This document provides the details of a mathematics assignment consisting of 40 questions with 4 marks each. The questions cover various topics in calculus, trigonometry, matrices and determinants. Some example questions are: 1) Find dy/dx if y = tan−1[√1+x2 + √1−x2/√1+x2 − √1−x2] 2) Evaluate: ∫(x2 + 5x)dx from 1 to 3 as limit of sum 3) Prove that: |(b + c)2 ba ca| = 2abc(a + b + c)3 4) Solve the system of equations: x + 2y

1. MATHEMATICS
ASSIGNMENT (CHAPTER :- 1 – 8) CLASS – XII
4 MARKS QUESTIONS:
1. y = (𝐭𝐚𝐧−𝟏
𝒙)
𝟐
, show (x2
+ 1)2
y2 + 2x (x2
+ 1)2
y1 = 2.
2. Prove using properties :- |
𝒃 + 𝒄 𝒒 + 𝒓 𝒚 + 𝒛
𝒄 + 𝒂 𝒓 + 𝒑 𝒛 + 𝒙
𝒂 + 𝒃 𝒑 + 𝒒 𝒙 + 𝒚
| = 𝟐 |
𝒂 𝒑 𝒙
𝒃 𝒒 𝒚
𝒄 𝒓 𝒛
| .
3. Evaluate :- ∫
𝟐
(𝟏+𝒙)(𝟏+ 𝒙 𝟐)
dx
4. If (cos x)y
= (cos y)x
, find
𝒅𝒚
𝒅𝒙
.
5. Prove :- 𝐭𝐚𝐧−𝟏
(
𝐜𝐨𝐬 𝒙
𝟏+𝐬𝐢𝐧 𝒙
) =
𝝅
𝟒
−
𝒙
𝟐
.
6. A = R – {3}, B = R – {1}. f : A → B. f(x) =
𝒙−𝟐
𝒙−𝟑
. Show that f is one – one & onto.
7. Prove :- ∫ (√ 𝒕𝒂𝒏 𝒙 + √ 𝒄𝒐𝒕 𝒙 )𝒅𝒙 = √𝟐
𝝅
𝟐
𝝅
𝟐
𝟎
.
8. If a + b + c ≠ 0 & |
𝒂 𝒃 𝒄
𝒃 𝒄 𝒂
𝒄 𝒂 𝒃
| = 0. Prove a = b = c
9. Evaluate :- ∫(𝟐 𝒔𝒊𝒏 𝟐𝒙 − 𝒄𝒐𝒔 𝒙)√𝟔 − 𝒄𝒐𝒔 𝟐 𝒙 − 𝟒 𝒔𝒊𝒏 𝒙 dx.
10. Differentiate :- log (xsinx
+ cot2
x), w.r.t. x
11. Evaluate :- ∫
𝟓𝒙
(𝒙+𝟏)( 𝒙 𝟐+ 𝟗)
dx
12. Prove :- y =
𝟒 𝒔𝒊𝒏 𝒙
(𝟐+𝒄𝒐𝒔 𝒙)
- x is an increasing function of x in [0,
𝝅
𝟐
]
13. Evaluate :- ∫ 𝑠𝑒𝑐3
𝑥 𝑑𝑥 .
14. Differentiate :- tan−1
(
𝑎 sin 𝑥+𝑏 cos 𝑥
𝑎 cos 𝑥−𝑏 sin 𝑥
) w.r.t. x
15. If x √ 𝟏 + 𝒚 + 𝒚 √ 𝟏 + 𝒙 = 𝟎. 𝐏𝐫𝐨𝐯𝐞
𝒅𝒚
𝒅𝒙
=
𝟏
(𝟏+𝒙) 𝟐 .
16. If A = [
𝟏 𝟎
−𝟏 𝟕
], find k : A2
= 8A + kI .
17. If f(x) = {
|𝒙|
𝒙
𝒙 ≠ 𝟎
𝟎 𝒙 = 𝟎
. check the continuity at x = 0.
18. If y = ecos x
, prove y2 + sin x y1+ y cos x = 0.
19. Evaluate :- ∫
𝟏
𝒙 𝟒+ 𝟏
dx.
20. Prove :- |
(𝒃 + 𝒄) 𝟐
𝒃𝒂 𝒄𝒂
𝒂𝒃 (𝒄 + 𝒂) 𝟐
𝒄𝒃
𝒂𝒄 𝒃𝒄 (𝒂 + 𝒃) 𝟐
| = 2abc (a + b + c)3
.
21. If 𝐜𝐨𝐬−𝟏
𝒙 + 𝐜𝐨𝐬−𝟏
𝒚 + 𝐜𝐨𝐬−𝟏
𝒛 = 𝝅 , Prove x2
+ y2
+ z2
+ 2xyz = 1.

2. 22. Find equation of tangent to the curve x2
+ 3y = 3 which is PARALLEL to line y – 4x + 5 = 0.
23. Find the matrix X : [
𝟐 −𝟏
𝟎 𝟏
−𝟐 𝟒
] 𝑿 = [
−𝟏 −𝟖 −𝟏𝟎
𝟑 𝟒 𝟎
𝟏𝟎 𝟐𝟎 𝟏𝟎
]
24. Evaluate :- ∫
sin−1 𝑥
𝑥2
dx.
25. Prove : ∫
𝒙 𝒅𝒙
𝒂 𝟐 𝒄𝒐𝒔 𝟐 𝒙+ 𝒃 𝟐 𝒔𝒊𝒏 𝟐 𝒙
𝝅
𝟎
=
𝝅 𝟐
𝟐𝒂𝒃
.
26. Using property of determinants, Prove : |
𝒂 + 𝒃 + 𝒄 −𝒄 −𝒃
−𝒄 𝒂 + 𝒃 + 𝒄 −𝒂
−𝒃 −𝒂 𝒂 + 𝒃 + 𝒄
| = 2 (a + b) (b + c) ( c + a)
27. Solve for x : 𝐬𝐢𝐧−𝟏
(
𝟐𝒂
𝟏+ 𝒂 𝟐) + 𝐬𝐢𝐧−𝟏
(
𝟐𝒃
𝟏+ 𝒃 𝟐) = 𝟐 𝐭𝐚𝐧−𝟏
𝒙
28. Evaluate :- ∫
𝒅𝒙
𝟓+𝟒 𝒄𝒐𝒔 𝒙
𝝅
𝟎
.
29. Prove :
𝑑
𝑑𝑥
(
𝑥
2
√𝑎2 − 𝑥2 +
𝑎2
2
sin−1 𝑥
2
) = √𝑎2 − 𝑥2 .
30. Show that the relation R in set R of real numbers, defined as R = {(a, b): a ≤ b2
} is neither reflexive nor
symmetric nor transitive.
31. Using elementary operation find A-1
, A = [
𝟐 𝟑
𝟒 𝟓
] .
32. Show that the curve 2x = y2
& 2xy = k cut at right angles if k2
= 8.
33. Find
𝒅𝒚
𝒅𝒙
if y = 𝐭𝐚𝐧−𝟏
[
√ 𝟏+ 𝒙 𝟐 + √ 𝟏− 𝒙 𝟐
√ 𝟏+ 𝒙 𝟐 – √ 𝟏− 𝒙 𝟐
] .
34. Evaluate : ∫ 𝒇(𝒙)𝒅𝒙, 𝒇(𝒙) = |𝒙| + |𝒙 + 𝟐| + |𝒙 + 𝟓|
𝟎
−𝟓
.
35. Find the intervals in which f(x) = sin x + cos x, 0 ≤ 𝒙 ≤ 𝟐𝝅 is strictly increasing or decreasing.
36. Show that the relation R on z defined by (a, b) ∈ R  a – b is divisible by 5 is an equivalence relation.
37. Solve by matrix method :- x + 2y + 5z = 10, x – y – z = -2, 2x + 3y – z = -11.
38. Evaluate :- ∫
𝒙 𝒅𝒙
𝟒− 𝒄𝒐𝒔 𝟐 𝒙
𝝅
𝟎
.
39. If y = Aemx
+ Benx
, show that y2
– (m + n) y1 + mny = 0.
40. Evaluate : ∫ ( 𝒙 𝟐
+ 𝟓𝒙)𝒅𝒙
𝟑
𝟏
as limit of sum.