ANSWERS

ANSWERS

1.3 EXERCISE

1. (i) {2}

(ii) {0, 1}

2. (i) {0, ¨C1, 1}

(ii)

3.

?11

3

(iii)

{?

3, ? 2, 2, 3

}

{1, 2, 22, 23, ...2P ¨C 1,(2p ¨C1}

4. (i) True

(ii) False

7. (i) {2, 4, 6, 8, ... , 98}

8. (i) {4, 8, 12}

9. (i) {4, 5, 6, ....10}

10.

13.

16.

24.

25.

28.

30.

34.

(iii) {1, p}

(ii) {7, 8, 9}

(iii) True

(iv) True

(ii) (1,4, 9, 16, 25, 36, 49, 64, 81,}

(iii)

(ii) {5}

1 3

,1,

2 2

(iv) {0, 1, 2}

(iii) {1, 2, 3, 4, 5}

11.

True

14. False

15. True

True

17. True

22. T = {10}

(i) 2 (ii) 3 (iii) 3 (iv) 9

25

26. 20

27. (a) 3300 (b) 4000

(i) 6, (ii) 3, (iii) 9, (iv) 1, (v) 2, (vi) 6, (vii) 30, (viii) 20 29. C

B

31. B

32. D

33. C

D

35. B

36. B

37. C

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304

38.

42.

46.

49.

51.

53.

57.

EXEMPLAR PROBLEMS ¨C MATHEMATICS

C

39.

B

43.

n (B)

47.

{0, 1, 2, 3, 4, 5, 6, 8}

A ¡È ¦¢¡ä

52.

True

54.

True

58.

C

40. A

41. B

C

44. [1,2]

45. 1

A ¡É B¡ä

48. {¦Õ, {1}, {2}, {1, 2}

50.(i) {1,5, 9, 10 } (ii) { 1, 2,3, 5, 6, 7, 9, 10 }

(i) ?(b) (ii) ?(c) (iii) ? (a) (iv) ?(f) (v) ?(d) (vi) ? (e)

False

55. False

56. True

False

2.3 EXERCISE

1.

(i) {(¨C 1, 1), (¨C1, 3), (2, 1), (2, 3), (3, 1), (3, 3)}

(ii) {(1, ¨C1), (1, 2), (1, 3), (3, ¨C1), (3, 2), (3, 3)}

(iii) {(1, 1), (1, 3), (3, 1), (3, 3)}

(iv) {(¨C1, ¨C1), (¨C1, 2), (¨C1, 3), (2, ¨C1), (2, 2), (2, 3), (3, ¨C1), (3, 2), (3, 3)}

2. {(0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)}

3. (i) {(0, 3), (1, 3)}

(ii) {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1,5)}

4.

5.

6.

7.

8.

9.

10.

(i) a =

2

11

and b =

3

3

(ii) a = 0 and b = ¨C 2

(i) {(1, 4), (2, 3), (3, 2), (4, 1) }

(ii) {(1, 1), (1, 2), (1,3),(2, 1), (2, 2), (3, 1)}

(iii) { (4, 5), (5, 4), (5, 5)}

Domain of R = {0, 3, 4, 5} = Range of R

Domain of R1 = [¨C5, 5 ] and Range of R1 = [¨C3, 17 ]

R2 = {(0, 8), (8, 0) (0,¨C 8), (¨C 8, 0)}

Domain of R3= R and range of R3 = R+ ¡È {0}

(i) h is not a function (ii) f is a function (iii) g is a function

function(v) t is a constant function

11. (a) 6

(b)

1364

4

12. (a) x = 4

(b) x > 4

2

13. (i) (f + g) x = x + 2x + 2

(c) 13

(iv) s is a

(d) t2 _ 4 (e) t + 5

(ii) (f ¨C g) x = 2x ¨C x2

18/04/18

ANSWERS

305

? f ?

2x + 1

(iv) ? ? x = 2

x +1

?g?

(iii) ( f g) x = 2x3 + x3 + 2x + 1

14. (i) f = {(¨C1, 0), (0, 1), (3, 28), (7, 344), (9, 730)}

15. x = ¨C1 ,

4

3

16. Yes, ¦Á = 2 , ¦Â = ?1

17. (i) R ¨C {2n¦Ð : n¡ÊZ}

(iv) R ¨C {¨C1, 1}

18. (i) [

3

,¡Þ)

2

(ii) (¨C ¡Þ, 1]

(ii) R +

(v) R ¨C {4}

(iii) R

(iii) [ 0 , ¡Þ)

(iv) [ ¨C 2, 4]

?2 x , ? 3 ¡Ü x < ? 2

19.

f ( x) =

4 , ? 2¡Ü x < 2

2 ,2¡Ü x¡Ü3

21. (i) (f + g) x =

x+x

(ii) (f ¨C g ) x =

? f ?

1

?x=

x

?g?

3

(iii) (fg) x = x 2

22.

24.

28.

32.

36.

38.

42.

x ¨Cx

(iv) ?

Domain of f = (5, ¡Þ) and Range of

D

25. D

B

29. B

C

33. A

{2, 3, 4, 5}

37. (a) ? (iii) (b)

False

39. False

True.

f = R+

26. B

27.

30. A

31.

34. B

35.

? (iv) (c) ? (ii) (d)

40. True

41.

C

C

A

? (i)

False

3.3 EXERCISE

4.

56

33

5.

2cos x

cos 2 x

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EXEMPLAR PROBLEMS ¨C MATHEMATICS

1

2 +1

8.

16. ¦È = 2n¦Ð +

18. ¦È =

15. ¦È = n¦Ð + (¨C1)n

7¦Ð

4

17. ¦È = 2n¦Ð ¡À

¦Ð 5¦Ð

,

3 3

19. x =

22. 1

25.

23

17

3 ?1 1

+

2

2

¦Ð

4

3

2

27. n¦Ð ¡À

28.

n¦Ð ¦Ð

¡À

2 8

29. ¦È = 2n¦Ð ¡À

30.

34.

38.

42.

46.

50.

54.

C

B

A

A

C

C

A

D

C

B

D

C

B

B

58. B

59. D

62. tan ¦Â

63.

x2 ?

32.

36.

40.

44.

48.

52.

56.

¦Ð ¦Ð

+

4 12

D

B

D

C

C

C

A

33.

37.

41.

45.

49.

53.

57.

C

C

D

B

B

C

B

60. 1

61.

1

8

1

[4 ¨C 3(a2 ¨C 1)2] ,

4

2

x + 1 65. 13

sin 2A

68. True

69. False

72. False

73. True

76. (a) ? (iv) (b) ? (i) (c) ? (ii) (d)

64.

¦Ð

3

¦Ð ¦Ð ¦Ð

, ,

6 4 2

26.

31.

35.

39.

43.

47.

51.

55.

¦Ð ¦Ð

?

4 4

2 ? a2

66. [¨C3, 3]

67. 2

70. False

74. True

? (iii)

71. True

75. True

18/04/18

ANSWERS

307

4.3 EXERCISE

1. P(n) : 2n < ¡Ïn

2. P(n) : 1 + 2 + 3 + ... + n =

26. A

29. 4

27. B

30. False

28.

n (n + 1)

2

A

5.3 EXERCISE

1.

2n

2. ¨C1 + i

5.

(1, 0)

6. icot

13. 1:3

2 ¡À i 2, ? 2 ¡À i 2

23.

2 cos

(a

¦È

2

11.

10

2

,0 ,

3

3

14.

21.

25. (i)

3. (0, ¨C 2)

3

¨C 2i

2

12.

15. 1

2

5

1

¨C 2i

2

18. 0

22. ¨C2 ¨C i

5¦Ð

5¦Ð

+ i sin

12

12

2

+ b2

)( z

1

(iii) ¨C2 (iv) 0 (v)

2

+ z2

2

)

(ii) ¨C15

1 i

?

2 2

(vi) z1 (vii) 0

(viii) 6 and 0 (ix) a circle

26. (i)

(v)

27. (a)

(d)

(g)

4.

F

T

? (v),

? (iv),

? (viii) and

(ii)

(vi)

(b)

(e)

(h)

(x) ¨C 2 3 + 2i

F

T

? (iii),

? (ii),

? (vii)

(iii)

(vii)

(c)

(f)

T

F

? (i),

? (vi),

(iv) T

(viii) F

18/04/18

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