21
ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. log log log x a a x a a e e e 2. 1 1 cos x 3. sin 1 sin x x 4. 1 tan (sec tan ) x x 5. 6 6 2 2 sin cos sin cos x x x x 6. 1 1 sin tan 1 sin x x 7. 1 3 4 3 1 x x 8. 3 2 x x 9. 3 3 sin cos x x 10. 4 cos x 11. cos 2 cos 4 x x 12. 4 2 3 1 x x 13. sin 4 cos 7 x x 14. 1 cos 1 cos x x 15. x x x x e e e e 16. 2 2 2 2 sin 2 sin cos x a x b x 17. sin sin x x a 18. 1 sin sin x a x b 19. 1 cot 1 cot x x 20. 1 1 x e 21. 1 1 x x 22. 2 sin 2 cos x a b x 23. 2 1 2 sec 2 tan 1 x x 24. 2 5 sin cos x x 25. 5 sin x 26. 3 tan x 27. 2 1 log x x x x 28. 5 cos sin x x 29. 2 1 9 25x 30. 4 2 1 1 x x 31. 2 1 3 2 x x 32. 2 1 8 20 x x 33. 2 6 5 x x x e e e 34. 1 1 n xx 35. 4 2 1 x x x 36. 2 1 9 8 x x 37. 2 1 16 6 x x 38. 2 4 2 1 x x x 39. 2 2 3 3 18 x x x 40. 2 2sin2 cos 6 cos 4sin 41. 2 2 2 6 5 x x x 42. 2 2 6 12 x x x 43. a x a x 44. 2 3 1 5 2 x x x 45. 2 2 2 2 1 sin cos a x b x 46. sin sin 3 x x 47. 1 2 3cos2 x 48. 4 4 sin 2 sin cos x x x 49. 1 1 2sin x 50. 1 5 4 cos x 51. 1 3 2sin cos x x 52. 3sin 2 cos 3cos 2sin x x x x 53. 2 log x 54. 1 sin x 55. sin 1 cos x x x 56. 3 sec x

ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

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Page 1: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

ASSIGNMENT CLASS XII INDEFINITE INTEGRALS

Evaluate the following Integrals:

1. log log logx a a x a ae e e 2. 11 cos x

3. sin1 sin

xx

4. 1tan (sec tan )x x

5. 6 6

2 2

sin cossin cos

x xx x 6. 1 1 sintan

1 sinxx

7. 13 4 3 1x x

8. 3

2x

x

9. 3 3sin cosx x 10. 4cos x 11. cos 2 cos 4x x 12. 4

2

31

xx

13. sin 4 cos7x x 14. 1 cos1 cos

xx

15. x x

x xe ee e

16. 2 2 2 2

sin 2sin cos

xa x b x

17.

sinsin

xx a

18.

1sin sinx a x b

19. 1 cot1 cot

xx

20. 11xe

21.

11x x

22. 2

sin 2cos

xa b x

23. 2 1

2

sec 2 tan1

xx

24. 2 5sin cosx x

25. 5sin x 26. 3tan x 27. 21 logx x xx

28.

5cossin

xx

29. 2

19 25x

30. 4

2

11

xx

31. 2

13 2x x

32. 2

18 20x x

33. 2 6 5

x

x xe

e e 34.

1

1nx x 35. 4 2 1

xx x

36. 2

19 8x x

37. 2

116 6x x

38. 2 4

21

xx x

39. 2

2 33 18x

x x

40. 2

2sin 2 cos6 cos 4sin

41. 2

22 6 5

xx x

42. 2

2 6 12x

x x 43. a x

a x

44. 2

3 15 2

xx x

45. 2 2 2 2

1sin cosa x b x

46. sinsin 3

xx

47. 12 3cos 2x

48. 4 4

sin 2sin cos

xx x

49. 11 2sin x

50. 15 4cos x

51. 13 2sin cosx x

52. 3sin 2cos3cos 2sin

x xx x

53. 2log x 54. 1sin x 55. sin1 cosx x

x

56. 3sec x

Page 2: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

57.

1

3 22

sin

1

x

x

58. 2 1tanx x 59. 1

2

2tan1

xx

60. 1

2

sin xx

61. 1 sin1 cos

x xex

62. 2

log1 log

xx

63. 2

21

xx ex

64.

21 1

log logx x

65. cosaxe bx 66. 27 10x x 67. 216 log xx

68. 23 2 1x x x

69. 21 1x x x 70.

2 11 2 3

xx x x

71. 1sin sin 2x x

72. 2

3 12 2

xx x

73. 2

82 4x x

74.

2

2 21 4x

x x 75.

3

3

tan tan1 tan

76.

sin 2

1 sin 2 sinx

x x

77. 5

11x x

78. 2

4 2

11

xx x

79. 2

4

416

xx

80. 4

11x

81. tan x 82. cot x 83. 4 4

1sin cosx x

84. 2

4

11

xx

85.

13 1x x

86. 2

14 1x x

87. 2

11 1x x

88. 2 2

11x x

89. 3 2 1x

x x x 90.

sinsin

xx

91. 2

12

xx ex

92.

1 tanlog cos

xx x

93. 2 2x ax

94. 25 4

x

x x

ee e

95. 4 2

15 16x x

96. 1 2 tan sec tanx x x

97. cos xx ee

x 98. cos log x 99. 2

2sin 2 cos6 cos 4sin

100. 1 1 cos 2tan

1 cos 2xx

Page 3: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

ANSWERS (INDEFINITE INTEGRALS) ( add a constant c to every answer)

1. 1

log 1

x aaa x a x

a a

2. cot cosx ecx 3. sec tanx x 4. 2

4 4x x 5. tan cot 3x x x

6. 2

4 2x x 7. 3 2 3 22 3 4 3 1

27x x 8.

32 4 8log 2

3x x x x 9. 1 3 1cos 2 cos6

32 2 6x x

10. 1 sin 43 2sin 28 4

xx x 11. 1 sin 6 sin 2

2 6 2x x

12. 3

14 tan3x x x 13. 1 1cos11 cos3

22 6x x

14. 2cot2x x 15. log x xe e 16.

2 2 2 2

2 2

1 log sin cosa x b xa b

17. sin log sin cosa x a x a a 18.

sincos .log

sinx a

ec a bx b

19. log cos sinx x

20. log 1 xe 21. 2 log 1x 22. 2

2 log coscosaa b x

b a b x

23. 11 tan 2 tan

2x

24. 3 5 7sin 2sin sin

3 5 7x x x 25.

53 73 s 1cos cos cos

5 7co xx x x

26. 21 tan log sec2

x x

27. 31 log3

x x 28. 4 21 sin sin log sin4

x x x 29. 11 5sin5 3

x

30. 3

12 tan3x x x

31. 1 1log4 3

xx

32. 11 4tan2 2

x

33. 1 1log4 5

x

xee

34. 1 log1

n

nx

n x 35.

211 2 1tan

3 3x

36. 1 4sin5

x

37. 1 3sin5

x

38. 2

1 2 1sin5

x

39. 2 2 3log 3 18 log3 6

xx xx

40. 2 12log sin 4sin 5 7 tan sin 2 41. 2 11 1log 2 6 5 tan 2 34 2

x x x

42. 2 1 33log 6 12 2 3 tan3

xx x x

43. 1 2 2sin xa a xa

44. 2 1 13 5 2 2sin6

xx x

45. 11 tantan a xab b

46. 1 3 tanlog2 3 3 tan

xx

47. 1 5 tan 1log2 5 5 tan 1

xx

48. 1 2tan tan x

49. 1 tan( / 2) 2 3log3 tan( / 2) 2 3

xx

50. 12 tan / 2tan3 3

x

51. 1tan 1 tan2x

52. 5 12 log 3cos 2sin13 13

x x x 53. 2log 2 logx x x x x 54. 1 2sin 1x x x

Page 4: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

55. cot2xx 56. 1 1sec tan log sec tan

2 2x x x x 57. 1 2

2

1sin | log 121

x x xx

58. 3

1 2 21 1tan log 13 6 6x x x x 59. 1 22 tan log 1x x x 60.

21 1 1sin logxx

x x

61. cot2

x xe 62. log 1

xx

63. 1

xex

64. log

xx

65. 2 2 cos sinaxe a bx b bx

a b

66. 2 11 9 2 72 7 7 10 sin4 8 3

xx x x

67. 2 21 log log 16 8log log log 162

x x x x

68. 3 22 2 27 1 3 11 1 log 12 2 8 2

x x x x x x x x

69. 3 22 2 11 1 5 2 11 2 1 1 sin3 8 16 5

xx x x x x

70. 1 1 1log 1 1 cos log 2 log 36 3 2

x x x x 71. 1 1 2log 1 cos log 1 cos log 1 2cos2 6 3

x x x

72.

5 7 5log 2 log 216 4 2 16

x xx

73. 2 11log 2 log 4 tan2 2

xx x 74. 1 11 2tan tan3 3 2

xx

75. 2 11 1 1 2 tan 1log 1 tan log tan tan 1 tan3 6 3 3

76.

4

2

2 sinlog

1 sin

xx

77. 5

5

1 log5 1

xx

78. 2

2

1 1log2 1

x xx x

79. 2

11 4tan2 2 2 2

xx

80. 2 2

12

1 1 1 2 1tan log2 2 2 4 2 2 1

x x xx x x

81. 1 tan 2 tan 11 tan 1 1tan log2 2 tan 2 2 tan 2 tan 1

82. 1 cot 2cot 11 cot 1 1tan log

2 2cot 2 2 cot 2cot 1

83. 2

11 tan 1tan2 2 tan

xx

84. 2

2

1 2 1log2 2 2 1

x xx x

85. 12 tanx x 86. 11 31 1log tan 124 3 1 3

xx

x

87. 11

xx

88. 21 x

x

89. 2 11 1 1log 1 tan log 14 2 2

x x x 90. cos 2 sin 2 .log sinx x

91. 2

xex

92. log log cosx x 93. 2

2 2 loga x x

x a ax

94. 1 2sin3

xe

95. 2 2

12

1 4 1 13 4tan log8 3 3 16 3 13 4

x x xx x x

96. 2log sec sec tanx x x 97. 2sin xe

98. sin log cos log2x x x 99. 2 12 log sin 4sin 5 7 tan sin 2 100.

2

2x

Page 5: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

ASSIGNMENT CLASS XII DEFINITE INTEGRALS

Evaluate the following:

1. 2

3

0

cos x dx

2. 4

0

1 sin 2x dx

3. 4

20

12 3

dxx x

4. 1

20

25 1

x dxx

5. 2

21

log x dxx 6.

2

21

11

dxx x 7.

2

4

cos 2 log sinx x dx

8.

2

21

1 xx e dxx

9.

2

0

cos1 sin 2 sin

dx

10.

1 2 1

3 220

sin

1

x dxx

11.

24

0

cos x dx

12. 2

0

tan cotx x dx

13. 0

15 4cos

dxx

14. 2

0

12cos 4sin

dxx x

15. 2

0

cos3cos sin

x dxx x

16. 2

4 40

sin 2sin cos

x dxx x

17. 1

20 1

x

xe dxe 18.

1 1

20

tan1

x dxx

19. 4

4

0

sec x dx

20. 1

0

11

x dxx

21.

2

21

11 log

dxx x 22.

0

cos x dx

23. 1

1

xe dx 24.

1

1

1 2 0( ) , where ( )

1 2 0x x

f x dx f xx x

25. 3

0

x dx 26. 2

2

0

x dx 27. 1

1

2 1x dx

28. 2

2

sin cosx x dx

29. 4

4

sin x dx

30.

2

1 3x dx

x x 31. 2

0

sinsin cos

x dxx x

32. 2 2

0

sinsin cos

x dxx x

33. 2

0

sinsin cos

n

n n

x dxx x

34. 2

0

sin 2 log tanx x dx

35. 4

3 4

4

sinx x dx

36.

a

a

a x dxa x

37. 0

tansec cos

x x dxx ecx

38. 1

1 2

0

cot 1 x x dx 39. 1

1

2log2

x dxx

40. 1

12

0

2sin1

x dxx

41. 1

20

log 11

xdx

x

42. 1 2

12

0

1cos1

x dxx

43. 2

0

11 cot

dxx

44.

21

3 220

tan

1

x x

x

45. 2

2

0

cos 2x x dx

46. 2

19

dxx

47. 1 2

20

11

xx dxx

48.

2

30

11 tan

dxx

49. 1

5

0

1x x dx 50. 2 2

0

1a

dxx a x 51.

2 2 2 20

1 dxx a x b

52. 2

0

cos4 2

x xe dx

Page 6: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

53. 0

5

2 5x x x dx

54. If 2

3

0 0

2 sina

x dx a x dx

, find the value of 1a

a

x dx

.

Evaluate the following integrals as limit of sums:

55. 2

0

2 1x dx 56. 4

2

2 1x dx 57. 2

2

0

3x dx 58. 3

2

1

2 5x dx 59. 3

2

1

x x dx

60. 3

2

2

2 1x dx 61. 3

2

0

2 3 5x x dx 62. b

x

a

e dx

ANSWERS

1. 23

2. 2 1 3. 5 3 3log1 3

4. 1 log 65

5. 1 log2 2

e

6. 3 1log 2 log52 2

7. 1 1log 24 8 4

8.

2

2e e 9. 4log

3

10. 1 log 24 2 11. 3

16 12. 2

13. 3 14. 1 3 5log

25

15. 3 1 log320 10 16.

2 17. 1tan

4e

18. 3 2112

19. 43

20. 12 21. log 2

1 log 2 22. 2 23. 2 2e 24. 4

25. 3 26. 5 2 3 27. 52

28. 4 29. 2 2 30. 12

31. 4

32. 1 log 2 12

33. 4 34. 0 35. 0 36. a 37.

2

4

38. log 22 39. 0 40. log 2

2 41. log 2

8 42. log 2

2

43. 4 44. 2 45.

4

46. 3 47. 1

4 2 48.

4 49. 1

42

50. 4 51.

2ab a b

52. 23 2 15

e 53. 63

2 54. 1 or

2 2 55. 6

56. 10 57. 263

58. 823

59. 383

60. 413

61. 932

62. b ae e

Page 7: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

ASSIGNMENT CLASS XII AREAS OF BOUNDED REGIONS

1. Sketch the region bounded by 22y x x and x - axis and find its area. 2. Find the area of the region included between the parabolas 2 24 and 4 , where 0y ax x ay a . 3. Find the smaller area bounded by the curves 2 2 8 andx y y x .

4. Find the area of the region 2, :x y x y x .

5. Find the area of the region 2, :x y x y x .

6. Find the area bounded by the curves 2 4y ax and the lines 2 and axisy a y . 7. Find the area of the region 2 2, : 1x y x y x y .

8. Find the area bounded by the curves 3,y x y x . 9. Using integration, find area of ABC whose vertices have the coordinates: (i) 2,5 , 4,7 and 6,2A B C (ii) 3,0 , 4,5 and 5,1A B C 10. Find the area of the region bounded by the following curves after making a rough sketch: 1 1 , 3, 3, 0y x x x y

11. Sketch the graph of 1y x . Evaluate 1

3

1x dx

. What does this value represent on the graph?

12. Sketch the region common to the circle 2 2 16x y and the parabola 2 6x y . Also, find the area of the region using integration. 13. Find the area bounded by the lines : (i) 4 5, 5 , 4 5y x y x y x (ii) 2 2, 1,2 7x y y x x y

14. Sketch the graph of 2

2 2 when 2( )

2 when 2x x

f xx x

. Evaluate

4

0

( )f x dx . What does this value represent

on the graph?

15. Find the area of the smaller region bounded by the ellipse 2 2

116 9x y and the line 1

4 3x y .

16. Find the area of the region enclosed between the circles 22 2 216 and 4 16x y x y .

17. Draw the rough sketch of 2 21 and 1y x y x and determine the area enclosed by them.

18. Find the area of the region bounded by the curve 21y x , line y x and the positive x axis .

ANSWERS

1. 4 sq. units3

2. 216 sq. units3

a 3. 2 sq. units 4. 1 sq. units6

5. 1 sq. units3

6. 22 sq. units3

a 7. 1 sq. units4 2

8. 1 sq. units2

9 (i). 7sq. units (ii) 9 sq. units2

10. 16sq. units 11. 4 12. 4 3 16 sq. units3 3

13 (i). 15 sq. units

2 (ii) 6 sq. units

14. 62 sq. units3

, This value represents the area of the region bounded by the given curve and x -axis between

0 to 4x . 15. 3 2 sq. units 16. 48 3 sq. units3

17. 8 sq. units

3 18. 1 sq. units

8

Page 8: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

ASSIGNMENT CLASS XII DIFFERENTIAL EQUATIONS

1. Determine the order and degree of each of the following differential equations:

2

2 2

1( ) 9 4 xd yi y ex dx

2 2( ) 1 1 0ii x y dx y x dy

2

( ) 1dy dyiii y x adx dx

2 32

2( ) 0d y dyivdx dx

22 2

22 2( ) 3 logd y dy d yv x

dx dx dx

2 22 2

2 2( ) sind y dy d yvi xdx dx dx

2. Form the differential equations from the following family of curves: 2( )i y c x c 2( )ii y a b x b x 2 2 2( ) 2iii y ay x a

2 2 2( ) 2iv x a y a ( ) sinv y a x b 2( ) x xvi xy Ae Be x 3. Find the differential equation of all the circles in the first quadrant which touch coordinate axes.

4. Show that 2 x xy ae be is a solution of the differential equation 2

2 2 0d y dy ydx dx

.

5. Show that cos siny A nx B nx is a solution of the differential equation 2

22 0d y n y

dx .

6. Show that 1cosm xy e

is a solution of the differential equation 2

2 221 0d y dyx x m y

dx dx .

7. Show that , 0By Ax xx

is a solution of the differential equation 2

22 0d y dyx x y

dx dx .

8. Show that 2x xy e e is a solution of the differential equation 2

'2 3 2 0 , (0) 1, (0) 3d y dy y y y

dx dx .

9. Solve the following differential equations:

2 2 1( ) 1 3 6 cosdyi x x xdx

2( ) x y ydyii e x edx

( ) 1dyiii x y xydx

( )cos 1 cos sin 1 sin 0iv x y dx y x dy ( ) cos logx xv x y dy xe x e dx

2 1( )2 1

dy x yvidx x y

2 2( ) 1 1 0vii x y dx y x dy 2 2( ) 1 1dyviii y x x y

dx

2( ) dy dyix y x a ydx dx

1( ) cos dxx x ydy

2 2( ) 1 1 0, given that 0, when 1xi x y dy y x dy y x

( ) sin 2 , given that (0) 1dyxii y x ydx

2( ) 1 1 log 0,given that when 1, 1xiii y x dx x dy x y

10. Solve the following differential equations:

3 2( )2 3

dy x yidx x y

2 2( ) 2dyii x xy y

dx 3 3 2( ) 0iii x y dy x y dx ( ) tandy yiv x y x

dx x

2 2( ) 3v yx y dx x xy dy 2 2( ) 2 2 0vi xy dx x y dy 2 2( )vii x dy y dx x y dx

( ) log log 1dyviii x y y xdx

2 2( ) 2 2 0 , (1) 2dyix xy y x ydx

( ) sin sin , (1)2

dy y yx x x y ydx x x

Page 9: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

11. Solve the following differential equations:

3( ) 4 8 5 xdyi y edx

3( ) 2 0dyii x y xdx

2 2 2( ) 1 2 2 1dyiii x xy x xdx

2 2( ) 1 2 4dyiv x xy xdx

22

2( ) 1 21

dyv x xydx x

2( )sin cos sin cosdyvi x y x x xdx

cos( )1 sin

dy x y xviidx x

( ) cos sin , 1

2dyviii x y x x x ydx

2( ) cot 2 cot , 02

dyix y x x x x ydx

2( ) 2 sin , (0) 0xdyx y e x ydx

ANSWERS

1. ( ) 2,1 ( )1,1 ( )1,2 ( ) 2,2 ( ) undefined,undefined ( ) 2,undefinedi ii iii iv v vi

2. 3

( ) 4 2dy dyi y x ydx dx

22

2( ) 0d y dy dyii xy x ydx dx dx

2

2 2 2( ) 2 4 0dy dyiii x y xy xdx dx

2 2( ) 2 4 dyiv x y xydx

2

2( ) 0d yv ydx

2

22( ) 2 2d y dyvi xy x x

dx dx

3. 2 2

2 1 dy dyx y x ydx dx

9. 23 1 11( ) 6sin cos

2i y x x x c

3

( )3

y x xii e e c

2

( ) log 12xiii y x c ( ) 1 sin 1 cosiv x y c ( )sin logxv y e x c

4( ) 2 log 3 6 13

vi y x x y c 2 2( ) 1 1vii x y c 2 2( ) 1 1viii x y c

( ) 1ix x a ay cy ( ) tan2

x yx y c

2 2( )1 2 1xi x y 1( ) log 1 cos 22

xii y x

1 1( ) tan 1 log4 2 2

xiii y x

10. 2 2 1( )3log 4 tan yi x y cx

( )ii y cx x y

3

3( ) log3xiii y cy

( ) sin yiv x cx

( ) log 3logyv y x cx

2 3( )3 2vi x y y c

1( ) sin logyvii x cx

( ) log logviii y x cx ( ) , 0,1 log

xix y x ex

( ) log cos yx xx

11. 3 25( )4

x xi y e ce 3( )ii y x cx 2 1 2 2( ) 1 tan 1 1iii y x x x x c x

2

2 24( ) 1 2log 4

2x x

iv y x x x c

2 1( ) 1 log1

xv y x cx

31( ) sin sin3

vi y x x c

22( )2 1 sin

c xvii yx

( ) sinviii y x

22( )

4sinix y x

x

2( ) 1 cosxx ye x

Page 10: ASSIGNMENT CLASS XII INDEFINITE INTEGRALS · 2012-06-20 · ASSIGNMENT CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x

ASSIGNMENT CLASS XII VECTOR ALGEBRA

1. In a regular hexagon ABCDEF, if andAB a BC b

, then express , , , , , ,CD DE EF FA AC AD AE

and CF

in terms of a and b

.

2. If , ,a i j b j k c k i , find a unit vector in the direction of a b c

.

3. The position vectors of the points P, Q and R are 2 3 , 2 3 5 ,7i j k i j k i k respectively. Prove that P, Q and R are collinear.

4. If 2 3 , 2 4 5a i j k b i j k represents two adjacent sides of a parallelogram, find unit vectors

parallel to the diagonals of the parallelogram.

5. Prove that the points , 4 3 , 2 4 5i j i j k i j k are the vertices of a right angled triangle.

6. If the position vectors of the vertices of a triangle ABC are 2 3 , 2 3 , 3 2i j k i j k i j k , prove that ABC is an equilateral triangle.

7. Write the position vector of a point dividing the line segment joining points A and B with position vectors anda b

externally in the ratio 1: 4, where 2 3 4 anda i j k b i j k

.

8. Find the projection of b c

on a

, where 2 2 , 2 2 and 2 4a i j k b i j k c i j k .

9. If 2 and 3 2a i j k b i j k , find the value of 3 . 2a b a b

.

10. Find a vector whose magnitude is 3 units and which is perpendicular to each of the vectors 3 4a i j k

and 6 5 2b i j k .

11. If , anda b c

be three vectors such that 0a b c

and 3, 5, 7a b c

, find angle between

anda b

.

12. If anda b

are vectors such that 2, 3 and . 4, find anda b a b a b a b

.

13. If a and b are unit vectors and is the angle between them, prove that 1sin2 2

a b and

1cos2 2

a b .

14. Show that the points , andA B C with position vectors 2 , 3 5 , 3 4 4i j k i j k i j k respectively, are the vertices of the right triangle. Also, find the remaining angles of the triangle.

15. If 2 3 and 3 2a i j k b i j k , then show that a b

is perpendicular to a b

.

16. Find the angle between the vectors a b

and a b

, if 2 3 and 3 2a i j k b i j k .

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17. Express the vectors 5 2 5a i j k

as sum of two vectors such that one is parallel to the vector 3b i k

and the other is perpendicular to b

.

18. The dot products of a vector with the vectors 3 , 3 2 and 2 4i j k i j k i j k are 0, 5, 8 respectively. Find the vector.

19. Find a unit vector perpendicular to each of the vectors 4 3 and 2 2a i j k b i j k .

20. If 26, 7 and 35, find .a b a b a b

.

21. Find the area of the triangle whose adjacent sides are determined by the vectors 2 5 anda i k

2b i j k .

22. Find the area of the parallelogram whose adjacent sides are determined by the vectors 3 2 anda i j k

3 4b i j k .

23. Find the area of the parallelogram whose diagonals are determined by the vectors 2 3 6 anda i j k

3 4b i j k .

24. Show that points whose position vectors are 5 6 7 , 7 8 9 , 3 20 5a i j k b i j k c i j k are

collinear. 25. Let , 3 , 7a i j b j k c i k

. Find a vector d

such that it is perpendicular to both anda b

,and . 1c d

26. If , ,a b c

are the position vectors of the vertices , and aA B C of ABC respectively, find an expression

for the area of ABC and hence deduce the condition for the points , andA B C to be collinear. 27. If a i j k

, c j k

are given vectors, then find a vector b

satisfying equations a b c

and . 3a b

. 28. If , ,a b c

are three vectors such that 0a b c

, then prove that a b b c c a

.

29. If anda b c d a c b d

, show that a d

is parallel to b c

, where ,a d b c

. 30. If . . and and 0a b a c a b a c a

, then show that b c

.

ANSWERS

1. , , , , , 2 , 2 , 2CD b a DE a EF b FA a b AC a b AD b AE b a CF a

2. 13

i j k 4. 1 3 6 27

i j k , 1 2 869

i j k 7. 113 53

i j k 8. 2 9. 15

10. 2 2i j k 11. 060 12. 5 , 21 14. 1 135 6cos ,cos41 41

16. 2

17. 6 2 , 2i k i j k 18. 2i j k 19. 1 2 23

i j k 20. 7 21. 1 165 sq.units2

22. 10 3 sq.units 23. 1 1274 sq.units2

25. 1 34

i j k 26.

12

ar ABC a b b c c a

; 0a b b c c a

27. 1 5 2 23

b i j k

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ASSIGNMENT CLASS XII THREE DIMENSIONAL GEOMETRY

1. Find the coordinates of the foot of the perpendicular drawn from the point to the line joining (0, 1,3)B and (2, 3, 1)C .

2. Find the vector equation of the line passing through the point (2, 1,1)A , and parallel to the line joining the points ( 1, 4,1)B and (1,2, 2)C . Also, find the Cartesian equation of the line.

3. The cartesian equations of a line are 6 2 3 1 2 2x y z . Find ( )i the direction ratios of the line, ( )ii the cartesian equation of the line parallel to this line and passing through the point (2, 1, 1) .

4. Find the equations of the line passing through the point ( 1, 3, 2) and perpendicular to each of the lines

1 2 3x y z and 2 1 1

3 2 5x y z

.

5. Show that the lines 5 7 34 4 5

x y z

and 8 4 5

7 1 3x y z

intersect each other. Also, find point of

their intersection.

6. Show that the lines 1 1 13 2 5

x y z and 2 1 1

4 3 2x y z

do not intersect each other.

7. Find the foot of perpendicular drawn from the point 1,6,3P on the line 1 21 2 3x y z . Also, find its

distance from P .

8. Find the image of the point 5,9,3 in the line 1 2 32 3 4

x y z .

9. A perpendicular is drawn from the point 0,2,7 to the line 2 1 31 3 2

x y z

. Find

( )i foot of the perpendicular ( )ii length of the perpendicular ( )iii image of the point in the line.

10. Find the coordinates of the point where the line 1 2 32 3 4

x y z meets the plane 4 6x y z .

11. Find the angle between the lines 1 2 3 61 3 2

x y z and 4 3 , 5

3 2x y z

.

12. Find the value of k for which the lines 1 2 33 2 2

x y zk

and 1 1 6

3 1 5x y z

k

are perpendicular to

each other.

13. Find the angles of a ABC whose vertices are 1,3 , 2 , 2,3,5 and 3,5, 2A B C .

14. Show that the angle between any two diagonals of a cube is 1 1cos3

.

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15. Find the shortest distance between the following pair of lines:

( )i 8 9 103 16 7

x y z

and 15 58 2 5

3 16 5x y z

( ) 1 1 1 and 1 2 2 1 2ii r i j k r i j k .

16. Find the shortest distance between the following pair of parallel lines:

1 2 3 2 1 1( ) and1 1 1 1 1 1

x y z x y zi

( ) 2 and 2 4 2 2ii r i j i j k r i j k i j k .

17. Find the equation of the plane passing through the points 0, 1, 1 , 4,5,1 and 3,9, 4A B C .

18. Show that the four points , , ,A B C D with position vectors 4 5 , , 3 9 4i j k j k i j k and

4 i j k respectively are coplanar.

19. A plane meets the coordinate axes at , andA B C such that the centroid of ABC is 3, 4, 6 . Find the equation of the plane.

20. Reduce the equation of the plane 12 3 4 52 0x y z to the normal form, and hence find the length of the perpendicular from the origin to the plane. Write down the direction cosines of the normal to the plane.

21. The position vectors of two points andA B are 3 2i j k and 2 4i j k respectively. Find the vector equation of the plane passing through B and perpendicular to AB

.

22. Find the vector equation of the plane passing through the point 1,2,3 and perpendicular to the line with direction ratios 2,3, 4 .

23. Find the vector equation of the plane through the intersection of the planes . 2 12 0r i j and

. 3 4 0r i j k , which is at a unit distance from the origin.

24. Find the vector equation of the plane passing through the intersection of the planes . 2 7 4 3r i j k

and . 3 5 4 11 0r i j k , and passing through the point 2, 1, 3 .

25. Find the equation of the plane passing through the intersection of the planes 2 3 1 0x y z and 2 3 0x y z , and perpendicular to plane 3 2 4 0x y z . Also find the inclination of this plane with

xy - plane.

26. Find the equation of the plane passing through the line of intersection of the planes 2 3x y z and

5 3 4 9 0x y z , and parallel to the line 1 3 52 4 5

x y z .

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27. Find the equation of the plane passing through the point (1,1,1) and perpendicular to each of the planes

2 3 7 and 2 3 4 0x y z x y z .

28. Find for which the planes . 2 7r i j k and . 3 2 2 9r i j k

are perpendicular to each other.

29. Find the equation of the plane passing through the point 1, 1,2 and 2, 2, 2P Q and perpendicular to the plane 6 2 2 9x y z .

30. Show that the line . 2 2 3 4r i j k i j k is parallel to the plane . 5 5r i j k

. Also, find

the distance between them.

31. Find the vector equation of a line passing through the point with position vector 2 3 5i j k and

perpendicular to the plane . 6 3 5 2 0r i j k . Also, find the point of intersection of this line and the

plane.

32. Find the angle between the line 2 1 33 1 2

x y z

and the plane 3 4 5 0x y z .

33. Find the equation of the plane passing through the points (1, 2,3) and (0, 1,0 ) and parallel to the line 1 2

2 3 3x y z

.

34. Find the equation of the plane passing through the line of intersection of the planes 2 3x y z ,

5 3 4 9 0x y z , and parallel to the line 1 3 52 4 5

x y z .

35. Find the distance of the point (2,3,4) from the plane . 3 6 2 11 0r i j k .

36. Find the distance between the parallel planes . 2 3 6 5r i j k and . 6 9 18 20 0r i j k

.

37. Find the length and the foot of perpendicular from the point (1,1,2) to the plane . 2 2 4 5 0r i j k .

38. Find the image of the point (1, 3, 4)P in the plane 2 3 0x y z .

39. Prove that the image of the point (3, 2,1) in the plane 3 4 2x y z lies on the plane 4 0x y z .

40. Find the distance of the point (2,3,4) from the plane 3 2 2 5 0x y z , measured parallel to the line 3 2

3 6 2x y z

.

41. Find equation of the plane which contains two parallel lines 3 4 13 2 1

x y z and 1 2

3 2 1x y z

.

42. Find the vector and cartesian forms of the equation of the plane containing two lines 2 4 2 3 6r i j k i j k

and 3 3 5 2 3 8r i j k i j k .

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43. Find the equation of the plane containing two lines 2r i j i j k

and

2r i j i j k . Find the distance of this plane from the origin and also from the point (1, 1, 1) .

44. Prove that the lines 1 2 32 3 4

x y z and 2 3 4

3 4 5x y z

are coplanar. Also, find the equation

of the plane containing these two lines.

ANSWERS

1. 5 2 19, ,3 3 3

D

2. 2 2 2r i j k i j k , 2 1 1

2 2 1x y z

3. 2 1 1( )1, 2,2 ( )1 2 3

x y zi ii

4. 1 3 22 7 4

x y z

5. 1,3,2 7. 1,3,5 ; 13 unitsN 8. 1, 1, 11 9. 3 1( ) , , 4

2 2i

70( ) units2

ii ( ) 3, 3,1iii 10. 1,1,1 11. 2 12. 10

7k

13. 0 1 11 290 , cos , cos33

A B C 15. 5 2( )1.4units ( ) units

2i ii 16. ( ) 26 unitsi

1( ) 66 units6

ii 17. 5 7 11 4 0x y z 19. 4 3 2 36x y z 20. 12 3 44, , ,13 13 13

21. . 2 3 6 28 0r i j k 22. . 2 3 4 4 0r i j k

23. . 2 2 3 0r i j k , . 2 2 3 0r i j k

24. . 15 47 28 7r i j k 25. 7 13 4 9x y z , 1 4cos

234

26. 7 9 10 27x y z

27. 17 2 7 12 0x y z 28. 2 29. 2 4 0x y z 30. 10 units3 3

31. 2 3 5 6 3 5r i j k i j k , 76 108 170, ,

35 35 35

32. 1 7sin52

33. 6 3 3x y z

34. 7 9 10 27 0x y z 35. 1 unit 36. 5 units3

37. 13 6 1 25 1units , , ,12 12 12 6

38. ( 3,5, 2)Q 40. 7 units 41. 8 26 6 0x y z 42. . 6 28 12 98 0r i j k ,

3 14 6 49 0x y z

43. 10 ; 0 units , units3

x y z 44. 2 0x y z

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ASSIGNMENT CLASS XII LINEAR PROGRAMMING

1. A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foos A and B are available at the cost of Rs 5 and Rs 4 perpendicular unit respectively. One unit of the food A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of food B contains 100 units of vitamins, 2 units of minerals and 40 units of calories. Find what combination of the foods A and B should be used to have least cost. 2. Every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates. The corresponding values for rice are 0.05 gm and 0.5 gm respectively. Wheat costs Rs. 4 per kg and rice Rs. 6 per kg. The minimum daily requirements of proteins and carbohydrates for an average child are 50 gms and 200 gms respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost. Frame an L.P.P. and solve it graphically. 3. A furniture firm manufactures chairs and tables, each requiring the use of three machines A, B and C. Production of one chair requires 2 hours on machine A, 1 hour on machine B and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as L.P.P. and solve it graphically. 4. A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are as follows:

Machine Area occupied (in m2)

Labour force Daily output (in units)

A 1000 12 men 60

B 1200 8 men 40

He has maximum area of 9000 m2 available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximize the daily output? 5. An oil company requires 12000, 20000 and 15000 barrels of high-grade, medium-grade and low-grade oil, respectively. Refinery A produces 100, 300 and 200 barrels perpendicular day of high-grade, medium-grade and low-grade oil, respectively, while refinery B produces 200, 400 and 100 barrels perpendicular day of high-grade, medium-grade and low-grade oil, respectively. If the refinery A costs Rs 400 per day and refinery B costs Rs 300 per day to operate, how many days should each be run to minimize costs. 6. A manufacturer produces two types of steel trunks. He has two machines A and B. The first type of trunk requires 3 hours on machine A and 3 hours on machine B. The second type of trunk requires 3 hours on machine A and 2 hours on machine B. Machines A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of Rs 30 and Rs 25 per trunk of the first type and second type respectively. How many trunks of each type must he make each day to make the maximum profit? 7. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5760 to invest and has space for at most 20 items. A fan costs him Rs 360 and a sewing machine costs him Rs 240. He expects to sell a fan a profit of Rs 22 and a sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he buys, how should he invest his money to maximize the profit? what is the maximum profit? 8. If a young man drives his vehicle at 25 km/hr, he has to spend Rs 2 perpendicular km on petrol. If he rides at a faster speed of 40 km/hr, the petrol cost increases at Rs 5 perpendicular km. He has Rs 100 to spend on petrol and wishes to find what is the maximum distance he can travel in one hour. Express this as an LPP and solve it graphically.

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9. A catering agency has two kitchens to prepare food at two places A and B. From these places ‘Mid-day Meal’ is to be supplied to three different schools situated at P, Q, R. The monthly requirements of the schools are respectively 40, 40 and 50 food packets. A packet contains lunch for 1000 students. Preparing capacity of kitchens A and B are 60 and 70 packets per month respectively. The transportation cost per packet from the kitchens to schools is given below:

How many packets from each kitchen should be transported to school so that the cost of transportation is minimum ? Also find the minimum cost.

10. A company has factories located at each of the two places andP Q . From these locations, a certain commodity is delivered to each of the depots situated at , andA B C . The weekly requirements of the depots are respectively 7, 6 and 4 units of the commodity while the weekly production capacities of the factories at

andP Q are respectively 9 and 8 units. The cost of transportation per unit is given below:

To

From

Cost(in Rs) A B C

P 16 10 15 Q 10 12 10

How many units should be transported from each factory to each depot in order that the transportation cost is minimum. Formulate the above LPP mathematically and the solve it.

ANSWERS

1. 5 units of A and 30 units of B; minimum cost is Rs 145 2. 400 g of wheat and 200 g of rice; minimum cost is Rs 2.80 3. 15 chairs and 25 tables; maximum profit is Rs 1,950 4. 6 machines of type A and no machine of type B OR 2 machines of type A and 6 machines of type B 5. A for 60 days, B for 30 days 6. 3 trunks of each type; maximum profit is Rs 165 7. 8 fans and 12 sewing machines; maximum profit is Rs 392 8. at 25 km/hr – 50/3 km; at 40 km/hr – 40/3 km; maximum distance is 30 km 9. 10, 0, 50 packets from A and 30, 40, 0 packets from B; minimum cost is Rs 400 10. 0, 6 and 3 unis from P and 7, 0 and 1 unis from Q

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ASSIGNMENT CLASS XII PROBABILITY

1. If andA B are two events such that:

1 1 5( ) ( ) , ( ) and ( ) , find ( )and ( )3 4 12

i P A P B P A B P A B P B A

'7 9 4( ) ( ) , ( ) and ( ) , find ( )13 13 13

ii P A P B P A B P A B .

2. A pair of dice is thrown. Find the probability of getting 7 as the sum, if it is known that the second die

always exhibits an odd number.

3. The probability that a student selected at random from a class will pass im mathematics is 45

, and the

probability that he will pass in mathematics and economics is 12

. What is the probability that he will pass

in economics if it is known that he has passed in mathematics? 4. A pair of dice is thrown. If the two numbers appearing on them are different, find the probability tha:

( )i the sum of the numbers is 6 ( )ii the sum of the numbers is 4 or less. 5. Find the probability of drqwing a diamond card in each of the two consecutive draws from a well shuffled

pack of cards, if the card drawn is not replaced after the first draw. 6. A bag contains 19 tickets, numbered from 1 to 19. A ticket is drawn and then another ticket is drawn

without replacement. Find the probability that both tickets will show even numbers. 7. Two cards are drawn without replacement from a pack of 52 cards. Find the probability that:

( )i both are kings ( )ii the first is a king and the second is an ace. 8. A bag contains 10 white and 15 black balls. Two balls are drawn succession without replacement. Find the

probability that the first is white and the second is black ball? 9. A die is rolled. If the outcome is an odd numbers, what is the probability that it is a prime? 10. A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional

probability that the number 5 has appeared atleast once? 11. Given two independent events andA B such that ( ) 0.3 and ( ) 0.6P A P B , find:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )i P A B ii P A B iii P A B iv P A B ( ) ( ) ( ) ( ) ( ) ( )v P A B vi P A B vii P B A 12. A coin is tossed thrice. Let the event E be ‘the first throw results in a hea’, and the event F be ‘the last

throw results in tail’. Find whether the events E and F are independent? 13. Sumit and Nishu appear for an interview for two vacancies in a company. The probabilities of their

selection are respectively 1 1and5 6

. what is the probability that:

( )i both of them are selected ( )ii only one of them is selected ( )iii none of them is selected? 14. A and B appear for an interview for two posts. The probabilities of their selection are respectively

1 2and3 5

. What is the probability that only one of them will be selected?

15. A can solve 90% of the problems given in a book, and B can solve 70%. What is the probability that atleast one of them will solve a problem selected at random from the book?

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16. A speaks the truth in 60% of the cases, and B in 90% of the cases. In what percentage of cases are they

likely to contradict each other in stating the same fact? 17. A problem in mathematics is given to three students whose chances of solving it correctly are

1 1 1, and2 3 4

respectively. What is the probability that only one of them solves it correctly?

18. An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the

probability of getting: ( )i 2 red balls ( )ii 2 blue balls ( )iii one red and one blue ball. 19. A can hit a target 4 times in 5 shots, B 3 times in 4 shots, and C 2 times in 3 shots. Find the probability

that: ( )i A, B, C all may hit ( )ii B, C may hit and A may lose ( )iii any two of A, B, and C will hit the target ( )iv none of them will hit the target? 20. Two persons A nad B throw a coin alternately till one of them gets a ‘head’ and wins the game. Find their

respective probabilities of winning if A starts first. 21. A speaks the truth 8 times out of 10 times. A die is thrown. He reports that it was 5. What is the

probability that it was actually 5? 22. In a bulb factory, machines A, B and C manufatures 60%, 30% and 10% bulbs respectively. 1%, 2% and

3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.

23. A candidate has to reach the examination centre in time. Probability of him going by bus or scooter or by

other means of transport are 3 1 3, ,10 10 5

respectively. The probability that he will be late is 1 1and4 3

respectively, if he travels by bus or scooter. But he reaches in time if he uses any other mode of transport. He reached late at the centre. Find the probability that he travelled by bus. 24. Two bags A and B contain 4 white 3 black balls and 2 white and 2 black balls respectively. From bag A two balls are transferred to bag B. Find the probability of drawing: ( )i 2 white balls from bag B ? ( )ii 2 black balls from bag B ? ( )iii 1 white & 1 black ball from bag B? 25. In a bolt factory machines, A, B and C manufacture respectively 25%, 35% and 40% of the total bolts. Of their output 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from product.

( i ) What is the probability that the bolt drawn is defective ? ( ii ) If the bolt is found to be defective find the probability that it is a product of machine B.

26. A letter is known to have come from either TATANAGAR or CALCUTTA. On the envelope just two consecutive letters TA are visible. What is the probabiity that the letter has come from: ( )i Tata nagar ( )ii Calcutta 27. Bag A contains 2 white and 3 red balls, and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and it is found to be red. Find probability that it was drawn from bag B. 28. Three urns A, B and C contain 6 red and 4 white; 2 red and 6 whit; and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the drawn ball is found to be red, Find the probability that the ball was drawn from the urn A. 29. A company has two plants to manufacture bicycles. The first plant manufatures 60% of the bicycles and the second plant, 40%. Also, 80% of the bicycles are rated of standard quality at the first plant and 90% of standard quality at the second plant. A bicycle is picked at random and found to be of standard quality. Find the probability that it comes from the second plant.

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30. A factory has three machines, X, Y and Z, producing 1000, 2000 and 3000 bolts perpendicular day respectively. The machine X produces 1% defective bolts, machine Y produces 1.5% defective bolts and machine Z produces 2% defective bolts. At the end of the day, a bolt is drawn at random and it is found to be defective. Find the probability that this defective bolt has been produced by the machine X? 31. A random variable X has the following probability distribution:

ix 2 1 0 1 2 3

ip 0.1 k 0.2 2 k 0.3 k ( )i Find the value of k ( )ii Find mean of X ( )iii Find variance of X. 32. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of successes. 33. A football match may be either won, drawn or lost by the host country’s team. So there are three ways of forecasting the result of any one match, one correct and two incorrect. Find the probability of forecasting at least three correct results for four matches. 34. Let X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that:

if 0or1

( ) 2 f 25(5 ) if 3or 4

kx xP X x kx i x

x x

k is a positive integer.

( )i Find the value of k ( )ii What is the probability that you will get admission in exactly 2 colleges? ( )iii Find the mean and variance of the probability distribution. 35. Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement from a bag containing 4 white and 6 red balls. Also find the mean and variance of the distribution.

36. A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of number of successes.

37. The probability of hitting a target by A is 15

. If he fires 5 times, find the probability that he will hit atleast

two times.

38. Two cards are drawn successively with replacement from a pack of 52 cards. Find the mean and variance of the number of kings.

39. A coin is tossed 4 times. Let X denote the number of heads. Find the mean and variance of X.

40. 3 defective bulbs are mixed with 7 good ones. Let X be the number of defective bulbs when 3 bulbs are drawn at random. Find the mean and variance of X.

41. An unbiased coin is tossed 8 times. Find, by using binomial distribution, the probability of getting atleast 3 heads.

42. The probability of a man hitting a target is 1/4. How many times must he fire so that the probability of his hitting the target at least once is greater than 2/3?

43. Six coins are tossed simultaneously. Find the probability of getting: ( )i 3 heads ( )ii no head ( )iii atleast one head ( )iv not more than 3 heads

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44. The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university: ( )i none will graduate ( )ii only one will graduate ( )iii all will graduate

45. A pair of dice is thrown 7 times If getting a total of 7 is considered as success, Find the probability of: ( )i no success ( )ii 6 successes ( )iii atleast 6 successes ( )iv atmost 6 successes

46. Find the binomial distribution for which the mean and variance are 12 and 3 respectively.

ANSWERS

2 1 51.( ) , ( )3 2 9

i ii 2. 16

3. 58

4. 2 2( ) ( )15 15

i ii 5. 117

6. 419

7. 1 4( ) ( )221 663

i ii 8. 14

9. 23

10. 25

11. ( )0.18 ( ) 0.12 ( )0.42i ii iii

( )0.28 ( )0.72 ( ) 0.3 ( ) 0.6iv v vi vii 12. yes 13. 1 3 2( ) ( ) ( )30 10 3

i ii iii 14. 715

15. 0.97 16. 42% 17. 1124

18. 16 49 56( ) ( ) ( )121 121 121

i ii iii 19. 2 1 13 1( ) ( ) ( ) ( )5 10 30 60

i ii iii iv

20. 2 1,3 3

21. 49

22. 25

23. 913

24. 5 4 4( ) ( ) ( )21 21 7

i ii iii 25. 28( )0.0345 ( )69

i ii

26. 7 4( ) ( )11 11

i ii 27. 2552

28. 3661

29. 37

30. 0.1 31. ( ) 0.1 ( )0.8 ( ) 2.16i ii iii

32.

X 0 1 2 3 4 P(X) 625

1296 500

1296 150

1296 20

1296 1

1296

33. 19

34. 1 1 19 47( ) ( ) ( ) ,8 2 8 64

i k ii iii 35.

37. 8213125

38. 24169

39. 2, 1 1.20, 0.56 36.

40. 0.9, 0.49 41. 219256

42. 4 times 43. 5 1 63 21( ) ( ) ( ) ( )16 64 64 32

i ii iii i 44. ( ) 0.216 ( )0.432( ) 0.064i ii iii

45. 7 7 5 75 1 1 1( ) ( )35 ( ) ( )1

6 6 6 6i ii iii iv

46. 16

16 3 1( ) , where 0,1,2,...,154 4

r r

rP X r C r

X 0 1 2 3 P(X) 1

6 1

2 3

10 1

30

X 0 1 2 P(X) 4/9 4/9 1/9