relation-function.pdf

20 NDA/NA Pathfinder PRACTICE EXERCISE 1. If A = { 1, 2, 5, 6} and B = { 1, 2, 3}, then what is 12. The function f( x )...

20 NDA/NA Pathfinder PRACTICE EXERCISE 1. If A = { 1, 2, 5, 6} and B = { 1, 2, 3}, then what is 12. The function f( x ) = log( x + x 2 + 1 ) is ( A × B) ∩ ( B × A) equal to? (a) an even function (b) an odd function (a) {(1, 1), (2, 1), (6, 1), (3, 2 )} (b) {(1, 1), (1, 2 ), (2, 1), (2, 2 )} (c) periodic function (d) None of these (c) {(1, 1), (2, 2 )} (d) {(1, 1), (1, 2 ), (2, 5), (2, 6)} 13. If A = { a , b, c} and R = {( a , a ),( a , b),( b, c),( b, b),( c, c), 2. Which one of the following is correct? ( c, a )} is a binary relation on A, then which one (a) A × (B − C ) = ( A − B) × ( A − C ) of the following is correct? (b) A × (B − C ) = ( A × B) − ( A × C ) (a) R is reflexive and symmetric, but not transitive (c) A ∩ (B ∪ C ) = ( A ∩ B) ∪ C (b) R is reflexive and transitive, but not symmetric (d) A ∪ (B ∩ C ) = ( A ∪ B) ∩ C (c) R is reflexive, but neither symmetric nor transitive 3. Let R = { x| x ∈ N , x is a multiple of 3 and x ≤ 100} (d) R is reflexive, symmetric and transitive S = { x| x ∈ N , x is a multiple of 5 and x ≤ 100}. 14. The values of b and c for which the identity What is the number of elements in f ( x + 1) − f ( x ) = 8x + 3 is satisfied, where ( R × S ) ∩ (S × R )? f ( x ) = bx 2 + cx + d , are (a) 36 (b) 33 (c) 20 (d) 6 (a) b = 2, c = 1 (b) b = 4, c = − 1 4. If φ ( x ) = a , then [φ ( p)] is equal to x 3 (c) b = − 1, c = 4 (d) None of these (a) φ (3p) (ax + a− x ) (b) 3φ ( p) (c) 6φ ( p) (d) 2φ ( p) 15. If the function f ( x ) = (where, a > 2), 2 5. If f ( x ) = x 2 − x −2 , then f   is equal to 1 then f ( x + y ) + f ( x − y ) is equal to  x f( x ) f( y) 1 (a) 2 f(x) ⋅ f( y) (b) f(x) ⋅ f( y) (c) (d) (a) f(x) (b) − f(x) (c) (d) [f(x)]2 f( y) f( x ) f(x) 16. Let f be a function with domain [− 3, 5] and let 1+ x f (x) ⋅ f (x2 ) 6. If f ( x ) = , then is equal to g( x ) =|3x + 4|, then the domain of fog ( x ) is 1− x 1 + [ f ( x )]2 (a)  − 3,  (b)  − 3,  1 1 1 1 1 1  3  3  (a) (b) (c) (d) 4 6 8 2 (c)  − 3,  1 (d) None of these  3 7. Which one of the following functions, f : R → R is injective? αx 17. Let f( x ) = , x ≠ −1. Then, for what value of α (a) f(x) = | x|, ∀ x ∈ R (b) f(x) = x , ∀ x ∈ R 2 x+1 (c) f(x) = 11, ∀ x ∈ R (d) f(x) = − x, ∀ x ∈ R is f [ f ( x )] = x? (a) 2 (b) − 2 8. The domain of the function f ( x ) = x − 1 + 6 − x (c) 1 (d) −1 is (a) [1, ∞ ) (b) (− ∞, 6) 18. The inverse of the function f ( x ) = loga ( x + x 2 + 1 ) (c) [1, 6] (d) None of these (where, a < 0, a ≠ 1) is 1 x 9. The period of the function f( x ) =|sin x| +|cos x| is (a) ( a − a− x ) (b) not defined for all x 2 (a) π / 2 (b) π (c) 2 π (d) π / 4 (c) defined for x > 0 (d) None of these 10. The domain of the function 19. If f( x ) = 3x + 10 and g( x ) = x 2 − 1, then ( fog)−1 is 1 f( x ) = + ( x + 2) is equal to log10(1 − x ) x − 7 1/ 2 x + 7 1/ 2 x− 1/ 2 x + 3 1/ 2 (a)  (b)  (c)  3    (d)  (a) ] − 3, − 2.5 [ ∪ ] − 2.5, − 2 [ (b) [−2,0 [ ∪ ] 01 ,[  3   3   7   7  (c) ] 0,1[ (d) None of these 1 20. Let f ( x ) = ( − 1)[ x ] (where [⋅] denotes the greatest 11. The range of the function f( x ) = is integer function), then ( 2 − sin 3x ) (a) range of f is {− 1, 1} (b) f is an even function (a)  , 1  (b)  , 1 (c)  , 1  (d) 1, 1 1 1 1  3   3   3   (c) f is an odd function (d) f is one-one function 3  MATHEMATICS Relations and Functions 21 21. The function f : R → R defined by f ( x ) = 4x + 4|x| is 29. If g( x ) = loge x 2, then range of the function g[ f ( x )] (a) one-one and into (b) many-one and into is (a)  − ∞, loge 11 (b) loge , ∞  (c) one-one and onto (d) many-one and onto 11  3   3  22. If f ( x ) satisfies the relation 2 f ( x ) + f (1 − x ) = x 2 (c)  − loge , loge  11 11 (d) None of these  3  for all real x, then f ( x ) is 3 x 2 + 2x − 1 x 2 + 2x − 1 (a) (b) 6 3 Directions (Q. Nos. 30-31) The following functions x 2 + 4x − 1 x 2 + 4x − 1 are defined for the set of variables x1, x2, K, xn (c) (d) 3 6  xi + j , if i + j ≤ n2 23. For real numbers x and y, define a relation R, f ( xi , xj ) =  and g ( xi , xj ) = xm xi + j − n , if i + j > n 2 xRy if only if x − y + 2 is an irrational number. where, m is the remainder when i × j is divided by n. Then the relation R is (a) reflexive (b) symmetric 30. Find the value of f [ f ( x2 , x3 ), f ( x5 , x6 )], if n = 3. (c) transitive (d) an equivalence relation (a) x5 (b) x10 (c) x13 (d) x8 24. Let A = {2, 3, 4, 5, …, 16, 17 18} and ‘*’ be the 31. Find the value of g [g( x2 , x3 ), g( x7 , x8 )], if n = 5. equivalence relation on A × A defined by ( a , b) * ( c, d ) if ad = bc. Then, the number of (a) x1 (b) x2 (c) x5 (d) All of these ordered pairs of the equivalence class of ( 3, 2) is [x ] (a) 5 (b) 6 (c) 7 (d) 8 Directions (Q. Nos. 32-33) Consider f ( x ) = and x 25. Consider the following with regard to a relation g( x ) = | x | , where [⋅] denotes the greatest integer R on a set of real numbers defined by xRy if and function. only if 3x + 4 y = 5 1 2 3 32. What is the value of fog( −2 / 3) − gof ( −2 / 3)? I. 0 R 1 II. 1 R III. R 2 3 4 (a) 1 (b) −1 (c) 0 (d) 2 Which of the above statement(s) is/are correct? 33. What is the value of fof ( −7 / 4) + gog ( −1)? (a) I and II (b) I and III (c) II and III (d) I, II and III 1 1 −x (a) 0 (b) −1 (c) (d) − 26. The function f : R → R is defined by f ( x ) = 3 4 8 I. f is one-one function. II. f is onto function. PREVIOUS YEARS’ QUESTIONS III. f is a decreasing function. Which of the above statement(s) is/are correct? 34. If f ( xy ) = f ( x ) f ( y ), then f ( t ) may be of the form e 2012 I (a) I and II (b) II and III (c) I and III (d) All of these (a) t + k (b) ct + k (c) t k + c (d) t k 27. The domain of the given function where k is a constant f ( x ) = log2 sin x is 35. Let A = { x ∈ W, the set of whole numbers and π 2π I. (4n + 1) II. (4n + 1) x < 3}, B = { x ∈ N , the set of natural numbers and 3 3 2 ≤ x < 4} and C = { 3, 4}, then how many elements π III. (4n + 1) where n ∈ N. will ( A ∪ B) × C contain? e 2012 II 2 (a) 6 (b) 8 (c) 10 (d) 12 Choose the correct option using the code given 36. Let P = {1, 2, 3} and a relation on set P is given by the set R = {(1, 2), (1, 3), (2, 1) (1, 1) (2, 2), below. (a) Only I (b) Only II (3, 3), (2, 3)}. Then, R is e 2012 II (c) Only III (d) I and II (a) reflexive, transitive but not symmetric (b) symmetric, transitive but not reflexive Directions (Q. Nos. 28-29) Consider the function (c) symmetric, reflexive but not transitive f ( x ) = 3x 2 − 4x + 5. (d) None of the above 37. The relation ‘has the same father as’ over the set 28. The domain of function f ( x ) is of children is e 2012 II (d)  , ∞  2 (a) only reflexive (b) only symmetric (a) R (b) (− ∞, 1) (c) (1, ∞ ) 3  (c) only transitive (d) an equivalence relation 22 NDA/NA Pathfinder x+5 49. If f be a function from the set of natural 38. If f : R → R be a function whose inverse is , 3 numbers to the set of even natural numbers then what is the value of f( x )? e 2012 II given by f( x ) = 2x. Then, f is e 2013 II (a) f (x) = 3x + 5 (b) f (x) = 3x − 5 (a) one-one but not onto (b) onto but not one-one (c) f (x) = 5x − 3 (d) Does not exist (c) Both one-one and onto (d) Neither one-one nor onto 39. If A = { x ∈ R|x ≥ 0} and a function f : A → A is 50. Let X be the set of all citizens of India. Elements defined by f( x ) = x 2, then which one of the x, y in X are said to be related, if the difference following is correct? e 2012 II of their age is 5 yr. Which one of the following is (a) The functions does not have inverse correct? e 2014 I (b) f is its own inverse (a) The relation is an equivalence relation on X (c) The functions has an inverse but is not its own inverse (b) The relation is symmetric but neither reflexive nor transitive (d) None of the above (c) The relation is reflexive but neither symmetric nor 40. Consider the following statements transitive I. If f (x) = x3 and g ( y) = y3 , then f = g. (d) None of the above II. Identity function is not always a bijection. 51. Let S denote set of all integers. Define a relation Which of the above statement(s) is/are correct? R on S as ‘aRb if ab ≥ 0, where a , b ∈ S. Then, R is e 2014 I (a) Only I (b) Only II e 2012 II (a) reflexive but neither symmetric nor transitive relation (c) Both I and II (d) Neither I nor II (b) reflexive, symmetric but not transitive relation 41. If A = { x , y }, B = { 2, 3} , C = { 3, 4}, then what is the (c) an equivalence relation number of elements in A × ( B ∪ C )? e 2013 I (d) symmetric but neither reflexive nor transitive relation (a) 2 (b) 4 (c) 6 (d) 8 52. Consider the following relations from A to B, 42. If A is a relation on a set R, then which one of where the following is correct? e 2013 I A = {u, v, w, x, y, z} and B = {p, q, r, s}. (a) R ⊆ A (b) A ⊆ R (c) A ⊆ (R × R ) (d) R ⊆ ( A × A) I. {(u, p), (v, p), (w, p), (x, q), (y, q), (z, q)} 43. Let N be the set of natural numbers and II. {(u, p), (v, q), (w, r), (z, s)} f : N → N be a function given by f ( x ) = x + 1 for III. {(u, (v, r), (w, q), (u, p), (v, q), (z, q)} x ∈ N. Which one of the following is correct? s), e 2013 I IV. {(u, q), (v, p), (w, s), (x, r), (y, q), (z, s)} (a) f is one-one and onto (b) f is one-one but not onto Which of the above relations are not functions? (c) f is only onto (d) f is neither one-one nor onto e 2014 I (a) I and II (b) I and IV (c) II and III (d) III and IV 44. What is the range of the function |x| 53. Let N denote the set of all non-negative integers f(x) = ,x ≠ 0? x e 2013 I and Z denote the set of all integers. The function (a) Set of all real numbers (b) Set of all integers f : Z → N given by f( x ) =| x | is e 2014 I (c) {−1, 1} (d) {−1 , 0, 1} (a) one-one but not onto (b) onto but not one-one (c) Both one-one and onto (d) Neither one-one nor onto 45. If A = { 1, 2}, B = { 2, 3} and C = { 3, 4}, then what is the cardinality of ( A × B) ∩ ( A × C )? e 2013 II 54. A and B are two sets having 3 elements in (a) 8 (b) 6 (c) 2 (d) 1 common. If n( A) = 5 and n( B) = 4, then what is n( A × B) equal to? e 2014 II 46. If A is a finite set having n elements, then the (a) 0 (b) 9 (c) 15 (d) 20 number of relations which can be defined in A is 55. If f ( x ) = ax + b and g( x ) = cx + d such that e 2013 II (a) 2 n (b) n2 (c) 2 n2 (d) nn f [g( x )] = g[ f ( x )], then which one of the following is correct? e 2014 II 47. Let A = { a , b, c, d } and B = { x , y , z }. What is the (a) f(c ) = g (a) (b) f(a) = g (c ) (c) f(c ) = g (d ) (d) f(d ) = g (b ) number of elements in A × B ? e 2013 II 56. The function f : N → N , N being the set of (a) 6 (b) 7 natural numbers, defined by f ( x ) = 2x + 3 is (c) 12 (d) 64 e 2014 II 48. The relation R in the set Z of integers given by (a) injective and surjective R = {( a , b) : a − b is divisible by 5} is e 2013 II (b) injective but not surjective (a) reflexive (b) reflexive but not symmetric (c) not injective but surjective (c) symmetric and transitive (d) an equivalence relation (d) neither injective nor surjective MATHEMATICS Relations and Functions 23 57. The relation S is defined on the set of integers Z 65. Consider the following functions as xSy, if integer x divides integer y. Then I. f (x) = x3 , x ∈ R (a) S is an equivalence relation e 2014 II II. f (x) = sin x, 0 < x < 2π (b) S is only reflexive and symmetric III. f (x) = ex , x ∈ R (c) S is only reflexive and transitive (d) S is only symmetric and transitive Which of the above functions have inverse defined on their ranges? e 2015 I (a) I and II (b) II and III Directions (Q. Nos. 58-60) Read the following (c) I and III (d) I, II and III information carefully and answer these questions given below. 66. Let Z be the set of integers and aRb, where a, x −1 b ∈ Z if and only if ( a − b) is divisible by 5. Consider the function f ( x ) =. x +1 e 2014 II Consider the following statements f(x) + 1 58. What is + x equal to? I. The relation R partitions Z into five equivalent f(x) − 1 classes. (a) 0 (b) 1 (c) 2x (d) 4x II. Any two equivalent classes are either equal or disjoint. 59. What is f ( 2x ) equal to? Which of the above statement(s) is/are correct? f( x ) + 1 f( x ) + 1 3f(x ) + 1 f( x ) + 3 (a) (b) (c) (d) e 2015 I f( x ) + 3 3f(x ) + 1 f( x ) + 3 3f(x ) + 1 (a) Only I (b) Only II 60. What is f [ f ( x )] equal to? (c) Both I and II (d) Neither I nor II (b) −x 1 (a) x 67. The domain of the function f ( x ) = is 1 |x|– x (c) − (d) None of these x e 2015 II 61. Let A = { x , y , z } and B = { p, q , r , s}, what is the (a) [0, ∞ ) (b) (− ∞, 0) (c) [1, ∞ ) (d) (− ∞, 0] number of distinct relations from B to A? e 2015 I 68. If f : R → R , g : R → R are two functions given by (a) 4096 (b) 4094 (c) 128 (d) 126 f ( x ) = 2x − 3 and g( x ) = x3 + 5, then ( fog)−1( x ) is 62. Let X be the set of all persons living in a city. equal to e 2015 II Persons x, y in X are said to be related as x < y, x + 7 1/ 3 x − 7 1/ 3 if y is atleast 5 yr older than x. Which one of the (a)   (b)    3   2  following is correct? e 2015 I 1/ 3 1/ 3 (c)  x −  (d)  x +  7 7 (a) The relation is an equivalence relation on X  2  2 (b) The relation is transitive but neither reflexive nor symmetric 69. Let X be the set of all persons living in Delhi. (c) The relation is reflexive but neither transitive nor The persons a and b in X are said to be related, symmetric if the difference in their ages is atmost 5 yr. The (d) The relation is symmetric but neither transitive nor relation is e 2015 II reflexive (a) an equivalence relation x (b) reflexive and transitive but not symmetric 63. For each non-zero real number x, let f ( x ) =. |x| (c) symmetric and transitive but not reflexive The range of f is e 2015 I (d) reflexive and symmetric but not transitive (a) a null set 1 (b) a set consisting of only one element 70. If g( x ) = and f ( x ) = x , x ≠ 0, then which one of f(x) (c) a set consisting of two elements the following is correct? e 2015 II (d) a set consisting of infinitely many elements (a) f(f(f(g (g (f(x )))))) = g (g (f(g (f(x )))))  1 + x 3x + x3 (b) f(f(g (g (g (f(x )))))) = g (g (f(g (f(x ))))) 64. If f ( x ) = loge   , g( x ) = and  1 − x 1 + 3x 2 (c) f(g (f(g (g (f(g (x ))))))) = g (g (f(g (f(x )))))  e − 1 (d) f(f(f(g (g (f(x )))))) = f(f(f(g (f(x ))))) gof ( t ) = g ( f ( t )), then what is gof   equal to?  e + 1 e 2015 I 71. f ( xy ) = f ( x ) + f ( y ) is true for all e 2015 II (a) 2 (b) 1 1 (a) polynomial functions f (b) trigonometric functions f (c) 0 (d) 2 (c) exponential functions f (d) logarithmic functions f 24 NDA/NA Pathfinder 72. Consider the following statements 76. Let R be a relation on the set N of natural Statement I The function f : R → R such that numbers defined by ‘nRm ⇔ n is a factor of m’. Then, which one of the following is correct? f ( x ) = x3 for all x ∈ R is one-one. e 2016 I Statement II f ( a ) = f ( b) ⇒ a = b for all a , b ∈ R , if (a) R is reflexive, symmetric but not transitive the function f is one-one. (b) R is transitive, symmetric but not reflexive (c) R is reflexive, transitive but not symmetric Which one of the following is correct in respect of (d) R is an equivalence relation the above statements? e 2015 II (a) Both the statements are true and Statement II is the Directions (Q. Nos. 77-78) Let f ( x ) be the greatest correct explanation of Statement I integer function and g( x ) be the modulus function. (b) Both the statements are true and Statement II is not e 2016 I the correct explanation of Statement I 77. What is ( gof )  −  − ( fog )  −  equal to? 5 5 (c) Statement I is true but Statement II is false  3  3 (d) Statement I is false but Statement II is true (a) −1 (b) 0 73. Suppose there is a relation * between the (c) 1 (d) 2 positive numbers x and y given by x * y if and only if x ≤ y 2. Then which one of the following is 78. What is ( fof )  −  + ( gog ) ( −2) equal to? 9 correct? e 2016 I  5 (a) * is reflexive but not transitive and symmetric (a) −1 (b) 0 (b) * is transitive but not reflexive and symmetric (c) 1 (d) 2 (c) * is symmetric and reflexive but not transitive (d) * is symmetric but not reflexive and transitive Directions (Q. Nos. 79-80) Consider the function x −x  27( x 2/3 − x ) 74. If f ( x1 ) − f ( x2 ) = f  1 2  for x1 , x2 ∈ ( −1, 1), then f (x ) =.  1 − x1x2  4 e 2016 I what is f ( x ) equal to? e 2016 I 79. How many solutions does the function f ( x ) = 1 1− x  2 + x  (a) ln   (b) ln   have? 1 + x   1− x  (a) One (b) Two 1− x  1 + x  (c) Three (d) Four (c) tan−1   (d) tan−1   1 + x  1− x  80. How many solutions does the function f ( x ) = − 1 x2 have? 75. What is the range of the function y = 1 + x2 (a) One (b) Two (c) Three (d) Four where x ∈ R ? e 2016 I (a) [0, 1) (b) [0, 1] (c) (0, 1) (d) (0, 1] ANSWERS 1 b 2 b 3 a 4 a 5 b 6 d 7 d 8 c 9 a 10 b 11 b 12 b 13 c 14 b 15 a 16 b 17 d 18 a 19 a 20 a 21 a 22 b 23 a 24 b 25 c 26 c 27 c 28 a 29 b 30 b 31 a 32 c 33 d 34 d 35 b 36 a 37 d 38 b 39 c 40 a 41 c 42 c 43 b 44 c 45 c 46 c 47 c 48 d 49 c 50 b 51 c 52 c 53 b 54 d 55 d 56 b 57 c 58 a 59 c 60 c 61 a 62 b 63 c 64 b 65 c 66 c 67 b 68 b 69 d 70 b 71 d 72 a 73 a 74 a 75 a 76 c 77 c 78 b 79 b 80 a