![CHAPTERS 1 – 8 ASSIGNMENT CLASS – XII
6 marks questions:
1. f : R → R & g : R → R two functions f(x) = 3x + 1 & g(x) = 4x – 2 find fog& gof and show fog & gofis both
one – one & onto.
2. UsingElementaryoperation, findinverse ofA = [
𝟏 𝟑 −𝟐
−𝟑 𝟎 −𝟏
𝟐 𝟏 𝟎
].
3. A = [
𝟏 −𝟏 𝟎
𝟐 𝟑 𝟒
𝟎 𝟏 𝟐
]& B = [
𝟐 𝟐 −𝟒
−𝟒 𝟐 −𝟒
𝟐 −𝟏 𝟓
], findAB. Hence solve the systemof equation
x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7.
4. Findthe matrix A : [
𝟐 𝟏
𝟑 𝟐
] A [
−𝟑 𝟐
𝟓 −𝟑
] = [
𝟏 𝟐
𝟐 −𝟏
] .
5. If xy
+ yx
+ xx
= loga find
𝒅𝒚
𝒅𝒙
.
6. Findall the local maximum/ minimumpoints& valuesof functionf(x) = sin4
x+ cos4
x inthe interval [0,
𝝅
𝟐
]
7. Show that the semi – vertical angle of cone maximum volume & ofgivenslant height is 𝐜𝐨𝐬−𝟏(
𝟏
√ 𝟑
) .
8. Show that ofall the rectanglesof givenarea, the square has the smallestperimeter.
9. If A = [
𝟏 −𝟐 𝟑
𝟎 −𝟏 𝟒
−𝟐 𝟐 𝟏
], find (AT
)-1
.
10. Evaluate :- ∫ ( √ 𝒕𝒂𝒏 𝒙 + √ 𝒄𝒐𝒕 𝒙 ) 𝒅𝒙
𝝅
𝟐
𝟎 .
11. Findthe area of that part of circle x2
+y2
=16 which isinterior to parabola y2
= 6x.
12. Evaluate :- ∫
𝒙 𝒅𝒙
𝟒− 𝒄𝒐𝒔 𝟐 𝒙
𝝅
𝟐
𝟎 .
13. Prove that :- ∫ √
𝑎−𝑥
𝑎+𝑥
= 𝑎𝜋
𝑎
−𝑎 .
14. Evaluate :- ∫
𝑑𝑥
sin 𝑥+sin2𝑥
.
15. Findthe area betweenx2
+ y2
= 1 & (x – 1)2
+ y2
= 1
16. Findthe area boundedby the line x = y & circle x2
+ y2
= 16 in firstquadrant above x-axis.
17. Solve by matrix method :- x – y + 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2.
18. Usingintegration,findthe area of ABC where A( 2, 3), B(4, 7) & C(6, 2).
19. Evaluate :- ∫ 𝒍𝒐𝒈 𝒔𝒊𝒏 𝒙 𝒅𝒙
𝝅
𝟐
𝟎 .
20. Evaluate b:- ∫ ( 𝑥2 + 𝑒2𝑥) 𝑑𝑥
2
0 as limitas sum.
21. Prove that the area between two parabolas y2
= 4ax & x2
= 4ay is
𝟏𝟔 𝒂 𝟐
𝟑
square units.
22. Prove that the radius of right circular cylinderof greatestCSA which can be inscribedin a cone is halfof that
of cone.
23. Findthe equationsof tangents & normal to the curve x = 1 – cos , y = - sin at =
𝝅
𝟒
.
24. Findthe local maximum& minimumof the functionf(x) = sin x – cos x (0, 2𝝅 ). Alsofind local maximum &
local minimumvalues.
25. If ( 𝐭𝐚𝐧−𝟏 𝒙)2
+ ( 𝐜𝐨𝐭−𝟏 𝒙)2
=
𝟓 𝝅 𝟐
𝟖
, findx.](https://image.slidesharecdn.com/chapters1-86marks-150828085345-lva1-app6892/85/Chapters-1-8-6-marks-1-320.jpg)
![26. The sum of perimeterofcircle & square is k. prove that the sum of theirareas is least whenside of square is
double the radius of circle.
27. Evaluate :- ∫( 𝟐𝒙 + 𝟑)√ 𝒙 𝟐 + 𝟓𝒙 + 𝟔 dx.
28. Evaluate :- ∫ 𝟐𝒙 (
𝟏+𝒔𝒊𝒏 𝒙
𝟏+ 𝒄𝒐𝒔 𝟐 𝒙
)
𝝅
−𝝅 dx.
29. Prove that the product of matrices :- [
𝒄𝒐𝒔 𝟐
𝒔𝒊𝒏 𝟐
𝟐
𝒔𝒊𝒏 𝟐
𝟐
𝒔𝒊𝒏 𝟐
] & [
𝒄𝒐𝒔 𝟐
𝒔𝒊𝒏 𝟐
𝟐
𝒔𝒊𝒏 𝟐
𝟐
𝒔𝒊𝒏 𝟐
], is the null matrix when &
differby an odd multiple of
𝝅
𝟐
.
30. Solve by matrix method :-
𝟏
𝒙
−
𝟏
𝒚
+
𝟏
𝒛
= 𝟒 ,
𝟐
𝒙
+
𝟏
𝒚
−
𝟑
𝒛
= 𝟎 ,
𝟏
𝒙
+
𝟏
𝒚
+
𝟏
𝒛
= 𝟐 .
31. Usingintegration,findthe area of region : {(x,y) : y2
4x, 4x2
+ 4y2
9}.
32. Usingproperties,prove that :- |
−𝒃𝒄 𝒃 𝟐 + 𝒃𝒄 𝒄 𝟐 + 𝒃𝒄
𝒂 𝟐 + 𝒂𝒄 −𝒂𝒄 𝒄 𝟐 + 𝒂𝒄
𝒂 𝟐 + 𝒂𝒃 𝒃 𝟐 + 𝒂𝒃 −𝒂𝒃
| = (ab + bc + ca)3
.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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- 1. CHAPTERS 1 – 8 ASSIGNMENT CLASS – XII 6 marks questions: 1. f : R → R & g : R → R two functions f(x) = 3x + 1 & g(x) = 4x – 2 find fog& gof and show fog & gofis both one – one & onto. 2. UsingElementaryoperation, findinverse ofA = [ 𝟏 𝟑 −𝟐 −𝟑 𝟎 −𝟏 𝟐 𝟏 𝟎 ]. 3. A = [ 𝟏 −𝟏 𝟎 𝟐 𝟑 𝟒 𝟎 𝟏 𝟐 ]& B = [ 𝟐 𝟐 −𝟒 −𝟒 𝟐 −𝟒 𝟐 −𝟏 𝟓 ], findAB. Hence solve the systemof equation x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7. 4. Findthe matrix A : [ 𝟐 𝟏 𝟑 𝟐 ] A [ −𝟑 𝟐 𝟓 −𝟑 ] = [ 𝟏 𝟐 𝟐 −𝟏 ] . 5. If xy + yx + xx = loga find 𝒅𝒚 𝒅𝒙 . 6. Findall the local maximum/ minimumpoints& valuesof functionf(x) = sin4 x+ cos4 x inthe interval [0, 𝝅 𝟐 ] 7. Show that the semi – vertical angle of cone maximum volume & ofgivenslant height is 𝐜𝐨𝐬−𝟏( 𝟏 √ 𝟑 ) . 8. Show that ofall the rectanglesof givenarea, the square has the smallestperimeter. 9. If A = [ 𝟏 −𝟐 𝟑 𝟎 −𝟏 𝟒 −𝟐 𝟐 𝟏 ], find (AT )-1 . 10. Evaluate :- ∫ ( √ 𝒕𝒂𝒏 𝒙 + √ 𝒄𝒐𝒕 𝒙 ) 𝒅𝒙 𝝅 𝟐 𝟎 . 11. Findthe area of that part of circle x2 +y2 =16 which isinterior to parabola y2 = 6x. 12. Evaluate :- ∫ 𝒙 𝒅𝒙 𝟒− 𝒄𝒐𝒔 𝟐 𝒙 𝝅 𝟐 𝟎 . 13. Prove that :- ∫ √ 𝑎−𝑥 𝑎+𝑥 = 𝑎𝜋 𝑎 −𝑎 . 14. Evaluate :- ∫ 𝑑𝑥 sin 𝑥+sin2𝑥 . 15. Findthe area betweenx2 + y2 = 1 & (x – 1)2 + y2 = 1 16. Findthe area boundedby the line x = y & circle x2 + y2 = 16 in firstquadrant above x-axis. 17. Solve by matrix method :- x – y + 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2. 18. Usingintegration,findthe area of ABC where A( 2, 3), B(4, 7) & C(6, 2). 19. Evaluate :- ∫ 𝒍𝒐𝒈 𝒔𝒊𝒏 𝒙 𝒅𝒙 𝝅 𝟐 𝟎 . 20. Evaluate b:- ∫ ( 𝑥2 + 𝑒2𝑥) 𝑑𝑥 2 0 as limitas sum. 21. Prove that the area between two parabolas y2 = 4ax & x2 = 4ay is 𝟏𝟔 𝒂 𝟐 𝟑 square units. 22. Prove that the radius of right circular cylinderof greatestCSA which can be inscribedin a cone is halfof that of cone. 23. Findthe equationsof tangents & normal to the curve x = 1 – cos , y = - sin at = 𝝅 𝟒 . 24. Findthe local maximum& minimumof the functionf(x) = sin x – cos x (0, 2𝝅 ). Alsofind local maximum & local minimumvalues. 25. If ( 𝐭𝐚𝐧−𝟏 𝒙)2 + ( 𝐜𝐨𝐭−𝟏 𝒙)2 = 𝟓 𝝅 𝟐 𝟖 , findx.
- 2. 26. The sum of perimeterofcircle & square is k. prove that the sum of theirareas is least whenside of square is double the radius of circle. 27. Evaluate :- ∫( 𝟐𝒙 + 𝟑)√ 𝒙 𝟐 + 𝟓𝒙 + 𝟔 dx. 28. Evaluate :- ∫ 𝟐𝒙 ( 𝟏+𝒔𝒊𝒏 𝒙 𝟏+ 𝒄𝒐𝒔 𝟐 𝒙 ) 𝝅 −𝝅 dx. 29. Prove that the product of matrices :- [ 𝒄𝒐𝒔 𝟐 𝒔𝒊𝒏 𝟐 𝟐 𝒔𝒊𝒏 𝟐 𝟐 𝒔𝒊𝒏 𝟐 ] & [ 𝒄𝒐𝒔 𝟐 𝒔𝒊𝒏 𝟐 𝟐 𝒔𝒊𝒏 𝟐 𝟐 𝒔𝒊𝒏 𝟐 ], is the null matrix when & differby an odd multiple of 𝝅 𝟐 . 30. Solve by matrix method :- 𝟏 𝒙 − 𝟏 𝒚 + 𝟏 𝒛 = 𝟒 , 𝟐 𝒙 + 𝟏 𝒚 − 𝟑 𝒛 = 𝟎 , 𝟏 𝒙 + 𝟏 𝒚 + 𝟏 𝒛 = 𝟐 . 31. Usingintegration,findthe area of region : {(x,y) : y2 4x, 4x2 + 4y2 9}. 32. Usingproperties,prove that :- | −𝒃𝒄 𝒃 𝟐 + 𝒃𝒄 𝒄 𝟐 + 𝒃𝒄 𝒂 𝟐 + 𝒂𝒄 −𝒂𝒄 𝒄 𝟐 + 𝒂𝒄 𝒂 𝟐 + 𝒂𝒃 𝒃 𝟐 + 𝒂𝒃 −𝒂𝒃 | = (ab + bc + ca)3 .