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 cos 7x x 14. 1 cos1 cos
xx
15. x x
x x
e 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. 7sin 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 x
ee e
34.
11nx x
35. 4 2 1x
x 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
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.
2
2
11
xx
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
Some More Problems:
1. Integrate the following functions with respect to x :
(i) sin 3sin 5 sin 2
xx x
(ii) cos sin1 sin 2
x xx
(iii) 5cos
sinx
x (iv) 1
2 3x x
(v) 2 sin 4 21 cos 4
x xex
(vi) 1 11 1
x xx x
(vii)
1
1 1x xe e (viii)
2
2
1.
1x
xe dx
x
2. Evaluate the following:
( ) tan tan 2 tan 3i x x x dx cos( )( )sin( )
x aii dxx b 1( ) sin xiii dx
a x
1( )1
xiv dxx
2
1( )sin sin 2
v dxx x 2( )vi x x x dx
2
3 3( )
1 2xvii dx
x x sin( )sin 4
xviii dxx
2
2
1( )1 2
x xix dxx x
2 2
2 2
1 4( )
3 5
x xx dx
x x
4 2
1( )5 16
xi dxx x
3
3( )
1
xx exii dx
x
2
2( )sin cos
xxiii dxx x x
ANSWERS (INDEFINITE INTEGRALS) ( add a constant c to every answer)
1. 1
log 1
x aaa x a x
a a
2. tan2x
3. sec tanx x x 4. 2
4 4x x 5. tan cot 3x x x
6. 2
4 4x 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. 2log 1x 22. 2
2 log coscos
aa b xb a b x
23. 11 tan 2 tan2
x
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 sec
2x 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
x
ee
34. 1 log1
n
n
xn x
35. 2
11 2 1tan3 3
x
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 x
a
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
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 sinxx
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. 21 2 log 11
x xx
96. 2log sec sec tanx x x 97. 2sin xe
98. sin log cos log2x x x 99. 2 12log sin 4sin 5 7 tan sin 2 100.
2
2x
Answers (Some More Problems):
1. (i) 1 1log sin 2 log sin 52 5
x x (ii) 1 log sec tan4 42
x x
(iii) 4
2 sinlog sin sin4
xx x (iv) 3 2 3 22 1 33
x x (v) 21 cot 22
xe x
(vi) 2 2 21 11 log 12 2
x x x x x (vii) 1 1log2 1
x
x
ee
(viii) 11
x xex
2. 1 1 1( ) log cos3 log cos 2 log cos3 2 2
i x x x ( )cos( ) log sin( ) sin( )ii a b x b a b x
1 1( ) tan tanx xiii x ax aa a
1( ) 2 1 1 siniv x x x x
1 tan( ) log2 tan 2
xvx
3
2 2 221 1 1 1( ) ( ) (2 1) log3 8 16 2
vi x x x x x x x x
3
3
1 1( ) log3 2
xviix
1 1 2 sin 1 1 sin( ) log log8 1 sin4 2 1 2 sin
x xviiixx
1( ) 2 log 1 3log 21
ix x xx
11 27 5( ) tan log4 3 3 8 5 5
x xx xx
2 21
2
1 4 1 13 4( ) tan log8 3 3 16 13 13 4
x x xxix x x
2( )
1
xexiix
sec( ) tansin cos
x xxiii xx x 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 2
0
1 dxx a x b
52. 2
0
cos4 2
x xe dx
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
Some More Problems: Evaluate the following definite integrals:
2
0
sin( )1 cosx xi dx
x
0
( )3 2sin cos
dxiix x
20
sin( )1 cos
x xiii dxx
3 10
5
sin( )sin cos
xiv dxx x
20
( )1 cos
xv dxx
2
0
( ) log sinvi x dx
2
2 20
( )9sin 4cos
dxviix x
1
99
0
( ) 1viii x x dx 2
30
cos( )cos sin
2 2
xix dxx x
cos
cos cos0
( )x
x x
ex dxe e
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 5log25
15. 3 1 log320 10 16.
2 17. 1tan
4e 18. 3 21
12 19. 4
3
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 1
2 33.
4
34. 0 35. 0 36. a 37. 2
4 38. log 2
2 39. 0 40. log 2
2 41. log 2
8
42. log 22
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. 26
3
58. 823
59. 383
60. 413
61. 932
62. b ae e
Answers (Some more problems):
2 2 1( ) log 2 ( ) ( ) ( ) ( ) ( ) log 2 ( ) ( ) ( ) 2 2 1 ( )
2 4 4 20 2 12 10100 22 2i ii iii iv v vi vii viii ix 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. 11. 4 12. 4 3 16 sq. units3 3
13 (i). 15 sq. units
2 (ii) 6 sq. units 14. 62 sq. units
3, 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
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 2x 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
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
12. Solve the following differential equations:
2 2( ) 1 1 0; (0) 1x xi e dy y e dx y ( ) 1 1dyii x ydx
2( ) 1 dyiii x xy axdx
2 2 2( ) 0iv x x y y dx xy dy
2( ) 2 ; (2) 1dyv x y y ydx
3 22 2
4 1( ) 01 1
dy xyvidx x x
2 2 2 2( ) 1 0dyvii x y x y xydx
( ) 1 1 ; 1x ydyviii x e xdx
2 2( ) log log 0y yix xy dx y x dyx x
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
12. 1 1( ) tan tan4
xi y e ( ) 2yii x Ce y 2( ) 1iii y a C x
2 2( ) log Civ x y xx
2( ) 2v x y 2
22 21 1( ) . 1 log 12 2
x xvi y x x x
22 21 1
( ) log 1 1x
vii x y cx
( ) 1y xviii e x e c 2 2( ) 1 2 log 4 log 0yix x y y c
x
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
12. 1 1( ) tan tan4
xi y e ( ) 2yii x Ce y 2( ) 1iii y a C x
2 2( ) log Civ x y xx
2( ) 2v x y 2
22 21 1( ) . 1 log 12 2
x xvi y x x x
22 21 1
( ) log 1 1x
vii x y cx
2sin( ) cos3
xviii y c ecx
( ) 1y xix e x e c 2 2( ) 1 2 log 4 log 0yx x y y cx
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 .
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
ˆ3 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
.
31. If , ,a b c
are three unit vectors such that . . 0a b a c
and the angle between andb c
is 6 , prove that
2a b c
.
32. If a b a b , prove that anda b
are perpendicular to each other.
33. If ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2 3 , 2 and 2a i j k b i j k c i j k , verify that . .a b c a c b a b c
.
34. If the position vectors of the vertices , , andA B C D of a quadrilateral are ˆ ˆˆ ˆ ˆ ˆ, 2 4 ,i j k i j k ˆ ˆˆ ˆ ˆ ˆ5 5 and 2 2 5i j k i j k respectively, then show that ABCD is a square.
35. The volume of the parallelopied whose co-terminus edges are ˆ ˆ ˆˆ ˆ ˆ ˆ12 , 3 and 2 15i k j k i j k is 546 cubic units, find the value of .
36. Find x such that the four points 3, 2,1 , 4, ,5 , 4, 2, 2 and 6,5, 1A B x C D are coplanar.
37. If the vectors ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2 , 2 3 and 3 5a i j k b i j k c i j k are coplanar, find .
38. Show that the four points whose poition vectors are ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ6 7 , 16 19 4 , 3 6 and 2 5 10i j i j k j k i j k are coplanar.
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 35. 3 36. 5x 37. 2
ASSIGNMENT CLASS XII THREE DIMENSIONAL GEOMETRY
1. Find the coordinates of the foot of the perpendicular drawn from the point 1,8,4 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
.
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 6 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 .
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 22 3 3
x 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 23 6 2
x 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
.
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.
45. Find distance of a point 1, 5, 10 from the point of intersection of the line
ˆ ˆˆ ˆ ˆ ˆ2 2 3 4 2r i j k i j k and the plane ˆˆ ˆ. 5r i j k
.
46. Find image of the line 1 3 43 1 5
x y z
in the plane 2 3 0x y z .
47. Find equation of plane through ˆˆ ˆ3 2i j k and parallel to line ˆ ˆˆ ˆ ˆ3 2 5r j k i j k
and
ˆˆ ˆ ˆ ˆ3 5 4r i j k i j .
48. Find the equation of the plane through the point 1,1,1 and containing the line
ˆ ˆˆ ˆ ˆ ˆ3 5 3 5r i j k i j k . Also, show that the plane contains the line ˆ ˆˆ ˆ ˆ ˆ2 5 2 5r i j k i j k
.
49. Find the equation of the plane mid-parallel to the planes3 4 12 26 0 and 3 4 12 13 0x y z x y z .
50. (1,5, 6) , (0, 11,7) and (2, 3,1)A B C are the vertices of a triangle and D is the foot of perpendicular
from onA BC . Find the coordinates of D .
51. Find the equation of a line passing through the point 1,3, 2 and perpendicular to the lines
2 1 1and1 2 3 3 2 5x y z x y z
.
52. A perpendicular is drawn from the point 1,6,3 to the line 1 21 2 3x y z . Find:
( )i foot of the perpendicular ( )ii length of the perpendicular ( )iii image of the point in the line.
53. Show that lines ˆ ˆ ˆ3 and 4 2r i j k i j r i k i k intersect. Find their point of
intersection.
54. Find the length and the equations of the line of shortest distance between the lines
1 2 3 2 4 5and2 3 4 3 4 5
x y z x y z .
55. Show that the four points (2,3, 4) , ( 3,5,1) , (4, 1, 2) and (2,0,1)A B C D are coplanar. Also find the
equation of the plane containing them.
56. Find the length and the foot of the perpendicular from the point (7,14,5) to the plane 2 4 2x y z .
57. Show that the line ˆ ˆ2 2 3 4r i j k i j k is parallel to the plane ˆ. 5 5r i j k
. Also find the
distance between them.
58. Find the equation of the plane parallel to the line 2 1 31 3 2
x y z , which contains the point 5, 2, 1
and passes through the origin.
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, 2,3 ( )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 4cos234
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 45. 13 units
46. 3 5 23 1 5
x y z
47. ˆˆ ˆ. 4 5 17 27 0r i j k
48. ˆˆ ˆ. 2 0r i j k
49. 6 8 24 13 0x y z 50. (4,5, 5)D 51. 1 3 22 7 4
x y z
52.
( ) 1,3,5 ( ) 13 units ( ) 1,0,7i ii iii 53. 4,0, 1 54. 6 3 5 3 3 13units ,6 3 2 3
x y z
55. 1x y z 56. (1, 2,8) ,3 21 units 57. 10 units3 3
58. 7 11 13 0x y z
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?
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.
30. A factory has three machines, X, Y and Z, producing 1000, 2000 and 3000 bolts per 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
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.
47. Two integers are selected at random from integers 1 through 11. If the sum is even, find the probability
that both the numbers selected are odd.
48. An urn contains 4 white and 6 red balls. Four balls are drawn at random (without replacement) from the
urn. Find the probability distribution of number of white balls.
49. In answering a question on MCQ test with 4 choices per questions, a student knows the answer, guesses or
copies the answer. Let 1 2 be the probability that he knows the answer, 1 4 be the probability that he guesses
and 1 4 that he copies it. Assuming that a student, who copies the answer will be correct with the probability
3 4 , what is the probability that the student knows the answer, given that he answered it correctly.
50. An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two balls are
drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the
probability that it is a white ball.
51. From a bag containing 20 tickets, numbered from 1 to 20, two tickets are drawn at random. Find the
probability that: ( )i both the tickets have the prime numbers on them
( )ii on one there is a prime number and on the other there is a multiple of 4.
52. A class consists of 10 boys and 8 girls. Three students are selected at random. What is the probability that
the selected group has:
( )i all boys ( )ii all girls ( )iii 2 boys and 1 girl ( )iv atleast 1 girl ( )v atmost 1 girl.
53. One bag contains 5 white and 6 black balls. Another bag contains 7 white and 3 black balls. One ball at
random is transferred from the first bag to the second bag and then a ball is drawn from the second bag. Find
the probability that the ball drawn is white.
54. For A, B and C, the chances of being selected as the managers of a firm are in the ratio 4:1:2 respectively.
The respective probabilities for them to introduce a radical change he market strategy are 0.3, 0.8 and 0.5
respectively. If the changes take place, find the probability that it is due to the appointment of B or C.
55. An urn contains 5 white, 7 red and 8 black balls. If four balls are drawn one by one with replacement,
what is the probability that:
( )i all are white? ( )ii only 3 are white? ( )iii none is white ( )iv atleast 3 are white?
56. Three urns I , II and III contains 1 white, 2 black, 3 red balls ; 2 white, 1 black, 1 red balls ; 4 white, 5
black, 3 red balls respectively. One urn is chosen at random and two balls are drawn. They happen to be white
and red. What is the probability that they come from urn I , II or III ?
57. Two urns contain 1 white, 6 red and 4 white, 3 red balls. One of the urns is selected at random and a ball is
drawn from it. Find :
(i) The probability of drawing a white ball
(ii) The probability of drawing the ball from first urn if ball drawn is white.
58. A and B take turn in throwing two dice. The first to throw sum 9 being awarded. Show that if A has the
first throw, their chances of winning are in the ratio 9 : 8.
59. An experiment succeeds twice as often as it fails. Find the probability that in next six trials, there will be
at least 4 successes.
60. A and B throw two dice simultaneously turn by turn. A will win if he throws a total of 5, B will win if he
throws a doublet. Find the probability that B will win the game, though A started it.
ANSWERS (PROBABILITY)
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
47. 3 5 48. 49. 2 3 50. 59 130
51. 14 4( ) ( )95 19
i ii 52. 5 7 15 29 10( ) ( ) ( ) ( ) ( )34 102 34 34 17
i ii iii iv v 53. 82121
54. 35
55. 1 3 81 13( ) ( ) ( ) ( )256 64 256 256
i ii iii iv 56. 33 55 30, ,118 118 118
57. (i) 514
(ii) 15
59. 496729
60. 47
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
X 0 1 2 3 4
P(X) 1 14 8 21 3 7 4 35 1 210
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 per 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? What should be the
qualities of good machine?
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 per km on petrol. If he rides at a faster
speed of 40 km/hr, the petrol cost increases at Rs 5 per 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.
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 & 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