assignment

1. CLASS XII ASSIGNMENT (CHAPTERS 1 TO 9)
1. Find the equation of tangent to the curve x = sin 3t , y = cos 2t, at t =
𝜋
4
.
2. Solve the following differential equation: cos2
x
𝑑𝑦
𝑑𝑥
+ y = tan x.
3. Using integration, find the area of the region bounded by the parabola y2
= 4x and the circle 4x2
+ 4y2
= 9.
4. Evaluate : ∫ √
𝑎−𝑥
𝑎+𝑥
𝑎
−𝑎
dx .
5. By using properties of determinants, prove the following: |
𝑥 + 4 2𝑥 2𝑥
2𝑥 𝑥 + 4 2𝑥
2𝑥 2𝑥 𝑥 + 4
| = (5x + 4) (4 – x)2
.
6. Evaluate : ∫
𝑒 𝑥
√5−4𝑒 𝑥−𝑒2𝑥
dx.
7. If y = 3 cos (log x) + 4 sin (log x), then show that x2 𝑑2
𝑦
𝑑𝑥2 + x
𝑑𝑦
𝑑𝑥
+ y = 0.
8. Using matrices, solve the following system of equations : 2x – 3y + 5z = 11 ; 3x + 2y – 4z = -5 ; x + y – 2z = -3.
9. Evaluate : ∫
𝑒𝑐𝑜𝑠 𝑥
𝑒𝑐𝑜𝑠 𝑥+ 𝑒− 𝑐𝑜𝑠 𝑥
𝜋
0
dx.
10. Let Z be the set of all integers and R be the relation on Z defined as
R = {(a, b) : a, b ∈ Z, and (a – b) is divisible by 5}. Prove that R is an equivalence relation.
11. Evaluate : ∫
𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥
√ 𝑠𝑖𝑛 2𝑥
𝜋
3
𝜋
6
𝑑𝑥.
12. Using integration, find the area of the region bounded by the curve x2
= 4y and the line x = 4y – 2.
13. Show that the right circular cylinder, open at the top, and of given surface area and maximum volume
is such that its height is equal to the radius of the base.
14. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a *b = min.{a, b}. Write the operation
table of the operation *.
15. Find the value of ‘a’ for which the function f defined as f(x) = {
𝑎 𝑠𝑖𝑛
𝜋
2
( 𝑥 + 1), 𝑥 ≤ 0
𝑡𝑎𝑛𝑥−𝑠𝑖𝑛𝑥
𝑥3
, 𝑥 > 0
is continuous at 0
16. Sand is pouring from a pipe at the rate of 12cm3
/s. The falling sand forms a cone on the ground in such
a way that the height of the cone is always one – sixth of the radius of the base. How is the height of the
sand cone increasing when the height is 4 cm?
17. Using elementary transformations, find the inverse of the matrix (
1 3 −2
−3 0 −1
2 1 0
) .
18. Evaluate : ∫ 2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥tan−1
( 𝑠𝑖𝑛 𝑥)
𝜋/2
0
𝑑𝑥dx.
19. If sin y = x sin(a + y), prove that
𝑑𝑦
𝑑𝑥
=
𝑠𝑖𝑛2
(𝑎+𝑦)
𝑠𝑖𝑛 𝑎
.
20. Evaluate : ∫
2
(1−𝑥)(1−𝑥2)
dx.
21. Find the point on the curve y = x3
– 11x + 5 at which the equation of tangent is y = x – 11.
22. Prove that : tan−1
(
cos 𝑥
1+sin𝑥
) =
𝜋
4
] −
𝑥
2
, 𝑥 ∈ (−
𝜋
2
,
𝜋
2
).
23. Using properties of determinants, prove that |
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
|
24. Let A = | 𝑅 − {3} 𝑎𝑛𝑑 𝐵 =| 𝑅 − {1}. Consider the function f : A → 𝐵 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓( 𝑥) = (
𝑥−2
𝑥−3
). Show
that f is one – one and onto and hence find f-1
.
25. Evaluate : ∫ (2𝑥2
+ 5𝑥
3
1
) dx as a limit of a sum.

2. 26. Prove that : 𝑡𝑎𝑛−1
(
1
12
) + tan−1
(
1
5
) + 𝑡𝑎𝑛−1
(
1
8
) + tan−1
(
1
7
) =
𝜋
4
27. If x = a sin t and y = a(cos t + log tan t/2), find
𝑑2 𝑦
𝑑𝑥2
.
28. Show that the function f(x) = | 𝑥 − 3|, 𝑥 ∈ | 𝑹 , is continuous but not differentiable at x = 3.
29. Evaluate : ∫
𝑠𝑖𝑛(𝑥−𝑎)
𝑠𝑖𝑛( 𝑥+𝑎)
dx.
30. Evaluate : ∫
𝑥2
(𝑥2+4)(𝑥2+9)
dx.
31. Using integration, find the area of the region enclosed between the two circles x2
+ y2
= 4 and (x – 2)2
+ y2
= 4.
32. Show that the differential equation 2ye x/y
dx + (y – 2x ex/y
) dy = 0 is homogeneous. Find the particular
Solution of this differential equation, given that x = 0 when y = 1.
33. Prove that 2 tan−1 (
1
5
) + sec−1(
5√2
7
) + 2 tan−1 (
1
8
) =
𝜋
4
34. Let A = {1, 2, 3,.......,9} and R be the relationinA  A definedby(a,b) R (c,d) if a + d = b + c for(a, b),(c, d) inA  A.
Prove that R isan equivalence relation.Alsoobtainthe equivalenceclass[(2,5)].
35. If y = xx
,prove that
𝑑2 𝑦
𝑑𝑥2
−
1
𝑦
(
𝑑𝑦
𝑑𝑥
)
2
−
𝑦
𝑥
= 0
36. Findthe area of the regioninthe firstquadrantenclosedbythe x- axis,the line y= x and the circle x2
+ y2
= 32.
37. Evaluate : ∫
𝑑𝑥
1+√ 𝑐𝑜𝑡𝑥
𝜋
3
𝜋
6
.
38. Evaluate: ∫
𝑑𝑥
𝑠𝑖𝑛𝑥−𝑠𝑖𝑛2𝑥
39. Verify Rolle’s Theorem for the function f(x) = x2
+ 2x – 8 x ∈ [-4, 2]
40. An open box with square base is to be made out of given quantity of cardboard of area c2
sq units. Show that the
maximum volume of the box is
𝑐3
6√3
cubic units.
41. Prove that volume of largest cone that can be inscribed in a sphere of radius ‘R’ is
8
27
of the volume of sphere. Also
find the height of cone.
42. If A = [
3 2 1
4 −1 2
7 3 −3
] find A-1
and by using A-1
solve : 3x+4y+7z=14, 2x-y+3z=4 , x+2y-3z=0
43. Find the particular solution of the differential equation : (x dy – y dx) y sin
𝑦
𝑥
= (y dx + x dy) x cos
𝑦
𝑥
, given that y =
𝜋, when x = 3.
44. Discuss the commutativity and associativity of the binary operation * on R defined by
a * b = a + b + ab, for all a,b ∈ R.
45. If (tan-1x)2 + (cot-1x)2 =
5𝜋2
8
, find x
46. If A = [
1 −1 0
2 3 4
0 1 2
] and B = [
2 2 −4
−4 2 −4
2 −1 5
] , find AB and solve the system x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7.
47. Findthe equationof normal tothe curve x2
= 4y whichpassesthroughthe point(1,2)
48. The radiusof a spherical diamondismeasuredas7 cm withan error of 0.04cm . findthe approximate errorin
calculatingitsvolume.If the costof 1 c𝒎 𝟑 diamondisRs 1000, What isthe losstobuyerof the diamond?
49. Evaluate:∫
𝑥2
( 𝑥 𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥)2 dx