REVISION WORK IN CLASS XII : MATHEMATICS
COMPILED BY M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI
1. If f(x) is an invertible function, find the inverse of 𝑓(𝑥) =
3𝑥−2
5
2. Let * be a binary operation defined by a*b = 3a + 4b - 2 find 4*5
5𝑥+3
3. If f(x) = 4𝑥−5, show that ff is an identity function.
4. Show that the f:RR defined by f(x) = |𝑥|is neither one-one nor onto.
5. * be a binary operation defined on Q, given by a *b = a + ab. Is * commutative.
6. Show that the relation defined on R by R = {(a, b) : a – b is divisible by 3: a , b  Z} is an
equivalence relation.
7. Show that the relation defined on R x R by (a ,b) R (c , d)  a + b = b + c on the set N X N is an
equivalence relation.
8. Show that f(x) = 7x – 5 is an invertible function. Find the inverse of f.
9. Consider f: R+  [5 ) given by f(x) = 9x2 + 6x – 5, show that f is invertible. Also find f-1
10. Let * be a binary operation on NXN defined by (a , b) * (c , d) = (a+c , b+d). Show that * is
commutative, associative. Find the identity element for * if exists.
√3
11. Find the principal value of 𝑠𝑖𝑛−1 [ ]
2
1
12. Find the principal value of 𝑐𝑜𝑠 −1 (− 2)
13. Find the principal value of 𝑠𝑖𝑛−1 [𝑠𝑖𝑛
3𝜋
]
5
1−cos 𝑥
14. Simplify : 𝑡𝑎𝑛−1 √1+cos 𝑥
15. Show that : 𝑠𝑖𝑛−1 (2𝑥 √1 − 𝑥 2 ) = 2 sin-1x
8
31
16. Solve for x : tan-1(x + 1) + tan-1(x – 1) = tan-1( )
1
17. Prove that : 2 𝑡𝑎𝑛−1 + 𝑡𝑎𝑛 −1
5
−1 𝑥−1
18. Solve for x : 𝑡𝑎𝑛
+
𝑥−2
−1 3
19. Prove that : 𝑠𝑖𝑛
20. Prove that : 𝑠𝑖𝑛
5
−1 12
13
+
𝑡𝑎𝑛
1
= 𝑡𝑎𝑛−1
8
−1 𝑥+1
𝑥+2
−1 8
𝑠𝑖𝑛
=
17
−1 4
+ 𝑐𝑜𝑠
5
+
=
4
7
𝜋
4
𝑠𝑖𝑛−1
44
85
−1 63
𝑡𝑎𝑛
16
= 𝜋
𝑥+3
4
5 4
)= (
) find x and y.
21. If (
𝑦−4 𝑥+𝑦
3 9
2𝑥 + 5 3
| = 0, find x.
22. If |
5𝑥 + 2 9
23. Find the area of the triangle whose vertices are (3 , 1), (4.3) and (-5,4)
1 𝑎 𝑏+𝑐
24. Using properties of determinants, prove that |1 𝑏 𝑐 + 𝑎 | = 0
1 𝑐 𝑎+𝑏
25. If A and B are symmetric matrices, show that AB is symmetric, if AB = BA.
3
4
), show thatA2 – 5A + 7 = 0
26. If A = (
−4 −3
REVISION WORK IN CLASS XII : MATHEMATICS
COMPILED BY M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI
𝑠𝑖𝑛𝑥
cos 𝑛𝑥 sin 𝑛𝑥
) show by induction that An = (
) for nN
𝑐𝑜𝑠𝑥
− sin 𝑛𝑥 cos 𝑛𝑥
6
1 −5
28. Express the matrix (−2 −5
4 ) as a sum of symmetric and skew-symmetric
−3 3 −1
matrix.
27. Let A = (
cos 𝑥
−𝑠𝑖𝑛𝑥
1 𝑎 𝑎2
29. Using properties of determinants: prove that |1 𝑏 𝑏 2 | = (a – b) (b – c) (c – a)
1 𝑐 𝑐2
𝑥
𝑦
𝑧
2
2
2
𝑥
𝑦
𝑧
30. Using properties of determinants: prove that: |
| = 𝑥𝑦𝑧 (𝑥 − 𝑦)(𝑦 − 𝑧)(𝑧 − 𝑥 )
𝑥3 𝑦3 𝑧3
31. Using matrices solve the system: 2 x – y = 4 ; 2y + z = 5 ; z + 2x = 7
1 −1 2
−2 0 1
32. Use the product (0 2 −3) ( 9 2 −3) to solve the system x – y + 2z = 1, 2y – 3z = 1,
3 −2 4
6 1 −2
3x – 2y + 4z = 2
33. Using matrices solve the system: x + y + z = 4 ; 2x – y + z = -1 ; 2x + y -3z = -9
1 2 2
34. If A = (2 1 2) prove that A2 – 4 A – 5I = 0. Hence fine A-1.
2 2 1
35. State the condition under which the following system has a unique solution, find the unique solution
using matrix method. x + 2y – 2z + 5 = 0 , -x + 3y + 4 = 0, -2y + z – 4 = 0
36. If f(x) =
2𝑥+3𝑠𝑖𝑛𝑥
3𝑥+2𝑠𝑖𝑛𝑥
is continuous at x = 0, find f(0)
2
37. Find the value of ‘k’ such that the function: 𝑓(𝑥) = { 𝑘(𝑥 − 2𝑥) 𝑥 < 0 is continuous at x = 0
cos 𝑥
𝑥≥0
𝑑𝑦
38. Find
𝑤ℎ𝑒𝑛 √𝑥 + √𝑦 = 5 at (4,9)
𝑑𝑥
39. Find the point on the curve y = x2 – 4x + 5 where the tangent to the curve is parallel to x – axis
40. Prove that f(x) = x3 + x2 + x + 1 does not have a maxima or minima
2𝑥 − 1 𝑥 < 0
41. Discuss the continuity of : 𝑓(𝑥) = {
2𝑥 + 1 𝑥  0
𝑥
𝑥0
2|𝑥|
42. Examine the continuity of : 𝑓(𝑥) = {1
at x = 0
𝑥=0
2
43. Determine the value of ‘k’ for which 𝑓(𝑥) = {
𝑘
1−cos 4𝑥
8𝑥 2
𝑥=0
𝑥 0
is continuous at x = 0
𝑎𝑥 2 + 𝑏
44. Determine the values of ‘a’ and ‘b’ such that the function: 𝑓(𝑥) = { 2
2𝑎𝑥 − 𝑏
continuous.
1− 𝑠𝑖𝑛3 𝑥
3𝑐𝑜𝑠2 𝑥
𝑎
45. Let f(x) =
𝑏(1−𝑐𝑜𝑠𝑥)
(𝜋−2𝑥)2
{
𝑥<
𝑥=
𝜋
2
𝜋
2
𝑥>
𝜋
. Find a and b is f(x) is continuous at x = 2
𝜋
2
𝑥>2
𝑥 = 2 is
𝑥<2
REVISION WORK IN CLASS XII : MATHEMATICS
COMPILED BY M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI
√1+𝑥− √1−𝑥
46. Find the derivative with respect to x : y = 𝑡𝑎𝑛−1 [
√1+𝑥+ √1−𝑥
−1
47. Find the derivative with respect to x: y = (sin x)x + 𝑠𝑖𝑛
48. If x = 2 cos - cos 2 and y = 2 sin - sin 2, find
49. If 𝑥 √1 + 𝑦 + 𝑦 √1 + 𝑥 = 0 𝑓𝑖𝑛𝑑
-1
2
]
√𝑥
𝑑𝑦
𝑑𝑥
at 𝜗 =
𝜋
2
𝑑𝑦
𝑑𝑥
2 2
50. If y = (tan x) , prove that (1+x ) y2 + 2x (1 + x2) y1 = 0
51. Verify Rolle’s theorem if f(x) = x2 – 4x – 3 in the interval [1 4]
52. Find the points on the curve y = x3 – 2x2 –x at which the tangent lines are parallel to
the line y = 3x – 2.
53. The volume of a spherical balloon is increasing at the rate of 25 cm3/sec. find the
rate of change of it surface area at the instant when its radius is 5 cm.
54. Find the interval in which f(x) = x4 – 4x3 + 4x2 + 15 is increasing or decreasing.
55. Find the equation of the tangent and normal to the curve y = x2 + 4x + 1 at the point
whose x coordinate is 3.
56. Evaluate: ∫
𝑥+cos 6𝑥
57. Evaluate : ∫
1
58. ∫0
𝑥
𝑑𝑥
3𝑥 2 + 𝑠𝑖𝑛6𝑥
𝑥
2
𝑒𝑥
𝑑𝑥
𝑑𝑥
1+𝑋 2
1 𝑥3
59. ∫−1
1+𝑥 2
𝑑𝑥
60. Find the area of the region bounded by the parabola y = 4x2, the axis of y and the lines y = 1 and
y = 4.
𝑥2
61. 𝐸𝑣𝑎𝑙𝑎𝑢𝑡𝑒 ∶ ∫ 2
𝑑𝑥
𝑥 −4𝑥+3
62. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒: ∫ log(1 + 𝑥 2 ) 𝑑𝑥
63. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫ 𝑥 2 𝑡𝑎𝑛−1 𝑥 𝑑𝑥
𝑥
64. 𝐸𝑣𝑎𝑙𝑎𝑢𝑡𝑒: ∫
𝑑𝑥
(𝑥+2)(3−2𝑥)
1−𝑐𝑜𝑠 2𝑥
65. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒: ∫ 𝑡𝑎𝑛−1 √
𝑑𝑥
1+𝐶𝑂𝑆 2𝑋
sin 𝑥
66. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∶ ∫ (1−cos
𝑑𝑥
𝑥)(2−cos 𝑥)
𝜋
67. Evaluate: ∫0
𝑥
𝑑𝑥
1+sin 𝑥
2
68. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒: ∫−2|2𝑥 + 3| 𝑑𝑥
𝜋 𝑥 sin 𝑥
69. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒: ∫0
𝑑𝑥
1+ 𝐶𝑂𝑆 2 𝑋
3
70. Evaluate : ∫1 (𝑥 2 + 3𝑥) 𝑑𝑥 using
limit of sums
REVISION WORK IN CLASS XII : MATHEMATICS
COMPILED BY M.SRINIVASAN, PGT(MATHS), ZIET, MUMBAI
71. Solve:
𝒅𝒚
𝒅𝒙
𝟏 − 𝒚𝟐
+ √
𝟏 − 𝒚𝟐
𝒅𝒚
72.Solve: (1 + x2)
73. Solve: (x – 1)
𝒅𝒙
𝒅𝒚
𝒅𝒙
𝒅𝒚
𝒅𝒙
76. Solve: x
77. Solve: x
=
𝒅𝒚
𝒅𝒙
𝒅𝒚
𝒅𝒙
= ey
= 2x3y
74. Solve: cos x cosy
75. Solve:
=0
𝒅𝒚
= sinx sin y, when y (1) = 2
𝒅𝒙
𝒙
𝒙 𝒆 𝐥𝐨𝐠 𝒙+𝒆𝒙
𝒙 𝐜𝐨𝐬 𝒚
= y (log y – log x + 1)
𝒚
= y – x tan ( )
𝒙
78. Solve: (y + x)
𝒅𝒚
𝒅𝒙
=y–x
79. Solve: 2xy dy + (x2 + 2y2) dy = 0
80. Solve: (3xy + y2) dx + (x2 + xy) dy = 0
81. Solve:
𝒅𝒚
𝒅𝒙
+ 𝒚 𝒄𝒐𝒕𝒙 = 𝟐 𝐜𝐨𝐬 𝒙
82. Solve: cos3x
𝒅𝒚
𝒅𝒙
83. Solve: (1 + x2)
84. Solve:
85. Solve:
𝒅𝒚
𝒅𝒙
𝑑𝑦
𝑑𝑥
+ y cosx = sin x
𝒅𝒚
𝒅𝒙
+ y = tan-1x
− 𝟑𝒚 𝒄𝒐𝒕𝒙 = 𝐬𝐢𝐧 𝟐𝒙 𝒈𝒊𝒗𝒆𝒏 𝒕𝒉𝒂𝒕 𝒚 = 𝟐 𝒘𝒉𝒆𝒏 𝒙 =
+ 2𝑦 = 𝑥 𝑒
𝝅
𝟐
4𝑥
86. Find a unit vector in the direction of: 𝑎⃗ = 3𝑖̂ − 2𝑗̂ + 6𝑘̂
87. If 𝑎⃗ = 𝑖̂ + 2𝑗̂ − 3𝑘̂ and 𝑎⃗ = 2𝑖̂ + 4𝑗̂ + 9𝑘̂ , find a unit vector parallel to 𝑎⃗ + 𝑏⃗⃗
88. For what values of ‘’ the following vectors 𝑎⃗ = 2𝑖̂ + 𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑖̂ − 2𝑗̂ + 3𝑘̂
are perpendicular.
⃗⃗⃗⃗⃗⃗
89. If P(1, 5, 4) and Q(4, 1, 2) find the direction rations of 𝑃𝑄
90. Find the projection of the vector 𝑎⃗ = 2𝑖̂ + 3𝑗̂ + 𝑘̂ on 𝑏⃗⃗ = 𝑖̂ − 2𝑗̂ + 3𝑘̂
91. Find the direction cosines of a line parallel to
2𝑥−1
√3
=
𝑦+2
2
=
𝑧−3
3
92. Find the distance of the point (2 , 3 , 4) from the plane 𝑟⃗ (2 𝑖̂ + 3𝑗̂ + 𝑘̂ ) = 11
93. Find the equation of the line parallel to the vector (2 𝑖̂ + 3𝑗̂ + 𝑘̂ ) and passing
through the point (-1, 1 ,1) in vector and Cartesian form.
94. Find the angle between the line
𝑥+2
4
=
𝑦−1
−5
=
𝑧
7
and the plane 3x – 2z + 4 = 0.
95. Find the direction cosines of the line normal to the plane 3x – y – 4z + 7 = 0.