Page 1 CBSE TEST PAPER-04 CLASS - XII MATHEMATICS (Relations and Functions) [ANSWERS] Topic:- Relations and Functions 1. A function f: X Y is said to be one – one and onto (bijective), if f is both one – one and onto. 2. Let x 1, x 2 ?R If f(x 1) = f(x 2) 4 4 1 2 2 2 1 1 1 2 x x x x x x = = ± = ± Not one – one 4 1/4 1/4 ( ) y x x y f y y = = ± = Not onto. 1/4 ( ) f y y - = 3. f is one – one and onto, Ao that f is invertible with f -1 = {(3,1) (2, 3) (1, 2)} 4. gof (x) = g[f(x)] = g (8x 3 ) = ( ) 1 3 3 8x = 2x 5. (f. g) oh (f. g) h (x) f[h(x)]. g[h(x)] foh. goh 6. 3 * 4 = 2 (3) + 4-3 = 7 7. (i) Each triangle is similar to at well and thus (T1, T1) ? R ? R is reflexive. Page 2 CBSE TEST PAPER-04 CLASS - XII MATHEMATICS (Relations and Functions) [ANSWERS] Topic:- Relations and Functions 1. A function f: X Y is said to be one – one and onto (bijective), if f is both one – one and onto. 2. Let x 1, x 2 ?R If f(x 1) = f(x 2) 4 4 1 2 2 2 1 1 1 2 x x x x x x = = ± = ± Not one – one 4 1/4 1/4 ( ) y x x y f y y = = ± = Not onto. 1/4 ( ) f y y - = 3. f is one – one and onto, Ao that f is invertible with f -1 = {(3,1) (2, 3) (1, 2)} 4. gof (x) = g[f(x)] = g (8x 3 ) = ( ) 1 3 3 8x = 2x 5. (f. g) oh (f. g) h (x) f[h(x)]. g[h(x)] foh. goh 6. 3 * 4 = 2 (3) + 4-3 = 7 7. (i) Each triangle is similar to at well and thus (T1, T1) ? R ? R is reflexive. (ii) (T1, T2) ? R ? T 1 is similar to T 2 ? T 2 is similar to T 1 (T 2, T 1) ? R R is symmetric (iii) T1 is similar to T2 and T2 is similar to T3 ? T1 is similar to T3 ? (T 1, T 3) ? R ? R is transitive. Hence R is equivalence (II) part T 1 = 3, 4, 5 T2 = 5, 12, 13 T3 = 6, 8, 10 3 4 5 1 6 8 10 2 = = = T 1 is relative to T 3. 8. (a) a * b = 1 b * a = 1 for all a, b ? N also (a * b) * c = 1 * c = 1 a * (b * c) = a * (1) = 1 for all, a, b, c R N Hence R is both associative and commutative (b) a * b = 2 a b + , b * a = 2 b a + Hence commutative. (a * b) * c = * 2 a b c + ? ? ? ? ? ? = 2 2 4 a b a b c c + + + ? ? + = ? ? ? ? = 2 *( * ) * 2 2 a b a a b a b c a + ? ? + ? ? + ? ? ? ? = = ? ? ? ? 2 4 a b c + + = * is not associative. Page 3 CBSE TEST PAPER-04 CLASS - XII MATHEMATICS (Relations and Functions) [ANSWERS] Topic:- Relations and Functions 1. A function f: X Y is said to be one – one and onto (bijective), if f is both one – one and onto. 2. Let x 1, x 2 ?R If f(x 1) = f(x 2) 4 4 1 2 2 2 1 1 1 2 x x x x x x = = ± = ± Not one – one 4 1/4 1/4 ( ) y x x y f y y = = ± = Not onto. 1/4 ( ) f y y - = 3. f is one – one and onto, Ao that f is invertible with f -1 = {(3,1) (2, 3) (1, 2)} 4. gof (x) = g[f(x)] = g (8x 3 ) = ( ) 1 3 3 8x = 2x 5. (f. g) oh (f. g) h (x) f[h(x)]. g[h(x)] foh. goh 6. 3 * 4 = 2 (3) + 4-3 = 7 7. (i) Each triangle is similar to at well and thus (T1, T1) ? R ? R is reflexive. (ii) (T1, T2) ? R ? T 1 is similar to T 2 ? T 2 is similar to T 1 (T 2, T 1) ? R R is symmetric (iii) T1 is similar to T2 and T2 is similar to T3 ? T1 is similar to T3 ? (T 1, T 3) ? R ? R is transitive. Hence R is equivalence (II) part T 1 = 3, 4, 5 T2 = 5, 12, 13 T3 = 6, 8, 10 3 4 5 1 6 8 10 2 = = = T 1 is relative to T 3. 8. (a) a * b = 1 b * a = 1 for all a, b ? N also (a * b) * c = 1 * c = 1 a * (b * c) = a * (1) = 1 for all, a, b, c R N Hence R is both associative and commutative (b) a * b = 2 a b + , b * a = 2 b a + Hence commutative. (a * b) * c = * 2 a b c + ? ? ? ? ? ? = 2 2 4 a b a b c c + + + ? ? + = ? ? ? ? = 2 *( * ) * 2 2 a b a a b a b c a + ? ? + ? ? + ? ? ? ? = = ? ? ? ? 2 4 a b c + + = * is not associative. 9. Let (a1 b1) and (a2, b2) ? A × B (i) f(a 1 b 1) = f(a 2, b 2) b 1 = b 2 and a 1 = a 2 (a 1 b 1) = (a 2, b 2) Then f(a 1 b 1) = f(a 2, b 2) (a 1 b 1) = (a 2, b 2) for all (a 1 b 1) = (a 2, b 2) ? A × B (ii) f is injective, Let (b, a) be an arbitrary Element of B × A. then b ? B and a ? A ? (a, b) ) ? (A × B) Thus for all (b, a) ? B × A their exists (a, b) ) ? (A × B) Hence that f(a, b) = (b, a) So f: A × B B × A Is an onto function. Hence bijectiveRead More

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