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12th Maths Sectionwise paper - 01 (R&F, Inv Trig, Matrices & Determinants)

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MATHEMATICS Class II PU Plus 2016-17 Topics Sectionwise Test - 01 Relation, Functions, Inverse Trigonometric Function, Matrices and Determinants Max. marks 100 Duration Date 3 hrs 15 min 22-12-2016 Instructions 1. The questions paper has four parts: A, B, C and D. All parts are compulsory. 2. Write balanced chemical equations and draw labeled diagrams wherever required. 3. Use log tables and the simple calculator if necessary. (Use of scientific calculators is not allowed). PART A I. Answer the following questions [10 1 = 10] 1. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2) (2, 2) (3, 3) (1, 2) (2, 3)} is not transitive. 2. Let be a binary operation defined on the set of rational numbers Q defined by a b = ab + 1 prove that is commutative. 3. Define one-one function. 1 4. Find the principal value of cot 1 . 3 1 5. Find the value of tan 1 2cos 2sin 1 2 6. Write the domain of f(x) = cos 1 x. 7. Define a scalar matrix. 8. Construct a 3 2 matrix whose elements are given by a ij 1 i 3j . 2 4 7 9. If A find | 3 A |. 3 5 10. If 3 x 3 2 find the value of x. x 1 4 1 PART B II. Answer any TEN of the following questions [10 2 = 20] 11. Find g f and f g if f : R R and g : R R are given by f(x) = cos x and g(x) = 3x 12. Check whether the function f: R R given by f(x) = x2 is injectivity and surjectivity. 13. Verify whether the operation defined on Q by a b ab is associative or not. 2 2P1617MSWT1 1 2 14. Show that sin 1 2x 1 x 2 2sin 1 x , 1 1 x 2 2 1 x 2 1 15. Write the simplest form of tan 1 ,(x 0) x 16. Prove that tan 1 x cot 1 x , x R 2 2 7 1 17. Prove that tan 1 tan 1 tan 1 11 24 2 4 5 63 18. Prove that sin 1 sin 1 sin 1 5 13 65 x 19. Find the values of x and y if 2 7 5 3 4 7 14 y 3 1 2 15 14 1 2 0 2 1 3 20. If A and B find AB 3 1 2 2 1 1 21. If the area of the triangle with vertices ( 2, 0), (0, 4) and (0, k) is 4 square units, find the values of k using determinants. 1 5 22. For the matrix A verify that A + A is a symmetric matrix. 6 7 a b b c c a 23. Using properties of determinants and without expanding show that b c c a a b 0 c a a b b c 2 1 5 24. If 6 x 15 is singular matrix, find x 1 2 3 PART C III. Answer any TEN of the following questions [10 3 = 30] 25. Prove that the relation R in the set of integers Z defined by R = {(x, y) : x y is an integer} is an equivalence relation. 26. If is a binary operation defined on A = N N by (a, b) (c, d) = (a + c, b + d) prove that is associative. Find the identity if it exists. 27. Consider f: N N, g : N N and h : N R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sinz x, y and z in N. Show that h (g f ) (h g) f 28. Let f : X Y and g : Y Z be two invertible functions then prove that g f is also invertible with (g f ) 1 f 1 g 1 . 2P1617MSWT1 2 1 29. Prove that 3cos 1 x = cos 1 (4x3 3x), x ,1 2 x 1 x 1 30. If tan 1 tan 1 , then find the value of x. x 2 x 2 4 31. Solve tan 1 2x tan 1 3x 4 32. If sin 1 x + sin 1 y + sin 1 z = , show that x 1 x 2 y 1 y 2 z 1 z 2 2xyz . 4 5 33. Using elementary transformations, find the inverse of the matrix 3 4 3 5 34. Express the matrix as sum of symmetric and skew symmetric matrix. 1 1 35. If A and B are invertible matrices of the same order, then prove that (AB) 1 = B 1A 1. 1 3 3 36. If A 1 4 3 then verify that A adj A = | A | I 1 3 4 3 1 37. If A show that A2 5A + 7I = O. Hence find A 1. 1 2 38. Prove that tan 1 x tan 1 3 2x 1 1 3x x tan , x 2 2 1 x 1 3x 3 PART D IV. Answer any SIX of the following questions [6 5 = 30] 39. Prove that the function f : R R defined by f(x) = 4x + 3 is invertible and find the inverse of f. 40. Consider f : R+ [ 5, ) given by f(x) = 9x2 + 6x 5. Show that f is invertible with y 6 1 . f 1 (y) 3 1 x 1 x 1 1 1 41. Prove that tan 1 cos x, x 1 2 1 x 1 x 4 2 1 1 1 42. Show that 2 tan 1 tan 1 2 tan 1 8 7 5 4 0 6 7 0 1 1 2 43. If A 6 0 8 , B 1 0 2 and C 2 verify the distributive property 7 8 0 1 2 0 3 (A + B)C = AC + BC. 2P1617MSWT1 3 1 2 3 3 1 2 4 1 2 44. If A 3 2 1 , B 4 2 5 and C 0 3 2 , the compute A + B and B C. Also verify 4 2 1 2 0 3 1 2 3 that A + (B C) = (A + B) C 1 2 3 45. If A 3 2 1 then show that A3 23A 40I = O 4 2 1 x y 2z x y 46. Prove that z y z 2x y 2(x y z)3 z x z x 2y 47. Solve by matrix method 2x + 3y + 3z = 5, x 2y + z = 4, 3x y 2z = 3 x 1 x 2 48. Solve for x, in 3 x 2 x 1 2 3 x 1 = 0. x 3 PART E IV. Answer any ONE of the following questions [1 10 = 10] (6 + 4) 3x 4 49. (a) Show that f g I A and g f I B , if f : B A is defined by f x and g : A B is 5x 7 7x 4 3 7 defined by g(x) , where A R , B R and IA(x) = x, x A, IB(x) = x, 5x 3 5 5 x B are called identity functions on set A and B, respectively. (b) If tan 1x + tan 1 y + tan 1 z = , show that x + y + z = xyz 2 3 5 50. (a) If A 3 2 4 , find A 1. Using A 1 solve the system of equations 1 1 2 2x 3y + 5z = 11, 3x + 2y 4z = 5 and x + y 2z = 3. 1 a 2 b2 2ab 2b 2 2 (b) Prove that 2ab 1 a b 2a (1 a 2 b 2 )3 2b 2a 1 a 2 b2 *** 2P1617MSWT1 4

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