Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Kendriya Vidyalaya (O.F),Dumdum Mathematics Worksheet Prepared by:- Mrs. S. Basu Class:-XII-B & C Relation & Function 1. Let R be a relation defined on the set of Natural numbers N as. R {( x, y ) : x, y N ,2 x y 4} Find the domain & range of R. Also verify whether R is (i) Reflexive, (ii)Symmitric (iii) transitive. 2. Prove that the relation a, bRc, d a d b c for R all on the set NXN a, b, c, d NXN is an defined by equivalence relation. Also find the equivalence classes 3. 2,3& 1,3 Let N denote the set of all natural numbers & R be the relation on NXN defined by (a, b) R(c, d ) ad (b c) bc(a d ) , Check whether R is an equivalance relation on NXN. 4. Let n be a fixed positive integer. Define a relation R on Z as follows : (a, b) R a b is divisible by n. Show that R is an equivalence relation on Z 5. Show that the function a, b R, a 0 is a bijection. f : R R given by f ( x) ax b . Where 6. Show that the function f : R R given by f ( x) Cosxx R is neither on-one nor on to. 7. Let A R {2} & B R {1} . If f x f : A B is a mapping defined by x 1 , show that f is bijective. x2 8. Show that the function f : R R given by f ( x) x 3 x is a bijection. 9. If the function f : R R be given by f ( x) x 2 2 & g : R R be given by g ( x) 10. x find gof & fog. x 1 If f , g : R R are defined Respectively by f ( x) x 2 3x 1, g ( x) 2 x 3, , find (i) fog (ii) gof (iii) fof, (iv) gog 11. x, if x Q Let A x R : 0 x 1. If f : A A is defined by f x then 1 x if x Q prove fof ( x) x that for all x A . 12. Let A R {3} & B R {1} . Consider the function f : A B defined by f ( x) 13. x2 . Show that f is one. One & onto & hence find f x3 . Consider the function f : R [9, ] given by f ( x) 5 x 2 6 x 9 . Prove that f is invertible with f 1 ( y ) 14. 1 54 5 y 3 . 5 Let f : N N be a function defined as f ( x) 9 x 2 6 x 5 . Show that f : N S where s is the range of f, is invertible. Find the inverse of f & hence find f 1 (43) & f 1 (163) . 15. Discuss the commutativity & associativity of binary operation * defined on Q by the, rule a * b a b ab for all a, b Q . 16. If the binary operation * on the set Z is defined by a * b a b 5 , then find the identity element with respect to *. 17. Let * be a binary operation Qo (Set of non zero rational numbers) defined by a * b 3ab a, b Q . Show that * is commutative as well as 5 associative. Also find its identity elemenet if it exists. 18. Let * be a binary operation, on the set of all non zero real numbers, given by a * b ab for all a, b R {0} . Write the value of k given by 5 2 * ( x * 5) 10 . 19. Let A = RoXR, where RO denote the set of all non zero real numbers. A binary operation 'O' is defined on A as follows : (a, b)O(c, d ) (ac, bc d ) for all (a, b), (c, d ) RoXR . (i) Show that 'O' is commutative & associative on A (ii) Find the indentity element in A. (iii) Find the invertible element in A. Kendriya Vidyalaya (O.F),Dumdum Mathematics Worksheet Prepared by:- Mrs. S. Basu Class:-XII-B & C Inverse Trigonometric Function 1. Evaluate each of the following : 7 1 0 1 0 (a) Cos 1 Cos (b) Sin Sin (600 ) (c) Cos Cos(680 ) 6 2. Cosx x . Express the following in simplest form tan 1 2 1 Sinx 2 3. 1 Sinx 1 Sinx x Prove that, Cot 1 0 x 2 1 Sinx 1 Sinx 2 4. 1 x2 1 x2 Prove that, tan 2 2 1 x 1 x 5. 3 4 3 x . Simplify : Cos 1 Cosx Sinx where 4 4 5 5 6. 1 4 Evaluate : Cos Sin 1 Sec 1 4 3 7. Prove that : tan 2 ( Sec 1 2) Cot 2 (Co sec 1 3) 11. 8. If Sin Cot 1 ( x 1) cos(tan 1 x) , then find x. 9. 3 Solve the following equation for x : Costan 1 x Sin Cot 1 4 10. 3 3 6 Prove that : Cos Sin 1 Cot 1 5 2 5 13 1 1 1 2 Cos x 1 x 1 2 2 11. Prove that : tan 1 1 1 1 tan 1 tan 1 2 5 8 4 12. Prove that : tan 1 3 3 8 tan 1 tan 1 4 5 19 4 13. 8 Solve for x : tan 1 x 2 tan 1 x 2 tan 1 x 0 79 14. Solve for x : tan 1 15. x2 x2 Solve for x : tan 1 tan x4 x4 4 x x tan 1 ,0 x 6 2 3 4 Kendriya Vidyalaya (O.F),Dumdum Mathematics Worksheet Prepared by:- Mrs. S. Basu Class:-XII B & C : Matrics : 1. 2 1 0 4 If A = &B= find 3 A2 2 B I . 3 2 1 7 2. Cos 2 If A = Sin 2 3. 1 0 2 1 If 1 1 x 0 2 1 1 0 , find x . 2 1 0 1 4. 3 1 IF A = , show that A2 5 A 7 I 2 0 1 2 5. 1 If A = 2 & B = 2 1 4 verify AB BA 3 6. 4 2 1 7 as the sum of a symmetric & skew Express the matrix A = 3 5 1 2 1 Sin 2 , find A2 . Cos 2 symmetric matrix. 7. 1 3 y 0 5 6 Find the values of x & y if 2 0 x 1 2 1 8 8. Cos If A = Sin Sin is identify matrix, then write the value of . Cos 9. Construct a 2 x 2 matrix A = [Cuj] whose element aij are given by 3i j if i j aij 2 (i j ) 2 if i j 2 2 1 0 by using elementary 10. Find the inverse of the matrix. A = 1 3 0 2 1 row transformations.