The document contains a list of 31 multi-part math problems involving matrices, integrals, functions, geometry, and equations. The problems cover topics like finding the composition and inverse of functions, evaluating definite integrals, solving systems of equations using matrices, finding maximums and minimums, and proving geometric properties.

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
.