This document contains questions from the topics of inverse trigonometry, matrices and determinants, and continuity and differentiability from Class XII Mathematics. For inverse trigonometry, there are 1 mark, 2 mark and 4 mark questions testing concepts like principal values, evaluating inverse trig expressions, and proving trigonometric identities. For matrices, there are 1 mark, 2 mark, 4 mark and 6 mark questions on topics such as matrix operations, determinants, and solving systems of equations using matrices. For continuity and differentiability, there are 2 mark, 4 mark questions involving checking continuity, evaluating derivatives, applying mean value and Rolle's theorems.

1. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
INVERSE TRIGONOMETRY
1 Mark Questions:
1. Find the principal value of following:
(i) sin−1 (
1
2
) ⁡ (ii)cos⁡−1(−1) ⁡ (iii)sin−1 (sin⁡
2π
3
) (iv) sec- 1
(2)
2. Evaluate the following:
(i) tan−1 [2⁡cos⁡(⁡2⁡sin−1 1
2
)] (ii) sin[
π
3
−⁡sin−1 (−
1
2
)]
2 Marks Questions:
1. Show that cos -1
(
1−x2
1+⁡x2
) = 2⁡ tan−1 x⁡.
2. Write the value of tan−1 1
5
+⁡tan−1 1
8
.
3. If sin (sin−1 1
5
+⁡ cos−1 x) = 1,find⁡x.
4. Find the value of cos (sec-1
x + cosec -1
x).
5. Simplify: tan−1 (
√1+x2−1
x
)
6. Simplify: sin−1 (
sin⁡x+cos⁡x
√2
)
7. Prove that: tan-1
2 + tan -1
3 =
3π
4
8. Solve for x: tan−1 (
1⁡−x
1+x
) =⁡
1
2
tan−1 x
9. Solve: sin -1(1 – x) – 2 sin -1 x =
π
2
10. Simplify: cos−1
(2𝑥√1 − 𝑥2): -
1
2
≤ 𝑥 ≤
1
2
MATRICES AND DETERMINANTS
1 MARK QUESTIONS.
1. If matrix [
0 6 − 5𝑥
𝑥2 𝑥 + 3
] issymmetricfindx
2. (
3𝑥 − 2𝑦 5
𝑥 −2
) = (
3 5
−3 −2
) findy .
3. Findx if |
𝑥 1
3 𝑥
| = |
1 0
2 1
|
4. For whatvalue of x, matrix [
3 − 2𝑥 𝑥 + 1
2 4
] is singular?
5. ‘A’is a square matrix of order4 : | 𝐴| = 1 find (i) |2𝐴| (ii) | 𝑎𝑑𝑗⁡𝐴| (iii) |−𝐴|
6. Findcofactor of a12 in |
2 −3 5
6 0 4
1 5 −7
|
7. |
𝑥 + 1 𝑥 − ⁡1
𝑥 − ⁡3 𝑥 + ⁡2
| = |
4 −1
1 3
| findx.
8. A matrix A is of order 2 x 2 has determinant 4. What is the value of |2𝐴| ?
9. A is a square matrix of order 3 : |A|= -1, |B|=3 find |3AB|
10. If A is a skewsymmetricmatrix of order3,whatwill be the value of det.(A).
11. Findx if |
2 4
5 −1
|=|
2𝑥 4
6 𝑥
|.
12. If A is a square matrix suchthat A2
=I, thenwrite 𝐴−1 ,
13. If A and B are square matricesof order 3 such that | 𝐴|=-1 and | 𝐵|=3 the findthe value of |2𝐴𝐵|

2. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
2 MARKS QUESTIONS.
1. In the matrix eqn.[
1 2
3 4
] [
4 3
2 1
] = [
8 5
20 13
]apply𝑅2 → 𝑅2 − 𝑅1andthenapply𝐶2 → 𝐶2 − 𝐶1.
2. For the matrix A=[
3 1
−1 2
],A2
-5A+7I=O,thenfind 𝐴−1.
3. Findthe matrix X forwhich [
1 −4
3 −2
]X=[
−16 −6
7 2
]
4. Prove that the diagonal elementsof askew symmetricmatrix are zero.
5. A and B are symmetricmatricesof same order, thenshow thatAB issymmetric iff A and B commute.
4 MARKS QUESTIONS
1. Expressthe matrix [
6 2 −5
−2 −5 3
−3 3 −1
] as sum of symmetricandskew symmetricmatrix.
2. Findx if [x 4 -1] [
2 1 −1
1 0 0
2 2 4
][
𝑥
4
−1
] = 0
3. UsingElementaryRowoperations &columnoperationsfindA-1
whose
a. A = [
2 0 −1
5 1 0
0 1 3
] b. A = [
−1 1 2
1 2 3
3 1 1
]
4. Usingpropertiesof determinants,prove that
i) |
 2 ⁡ + ⁡
 2
⁡ + ⁡
 2 ⁡+ ⁡
| = (  - ) ( - ) (- ) ( +  + )
ii) |
1 1 + 𝑝 1 + 𝑝 + 𝑞
2 3 + 2𝑝 4 + 3𝑝 + 2𝑞
3 6 + 3𝑝 10 + 6𝑝 + 3𝑞
|= 1
iii) |
𝑎2 𝑏𝑐 𝑎𝑐 +⁡ 𝑐2
𝑎2 + ⁡𝑎𝑏 𝑏2 𝑎𝑐
𝑎𝑏 𝑏2 + ⁡𝑏𝑐 𝑐2
| = 4a2
b2
c2
iv) |
𝑎2 + ⁡1 𝑎𝑏 𝑎𝑐
𝑎𝑏 𝑏2 + ⁡1 𝑏𝑐
𝑎𝑐 𝑏𝑐 𝑐2 + ⁡1
| = (1 + a2
+ b2
+ c2
)
v) |
1 𝑎 𝑎3
1 𝑏 𝑏3
1 𝑐 𝑐3
| = (a– b) (b – c) (c – a) (a + b + c)
vi) |
3𝑎 −𝑎 + 𝑏 −𝑎 + 𝑐
−𝑏 + 𝑎 3𝑏 −𝑏 + 𝑐
−𝑐 + 𝑎 −𝑐 + 𝑏 3𝑐
| = 3(a + b + c) ( ab + bc + ca)
vii) |
1 +⁡ 𝑎2 − 𝑏2 2𝑎𝑏 −2𝑏
2𝑎𝑏 1 − 𝑎2 + 𝑏2 2𝑎
2𝑏 −2𝑎 1 − 𝑎2 − 𝑏2
| = (1+a2
+b2
)3
.
viii) |
𝑎 𝑏 𝑐
𝑎 − 𝑏 𝑏 − 𝑐 𝑐 − 𝑎
𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏
| = a3
+ b3
+ c3
– 3abc .
5. Findthe matrix A satisfyingthe matrix equation [
2 1
3 2
] A [
−3 2
5 −3
] = [
1 0
0 1
] .

3. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
6. Solve the equation: |
𝑥 + 𝑎 𝑥 𝑥
𝑥 𝑥 + 𝑎 𝑥
𝑥 𝑥 𝑥 + 𝑎
| = 0, a≠0
7. Usingpropertiesof determinantprove :- |
𝑥 𝑥2 1 + 𝑝𝑥3
𝑦 𝑦2 ⁡1 + 𝑝𝑦3
𝑧 𝑧2 1 + 𝑝𝑧3
| = (1 + pxyz)(x –y) (y – z) ( z – x)
6 Marks Questions:
8. A = [
1 2 −3
2 3 2
3 −3 −4
] , find A-1
,solve the equation x + 2y – 3z = - 4, 2x + 3y + 2z = 2, 3x – 3y – 4z = 11.
9. If A = [
1 −1 1
2 1 −3
1 1 1
] , find A-1
and use it to solve x + 2y + z = 4, -x + y + z = 0, x – 3y + z = 2
10. If A = [
−4 4 4
−7 1 3
5 −3 −1
] and B = [
1 −1 1
1 −2 −2
2 1 3
] , find AB and use it to solve the system of equations
x – y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1.
11. If A = [
1 1 1
1 2 −3
2 −1 3
] show that A3
– 6A2
+ 5A + 11I = 0. Hence find A-1
.
CONTINUITY AND DIFFERENTIABILITY
2 MARKS QUESTIONS.
1. Write the value of k for which 𝑓( 𝑥) = {
𝑘𝑥2, 𝑥 < 2
3, 𝑥 ≥ 2
iscontinuousatx=2
2. Write the value of k for which 𝑓( 𝑥) = {
3𝑠𝑖𝑛𝑥
2𝑥
+ 𝑐𝑜𝑠𝑥, 𝑥 ≠ 0
𝑘, 𝑥 = 0
iscontinuousatx=0
3. Write two pointsatwhich 𝑓( 𝑥) =
1
𝑥−[ 𝑥]
isnot continuous.
4. Write one pointwhere f(x) =| 𝑥| − | 𝑥 + 1| is not differentiable
5. Findthe value of k so that 𝑓( 𝑥) = {
1−𝑐𝑜𝑠4𝑥
8𝑥2
, 𝑥 ≠ 0
𝑘, 𝑥 = 0
iscontinuousat x=0.
6. If y = tan−1 5𝑥
1−6𝑥2
,-
1
√6
< x <
1
√6
,thenshowthat
𝑑𝑦
𝑑𝑥
=
2
1+4𝑥2
+
3
1+9𝑥2
.
7. If it is giventhatforthe functionf(x)=x3
-5x2
-3x,Meanvalue theoremisapplicable in[1,3],findall valuesof c.
8. f(x) = {
𝐾𝑥 + 1 𝑖𝑓 𝑥 ≤ 𝜋
𝑐𝑜𝑠⁡𝑥 𝑖𝑓 𝑥 > 𝜋
⁡𝑎𝑡⁡𝑥 = 𝜋, find k if function is continuous at π,
9. If x = √ 𝑎sin−1 𝑡⁡, 𝑦 =⁡√ 𝑎cos−1 𝑡⁡, 𝑠ℎ𝑜𝑤⁡𝑡ℎ𝑎𝑡⁡
𝑑𝑦
𝑑𝑥
=⁡−
𝑦
𝑥
.
4 MARKS QUESTIONS
10. Differentiate log( xsin x
+cot2
x) withrespecttox.
11. If y = log[ x + √𝑥2 +⁡ 𝑎2 ],showthat ( x2
+ a2
)
𝑑2⁡𝑦
𝑑𝑥2
+ x
𝑑𝑦
𝑑𝑥
= 0.
12. . If √1 − 𝑥2 + √1 − 𝑦2 = a (x – y), Prove
𝑑𝑦
𝑑𝑥
= √
1−𝑦2
1−𝑥2
13. If x = a sint andy = a ( cos t + logtan
𝑡
2
), find
𝑑2⁡𝑦
𝑑𝑥2
.
14. If x = a(cos t + t sin t), y = b(sin t – t cos t), Prove that
𝑑𝑦2
𝑑𝑥2
=
𝑏⁡𝑠𝑒𝑐3 𝑡
𝑎2 𝑡
.

4. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
15. Differentiate the following functions w.r.t.x :
(i) sin−1 (
2𝑥
1+𝑥2
) (ii) tan−1 (
1−𝑐𝑜𝑠⁡𝑥
𝑠𝑖𝑛⁡𝑥
) (iii) tan−1 (
𝑐𝑜𝑠⁡𝑥
1+𝑠𝑖𝑛⁡𝑥
) (iv) tan−1 (
5⁡𝑥
1−6𝑥2
)
(v) tan-1[
√1+𝑥2
√1+𝑥2
+√1−𝑥2
−√1−𝑥2
] (vi) tan-1
[
√1+𝑠𝑖𝑛⁡𝑥
√1+𝑠𝑖𝑛⁡𝑥
+√1−𝑠𝑖𝑛⁡𝑥
−√1−𝑠𝑖𝑛⁡𝑥
](vii) cot-1
(
1−𝑥
1+𝑥
)
16. Differentiate tan-1[
√1+⁡𝑥2−⁡√1−⁡𝑥2
√1+⁡𝑥2+⁡√1−⁡𝑥2
] withrespecttocos-1
x2
.
17. Verify lagrange’s Mean value theorem: (i) f(x) = x(x – 1) (x – 2) [ 0, ½] (ii) f(x) = x3
-5x2
– 3x [1, 3]
18. Verify Rolle’s theorem for the following functions :
(i) f(x) = x2
+ x – 6[-3, 2] (ii) f(x) = x (x – 1) (x – 2) [0, 2] (iii) f(x) = sin x + cos x [0,
𝜋
2
]
19. Differentiate sin-1
(
2𝑥
1+𝑥2
) w.r.t. tan-1
x .
20. If y =
sin−1 𝑥
√1−𝑥2
, show that (1-𝑥2)
𝑑2 𝑦
𝑑𝑥2
− 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦 = 0⁡.
21. Differentiate cos-1
{
1−⁡𝑥2
1+⁡𝑥2
} withrespectof tan-1
{
3𝑥⁡−⁡𝑥3
1−3⁡𝑥2
} .
22. If x = a sin 2t(1 + cos 2t), y = b cos 2t( 1 – cos 2t) Show that (
𝑑𝑦
𝑑𝑥
)
𝑡=⁡
𝜋
4
=
𝑏
𝑎
23. Find
𝑑𝑦
𝑑𝑥
, if y = sin-1
[x √1 − 𝑥 − √ 𝑥√1 − 𝑥2]⁡.
24. If y = log [x + √𝑥2 + 1], prove that (𝑥2 + 1)
𝑑2 𝑦
𝑑𝑥2
+ ⁡𝑥
𝑑𝑦
𝑑𝑥
= 0⁡.
25. If y =
sin−1 𝑥
√1−𝑥2
, show that (1-𝑥2)
𝑑2 𝑦
𝑑𝑥2
− 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦=0
APPLICATION OF DERIVATIVES
2 Marks Questions:
1. The volume of a cube is increasingata rate of 9 cm3
/s,how fast isthe surface areaincreasingwhenlengthof an
edge is10 cm.
2. Showthat the functionf(x)=x3
-3x2
+4x isstrictlyincreasingonR.
3. If the radiusof a sphere ismeasuredas9m withan errorof 0.03m,findthe approximate errorincalculatingthe
surface area.
4. The side of an equilateral triangle isincreasingatthe rate of 2cm/s. at whatrate is itsarea increasingwhenthe side
of triangle is20cm?
5. Usingdifferentials,findapproximatevalue of √25.2
6. The total cost C(x) inrupeesassociatedwiththe productionof x unitsof an itemisgivenby
C(x) = 0.007 x3
– 0.003 x2
+ 15x + 4000. Findthe marginal cost when17 unitsare produced.
7. The radiusof a spherical diamondismeasuredas7 cm withan error of 0.04 cm. Findthe approximate errorin
calculatingitsvolume.If the costof 1 cm3
diamondisRs.1000, what isthe lossto the buyerof the diamond?What
lessonyouget?
4 Marks Questions:
1. Separate the interval [0,
𝜋
2
]intosub – intervalsinwhichf(x) =sin4
x + cos4
x is increasingordecreasing.
2. Showthat the curves4x = y2
and4xy = K cut at rightanglesif K2
= 512 .
3. Findthe intervalsinwhichthe functionf givenbe f(x) –sinx –cosx,0  x  2 is strictlyincreasingorstrictly
decreasing.
4. Findall pointsonthe curve y = 4x3
– 2x5
at whichthe tangentspassesthroughthe origin.
5. Findthe equationof Normal tothe curve y = x3
+ 2x + 6 whichare parallel toline x+14y+4=0.
6. Showthat the curves y = aex
andy = be –x
cut at rightanglesif ab = 1.

5. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
7. Findthe intervalsinwhichthe function f f(x) =x3
+
1
𝑥3
, x ≠ 0 is increasingordecreasing.
8. Prove that y =
4⁡𝑠𝑖𝑛⁡𝜃
(2+𝑐𝑜𝑠⁡𝜃)
– 𝜃 is an increasingfunctionof 𝜃 in[0,
𝜋
2
]
9. Findthe area of the greatestrectangle thatcan be inscribedinanellipse
𝑥2
𝑎2
+⁡
𝑦2
𝑏2
= 1.
10. Findthe equationof tangenttothe curve y =
𝑋−7
( 𝑋−2)(𝑋−3)
at the pointwhere itcuts x-axis. [x-20y=7]
11. A helicopterif flyingalongthe curve y= x2
+ 2. A soldierisplacedatthe point(3,2) .findthe nearestdistance
betweenthe soliderandthe helicopter.
12. Showthat the rightcircular cylinderof givensurface andmaximumvolume issuchthatitsheightisequal to
diameterof base.
13. Showthat the semi – vertical angle of cone of maximumvolume andof givenslantheightistan-1
√2.
14. The sum of perimeterof circle andsquare isK. Prove thatthe sumof theirareasisleastwhen side of square is
double the radiusof circle.
15. Findthe value of x forwhich f(x) = [x(x – 2)]2
is an increasingfunction.Also,findthe pointsonthe curve,where the
tangentisparallel tox- axis.
16. A tankwithrectangularbase and rectangularsides,openatthe topis to be constructedsothat its depthin2 m and
volume is8 m3
.If buildingof tankcostsRs. 70 persq. meterforthe base and
Rs. 45 persq. Meterfor sides,whatisthe cost of leastexpensive tank?
17. Showthat the functionf defined f(x) = tan-1
(sinx + cos x) isstrictlyincreasingin(0,
𝜋
4
) .
18. Findthe intervalsinwhichf(x)=
3
2
x4
-4x3
-12x2
+5isstrictlyincreasingordecreasing.
19. Findthe intervalsinwhichf(x)=
3
10
x4
-
4
⁡5
x3
-3x2
+
36
5
𝑥 + 11 is strictlyincreasing ordecreasing.
20. Findthe minimumvalue of (ax+by),where xy=c2
.
21. Findthe pointon the curve x2
=4ywhichisnearesttothe point(-1,2).
INTEGRALS
2 MARKS QUESTIONS.
1. (i) ∫ 𝑡𝑎𝑛8 𝑥𝑠𝑒𝑐4 𝑥 𝑑𝑥 (ii)∫
1
4+9𝑥2
𝑑𝑥 (iii) ∫(
1
𝑥2
−
1
𝑥
) 𝑒 𝑥 𝑑𝑥 (iv)∫ 𝑠𝑖𝑛7 𝑥
𝜋
0 𝑑𝑥
(v) ∫
1
𝑥2+2𝑥+2
𝑑𝑥 (vi)∫
1
𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠2 𝑥
𝑑𝑥 (vii). ∫
𝑐𝑜𝑠2𝑥
( 𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥)2
𝑑𝑥
2. (i). ∫
1
𝑥2−4𝑥+9
𝑑𝑥 (ii). ∫√10 + 4𝑥 − 2𝑥2 𝑑𝑥 (iii). ∫
1
1+𝑐𝑜𝑡7 𝑥
𝜋
2
0 𝑑𝑥 (iv). ∫(
1−𝑥
1+𝑥2
)
2
𝑒 𝑥 𝑑𝑥
3. Evaluate:∫
𝑐𝑜𝑠2𝑥
𝑐𝑜𝑠2 𝑥⁡⁡𝑠𝑖𝑛2 𝑥
dx
4. Evaluate:∫
𝑑𝑥
√7−6𝑥−𝑥2
5. Evaluate:∫
2+𝑠𝑖𝑛⁡2𝑥
1+𝑐𝑜𝑠⁡2𝑥
𝑒 𝑥 dx
6. Evaluate:∫
𝑑𝑥
𝑥[6( 𝑙𝑜𝑔⁡𝑥)2+7⁡𝑙𝑜𝑔⁡𝑥+2]
dx
7. Evaluate: ∫
√ 𝑥
√1−𝑥3
𝑑𝑥
8. Evaluate: ∫
𝑠𝑖𝑛⁡𝑥
𝑠𝑖𝑛⁡(𝑥+𝑎)
𝑑𝑥
9. Evaluate:∫
(2𝑥−5)𝑒2𝑥
(2𝑥−3)3
dx
4 MARKS QUESTIONS.
10. Evaluate :-∫
𝑥2+⁡1
(⁡𝑥−1)2(⁡𝑥+3)
dx.

6. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
11. Evaluate :-∫
sin(⁡𝑥−𝑎)
sin(⁡𝑥+𝑎)
dx.
12. Evaluate :-∫
5𝑥⁡⁡2
1+2𝑥+3𝑥2
dx.
13. Evaluate :-∫(⁡2sin 2𝑥 − cos 𝑥)⁡√6 −⁡ 𝑐𝑜𝑠2 𝑥 − 4sin 𝑥⁡ dx.
14. Evaluate :- ∫
2
(⁡1−𝑥)(⁡1+⁡𝑥2⁡)
dx
15. Evaluate :- ∫
𝑑𝑥
𝑠𝑖𝑛𝑥−⁡sin2𝑥
dx
16. Evaluate :- ∫
𝑥2
𝑥2+⁡3𝑥−3
⁡⁡𝑑𝑥⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡
17. Evaluate :- ∫ 𝑒 𝑥 (
sin4𝑥−4
1−cos4𝑥
) dx
18. Evaluate :- ∫
𝑥2
( 𝑥−1)3⁡⁡(𝑥+1)
dx
19. Evaluate :- ∫
1
𝑠𝑖𝑛𝑥⁡(⁡5−4cos𝑥)
dx
20. Evaluate:∫
𝑐𝑜𝑠𝑥⁡⁡⁡⁡𝑑𝑥
(𝑠𝑖𝑛2 𝑥+1)(𝑠𝑖𝑛2 𝑥+4)
21. Evaluate:∫
𝑠𝑖 𝑛4 𝑥⁡𝑐𝑜𝑠𝑥⁡𝑑𝑥
( 𝑠𝑖𝑛𝑥+1)(𝑠𝑖𝑛𝑥+4)2
22. Evaluate:∫ 𝑠𝑖𝑛3𝑥⁡𝑒5𝑥⁡𝑑𝑥
23. Evaluate:∫
2𝑥−1
( 𝑥−1)( 𝑥+2) 𝑥−3)
dx
24. Evaluate:∫
1−𝑥2
𝑥(1−2𝑥)
dx
25. Evaluate: ∫
𝑥4
( 𝑥+1)(𝑥+2)4
⁡𝑑𝑥
26. Evaluate: ∫
𝑥4
( 𝑥−1)(𝑥2+1)
⁡𝑑𝑥
27. Evaluate: ∫
𝑥3+𝑥+1
(𝑥2−1)
dx
6 MARKS QUESTIONS.
28. Evaluate :-∫
𝑥 sin 𝑥cos𝑥
𝑠𝑖𝑛4 𝑥+⁡𝑐𝑜𝑠4 𝑥
𝜋
2
0 dx.
29. Show that ∫ (√ 𝑡𝑎𝑛⁡𝑥 +⁡√ 𝑐𝑜𝑡⁡𝑥)⁡𝑑𝑥 =⁡√2⁡𝜋
𝜋
2
0
30. Prove that : ∫ 𝑓(𝑥)𝑑𝑥
𝑎
0 = ∫ 𝑓(𝑎 − 𝑥)
𝑎
0 𝑑𝑥and hence Evaluate :∫
𝑥⁡𝑑𝑥
25⁡𝑠𝑖𝑛2⁡𝑥+16⁡𝑐𝑜𝑠2⁡𝑥
𝜋
0 .
31. Prove that : ∫ 𝑓(𝑥)𝑑𝑥
𝑏
𝑎 = ∫ 𝑓(𝑎 + 𝑏 − 𝑥)
𝑏
𝑎 𝑑𝑥 and hence Evaluate: ∫
√10−𝑥
√ 𝑥+⁡√10−𝑥
8
2 dx
32. Evaluate:∫
𝑥⁡𝑠𝑖𝑛⁡𝑥
1+⁡𝑐𝑜𝑠2 𝑥
𝜋
2
0 ⁡𝑑𝑥
33. Evaluate:∫
⁡𝑥⁡𝑑𝑥
4−𝑐𝑜𝑠2⁡𝑥
𝜋
0
34. Evaluate:∫
⁡𝑥⁡𝑑𝑥
1+𝑠𝑖𝑛𝑥
𝜋
0 ⁡
35. Evaluate:∫ (| 𝑥 − 1| +⁡| 𝑥 − 2| + | 𝑥 − 4|)
4
1 dx
36. Evaluate: ∫
𝑠𝑖𝑛2 𝑥
𝑠𝑖𝑛⁡𝑥+𝑐𝑜𝑠⁡𝑥
𝜋
2
0 dx
37. Evaluate: ∫
𝑒 𝑐𝑜𝑠⁡𝑥
𝑒 𝑐𝑜𝑠⁡𝑥+𝑒−𝑐𝑜𝑠⁡𝑥
𝜋
0 dx
38. Evaluate as the limit of sums :
(i)∫ (2𝑥2 − 5) 𝑑𝑥
3
0 (ii) ∫ ( 𝑥2 + 5𝑥) 𝑑𝑥
3
1 (iii) ∫ ( 𝑥2 + 𝑥 + 1) 𝑑𝑥
2
0 (iv) ∫ (3𝑥2 + 2𝑥) 𝑑𝑥
3
1

7. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
APPLICATION OF INTEGRATION (6 MARKS)
6 MARKS QUESTIONS.
1. Usingintegration,findthe areaof the regionenclosedbetweenthe twocirclesx2
+y2
= 4 and (x – 2)2
+ y2
= 4.
2. Findthe area of the region{ (x,y) : y2
 6ax and x2
+ y2
 16a2
} usingmethodof integration.
3. Prove that the area betweentwoparabolasy2
4ax andx2
= 4ay is16 a2
/ 3 sq units.
4. Usingintegration,findthe areaof the followingregion. {( 𝑥, 𝑦):
𝑥2
9
+⁡
𝑦2
4
⁡≤ 1⁡ ≤⁡
𝑥
3
+⁡
𝑦
2
}.
5. Findthe area of the region{(x,y) :x2
+ y2
≤ 4, 𝑥 + 𝑦 ≥ 2}.
6. Findthe area lyingabove x – axisand includedbetweenthe circle x2
+y2
= 8x andthe parabolay2
= 4x.
7. Findthe area of the regionincludedbetweenthe curve 4y = 3x2
and line 2y = 3x + 12
8. Sketchthe graph of y = | 𝑥 + 3| and Evaluate ∫ | 𝑥 + 3|
0
−6 dx .
9. Usingthe methodof integration,findthe areaof the regionboundedbythe lines3x – 2y + 1 = 0,
2x + 3y – 21 = 0 and x – 5y + 9 = 0.
10. Usingintegration,findthe areaof ∆ ABC where A (2,3),B (4,7),C (6,2).
11. Usingintegration,findthe areaof the triangle formedbypositive x-axisandtangentandnormal tothe circle
x2
+y2
=4 at (1,√3)
12. Findthe area of the regionincludedbetweenthe parabolay=
3⁡𝑥2
4
and the line 3x – 2y + 12 = 0.
13. Findthe area boundedbythe circle x2
+ y2
= 16 and the line y= x in the firstQuadrant. [2𝜋]
DIFFERENTIAL EQUATIONS (6MARKS)
2 MARKS QUESTIONS.
1. What is the degree and order of following differential equation?
(i) y
𝑑2 𝑦
𝑑𝑥2
+ (
𝑑𝑦
𝑑𝑥
)
3
= 𝑥 (
𝑑3 𝑦
𝑑𝑥3
)
2
. (ii)(
𝑑𝑦
𝑑𝑥
)
4
+ 3y
𝑑2 𝑦
𝑑𝑥2
= 0. (iii)
𝑑3 𝑦
𝑑𝑥3
+y2
+ 𝑒
𝑑𝑦
𝑑𝑥 = 0
2. Write the integratingfactorof
𝑑𝑦
𝑑𝑥
+ 2y tan x = sin x
3. Form the differential equation of family of straight lines passing through origin.
4. Form the differential equation of family of parabolas axis along x-axis.
5. Form the differential equationof familyof curves;(x-a)2
+2y2
=a2
.
6. Form the differential equationof familyof curves;y2
=a(b-x2
)
4 MARKS QUESTIONS.
1. Showthat the differential equation xdy – ydx = √𝑥2 + 𝑦2 dx is homogeneousandsolve it.
2. Show that the differential equation [𝑥⁡𝑠𝑖𝑛2 (
𝑦
𝑥
) − ⁡𝑦] dx + x dy= 0 ishomogeneous.Findthe particularsolutionof
thisdifferential equation,giventhaty=
𝜋
4
whenx = 1.
3. Showthat the differential equationx
𝑑𝑦
𝑑𝑥
sin (
𝑦
𝑥
) + ⁡𝑥 − 𝑦 sin (
𝑦
𝑥
) = 0 ishomogeneous.Findthe particularsolution
of thisdifferential equation,giventhatx = 1 wheny=
𝜋
2
.
4. Solve the followingdifferentialequation:- (1+ y + x2
y) dx + (x + x3
)dy= 0, where y= 0 whenx = 1.
5. Solve the followingdifferentialequation:√1 +⁡ 𝑥2 +⁡ 𝑦2 +⁡ 𝑥2 𝑦2 + xy
𝑑𝑦
𝑑𝑥
= 0 .
6. Findthe particularsolutionof the differential equation:( xdy –ydx) ysin
𝑦
𝑥
= (⁡𝑦𝑑𝑥 + 𝑥𝑑𝑦) 𝑥 cos
𝑦
𝑥
,
giventhaty =  whenx = 3.
7. 𝑆𝑜𝑙𝑣𝑒 ∶⁡ 𝑥𝑒
𝑦
𝑥 − ⁡𝑦 sin
𝑦
𝑥
+ ⁡𝑥⁡
𝑑𝑦
𝑑𝑥
sin
𝑦
𝑥
= 0 . giventhaty = 0 where x = 1, i.e.,y(1) = 0
8. Solve :(x2
+ xy) dy= (x2
+ y2
) dx.

8. CLASS XII MATHEMATICS NITISH GUPTA
A-1/53, 54, SECTOR-6 01141424079, 8376074222
9. Show that the differential equation : 2y ex/y
dx + (y – 2x ex/y
) dy = 0 is homogenous and find its
particular solution, given x = 0 when y = 1.
10. Solve the equation :(𝑥⁡𝑐𝑜𝑠⁡
𝑦
𝑥
+ 𝑦⁡𝑠𝑖𝑛⁡
𝑦
𝑥
)⁡𝑦⁡𝑑𝑥 =⁡(𝑦⁡𝑠𝑖𝑛⁡
𝑦
𝑥
− ⁡𝑥⁡𝑐𝑜𝑠⁡
𝑦
𝑥
) x dy.
11. Find the particular solution of the differential equation (1 + x3
)
𝑑𝑦
𝑑𝑥
+ 6x2
y = (1 + x2
), given that x = y = 1.
12. Solve :(1 + 𝑒
𝑥
𝑦)⁡𝑑𝑥 +⁡ 𝑒
𝑥
𝑦 (1 −
𝑥
𝑦
) ⁡𝑑𝑦 = 0 .
13. Solve the differential equation : (tan-1
y – x) dy = (1 + y2
) dx .
14. Find the particular solution of the differential equation :
15. Solve: (1 + e2x
) dy+ (1 + y2
) ex
dx = 0, given that y = 1 when x = 0.
16. Findthe general solutionof (x +2y3
)
𝑑𝑦
𝑑𝑥
= y.
17. Findthe equationof a curve passingthroughoriginandsatisfyingthe differential equation
(1 + x2
)
𝑑𝑦
𝑑𝑥
+ 2xy= 4x2
.
18. Findthe general solutionof
𝑑𝑦
𝑑𝑥
- 3y = sin 2x.