Sets, Relations & Functions PYQs Set 19 (PDF)

Set 19 of 25 · Questions 541-570 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 19 (Q541-570)
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541. $If function f(x) is differentiable at x = a, then lim_{x \to a} \frac{x ^{2} f ( a ) - a ^{2} f ( x )}{x - a} is :$ (1) $-a2f ’(a)$ (2) $af(a) - a2f ’ (a)$ (3) $2af(a) - a2f ’ (a)$ (4) $2af(a) + a2f ’ (a)$ 542. $The function f (x) = ∣ x^{2} - 2 x - 3∣. e^{∣9 x^{2} - 12 x + 4∣} is not differentiable at exactly:$ (1) $four points$ (2) $three points$ (3) $two points$ (4) $one point$ 543. $The function represented by the following graph is$ (1) $Continuous but not differentiable at x = 1$ (2) $Differentiable but not continuous at x = 1$ (3) $Continuous and differentiable at x = 1$ (4) $Neither continuous nor differentiable at x = 1$ 544. $Two sets A and B are as under : A = {(a, b) \in R \times R : \|a -5\| < 1 and \|b - 5\| < 1 } ; B = {(a, b ) \in R \times R : 4 (a -6)^{2} + 9 (b -5)^{2} \leq 36 }. Then$ (1) $A \cap B = φ ( an empty set)$ (2) $Neither A \subset B nor B \subset A$ (3) $B \subset A$ (4) $A \subset B$ 545. $If the function f : R - (- 1, - 1) \to A defined by f (x) = \frac{x ^{2}}{1 - x ^{2}}, is surjective, then A is equal to :$ (1) $R - (- 1)$ (2) $(0, \infty)$ (3) $R - (- 1, 0)$ (4) $R--1, 1$ 546. $Let f: (- \frac{π}{2}, \frac{π}{2}) \to R be given by f(x) = (log(sec x + tanx))3 . Then$ (1) $f(x) is an odd function$ (2) $f(x) is a one-one function$ (3) $f(x) is an onto function$ (4) $f(x) is an even function$ 547. $In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is α and the number of persons who speaks only Hindi is β, then the eccentricity of the ellipse 25(β 2x2 + α 2y2) = α 2 β 2 is$ (1) $\frac{119}{12}$ (2) $\frac{117}{12}$ (3) $\frac{3 15}{12}$ (4) $\frac{129}{12}$ 548. $The number of real solution(s) of the equation x^{2} + 3 x + 2 = min \| x-3 \|, \| x+2 \| is:$ (1) $1$ (2) $0$ (3) $2$ (4) $3$ 549. $The number of functions f from {1, 2, 3, \dots., 20}, onto {1,2,3,\dots\dots\dots..,20} such that f(k) is a multiple of 3, wherever k is a multiple of 4, is$ (1) $5! \times 6!$ (2) $(15)! \times 6!$ (3) $6^{5} \times (15)!$ (4) $5^{6} \times 15$ 550. $Let S = N \cup 0 . Define a relation R from S to R by : R = ((x, y) : log_{e} y = x log_{e} (\frac{2}{5}), x \in S, y \in R) Then, the sum of all the elements in the range of R is equal to:$ (1) $\frac{10}{9}$ (2) $\frac{3}{2}$ (3) $\frac{5}{2}$ (4) $\frac{5}{3}$ 551. $Let g : N \to N be defined as, g(3n + 1) = 3n + 2 g(3n + 2) = 3n + 3 g(3n + 3) = 3n + 1 for all n \geq 0 Then which of the following statements is true ?$ (1) $There exists an onto function f : N \to N such that fog=f$ (2) $There exists a one-one function f : N \to N such that fog=f$ (3) $gogog = g$ (4) $There exists a function f : N \to N such that gof =f$ 552. $The function f : R - > R, f (x) = \frac{x ^{2} + 2 x - 15}{x ^{2} - 4 x + 9}, x \in R is$ (1) $one-one but not onto.$ (2) $both one-one and onto.$ (3) $onto but not one-one.$ (4) $neither one-one nor onto.$ 553. $Let f (x) = \frac{1 - x ( 1 + ∣1 - x ∣ )}{∣1 - x ∣} cos (\frac{1}{1 - x}) for x \neq 1. Then$ (1) $lim_{x \to 1}_{+} f (x) = 0$ (2) $lim_{x \to 1}_{-} f (x) does not exist$ (3) $lim_{x \to 1}_{-} f (x) = 0$ (4) $lim_{x \to 1}_{+} f (x) does not exist$ 554. $If x^{m} y^{n} = (x + y)^{m + n} then \frac{dy}{dx} is equal to$ (1) $x+yxy$ (2) $xy$ (3) $0$ (4) $\frac{y}{x}$ 555. $Let f(x) = sin (\frac{π}{6} sin (\frac{π}{2} sin x)) for all x \in R and g(x) = \frac{π}{2} sin x for all x \in R. Let (fog) (x) dentoe f(g(x)) and (gof) (x) denote g(f(x)). Then which of the following is(are)true?$ (1) $Range of f is$ (2) $Range of fog is$ (3) $lim_{x \to 0} \frac{f ( x )}{g ( x )} = \frac{π}{6}$ (4) $There is an x \in R such that (gof)(x) = 1$ 556. $For x \in R - {0, 1}, let f_{1} (x) = \frac{1}{x}, f_{2} (x) = 1 - x and f_{3} (x) = \frac{1}{1 - x} be three given functions. If a function, J(x) satisfies (f_{2} \circ J \circ f_{1}) (x) = f_{3} (x) is equal to :$ (1) $\frac{1}{x} f_{3} (x)$ (2) $f_{1} (x)$ (3) $f_{3} (x)$ (4) $f_{2} (x)$ 557. $Let a function f : R \to R be defined as f (x) = ⎩ ⎨ ⎧ sin x - e^{x} a + [- x] 2 x - b if x \leq 0 if 0 < x < 1 if x \geq 1 Where [x] is the greatest integer less than or equal to x. If f is continuous on R, then (a+b) is equal to:$ (1) $4$ (2) $3$ (3) $2$ (4) $5$ 558. $If the domain of the function f (x) = log_{e} (\frac{2 x - 3}{5 + 4 x}) + sin^{- 1} (\frac{4 + 3 x}{2 - x}) is [α, β), then α^{2} + 4 β is equal to$ (1) $4$ (2) $3$ (3) $7$ (4) $5$ 559. $Let Z be the set of all integers, A = {(x, y) \in Z \times Z : (x - 2)2 + y2 \leq 4} B = {(x, y) \in Z \times Z : x^{2} + y2 \leq 4} and C = {(x, y) \in Z \times Z : (x - 2) 2 + (y - 2) 2 \leq 4} If the total number of relation from A \cap B to A \cap C is 2^{p}, then the value of p is:$ (1) $16$ (2) $25$ (3) $49$ (4) $9$ 560. $Let the domains of the functions f (x) = log_{4} log_{3} log_{7} (8 - log_{2} (x^{2} + 4 x + 5)) and g (x) = sin^{- 1} (\frac{7 x + 10}{x - 2}) be (α, β) and [γ, δ], respectively. Then α^{2} + β^{2} + γ^{2} + δ^{2} is equal to:$ (1) $15$ (2) $13$ (3) $16$ (4) $14$ 561. $If the domain of the function log_{5} (18 x - x^{2} - 77) is (α, β) and the domain of the function log_{(x - 1)} (\frac{2 x ^{2} + 3 x - 2}{x ^{2} - 3 x - 4}) is (γ, δ), then α^{2} + β^{2} + γ^{2} is equal to :$ (1) $195$ (2) $179$ (3) $186$ (4) $174$ 562. $If α \in R - -1 and f (x) = \| (\| x \| + α) (\| x \| - 1) \|, then the number of points at which f (x) is not differentiable, is$ (1) $3, when α < 0$ (2) $5 ; when α > 0$ (3) $4 , when α > 0$ (4) $5 , when α < 0$ 563. $Let [x] denote the greatest integer \leq x, where x \in R. If the domain of the real valued function is (-\infty, a) \cup [b, c) \cup [4,\infty), a < b < c, then the value of a + b + c is:$ (1) $8$ (2) $1$ (3) $-2$ (4) $-3$ 564. $Let [t] denote the greatest integer less than or equal to t. Let f(x) = x - [x], g(x) = 1 - x + [x], and h(x) = min{f(x), g(x)}, x \in [-2, 2]. Then h is :$ (1) $continuous in [-2,2] but not differentiable at more than four points in (-2,2)$ (2) $not continuous at exactly three points in [-2,2]$ (3) $continuous in [-2,2] but not differentiable at exactly three points in (-2,2)$ (4) $not continuous at exactly four points in [-2,2]$ 565. $Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S \to S such that f(m \cdot n) = f(m) \cdot f(n) for every m, n \in S and m \cdot n \in S is equal to$ 566. $Let α, β \in R be such that lim_{x \to 0} \frac{x ^{2} s in ( β x )}{α x - s in x} = 1 . Then 6(α + β) equals$ 567. $The number of distinct real roots of the equation \| x \| \| x + 2 \| - 5 \| x + 1 \| - 1 = 0 is ____$ 568. $Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A \to A such that f(1) + f(2) = 3 - f(3) is equal to$ 569. $Let c, k \in R . If f (x) = (c + 1) x^{2} + (1 - c^{2}) x + 2 k and f (x + y) = f (x) + f (y) - x y, for all x, y \in R, then the value of ∣2 (f (1) + f (2) + f (3) + \dots + f (20)) ∣ is equal to_______.$ 570. $Let a function g : [0, 4] \to R be defined as g (x) = {max_{0 \leq t \leq x} {t^{3} - 6 t^{2} + 9 t - 3}, 4 - x, 0 \leq x \leq 3 3 < x \leq 4 then the number of points in the interval (0,4)where g(x) is NOT differentiable, is$