Sets, Relations & Functions PYQs Set 2 (PDF)

Set 2 of 25 · Questions 31-60 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 2 (Q31-60)
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31. $Let R = {P, Q) ∣ P and Q are at the same distance from the origin} be a relation, then the equivalence classof (1, - 1) is the set:$ (1) $S = ((x, y) ∣ x^{2} + y^{2} = 1)$ (2) $S = ((x, y) ∣ x^{2} + y^{2} = 4)$ (3) $S = ((x, y) ∣ x^{2} + y^{2} = 2)$ (4) $S = ((x, y) ∣ x^{2} + y^{2} = 2)$ 32. $If the function f (x) = (\frac{( e ^{kx} - 1 ) tan kx}{4 x ^{2}}, x \neq = 016, x = 0), is continuous at x = 0, then k =$ (1) $\pm \frac{1}{8}$ (2) $\pm 4$ (3) $\pm 2$ (4) $\pm 8$ 33. $lim n \to \infty \frac{2 ^{2} + 4 ^{2} + 6 ^{2} + \dots + ( 2 n ) ^{2}}{n ^{3}} =$ (1) $\frac{2}{3}$ (2) $\frac{4}{3}$ (3) $\frac{3}{2}$ (4) $\frac{8}{7}$ 34. $lim_{x \to \infty} (\frac{x ^{2} + x + 3}{x ^{2} - x + 2})^{x} is equal to$ (1) $\infty$ (2) $e$ (3) $e^{4}$ (4) $e^{2}$ 35. $If the parametric equation of curve is given by x = cos θ + log tan \frac{θ}{2} and y = sin θ, then the points for which \frac{dy}{dx} = 0 are given by$ (1) $θ = \frac{nπ}{2}, n \in Z$ (2) $θ = (2 n + 1) \frac{π}{2}, n \in Z$ (3) $θ = (2 n + 1) π, n \in Z$ (4) $θ = nπ, n \in Z$ 36. $If 8 f (x) + 6 f (\frac{1}{x}) = x + 5 and y = x^{2} f (x), then \frac{dy}{dx} at x = - 1 equals$ (1) $0$ (2) $\frac{1}{14}$ (3) $\frac{- 1}{14} .$ (4) $1$ 37. $Let the number of elements in sets A and B be five and two respectively. Then the number of subsets of A x B each having at least 3 and at most 6 elements is$ (1) $752$ (2) $782$ (3) $792$ (4) $77$ 38. $Let ⋃_{i = 1}^{50} X_{i} = ⋃_{i = 1}^{n} Y_{i} = T, where each X_{i} contains 10 elements and each Y_{i} contains 5 elements. If each element of the set T is an element of exactly 20 of sets X_{i}'s and exactly 6 of sets Y_{i}'s then n is equal to:$ (1) $50$ (2) $15$ (3) $30$ (4) $45$ 39. $If f (x) = {x, 7, if x \leq 1 if x > 1 then at x = 1$ (1) $f is continuous$ (2) $lim_{x \to 1^{-}} f (x) = 7$ (3) $lim_{x \to 1^{+}} f (x) = 1$ (4) $f is discontinuous$ 40. $Domain of f (x) = \frac{x}{1 - \| x \|} is$ (1) $R - [- 1, 1]$ (2) $(- \infty, 1)$ (3) $(- \infty, 1) \cup (0, 1)$ (4) $R - {- 1, 1}$ 41. $The derivative of cos^{- 1} (2 x^{2} - 1) w.r.t. cos^{- 1} x is$ (1) $2$ (2) $\frac{2}{x}$ (3) $1 - x^{2}$ (4) $-12 1 - x^{2}$ 42. $f (x) = 2 a - x in - a < x < a = 3 x - 2 a in a \leq x . Then which of the following is true?$ (1) $f(x) is differentiable at all x\geqa$ (2) $f(x) is continuous at all x < a$ (3) $f (x) is discontinuous at x = a$ (4) $f (x) is not differentiable at x = a$ 43. $The domain and range of y (x) = cos x - 3 are respectively$ (1) $R and [- 1, 1]$ (2) $R and [- 4, - 2]$ (3) $R/{0} and [0, 1]$ (4) $R / ((2 n + 1) \frac{π}{2}) and [-1,1]$ 44. $Let f (x) = \frac{x + 1}{x + 3} and g (x) = \frac{2 - x}{x + 3} be two real valued functions. Then the domain of f / g is$ (1) $(- \infty, - 3) \cup [- 1, \infty)$ (2) $[- 1, 2)$ (3) $(- 3, 2)$ (4) $(- \infty, - 3) \cup [2, \infty)$ 45. $How many bijections f : Z \to Z are there such that f (x + y) = f (x) + f (y) for all x, y \in Z ?$ (1) $One$ (2) $Two$ (3) $Three$ (4) $Infinitely many$ 46. $If a set A has n elements, then the number of functions defined from A to A that are not one-one is$ (1) $(n)^{n^{2}}$ (2) $n! - (^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + \dots +^{n} C_{n})$ (3) $n^{n} - n!$ (4) $n^{n}$ 47. $Consider the following statements: Statement 1: lim_{x \to 1} \frac{a x ^{2} + bx + c}{c x ^{2} + bx + a} is 1 (where a + b + c \neq = 0) Statement 2: lim_{x \to - 2} \frac{\frac{1}{x} + \frac{1}{2}}{x + 2} is \frac{1}{4}$ (1) $Only statement 2 is true$ (2) $Only statement 1 is true$ (3) $Both statements 1 and 2 are true$ (4) $Both statements 1 and 2 are false$ 48. $If f (x) and g (x) are two functions with g (x) = x - \frac{1}{x} and f \circ g (x) = x^{3} - \frac{1}{x ^{3}}, then f^{’} (x) =$ (1) $3 x^{2} + 3$ (2) $x^{2} - \frac{1}{x ^{2}}$ (3) $1 + \frac{1}{x ^{2}}$ (4) $3 x^{2} + 3 x4$ 49. $If xcos (k + y) = cos y then \frac{dy}{dx} at y = \frac{π}{2} is$ (1) $sin k$ (2) $cos k$ (3) $1$ (4) $0$ 50. $f (x) = {3 x - 8, 2 k, if x \leq 5 if x > 5 is continuous, find k .$ (1) $\frac{3}{7}$ (2) $\frac{7}{2}$ (3) $\frac{2}{7}$ (4) $\frac{4}{7}$ 51. $Let R be a relation from the set {1, 2, 3.........,60} to itself such that R = {(a, b) : b = pq, where p, q \geq 3 are prime numbers}. Then, the number of elements in R is:$ (1) $600$ (2) $660$ (3) $540$ (4) $720$ 52. $Let f (x) = log_{e} x and g (x) = \frac{x ^{4} - 2 x ^{3} + 3 x ^{2} - 2 x + 2}{2 x ^{2} - 2 x + 1} Then the domain of f \circ g is$ (1) $[0, \infty)$ (2) $[1, \infty)$ (3) $(0, \infty)$ (4) $R$ 53. $Let f : (1, 3) \to R be a function defined by f (x) = \frac{x [ x ]}{1 + x ^{2}} where [x] denotes the greatest integer \leq x Then the range of f is:$ (1) $(0, \frac{1}{2}) \cup (\frac{3}{5}, \frac{7}{5})$ (2) $(\frac{2}{5}, \frac{1}{2}) \cup (\frac{3}{5}, \frac{4}{5})$ (3) $(\frac{2}{5}, 1) \cup (1, \frac{4}{5})$ (4) $(0, \frac{1}{3}) \cup (\frac{2}{5}, \frac{4}{5})$ 54. $On the set of positive rationals, a binary operation * is defined by a * b = \frac{2 ab}{5} . If 2 * x = 3^{- 1}, then x =$ (1) $\frac{1}{6}$ (2) $\frac{2}{5}$ (3) $\frac{5}{12}$ (4) $\frac{125}{48}$ 55. $If f (x) = e^{x}; g (x) = ln (x) for all x \in [1, \infty), then fog is\dots\dots.$ (1) $a one-one function$ (2) $an onto function$ (3) $not a function$ (4) $bijective$ 56. $If u = sin^{- 1} (\frac{2 x}{1 + x ^{2}}) and v = tan^{- 1} (\frac{2 x}{1 - x ^{2}}) then \frac{du}{dv} is$ (1) $\frac{1 - x ^{2}}{1 + x ^{2}}$ (2) $\frac{1}{2}$ (3) $1$ (4) $2$ 57. $Let A: {1, 2, 3, 4, 5, 6, 7}. Define B = {TA : either 1 T or 2 T } and C = TA : T the sum of all the elements of T is a prime number}. Then the number of elements in the set BC is_______________$ 58. $Let X = (n \in N : 1 \leq n \leq 50) . If A = {x \in X : n is a multiple of 2}; B = {n \in X : n is a multiple of 7}, then the number of elements in the smallest subset X containing both A and B is \dots\dots.$ 59. $If the function f (x) = \frac{cos ( sin x ) - cos x}{x ^{4}} is continuous at each point in its domain and f (0) = \frac{1}{k}, then k is ____$ 60. $Let S be the set of the first 11 natural numbers. Then the number of elements in A = {B \subseteq S : n (B) ⩾ 2 and the product of all elements of B is even} is____.$