ANSWERS 287
ANSWERS
1.3 EXERCISE
1. (b,b), (c,c), (a,c)
2. [-5,5]
3. 4 x 2 + 4 x – 1
x+3
4. f −1 ( x ) =
2
5. f −1 {( b, a ) , ( d , b ) , ( a, c ) ,(c, d )}
6. f ( f ( x ) ) = x 4 – 6 x3 + 10 x 2 – 3x
7. = 2, = – 1
8. (i) represents function which is surjective but not injective
(ii) does not represent function.
9. fog = {( 2,5) , ( 5, 2) , (1,5)}
12. (i) f is not function (ii) g is function (iii) h is function (iv) k is not function
1
14. ,1
3
17. Domain of R = {1,2,3,4, ..... 20} and
Range of R = {1,3,5,7,9, ..... 39}. R is neither reflective, nor symmetric and nor
transitive.
21. (i) f is one-one but not onto , (ii) g is neither one-one nor onto (iii) h is bijective,
(iv) k is neither one-one nor onto.
22. (i) transitive (ii) symmetric (iii) reflexive, symmetric and transitive (iv) transitive.
23. ( 2,5 ) = {(1, 4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) (5,8),(6,9)}
, 288 MATHEMATICS
25. (i) ( fog )( x ) = 4 x 2 – 6 x + 1
(ii) ( gof )( x ) = 2 x 2 + 6 x – 1
(iii) ( fof )( x ) = x 4 + 6 x 3 + 14 x 2 + 15 x + 5
(iv) ( gog )( x ) = 4 x – 9
26. (ii) & (iv)
27. (i) 28. C 29. B 30. D
31. B 32. B 33. A 34. C
35. C 36. B 37. D 38. A
39. B 40. B 41. A 42. A
43. C 44. B 45. D 46. A
47. B 48. R = {( 3,8) , ( 6,6) , (9, 4), (12,2)}
49. R = {(1,1) , (1, 2 ) , ( 2,1) ,(2, 2),(2,3), (3,2), (3,3), (3,4), (4,3), (4,4), (5,5)}
50. gof = {(1,3) , ( 3,1) , ( 4,3)} and fog = {( 2,5 ) , ( 5, 2 ) , (1,5 )}
1
x
51. ( fofof )( x) = 52. f –1
( x) = 7 + (4 – x)
3
3x 2 + 1
53. False 54. False 55. False 56. False
57. True 58. False 59. False 60. True
61. False 62. False
2.3 EXERCISE
–π π
1. 0 2. – 1 4. 5. –
12 3
14 –3 3
7. 0, –1 8. 11. ,
15 4 4
, ANSWERS 289
–1 4 an – a1
13. tan –x 17. 19. 1 + a a
3 4 1 n
20. C 21. D 22. B 23. D
24. A 25. A 26. B 27. C
28. A 29. B 30. A 31. D
32. D 33. B 34. A 35. C
36. A 37. A
2π 2π
38. 39. 40. 3 41. φ
3 5
π 2π
42. 43. 44. 0 45. 1
3 3
46. –2π, 2π 47. xy > – 1 48. π – cot –1 x
49. False 50. False 51. True 52. True
53. True 54. False 55. True
3.3 EXERCISE
1. 28 × 1, 1 × 28, 4 × 7, 7 × 4, 14 × 2, 2 × 14. If matrix has 13 elements then its order
will be either 13 × 1 or 1 × 13.
2. (i) 3×3, (ii) 9, (iii) a23 = x 2 – y , a31 = 0, a12 = 1
1 9
1 4
3. (i) 2 2 (ii)
0 2 –1 2
e x sin x e x sin 2 x
2x 2x
4. e sin x e sin 2 x 5. a = 2, b = 2 6. Not possible
3x 3x
e sin x e sin 2 x
5 2 –2 0 –1 1
(i) X + Y = (ii) 2X − 3Y =
– 10 –18
7.
12 0 1 –11
ANSWERS
1.3 EXERCISE
1. (b,b), (c,c), (a,c)
2. [-5,5]
3. 4 x 2 + 4 x – 1
x+3
4. f −1 ( x ) =
2
5. f −1 {( b, a ) , ( d , b ) , ( a, c ) ,(c, d )}
6. f ( f ( x ) ) = x 4 – 6 x3 + 10 x 2 – 3x
7. = 2, = – 1
8. (i) represents function which is surjective but not injective
(ii) does not represent function.
9. fog = {( 2,5) , ( 5, 2) , (1,5)}
12. (i) f is not function (ii) g is function (iii) h is function (iv) k is not function
1
14. ,1
3
17. Domain of R = {1,2,3,4, ..... 20} and
Range of R = {1,3,5,7,9, ..... 39}. R is neither reflective, nor symmetric and nor
transitive.
21. (i) f is one-one but not onto , (ii) g is neither one-one nor onto (iii) h is bijective,
(iv) k is neither one-one nor onto.
22. (i) transitive (ii) symmetric (iii) reflexive, symmetric and transitive (iv) transitive.
23. ( 2,5 ) = {(1, 4 ) , ( 2,5 ) , ( 3,6 ) , ( 4,7 ) (5,8),(6,9)}
, 288 MATHEMATICS
25. (i) ( fog )( x ) = 4 x 2 – 6 x + 1
(ii) ( gof )( x ) = 2 x 2 + 6 x – 1
(iii) ( fof )( x ) = x 4 + 6 x 3 + 14 x 2 + 15 x + 5
(iv) ( gog )( x ) = 4 x – 9
26. (ii) & (iv)
27. (i) 28. C 29. B 30. D
31. B 32. B 33. A 34. C
35. C 36. B 37. D 38. A
39. B 40. B 41. A 42. A
43. C 44. B 45. D 46. A
47. B 48. R = {( 3,8) , ( 6,6) , (9, 4), (12,2)}
49. R = {(1,1) , (1, 2 ) , ( 2,1) ,(2, 2),(2,3), (3,2), (3,3), (3,4), (4,3), (4,4), (5,5)}
50. gof = {(1,3) , ( 3,1) , ( 4,3)} and fog = {( 2,5 ) , ( 5, 2 ) , (1,5 )}
1
x
51. ( fofof )( x) = 52. f –1
( x) = 7 + (4 – x)
3
3x 2 + 1
53. False 54. False 55. False 56. False
57. True 58. False 59. False 60. True
61. False 62. False
2.3 EXERCISE
–π π
1. 0 2. – 1 4. 5. –
12 3
14 –3 3
7. 0, –1 8. 11. ,
15 4 4
, ANSWERS 289
–1 4 an – a1
13. tan –x 17. 19. 1 + a a
3 4 1 n
20. C 21. D 22. B 23. D
24. A 25. A 26. B 27. C
28. A 29. B 30. A 31. D
32. D 33. B 34. A 35. C
36. A 37. A
2π 2π
38. 39. 40. 3 41. φ
3 5
π 2π
42. 43. 44. 0 45. 1
3 3
46. –2π, 2π 47. xy > – 1 48. π – cot –1 x
49. False 50. False 51. True 52. True
53. True 54. False 55. True
3.3 EXERCISE
1. 28 × 1, 1 × 28, 4 × 7, 7 × 4, 14 × 2, 2 × 14. If matrix has 13 elements then its order
will be either 13 × 1 or 1 × 13.
2. (i) 3×3, (ii) 9, (iii) a23 = x 2 – y , a31 = 0, a12 = 1
1 9
1 4
3. (i) 2 2 (ii)
0 2 –1 2
e x sin x e x sin 2 x
2x 2x
4. e sin x e sin 2 x 5. a = 2, b = 2 6. Not possible
3x 3x
e sin x e sin 2 x
5 2 –2 0 –1 1
(i) X + Y = (ii) 2X − 3Y =
– 10 –18
7.
12 0 1 –11
