CLASS XII CBSE MATHEMATICS RELATIONS AND FUNCTIONS 1 Mark/2 Marks Questions

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1 CLASS XII CBSE MATHEMATICS RELATIONS AND FUNCTIONS 1 Mark/ Marks Questions (1) Let be a binary operation defined by a b = a + b 3. Find3 4. () The binary operation : R R R is defined as a b = a + b. Find ( 3) 4. (3) Let be a binary operation on N given by a b = HCF (a, b), a, b N. Write the value of 4. (4) If the binary operation on the set of integers Z, is defined by a b = a + 3b, then find the value of 8 3. (5) Let be a binary operation defined by a b = 3a + 4b. find 4 5. (6) If f(x) = x + 7 and g(x) = x 7, x R find (fog) (7). (7) If f(x) is an invertible function, find the inverse of f(x) = 3x 5. (8) Let A = {1,,3}, B = {4, 5, 6, 7} and let f = {(1, 4), (, 5), (3, 6)} be a function from A to B. whether f is one-one or not (9) If f: R R is defined by f(x) = (3 x 3 ) 1 3, then find fof (x). (10) Let f: R R be defined as f(x) = 10x +7. Find the function g R R such that g o f = f o g =IR (11) If R = {(x, y): x + y = 8} is a relation on N, write the range of R. 4 Marks/6 Marks Questions Equivalence Relations (1) Let T be the set of all triangles in a plane with R as a relation in T given by R = {(T1, T):T1 T}. Show that R is an equivalence relation. () Prove that the relation R in the set A ={1,, 3, 4, 5} given by R = {(a, b): a b is even}, is an equivalence relation.

2 (3) Show that the relation S in the set A={x Z: 0 x 1}given by S ={(a, b): a, b A, a b is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. (4) Let Z be the set of all integers and R be relation on Z defined as R = {(a, b): a, b Z and (a b)is divisible by 5}. Prove that R is an equivalence relation. (5) Determine whether the relation R defined on the set R of all real numbers as R = {(a, b): a, b R and a b + 3 S, where S is the set of all irrational numbers}, is reflexive, symmetric and transitive. (6) Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d) if ad(b + c) = bc(a + d). Show that R is an equivalence relation (7) Show that the relation R defined by (a, b)r(c, d) a + d = b + c on the A x A, where A = {1,,3,,10} is an equivalence relation. Hence write the equivalence class [(3,4)]; a, b, c, d A. (8) Show that the relation R in the set R of real numbers, defined as R = {(a, b): a b } is neither reflexive nor symmetric nor transitive. (9) Check whether the relation R defined in the set {1,, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. (10) Check whether the relation R in R defined as R = {(a, b): a b 3 } is reflexive, symmetric or transitive. (Hint: R is neither reflexive, nor symmetric, nor transitive.) (11) Show that the relation R in the set A = {1,,3,4,5} given by R = {(a, b): a b is divisible by } is an equivalence relation. Show that all the elements of {1,3,5} are related to each other and all the elements of {,4} are related to each other, but no element of {1,3,5} is related to any element of {,4}.

3 One-one and Onto functions, Invertible Function and Composition of functions (1) Let A = R - {3} and B = R {1}. Consider the function f : A B defined by f(x) = ( x ). Is f one-one and onto? Justify your answer (Ans: f is one-one x 3 and onto) () Let A = R {3}, R {1}. Let f: A B be defined by f(x) = x, X A. Show that f is bijective. Also, find (a) x, if f 1 (x) = 4 (b) f 1 (7) (3) Show that f: R { 1} R {1} given by f(x) = x find inverse of f. x+1 x 3 is invertible. Also, (4) Consider f: R + [4, )given by f(x) = x + 4. Show that f is invertible with the inverse f 1 of f given by f 1 (y) = y 4, where R + is the set of all non-negative real numbers. (5) Let A = { 1,0,1,}, B = { 4,,0,}and f, g: A B be functions defined by f(x) = x x, x A and g(x) = x 1 1, x A. Find gof (x) and hence show that f = g = gof. (6) In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f: R R defined by f(x) = 3 4x (ii) f: R R defined by f(x) = 1 + x x + 1, if x is odd (7) Show that f: N N, given by f(x) = { x 1 if x is even one and onto. (Ans: i) f is bijective. ii) f is neither) is both one (8) If the function f: R R, is given by f(x) = x + and g: R R, is given by g(x) = x x 1, x 1 find f0g and g0f. Also find f0g()and g0f( 3). (9) Let f: N N be defined by f(n) = { n+1 n, if n is odd, if n is even State whether the function f is bijective. Justify your answer. (10) how that f: [ 1, 1] R, given byf(x) = x x+ is one-one. Find the inverse of the function f: [ 1, 1] Range f.

4 (Hint: For y Range f, y = x x+, for some x in [ 1, 1], i.e.,x = y 1 y ) (11) Consider f: R+ [ 5, ) given by f(x) = 9x + 6x 5. Show that f is invertible with f 1 ( y+6 1 ) and hence find f 1 (43)and f 1 (163). 3 (1) Consider f: R + [ 9, ]given by f(x) = 5x + 6x 9.prove that f is invertible with f 1 (y) = ( 54+5y 3 ) 5 (13) Let f; N N be a function defined as f(x) = 4x + 1x Show that f: N S is invertible (where S is range of f). find the inverse of f and hence find f 1 (31)andf 1 (87). (14) If the function f: R R be defined by f(x) = x 3 and g: R R by g(x) = x 3 + 5, then find he value of (fog) 1 (x). (15) Let f: W W be defined as f(n) = n 1, if is n odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers. (Ans: f is invertible and the inverse of f is the same as f.) Binary Operation (1) (i) Is the binary operation, defined on set N, given by a b = a+b for all a, b N,commutative? (ii) is the above binary operation associative? () Consider the binary operations : R R R and 0 R R R defined as a b = a b and a 0 b = a all a, b R. Show that * is commutative but not associative, 0 is associative but not commutative. (3) Check whether the operation * defined in the set A = R R as (a, b) (c, d) = (a + c, b + d) is a binary operation or not, where R is the set of all real numbers. If it is a binary operation, is it commutative and associative too? Also find the identify elements of *. (4) Let A = R R and let * be a binary operation on A defined by (a, b) (c, d) = (ad + bc, bd) for all (a, b), (c, d) R R. (i) Show that * is commutative on A. (ii) Show that * is associative on A. (iii) Find the identity element of * in A.

5 (5) Show that the binary operation* on A=R-{-1} defined as a b = a + b + ab for all a, b A is commutative and associative on A. also find the identity element of * in A and prove that every element of A is invertible. (6) Let A=R x R* be the binary operation on A defined by (a, b) (c, d) = (a + c, b + d).prove that *is commutative and associative. Find the identity element for* on A. Also write the inverse element of the element (3, 5) in A. (7) Let A=Q X Q and let be a binary operation on A defined by (a, b) (c, d) = (ac, b + ad)for (a, b), (c, d) A,then with respect to on A (i) find the identity element in A (ii) find the invertible element in A. (8) Let A = Q Q, where Q is the set of all rational numbers, and * be a binary operation on A defined by (a, b) (c, d) = (ac, b + ad) for(a, b), (c, d) A. Then find i) The identity element of * in A. ii) Invertible elements of A, and hence write the inverse of elements (5,3)and ( 1, 4). (9) Let A = R x R and * be a binary operation on A defined by (a, b) (c, d) = (a + c, b + d) Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element(a, b) A. (10) Above question use A = N N (11) A binary operation * is defined on the set X = R { 1} by x y = x + y + xy, x, y X. Check, whether * is commutative and associative. Find its identity element and also find the inverse of each element of X (1) Define a binary operation *on the set {0, 1,, 3, 4, 5} as a + b, if a + b < 6 a b = { a + b 6, if a + b 6 Show that zero is the identity for this operation and each element a 0 of the set is invertible with 6 a being the inverse of a.

RELATIONS AND FUNCTIONS

For more important questions visit : www.4ono.com CHAPTER 1 RELATIONS AND FUNCTIONS IMPORTANT POINTS TO REMEMBER Relation R from a set A to a set B is subset of A B. A B = {(a, b) : a A, b B}. If n(a)