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. 7 sin x 26. 3 tan x 27. 2 1 log x x x x 28. 5 cos sin x x 29. 2 1 9 25 x 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 2sin 2 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 2cos 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
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ASSIGNMENT CLASS XII INDEFINITE INTEGRALS CLASS XII INDEFINITE INTEGRALS Evaluate the following Integrals: 1. e e ex a a x a alog log log 2. 1 1 cos x 3. sin 1 sin x x 4. tan (sec
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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 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