Sets, Relations & Functions PYQs Set 18 (PDF)

Set 18 of 25 · Questions 511-540 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 18 (Q511-540)
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511. $lim x \to 0 (\frac{4 !}{x ^{8}} (1 - cos \frac{x ^{2}}{3} - cos \frac{x ^{2}}{4} + cos \frac{x ^{2}}{3} cos \frac{x ^{2}}{4})) =$ (1) $8$ (2) $\frac{1}{6}$ (3) $\frac{1}{24}$ (4) $\frac{2}{3}$ 512. $If y = (x + 1) (x^{2} + 1) (x^{4} + 1) (x^{8} + 1), then lim_{x \to - 1} \frac{dy}{dx} =$ (1) $0$ (2) $2$ (3) $-4$ (4) $8$ 513. $If m sin^{- 1} x = log_{e} y, then (1 - x^{2}) y^{’ ’} - x y^{’} =$ (1) $- 2 y$ (2) $- m^{2} y$ (3) $m^{2} y$ (4) $2 y$ 514. $At x = \frac{π ^{2}}{4}, \frac{d}{dx} (tan^{- 1} (cos x) + sec^{- 1} (e^{x})) =$ (1) $\frac{1}{e ^{\frac{π ^{2}}{2}} - 1} - \frac{1}{π}$ (2) $\frac{π}{4} + \frac{1}{e ^{π^{2}} + e ^{π^{2} /2}}$ (3) $\frac{1}{e ^{π^{2}} + e ^{π^{2} /2}} + \frac{2}{π} cot (\frac{π}{2})$ (4) $\frac{1}{e ^{π}} + \frac{1}{π}$ 515. $If R denotes the set of all real numbers then the function f : R \to R defined by f (x) = \| x \| is$ (1) $surjective$ (2) $injective and surjective$ (3) $neither injective nor surjective$ (4) $injective$ 516. $If y = tan^{- 1} (\frac{1 + x ^{2} + 1 - x ^{2}}{1 + x ^{2} - 1 - x ^{2}}), where x^{2} \leq 1 . Then, find \frac{dy}{dx} is equal to$ (1) $\frac{π}{4} + \frac{1}{2} cos^{- 1} (x^{2})$ (2) $\frac{π}{4} - \frac{1}{2} cos^{- 1} (x^{2})$ (3) $\frac{- x}{1 - x ^{4}}$ (4) $\frac{- 2 x}{1 - x ^{4}}$ 517. $The derivative of sin (x^{3}) w.r.t. cos (x^{3}) is$ (1) $- cot (x^{3})$ (2) $cot (x^{3})$ (3) $- tan (x^{3})$ (4) $tan (x^{3})$ 518. $The function f : N - 1 \to N; defined by f (n) = the highest prime factor of n, is :$ (1) $both one-one and onto$ (2) $one-one only$ (3) $onto only$ (4) $neither one-one nor onto$ 519. $lim_{x \to \frac{π}{2}} \frac{cot x - cos x}{( π - 2 x ) ^{3}} equals$ (1) $\frac{1}{24}$ (2) $\frac{1}{16}$ (3) $\frac{1}{8}$ (4) $\frac{1}{4}$ 520. $If lim_{n \to \infty} x_{n} exists and is finite, x_{1} = 2, x_{n + 1} = \frac{a + b x _{n}}{b + c x _{n}} \forall n \in N and c > b > a > o then lim_{n \to \infty} x_{n} =$ (1) $\frac{ab}{c}$ (2) $\frac{a}{c}$ (3) $\frac{a}{b}$ (4) $a2b$ 521. $Consider a function f : N \to R, satisfying f (1) + 2 f (2) + 3 f (3) + \dots xf (x) = x (x + 1) f (x); x \geq 2 with f (1) = 1 . Then \frac{1}{f ( 2022 )} + \frac{1}{f ( 2028 )} is equal to$ (1) $8200$ (2) $8000$ (3) $8400$ (4) $8100$ 522. $Let the sum of the maximum and the minimum values of the function f (x) = \frac{2 x ^{2} - 3 x + 8}{2 x ^{2} + 3 x + 8} be \frac{m}{n}, where gcd (m, n) = 1 . Then m + n is equal to:$ (1) $195$ (2) $201$ (3) $217$ (4) $182$ 523. $If f : R \to R is defined by f (x) = [2 x] - 2 [x] for x \in R, then the range of f is (Here [x] denotes the greatest integer not exceeding x)$ (1) $Z , the set of all integers$ (2) $N , the set of all natural numbers$ (3) $R , the set of all real numbers$ (4) ${0, 1}$ 524. $Let f (x) = (x, if x is irrational 0, if x is rational) then f is$ (1) $continuous everywhere$ (2) $discontinuous everywhere$ (3) $continuous only at x = 0$ (4) $continuous at all rational numbers$ 525. $Let A = 2, 3, 4, 5, \dots, 30 and ‘ ≃ ’ be an equivalence relation on A \times A, defined by (a, b) ≃ (c, d), if andonly if ad = bc . Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) isequal to:$ (1) $5$ (2) $6$ (3) $8$ (4) $7$ 526. $Let f : [0, 3] \to A be defined by f (x) = 2 x^{3} - 15 x^{2} + 36 x + 7 and g : [0, \infty) \to B be defined by g (x) = \frac{x ^{2025}}{x ^{2025} + 1} . If both the functions are onto and S = {x \in Z : x \in A or x \in B}, then n (S) is equal to:$ (1) $29$ (2) $30$ (3) $31$ (4) $36$ 527. $lim_{x \to 0} \frac{x e ^{x} - sin x}{x} is equal to$ (1) $3$ (2) $1$ (3) $0$ (4) $2$ 528. $If f (x) = (k, for x = 1 \frac{( 9 x - 1 ) ( x - 1 )}{3 x ^{2} + 2 x - 5}, for x \neq = 1) is continuous on [0, \infty), then k =$ (1) $\frac{1}{16}$ (2) $\frac{1}{8}$ (3) $\frac{1}{4}$ (4) $\frac{1}{2}$ 529. $f : R \to R and g : [0, \infty) \to R is defined by f (x) = x^{2} and g (x) = x . Which one of the following is not true?$ (1) $gof (4) = 4$ (2) $f o g (2) = 2$ (3) $fog (- 4) = 4$ (4) $gof (- 2) = 2$ 530. $Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A \times B, each having at least three elements is:$ (1) $219$ (2) $256$ (3) $275$ (4) $510$ 531. $If y = log (\frac{1 - x ^{2}}{1 + x ^{2}}), then \frac{dy}{dx} is equal to$ (1) $\frac{4 x ^{3}}{1 - x ^{4}}$ (2) $\frac{- 4 x ^{3}}{1 - x ^{4}}$ (3) $\frac{- 4 x}{1 - x ^{4}}$ (4) $\frac{1}{4 - x ^{4}}$ 532. $Let A = -4, -2, -1, 0, 3, 5 and f : A \to R be defined by f (x) = (3 x - 1 for x > 3 x^{2} + 1 for - 3 \leq x \leq 32 x - 3 for x < - 3) Then the range of f is$ (1) ${- 11, 5, 2, 1, 10, 14}$ (2) ${- 11, - 7, 2, 1, 8, 14}$ (3) ${- 11, 5, 2, 1, 8, 14}$ (4) ${- 11, - 7, - 5, 1, 10, 14}$ 533. $Let f : R \to R be a function such that f(x + y) = f(x) + f(y), \forall x, y \in R. If f(x) is differentiable at x = 0, then$ (1) $f(x) is differentiable only in a finite interval containing zero$ (2) $f(x) is continuous \forall x \in R$ (3) $f'(x) is constant \forallx \in R$ (4) $f(x) is differentiable except at finitely many points$ 534. $Let [x] denote the greatest integer less than or equal to x . Then the domain of f (x) = sec^{- 1} (2 [x] + 1) is:$ (1) $(- \infty, - 1] \cup [0, \infty)$ (2) $(- \infty, - 1] \cup [1, \infty)$ (3) $(- \infty, \infty)$ (4) $(- \infty, \infty) -{0}$ 535. $Let A = {(α, β) \in R \times R : ∣ α - 1∣ \leq 4 and ∣ β - 5∣ \leq 6} and B = {(α, β) \in R \times R : 16 (α - 2)^{2} + 9 (β - 6)^{2} \leq 144} . Then$ (1) $A \subset B$ (2) $B \subset A$ (3) $neither A \subset B nor B \subset A$ (4) $A \cup B = {(x, y) : - 4 ⩽ x ⩽ 4, - 1 ⩽ y$ 536. $The number of relations, on the set {1, 2, 3} containing (1,2) and (2, 3) which are reflexive and transitive but not symmetric, is_________.$ 537. $Let f : R \to R and g : R \to R be defined as f (x) = (x + a, x < 0 (x - 1), x \geq 0) and g (x) = (x + 1, x < 0 (x - 1)^{2} + b, x \geq 0) where a, b are non-negative real numbers. If(gof) (x) is continuous for all x \in R thena+bis equal to ____$ 538. $Let A = 1, 2, 3, \dots 20 . Let R_{1} and R_{2} two relation on A such that R_{1} = {(a, b) : b is divisible by a} R_{2} = {(a, b) : a is an integral multiple of b} .Then, number of elements in R_{1} - R_{2} is equal to$ 539. $Let R = {a, b, c, d, e} and S = {1, 2, 3,4}. Total number of onto functions f: R S such that f(a) 1, is equal to________.$ 540. $Let f(x) be a polynomial of degree 3 such that f (k) = - \frac{2}{k} for k=2,3,4,5. Then the value of 52-10f(10) is equal to :$