assignment 6 markers from chapters 1 to 8 for prepration of first term exam

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
.