_______.$