300 questions

1. CLASS XII MATHEMATICS LONG ASSIGNMENT
1. Write the smallestequivalence relationRonsetA = {1, 2, 3} .
2. | 𝑎⃗| = 2 , | 𝑏⃗⃗| = √3 , | 𝑎⃗| .| 𝑏⃗⃗| = √3 . findangle between 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ .
3. Evaluate :- tan -1
(√3 ) – sec -1
( -2).
4. If A = (
4 6
7 5
) , thenwhat isA. ( Adj A)?
5. For whatvalue of k, the matrix (
2𝑘 + 3 4 5
−4 0 −6
−5 6 −2𝑘 − 3
) isskew - symmetric?
6. If |
sin 𝛼 cos 𝛽
cos 𝛼 sin 𝛽
| =
1
2
, where , are acute angles,thenwrite the value of  + .
7. Write the principal value of tan-1
(1) + cos-1
( - ½ ) .
8. Write the value of tan ( 2 tan-1 1
5
).
9. Findthe value of a if [
𝑎 − 𝑏 2𝑎 + 𝑐
2𝑎 − 𝑏 3𝑐 + 𝑑
] = [
−1 5
0 13
] .
10. If [
9 −1 4
−2 1 3
] = 𝐴 + [
1 2 −1
0 4 9
] , thenfindthe matrix A.
11. If |
𝑥 + 1 𝑥 − 1
𝑥 − 3 𝑥 + 2
| = |
4 −1
1 3
| , thenwrite the value of x.
12. If a unitvector 𝑎⃗ makesangles
𝜋
3
with 𝑖̂ ,
𝜋
4
with 𝑗̂ andan acute angle  with 𝑘̂ , thenfindthe value of .
13. For whatvalue of x, isthe matrix A = [
0 1 −2
−1 0 3
𝑥 −3 0
] a skew – symmetricmatrix ?
14. If matrix A = [
1 −1
−1 1
] and A2
= kA,thenwrite the value of k.
15. If A ij isthe cofactorof the elementaij of the determinant |
2 −3 5
6 0 4
1 5 −7
|,thenwrite the value of a32 . A32.
16. Write the value of tan-1
[ 2 sin( 2 cos-1 √3
2
)].
17. Write the principal value of tan-1
(√3) – cot-1
( -√3 ).
18. Find| 𝑥⃗| , if for a unitvector 𝑎⃗ ,( 𝑥⃗ − 𝑎⃗ ). ( 𝑥⃗ + 𝑎⃗ ) = 15
19. Write the inverse of the matrix [
cos 𝜃 sin 𝜃
−sin 𝜃 cos 𝜃
] .
20. Write the value of x + y + z if [
1 0 0
0 1 0
0 0 1
] [
𝑥
𝑦
𝑧
] = [
1
−1
0
].
21. Usingprincipal values,write the valueof 2cos-1
½ + 3 sin-1
½ .
22. If A is a square matrix of order 3 such that | 𝐴𝑑𝑗 𝐴| = 225, find | 𝐴′|.
23. Write the distance betweenthe parallelplanes2x –y + 3z = 4 and 2x – y + 3z = 18.
24. Evaluate tan-1
[2 cos(2 sin−1 1
2
)] .

2. 25. What isthe principal value of cos-1
(cos
2𝜋
3
) + sin-1
(sin
2𝜋
3
) ?
26. Write the value of sin [
𝜋
3
− sin−1 (
1
2
)] .
27. For whatvalue of x, the matrix [
5 − 𝑥 𝑥 +
2 4
] issingular?
28. For whatvalue of ‘a’ the vectors2𝑖̂ − 3𝑗̂ + 4𝑘̂ and a𝑖̂ + 6𝑗̂ − 8𝑘̂ are collinear?
29. If A = [
2 3
5 −2
] , write A-1
in termsof A.
30. Findthe value of x andy if :- 2 [
3 4
5 𝑥
] + [
1 𝑦
0 1
] = [
7 0
10 5
] .
31. What isthe principal value of cos-1
(cos
3𝜋
4
) + sin-1
(sin
3𝜋
4
) ?
32. Evaluate cos [
𝜋
6
+ cos−1(
1
2
)] .
33. For whatvalue of x, the matrix [
5 − 𝑥 𝑥 +
2 4
] issingular?
34. If A = [
1 1
1 1
] satisfiesA4
=A, thenwrite the value of  A.
35. Findthe value of x + y from the followingequation :- 2 [
𝑥 5
7 𝑦 − 3
] + [
3 −4
1 2
] = [
7 6
15 14
] .
36. Findthe scalar componentsof the vector 𝐴𝐵⃗⃗⃗⃗⃗⃗ withinitial pointA(2,1) andterminal pointB(-5,7).
37. What isthe principal value of cos-1
(cos
2𝜋
3
) + sin-1
(sin
2𝜋
3
) ?
38. Evaluate :- cos (tan−1 3
4
) .
39. For any vector 𝑟⃗ , evaluate :- ( 𝑟⃗ . 𝑖̂)𝑖̂ + ( 𝑟⃗ . 𝑗̂) 𝑗̂ + ( 𝑟⃗ . 𝑘̂) 𝑘̂.
40. If A = [
2 4
4 3
] , 𝑋 = [
𝑛
1
] , B = [
8
11
] and AX= B, thenfindn.
41. For whatvalue of x, the followingmatrix issingular? [
7 − 𝑥 𝑥 + 2
5 4
]
42. Evaluate : |
102 18 36
1 3 4
17 3 6
|
43. What isthe principal value of sin-1
(sin
5𝜋
6
) + cos-1
(cos
𝜋
6
) ?
44. What isthe principal value of cos-1
(−
√3
2
) ?
45. Evaluate : |
𝑎 𝑏 𝑐
𝑎 + 2𝑥 𝑏 + 2𝑦 𝑐 + 2𝑧
𝑥 𝑦 𝑧
|
46. Write the positionvectorof a pointdividingthe linesegmentjoiningpointsA andB withpositionvectors 𝑎⃗ and
𝑏⃗⃗ externallyinthe ratio1 : 4, where 𝑎⃗ = 2𝑖̂ + 3𝑗̂ + 4𝑘̂ and 𝑏⃗⃗ = −𝑖̂ + 𝑗̂ + 𝑘̂ .
47. If Adj A = [
3 5
7 −2
] and Adj B = [
2 −3
−5 2
] , findAdj AB .

3. 48. Write the value of x – y + z from the followingequation: [
𝑥 + 𝑦 + 𝑧
𝑥 + 𝑧
𝑦 + 𝑧
] = [
9
5
7
] .
49. Evaluate :- |
cos15° sin 15°
sin 75° cos75°
| .
50. Write the projectionof the vector 𝑖̂ − 𝑗̂ onthe vector 𝑖̂ + 𝑗̂ .
51. What isthe principal value of cos-1
(−
1
√3
) ?
52. For a 2  2 matrix,A = [ aij ],whose elements are givenbyaij =
𝑖
𝑗
, write the value of a12 .
53. Write A -1
forA = [
2 5
1 3
] .
54. Write the value of tan−1 3
4
+ tan−1 3
5
.
55. If [
𝑥 + 3 4
𝑦 − 4 𝑥 + 𝑦
] = [
5 4
3 9
], findx andy.
56. What isthe principal value of sin-1
(sin
5𝜋
6
) + tan-1
(tan
𝜋
6
) ?
57. Findx if |
2 5
−1 4
| = |
5𝑥 − 2 1
−4 3
| .
58. If 𝑎⃗ = 𝑖̂ + 2 𝑗̂ − 3 𝑘̂ and 𝑏⃗⃗ = 2 𝑖̂ + 4 𝑗̂ + 9 𝑘̂ , finda unitvector parallel to 𝑎⃗ + 𝑏⃗⃗ .
59. Findthe value of cos [
𝜋
6
+ cos−1(
1
2
)] .
60. Suppose thatVijaygoestoa grocery store and purchase the followingitems:Vijay:tenapples,1 dozeneggs,
twodozenoranges.Constructthe 1 x 3 matrix.
61. Findthe principal value of tan-1
√3 - sec-1
(-2).
62. Let A be a square matrix of order 3  3. Write the value of |2𝐴|, 𝑤ℎ𝑒𝑟𝑒 | 𝐴|= 4.
63. Write the value of (𝑖̂ × 𝑗̂) . 𝑘̂ + 𝑖̂ . 𝑗̂
64. Write the order of the product matrix : [
1
2
3
] [2 3 4] .
65. What isthe principal value of tan-1
(tan
7𝜋
6
) + cot-1
(cot
7𝜋
6
) ?
66. Simplify:cos  [
cos 𝜃 sin 𝜃
− sin 𝜃 cos 𝜃
] + sin 𝜃 [
sin 𝜃 − cos 𝜃
cos 𝜃 sin 𝜃
]
67. What isthe value of the followingdeterminant? ∆ = |
4 𝑎 𝑏 + 𝑐
4 𝑏 𝑐 + 𝑎
4 𝑐 𝑎 + 𝑏
|
68. If 𝑎⃗ and 𝑏⃗⃗ are two vectorsuch that | 𝑎⃗ . 𝑏⃗⃗| = | 𝑎⃗  𝑏⃗⃗| , write the angle between 𝑎⃗ and 𝑏⃗⃗ .
69. From the followingmatrix equation,findthe value of x :(
𝑥 + 𝑦 4
−5 3𝑦
) = (
3 4
−5 6
) .
70. What isthe principal value of sin-1
(sin
2𝜋
3
) ?
71. If sin-1
x – cos-1
x =
𝜋
6
, thensolve forx

4. 72. Prove that cot -1
7 + cot-1
8 + cot -1
18 = cot-1
3
73. If a + b + c  0 and |
𝑎 𝑏 𝑐
𝑏 𝑐 𝑎
𝑐 𝑎 𝑏
| = 0, thenusingpropertiesof determinants,prove thata= b = c.
74. Showthat the functiong(x) =| 𝑥 − 2| , x  R, iscontinuousbutnot differentiableatx = 2.
75. Differentiate log( x sin x
+ cot2
x) withrespecttox.
76. Showthat the curvesxy= a2
and x2
+ y2
= 2a2
toucheach other.
77. Separate the interval [0,
𝜋
2
]intosub – intervalsinwhichf(x) =sin4
x + cos4
x is increasingordecreasing.
78. Showthat the differential equationxdy –ydx = √𝑥2 + 𝑦2 dx ishomogeneousandsolve it.
79. Findthe particularsolutionof the differential equation:-
cos x dy = sinx ( cos x – 2y) dx,giventhaty = 0, whenx =
𝜋
3
.
80. Finda unitvectorperpendiculartothe plane of triangle ABC,verticesare A (3, -1, 2), B ( 1, -1, -3) and C ( 4, -3, 1).
81. Evaluate :- ∫( 2 sin 2𝑥 − cos 𝑥) √6 − 𝑐𝑜𝑠2 𝑥− 4 sin 𝑥 dx.
82. Evaluate :- ∫
5𝑥
( 𝑥+1)(𝑥2+ 9)
dx
83. Evaluate :- ∫ 𝑥 (tan−1 𝑥)21
0 dx.
84. Evaluate :- ∫ cot−1( 1 − 𝑥 + 𝑥2) 𝑑𝑥.
1
0
85. Findthe equationof the plane throughthe pointsA (1, 1, 0), B (1, 2, 1) andC ( -2, 2, -1) and hence findthe
distance betweenthe plane andthe line
𝑥−6
3
=
𝑦−3
−1
=
𝑧+2
1
.
86. A givenrectangularareaisto be fencedoff ina fieldwhose lengthliesalongastraightriver.If nofencingis
neededalongthe river,showthatleastlengthof fencingwillbe requiredwhenlengthof the fieldistwice its
breadth.
87. From the pointP( 1, 2, 4) perpendicularisdrawnonthe plane
2x + y – 2z + 3 = 0. Findthe equation,the length&the co – ordinatesof footof perpendiculars.
88. Usingpropertiesof determinants,prove |
1 𝑥 𝑥2
𝑥2 1 𝑥
𝑥 𝑥2 1
|= ( 1 – x3
)2
.
89. Prove that :- tan−1 (
1
2
) + tan−1 (
1
5
) + tan−1 (
1
8
) =
𝜋
4
90. Showthat the functionf inA = IR - {
2
3
} definedasf(x) =
4𝑥+3
6𝑥−4
isone – one and onto.Hence findf-1
.
91. Differentiate the followingfunctionwithrespecttox : ( logx)x
+ x log x
.
92. If y = log[ x + √𝑥2 + 𝑎2 ],showthat ( x2
+ a2
)
𝑑2 𝑦
𝑑𝑥2
+ x
𝑑𝑦
𝑑𝑥
= 0.
93. If x = a sint andy = a ( cos t + logtan
𝑡
2
), find
𝑑2 𝑦
𝑑𝑥2
.
94. Evaluate :- ∫
sin( 𝑥−𝑎)
sin( 𝑥+𝑎)
dx.

5. 95. Evaluate :- ∫
5𝑥 2
1+2𝑥+3𝑥2
dx.
96. Evaluate :- ∫
𝑥2
( 𝑥2+ 4)( 𝑥2+ 9)
dx.
97. A companymanufacturestwotypesof sweaters,type A andtype B,it costs Rs.360 to make one unitof type A
and Rs.120 to make a unitof type B. the companycan make at most300 sweaterscanspendamountRs.72000 a
day.The numberof sweatersof type A cannot exceedthe numberof type Bby more than100. The company
makesa profitof Rs. 200 oneach unitof type A but consideringthe difficultiesof acommonmanthe company
chargesa nominal profitof Rs.20 ona unitof type B. Using LPPsolve the problemformaximumprofit.
98. Bag I contains3 redand 4 blackballsandBags II contains4 red and 5 black balls.One ball istransferredfromBag
I to bag II andthentwo ballsare drawn at random( withoutreplacement)fromBagII.The ballssodrawn are
foundto be bothred incolour.Findthe probabilitythatthe transferredball isred.
99. Evaluate :- ∫ ( | 𝑥| + | 𝑥 − 2| + | 𝑥 − 4| ) 𝑑𝑥 .
4
0
100. Usingvectors,findthe area of the triangle ABCwithvertices
A (1, 2, 3), B ( 2, -1, 4) andC ( 4, 5, -1) .
101. A speaktruthin 60% of the cases,while Bin90% of the cases.In whatpercentof casesare theylikelyto
contradicteach otherinstatingthe same fact?
102. Showthat the differential equation 2𝑦𝑒
𝑥
𝑦 dx + ( y - 2𝑥𝑒
𝑥
𝑦 ) dy= 0 ishomogeneous.Findthe particularsolutionof
thisdifferential equation,giventhatx = 0 wheny = 1.
103. Findthe intervalsinwhichthe functionf (x) =
4 𝑥2+ 1
𝑥
is(a) strictlyincreasing(b) strictlydecreasing.
104. Findthe pointsonthe curve y = x3
at whichthe slope of the tangentisequal to the y – coordinate of the point.
105. Findthe general solutionof the differential equation:- x logx.
𝑑𝑦
𝑑𝑥
+ 𝑦 =
2
𝑥
. 𝑙𝑜𝑔 x
106. Findthe coordinatesof the point,where the line
𝑥−2
3
=
𝑦+1
4
=
𝑧−2
2
intersectsthe plane x – y + z – 5 = 0. Also,
findthe angle betweenthe line andthe plane.
107. Findthe vectorequationof the plane whichcontainsthe line of intersectionof the planes
𝑟⃗ .( 𝑖̂ + 2𝑗̂ + 3 𝑘̂ ) − 4 = 0 and 𝑟⃗ .( 2𝑖̂ + 𝑗̂ − 𝑘̂ ) + 5 = 0 and
whichis perpendiculartothe plane 𝑟⃗ .(5𝑖̂ + 3𝑗̂ − 6 𝑘̂ ) + 8 = 0 .
108. A school wantsto awardits studentsforthe valuesof Honesty,RegularityandHardworkwithatotal cash award
of Rs.6000. Three timesthe awardmoneyforHardwork addedto that givenforHonestyamountstoRs. 11000.
Te awardmoneygivenforHonestyandHardworktogetherisdouble the one givenforRegularity.Representthe
above situationalgebraicallyandfindthe awardmoneyforeachvalue,usingmatrix method
109. Showthat the heightof the cylinderof maximumvolume,thatcanbe inscribedina sphere of radiusR is
2𝑅
√3
.
Also findthe maximumvolume.
110. Usingintegration,findthe areaboundedbythe curve x2
= 4y andthe line x = 4y – 2.

6. 111. Usingintegration,findthe areaof the regionenclosedbetweenthe two circlesx2
+y2
= 4 and (x – 2)2
+ y2
= 4.
112. Assume thatthe chancesof a patienthavinga heartattack is40%. Assumingthata meditationandyogacourse
reducesthe riskof heart attack by 30% and prescriptionof certaindrugreducesitschancesby25%. Ata time a
patientcan choose anyone of the twooptionswithequal probabilities.Itisgiventhataftergoingthroughone of
the two options,the patientselectedatrandomsuffersaheartattack. Findthe probabilitythatthe patient
followedacourse of meditationandyoga.Interpretthe resultandstate whichof the above statedmethodsis
more beneficialforthe patient.
113. Showthat :- tan (
1
2
sin−1 3
4
) =
4− √7
3
.
114. Considerf : R+ → [ 4, ∞ ) givenbyf (x) = x2
+ 4. Show that f isinvertiblewiththe inverse f-1
of f givenbyf-1
(y) =
√ 𝑦 − 4 , where R+ isthe set of all non – negative real numbers.
115. Usingpropertiesof determinants,prove the following:- |
𝑥 𝑥 + 𝑦 𝑥 + 2𝑦
𝑥 + 2𝑦 𝑥 𝑥 + 𝑦
𝑥 + 𝑦 𝑥 + 2𝑦 𝑥
| = 9y2
( x + y) .
116. Findthe value of k, forwhich f(x) = {
√1+𝑘𝑥− √1−𝑘𝑥
𝑥
, 𝑖𝑓 − 1 ≤ 𝑥 < 0
2𝑥+1
𝑥−1
, 𝑖𝑓 0 ≤ 𝑥 < 1
is continuousatx = 0.
117. If x = a cos3
 andy = a sin3
,thenfindthe value of
𝑑2 𝑦
𝑑𝑥2
at  =
𝜋
6
.
118. Evaluate :- ∫
cos2𝑥−cos2𝛼
cos𝑥−cos𝛼
dx .
119. Evaluate :- ∫
𝑥+2
√𝑥2+ 2𝑥+3
dx.
120. Evaluate :- ∫
1
1+ 𝑒sin𝑥
2𝜋
0 dx.
121. Findthe particularsolutionof the differential equation(tan -1
y– x) dy= ( 1 + y2
) dx,giventhatwhenx = 0, y=0 .
122. Show that the differential equation [𝑥 𝑠𝑖𝑛2 (
𝑦
𝑥
) − 𝑦] dx + x dy= 0 ishomogeneous.Findthe particularsolution
of thisdifferential equation,giventhaty=
𝜋
4
whenx = 1.
123. Evaluate :- ∫
𝑥2+ 1
( 𝑥+1)2
𝑒 𝑥 𝑑𝑥.
124. Findthe equationsof the normalstothe curve y = x3
+ 2x + 6 whicha re parallel tothe line x + 14y + 4 = 0.
125. If 𝑎⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ , 𝑏⃗⃗ = 4 𝑖̂ − 2𝑗̂ + 3𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 𝑖̂ − 2𝑗̂ + 𝑘̂, finda vectorof magnitude 6 unitswhichisparallel
to the vector2 𝑎⃗ - 𝑏⃗⃗ + 3 𝑐⃗.
126. Let 𝑎⃗ = 𝑖̂ + 4𝑗̂ + 2𝑘̂ , 𝑏⃗⃗ = 3 𝑖̂ − 2𝑗̂ + 7𝑘̂ 𝑎𝑛𝑑 𝑐⃗ = 2𝑖̂ − 𝑗̂ + 4𝑘̂, finda vector 𝑑⃗ whichisperpendicularto
both 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ and 𝑐⃗ . 𝑑⃗ = 18.
127. Findthe area of the greatestrectangle thatcan be inscribedinanellipse
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1.

7. 128. A square tankof capacity250 cubicmetershas to be dug out.The cost of the landis Rs.50 persquare meter.
The cost of diggingincreaseswiththe depthandforthe whole tank,itisRs.( 400 x h2
),where hmetersisthe
depthof the tank. What shouldbe the dimensionsof the tanksothat the cost id minimum?
129. Findthe area of the region{ (x,y) : y2
 6ax andx2
+ y2
 16a2
} usingmethodof integration.
130. Findthe area of the regionboundedbythe parabolay= x2
and y = | 𝑥|
131. Findthe vectorequationof the plane throughthe points( 2, 1, -1) and ( -1, 3, 4) andperpendicularto the plane
x – 2y + 4z = 10.
132. Showthat the lines 𝑟⃗ = ( 𝑖̂+ 𝑗̂ − 𝑘̂ ) +  ( 3𝑖̂ − 𝑗̂ ) and 𝑟⃗ = (4 𝑖̂ − 𝑘̂ ) +  ( 2𝑖̂ + 3𝑘̂ ) are coplanar.Also,find
the plane containingthese twolines.
133. If y =
sin−1 𝑥
√ 1− 𝑥2
, showthat ( 1 – x2
)
𝑑2 𝑦
𝑑𝑥2
− 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦 = 0
134. Differentiate tan-1 [
√1+ 𝑥2− √1− 𝑥2
√1+ 𝑥2+ √1− 𝑥2
] withrespecttocos-1
x2
.
135. The functionf(x) isdefinedasf(x) = {
𝑥2 + 𝑎𝑥 + 𝑏, 0 ≤ 𝑥 < 2
3𝑥 + 2, 2 ≤ 𝑥 ≤ 4
2𝑎𝑥 + 5𝑏, 4 < 𝑥 ≤ 8
.
136. Findthe equationof tangentandnormal to the curve y =
𝑥−7
( 𝑥−2)(𝑥−3_
at the pointwhere itcuts the x – axis.
137. VerifyRolle’stheoremforfunctionf,givenbyf(x)=ex
( sin x – cos x) on [
𝜋
4
5 𝜋
4
] .
138. Showthat sin[ cot-1
{ cos ( tan-1
x)}] = √
𝑥2+ 1
𝑥2+ 2
.
139. Prove that the functionf : N → N, definedbyf(x)=x2
+ x + 1 is one – one but not onto.
140. Solve forx : 3 sin-1
(
2𝑥
1+ 𝑥2
) − 4 cos−1 (
1− 𝑥2
1+ 𝑥2
) + 2 tan−1(
2𝑥
1− 𝑥2
) =
𝜋
3
141. .Evaluate :- ∫
𝑥3+ 𝑥+1
𝑥2− 1
dx.
142. Evaluate :- ∫
2𝑥
( 𝑥2+ 1)( 𝑥2+ 2)
dx.
143. Evaluate :- ∫
𝑥sin 𝑥
1+ 𝑐𝑜𝑠2 𝑥
𝜋
0 dx.
144. Evaluate :- ∫ log( 1 + tan 𝑥) 𝑑𝑥
𝜋
4
0 , usingpropertiesof definite
145. There isa groupof 50 people whoare patrioticoutof which20 believe innon –violence.Twopersonsare
selectedatrandomoutof them,write the probabilitydistributionforthe selectedpersonswhoare non – violent.
Alsofindthe meanof the distribution
146. Showthat the differential equationx
𝑑𝑦
𝑑𝑥
sin (
𝑦
𝑥
) + 𝑥 − 𝑦 sin (
𝑦
𝑥
) = 0 ishomogeneous.Findthe particular
solutionof thisdifferential equation,giventhatx = 1 wheny =
𝜋
2
.
147. If the sumof twounitvectorsis a unitvector,show that the magnitude of theirdifference is √3 .
148. Usingpropertiesof determinants,prove the following:-

8. |
3𝑥 −𝑥 + 𝑦 −𝑥 + 𝑧
𝑥 − 𝑦 3𝑦 𝑧 − 𝑦
𝑥 − 𝑧 𝑦 − 𝑧 3𝑧
| = 3( 𝑥 + 𝑦 + 𝑧)(𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥) .
149. Solve the differential equation:- (1+ y + x2
y) dx + (x + x3
)dy= 0, where y= 0 whenx = 1.
150. Findthe distance betweenthe pointP(5,9) and the plane determinedbythe pointsA(3, -1,2), B(5, 2, 4) and
C( -1, -1, 6).
151. Findthe equationof the perpendiculardrawnfromthe point P( 2, 4, -1) to the line
𝑥+5
1
=
𝑦+3
4
=
𝑧−6
−9
. Also,
write downthe coordinatesof the footof the perpendicularfromPtothe line.
152. Findthe vectorand Cartesianequationof the plane containingthe twolines:-
𝑟⃗ = (2 𝑖̂ + 𝑗̂ − 3𝑘̂ ) +  ( 𝑖̂ + 2𝑗̂ + 5𝑘̂) and 𝑟⃗ = (3 𝑖̂ + 3𝑗̂ + 2 𝑘̂ ) +  ( 3𝑖̂ − 2𝑗̂ + 5𝑘̂ )
153. In answeringaquestiononaMCQ testwith4 choicesperquestion,astudentknowsthe answer,guessesor
copiesthe answer.Let½ be the probabilitythathe knowsthe answer,¼ be the probabilitythathe guessesand¼
that he copiesit.Assuming thata student,whocopiesthe answer,will be correctwiththe probability¾,what is
the probabilitythatthe studentknowsthe answer,giventhathe answereditcorrectly?
154. If A = [
1 2 −3
2 3 2
3 −3 −4
], findA-1
. hence ,solve the followingsystemof equation: x + 2y – 3z = -4, 2x + 3y + 2z = 2,
3x – 3y – 4z = 11.
155. Prove that the radiusof the basof rightcircular cylinderof greatest curvedsurface areawhichcanbe inscribed
ina givencone ishalf thatof the cone.
156. Findthe area of the regionenclosedbetweenthe twocircles
x2
+ y2
= 1 and ( x – 1)2
+ y2
= 1.
157. One kindof cake requires300 g of flourand15g of fat,anotherkindof cake requires150g of flourand30g of
fat.Findthe maximumnumberof cakeswhichcanbe made from 7.5kg of flourand600g of fat, assumingthat
there isno shortage of the otheringredientsusedinmakingthe cakes.Make itas an LPPand solve itgraphically.
158. Usingpropertiesif determinants,solve the followingforx : |
𝑥 − 2 2𝑥 − 3 3𝑥 − 4
𝑥 − 4 2𝑥 − 9 3𝑥 − 16
𝑥 − 8 2𝑥 − 27 3𝑥 − 64
| = 0
159. Findthe relationshipbetween‘a’and‘b’so that the function‘f’definedby:
f(x) = {
𝑎𝑥 + 1, 𝑖𝑓 𝑥 ≤ 3
𝑏𝑥 + 3, 𝑖𝑓 𝑥 > 3
is continuousatx = 3.
160. If xy
– ex – y
, showthat
𝑑𝑦
𝑑𝑥
=
𝑙𝑜𝑔 𝑥
{log( 𝑥𝑒)}2
.
161. If 𝑖̂ + 𝑗̂ + 𝑘̂ , 2𝑖̂ + 5𝑗̂ , 3𝑖̂ + 2𝑗̂ − 3𝑘̂ and 𝑖̂ − 6𝑗̂ − 𝑘̂ are the positionvectorsof the pointsA,B, C andD, find
the angle between 𝐴𝐵⃗⃗⃗⃗⃗⃗ and 𝐶𝐷⃗⃗⃗⃗⃗⃗ . Deduce that 𝐴𝐵⃗⃗⃗⃗⃗⃗ and 𝐶𝐷⃗⃗⃗⃗⃗⃗ are collinear.

9. 162. 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?
163. Evaluate :- ∫(5𝑥 − 1)√6 + 5𝑥 − 2𝑥2 dx.
164. Evaluate :- ∫
𝑥+sin 𝑥
1+cos𝑥
𝜋
2
0 dx .
165. Evaluate :- ∫
𝑥sin 𝑥 cos𝑥
𝑠𝑖𝑛4 𝑥+ 𝑐𝑜𝑠4 𝑥
𝜋
2
0 dx.
166. Evaluate :- ∫
𝑥2+ 1
( 𝑥−1)2 ( 𝑥+3)
dx.
167. Prove the following: cos(sin−1 3
5
+ cot−13
2
) =
6
5√13
.
168. Solve the differential equation:- (y+ x)
𝑑𝑦
𝑑𝑥
= 𝑦 − 𝑥 .
169. A companyhas twoplantsto manufacture motorcycles.70%motorcyclesare manufacturedatthe firstplant,
while 30%are manufacturedatthe secondplant.At the firstplant,80% motor cyclesare rated of the standard
qualitywhile atthe secondplant,90%are ratedof standardquality.A motorcycle,randomlypickedup,isfound
to be of standard quality.Findthe probabilitythatithascome outfrom the secondplant.
The probabilitythata studententeringauniversitywill graduate is0.4.findthe probabilitythatoutof 3
studentsof the university:
170. Findthe area of the region{(x,y):y2
 4x , 4x2
+ 4y2
 9}.
171. Showthat the altitude of the rightcircularcone of maximumvolume thatcanbe inscribedina sphere of radiusr
is
4𝑟
3
.
172. If the sumof the lengthsof the hypotenuse andaside of a righttriangle isgiven,show thatthe area of the
triangle ismaximumwhenthe angle betweenthemis

3
.
173. Findthe vectorequationof a line passingthroughthe pointwithpositionvector( 2𝑖̂ − 3𝑗̂ − 5𝑘̂ ) and
perpendiculartothe plane 𝑟̂ . ( 6𝑖̂− 3𝑗̂ − 5𝑘̂ ) + 2 = 0 . also,findthe pointof intersectionof thislineandthe
plane.
174. A retiredpersonhasRs.70,000 to investandtwotypesof bondsare available inthe marketforinvestment.First
type of bondsyieldsanannual income of 8% on the amountinvestedandthe secondtype of bondyields10%per
annum.Aspee norms,he has to investaminimumof Rs.10,000 inthe firsttype and notmore thanRs. 30,000 in
the secondtype.Howshouldhe planhisinvestment,soasto get maximumreturn,afterone yearof investment?
175. Findthe equationof the plane passingthroughthe point(1,1, 1) and containingthe line
𝑟⃗ = ( −3𝑖̂ + 𝑗̂ + 5𝑘̂ ) +  ( 3𝑖̂ − 𝑗̂ − 5𝑘̂ ) . Also,show thatthe plane containsthe lines
𝑟⃗ = ( −𝑖̂+ 2𝑗̂ + 5𝑘̂ ) +  ( 𝑖̂ − 2𝑗̂ − 5𝑘̂ ) .

10. 176. Usingpropertiesof determinants,prove that: |
𝑏 + 𝑐 𝑎 𝑎
𝑏 𝑐 + 𝑎 𝑏
𝑐 𝑐 𝑎 + 𝑏
| = 4abc.
177. Findthe value of ‘a’ forwhichthe functionf definedasf(x) = {
𝑎 sin
𝜋
2
( 𝑥 + 1), 𝑥 ≤ 0
tan𝑥−sin𝑥
𝑥3
, 𝑥 > 0
iscontinuousatx=0.
178. Differentiate xx cos x
+
𝑥2+ 1
𝑥2− 1
w.r.t. x.
179. Findthe intervalsin whichf(x) =sinx – cosx, 0  x  2 isstrictlyincreasingorstrictlydecreasing.
180. Solve forx : tan−1 (
𝑥−1
𝑥−2
) + tan−1 (
𝑥+ 1
𝑥+2
) =
𝜋
4
.
181. Evaluate :- ∫
𝑠𝑖𝑛𝑥+cos𝑥
√ 𝑠𝑖𝑛𝑥cos𝑥
dx
182. Evaluate :- ∫ | 𝑥cos(𝜋𝑥)|
1
2
−1 𝑑𝑥.
183. If 𝑎⃗ , 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ are three unitvectorssuchthat 𝑎⃗ . 𝑏⃗⃗ = 𝑎⃗ 𝑐⃗ = 0 andangle between 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ is
𝜋
6
,prove that
𝑎⃗ =  2( 𝑏⃗⃗  𝑐⃗) .
184. A coinis biasedsothatthe headis3 timesaslikelytooccur as tail.If the coin istossedthree times,findthe
probabilitydistributionof numberof tails.
185. Findthe lengthandthe equationof the line of shortestdistance betweenthe lines:
𝑥−3
3
=
𝑦−8
−1
=
𝑧−3
1
and
𝑥+3
−3
=
𝑦+7
2
=
𝑧−6
4
186. Showthat of all the rectanglesinscribedinagivenfixedcircle,the square hasthe maximumarea.
187. Usingintegrationfindthe areaof the triangularregionwhose sides are y = 2x + 1, y = 3x + 1 andx = 4.
188. A factorymakestennisracketsandcricketbats. A tennisrackettakes1.5 hoursof machine time and3 hours of
craftsman’stime initsmakingwhile acricketbat takes3 hoursof machine time time and1 hour of craftsman’s
time.Ina day, the factory has the availabilityof notmore than42 hoursof machine time and24 hoursof
craftsman’stime.If the profitona racket and on a bat isRs. 20 and Rs. 10 respectively,findthe numberof tennis
racketsand cricketbats that the factorymust manufacture toearn the maximumprofit.Make itas an LPPand
solve graphically
189. Suppose 5%of menand 0.25% of womenhave greyhair.A greyhairedpersonisselectedatrandom.Whatisthe
probabilityof thispersonbeingmale?Assume thatthere are equal numberof malesandfemales?
190. A man isknownto speaktruth3 outof 4 times.He throwsa die andreportsthat it isa six.Findthe probability
that itis actually.Isit possible tospeaktruthineachand everycase?
191. Three friendsA,Band C visitedaSuperMarket forpurchasingfreshfruits.A purchased1 kg apples,3 kggrapes
and 4 kgorangesand paidRs. 800. B purchased2 kg apples,1kg grapesand 2kg orangesand paidRs. 500, while
C paidRs. 700 for5 kg apples,1 kggrapes and1 kg oranges.Findthe cost of each fruitperkg by matrix method.
Why fruitsare good forhealth?
192. Prove that tan−1 (
cos𝑥
1+sin 𝑥
) =
𝜋
4
−
𝜋
2
, 𝑥 ∈ (−
𝜋
2
,
𝜋
2
) .

11. 193. Let A = R – {3} and B = R – {1} . considerthe functionf :A → B definedbyf(x)=(
𝑥 − 2
𝑥−3
) . Show that f isone -one
and ontoand hence findf-1
.
194. If y = cosec -1
x, x > 1, thenshow: x ( x2
– 1)
𝑑2 𝑦
𝑑𝑥2
+ ( 2𝑥2 − 1)
𝑑𝑦
𝑑𝑥
= 0 .
195. Usingproperties, 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
| .
196. If ( cos x)y
= (cos y) x
, find
𝑑𝑦
𝑑𝑥
.
197. If sin y = x sin( a + y),prove that
𝑑𝑦
𝑑𝑥
=
𝑠𝑖𝑛2 ( 𝑎+𝑦)
sin𝑎
.
198. Showthat the curves2x = y2
and2xy = k cut eachat right anglesif k2
= 8.
199. For the curve y = 4x3
– 2x5
, findall pointsatwhichthe tangentpassesthroughthe origin.
200. Prove that ∫ ( √tan 𝑥 + √cot𝑥 ) 𝑑𝑥 = √2 .
𝜋
2
𝜋
4
0 .
201. If 𝑎⃗ , 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ are three unitvectorssuchthat | 𝑎⃗| = 5, | 𝑏⃗⃗| = 12 and | 𝑐⃗| = 13 , and 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗= 0⃗⃗ , findthe value
of 𝑎⃗ . 𝑏⃗⃗ + 𝑏⃗⃗ . 𝑐⃗+ 𝑐⃗ . 𝑎⃗ .
202. Evaluate :- ∫ ( 2𝑥2 + 5𝑥 ) 𝑑𝑥
3
1 as a limitof a sum.
203. Evaluate :- ∫ sin 𝑥sin 2𝑥 sin 3𝑥 𝑑𝑥 .
204. Evaluate :- ∫
2
( 1−𝑥)( 1+ 𝑥2 )
dx
205. Findthe particularsolutionof the differential equation:- 2xy+ y2
– 2x2 𝑑𝑦
𝑑𝑥
= 0 . Giventhaty = 2 whenx =1.
206. Evaluate :- ∫
log(1+𝑥)
1+ 𝑥2
1
0 dx
207. An aeroplane cancarry a maximumof 200 passengers.A profitof Rs.1000 is made on eachexecutive classticket
and a profitof Rs. 600 ismade on eacheconomyclassticket.The airline reservesatleast20 seatsfor executive
class.However,atleast4 timesas manypassengersprefertotravel byeconomyclassthanby the executiveclass.
Determine howmanyticketsof eachtype mustbe soldinorderto maximise the profitforthe airline.Whatisthe
maximumprofit?.
208. Showthat the semi – vertical angle of the rightcircular cone of giventotal surface area andmaximumvolume is
sin-1 1
3
.
209. Two bagsA and B contain4 white and3 blackballsand2 white and2 blackballsrespectively.FrombagA,two
ballsare drawn at randomand thentransferredtobag B. A ball isthendrawnfrombag B and isfoundto be a
blackball.What isthe probabilitythatthe transferredballswere 1white and1 black?
210. Usingthe methodof integration,findthe areaof the regionboundedbythe lines: 2x + y = 4, 3x – 2y = 6, x – 3y
+ 5 = 0.
211. Findthe equationof plane passingthroughthe point(1,2, 1) and perpendiculartothe line joiningthe points(1,
4, 2) and( 2, 3, 5) . Also,findthe perpendiculardistance of the plane fromthe origin.

12. 212. Use product[
1 −1 2
0 2 −3
3 −2 4
] [
−2 0 1
9 2 −3
6 1 −2
] tosolve the equations: x – y+ 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2
213. Findthe shortestdistance betweenthe lines:
𝑟⃗ = 6𝑖̂ + 2𝑗̂ + 2𝑘̂ +  ( 𝑖̂ − 2𝑗̂ + 2𝑘̂ ) 𝑎𝑛𝑑 𝑟⃗ = −4𝑖̂− 𝑘̂ +  (3 𝑖̂ − 2𝑗̂ − 2𝑘̂ ) .
214. Evaluate ∫ ( 𝑥2 − 𝑥)
4
1 dx as a limitof sums.
215. Evaluate :- ∫
sin𝑥+cos𝑥
9+16sin 2 𝑥
𝜋
4
0 dx
216. Evaluate :- ∫ 𝑒2𝑥 sin 𝑥 𝑑𝑥 .
217. Evaluate :- ∫ | 𝑥3 − 𝑥|
2
1 dx
218. Evaluate :- ∫ log( 1 + tan 𝑥)
𝜋
4
0 dx
219. Two cards are drawnsimultaneously( withoutreplacement) fromawell –shuffledpackof 52 cards. Findthe
meanand variance of the numberof red cards?
220. If x = a ( cos t + t sint) and y = a (sint – t cos t), 0 < t <
𝜋
2
, 𝑓𝑖𝑛𝑑
𝑑2 𝑥
𝑑𝑡2
,
𝑑2 𝑦
𝑑𝑡2
𝑎𝑛𝑑
𝑑2 𝑦
𝑑𝑥2
.
221. If x = √ 𝑎sin−1 𝑡 , 𝑦 = √ 𝑎cos−1 𝑡 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
𝑑𝑦
𝑑𝑥
= −
𝑦
𝑥
.
222. Prove that :-
𝑑
𝑑𝑥
[
𝑥
2
√𝑎2 − 𝑥2 +
𝑎2
2
sin−1 (
𝑥
𝑎
)] = √𝑎2 − 𝑥2 .
223. If any three vectors 𝑎⃗ , 𝑏⃗⃗ and 𝑐⃗ are coplanar,prove that the vectors 𝑎⃗ + 𝑏⃗⃗ , 𝑏⃗⃗ + 𝑐⃗ and 𝑐⃗ + 𝑎⃗ are also
coplanar.
224. The two equal sidesof anisoscelestriangle withfixedbase bare decreasingatthe rate of 3 cm persecond.How
fastis the area decreasingwhenthe twoequal sidesare equal tothe base?
225. Showthat the curvesy = aex
and y = be-x
cut at right anglesif ab= 1.
226. Showthat the relationRon Z definedby(a,b)  R  a – b is divisible by5is an equivalence relation.
227. Showthat sin−1 12
13
+ cos−1 4
5
+ tan−1 63
16
= 𝜋 .
228. Findwhetherthe lines 𝑟⃗ = ( 𝑖̂ − 𝑗̂ − 𝑘̂ ) +  ( 𝑖̂ + 𝑗̂ ) and 𝑟⃗ = ( 2𝑖̂ − 𝑗̂ ) +  ( 𝑖̂ + 𝑗̂ − 𝑘̂ ) intersectornot.If
intersecting,findtheirpointof intersection.
229. Findthe coordinatesof the footof the perpendiculardrawnfromthe pointA (1, 8, 4) to the line joiningthe
pointB (0, -1, 3) and C ( 2, -3, -1).
230. Usingintegration,find the areaof the region
{ (x,y) : | 𝑥 − 1| ≤ 𝑦 ≤ √5 − 𝑥2 } .
231. A square tankof capacity250 cubicmetreshas to be dug out.The cost of the landis Rs.50 persquare metre.
The cost of diggingincreaseswiththe depthandforthe whole tank,itisRs.( 400 x h2
),where hmetres is the
depthof the tank. What shouldbe the dimensionsof the tanksothat the cost is minimum?

13. 232. If A = [
4 −5 −11
1 −3 1
2 3 −7
] , findA-1
. hence solve the equations: 4x – 5y – 11z = 12, x – 3y + z = 1, 2x + 3y – 7z = 2
233. In answeringaquestiononamultiple choice test,astudenteitherknowsthe answerorguesses.Let¾be the
probabilitythathe knowsthe answerand¼ be the probabilitythathe guesses.Assumingthatastudentwho
guessesatthe answerwill be correctwithprobability ¼ .What isthe probabilitythatthe studentknowsthe
answergiventhathe answereditcorrectly?
234. Findthe vectorequationof the line passingthroughthe point(2,3, 2) and parallel tothe line
𝑟⃗ = (−2 𝑖̂ + 3𝑗̂) +  (2 𝑖̂− 3 𝑗̂ + 6 𝑘̂) . Alsofindthe distance betweenthe lines.
235. Evaluate : ∫ 2 sin 𝑥cos 𝑥 tan−1(sin 𝑥) 𝑑𝑥
𝜋
2
0 .
236. Evaluate :- ∫
𝑥3− 1
𝑥3+ 𝑥
dx
Evaluate :- ∫ 𝑥2 tan−1 𝑥 𝑑𝑥 .
237. If tan−1 𝑥−1
𝑥−2
+ tan−1 𝑥+1
𝑥+2
=
𝜋
4
, thenfindthe value of x.
238. Prove that |
𝑥 𝑥2 1 + 𝑝𝑥3
𝑦 𝑦2 1 + 𝑝𝑦3
𝑧 𝑧2 1 + 𝑝𝑧3
| = (1 + pxyz) (x – y) ( y – z) (z – x) .
239. Findthe value of k for whichf(x) = {
log(1+𝑎𝑥)−log(1−𝑏𝑥)
𝑥
, 𝑖𝑓 𝑥 ≠ 0
𝑘 𝑖𝑓 𝑥 = 0
240. If y = (x)sin x
+ (sinx)x
, find
𝑑𝑦
𝑑𝑥
.
241. If y = 𝑒 𝑎 cos−1 𝑥 [cos 𝑥 log 𝑥 +
sin 𝑥
𝑥
] + (sin 𝑥) 𝑥 [logsin 𝑥 + 𝑥cot 𝑥] .
242. If x = a [cos 𝑡 + log |tan
𝑡
2
|] 𝑎𝑛𝑑 𝑦 = 𝑎 sin 𝑡 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑
𝑑𝑦
𝑑𝑥
𝑎𝑡 𝑡 =
𝜋
4
.
Usingdifferentials,findthe approximatevalue of √25.2 .
243. Considerf : R+ → [ 4,∞] givenbyf(x) = x2
+ 4. Show that f is invertible withthe inverse( f-1
) of f givenbyf-1
(y) =
√ 𝑦 − 4 , where R+ is the setof all non – negative real numbers.
244. Usingpropertiesof definite integrals,evaluate : ∫
𝑥 𝑑𝑥
25 𝑠𝑖𝑛2 𝑥+16 𝑐𝑜𝑠2 𝑥
𝜋
0 .
245. Evaluate :- ∫
𝑥 tan 𝑥
sec 𝑥+tan𝑥
𝜋
0 dx
246. The dot product of a vector withthe vectors2𝑖̂ + 3𝑗̂ + 𝑘̂ , 4𝑖̂ + 𝑗̂ and 𝑖̂ − 3𝑗̂ − 7𝑘̂ are respectively9,7 and6.
Findthe vector.
247. Findthe image of point(1, 6, 3) inthe line
𝑥
1
=
𝑦−1
2
=
𝑧−2
3
.
Findthe vectorequationof the plane passingthroughthe points 2𝑖̂ + 𝑗̂ − 𝑘̂ and -𝑖̂+ 3𝑗̂ + 4 𝑘̂ and
perpendiculartothe plane 𝑟⃗ . ( 𝑖̂− 2𝑗̂ + 4𝑘̂ ) = 10.
248. Findthe distance of the point(3, 4, 5) from the plane x + y + z = 2 measuredparalleltothe line 2x = y = z.

14. 249. Given, the curvedsurface of a rightcircularcone.Show that whenthe volume of the cone ismaximum,thenthe
heightof the cone is equal to √2 timesthe radiusof the base.
250. For the matrix A = [
1 1 1
1 2 −3
2 −1 3
] showthat A3
– 6A2
+ 5A + 11I = 0 . hence findA -1
.
251. In a competitive examination,anexamineeeitherguessesorcopiesknowsthe answertoamultiple choice
questionwithfourchoices.The probabilitythathe makesa guessis1/3 and the probabilitythathe copiesthe
answeris1/6. The probabilitythatthe answeriscorrect,giventhathe copiedit,is1/8. Findthe probabilitythat
he knowsthe answerto the question,giventhathe correctlyansweredit.Whycopyingispunishable butguess
workis not? explain.
252. Of the studentsina college,itknownthat 60% reside inhostel and40% are day scholars( not residinginhostel).
Previousyearresultsreportthat30% of the studentswhoreside inhostelattain‘A’grade and20% of day
scholarsattain‘A’grade intheirannual examination.Atthe endof the year,one studentischosenat random
fromthe college andhe hasan ‘A’grade,what isthe probabilitythatthe studentisa hostlier?
253. Usingthe methodof integration,findthe areaof the regionboundedbythe lines3x – 2y + 1 = 0, 2x + 3y – 21 = 0
and x – 5y + 9 = 0.
254. Prove that :- ∫ sin−1 (√
𝑥
𝑎+𝑥
)
𝑎
0 𝑑𝑥 =
𝑎
2
( 𝜋 − 2).
255. Evaluate :- ∫
1
𝑠𝑖𝑛𝑥 ( 5−4cos𝑥)
dx
256. Evaluate :- ∫ √
1− √ 𝑥
1− √ 𝑥
dx
257. Solve forx : 2 tan-1
( sinx) = tan-1
(2 sec x),0 < x <
𝜋
2
.
258. Solve forx : tan -1
( x + 1) + tan-1
(x – 1) = tan-1 8
31
.
259. If √1 − 𝑥2 + √1 − 𝑦2 = 𝑎( 𝑥 − 𝑦), 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡
𝑑𝑦
𝑑𝑥
= √
1− 𝑦2
1− 𝑥2
.
260. If y = (x + √ 𝑥2 − 1 )m
, thenshowthat (𝑥2 − 1)
𝑑2 𝑦
𝑑𝑥2
= x
𝑑𝑦
𝑑𝑥
- m2
y = 0.
261. Showthat the relationRin the setA = { x : x  Z, 0  x  12} givenbyR = {(a,b) : | 𝑎 − 𝑏| iseven} is an
equivalence relation.Findthe setof elementsrelatedto1.
262. Solve the followingdifferential equation:
𝑑𝑦
𝑑𝑥
+ y cot x = 4x cosec x , giventhaty = 0 whenx =
𝜋
2
.
263. The probabilityof twostudentsA andB comingto the school intime are
3
7
and
5
7
respectively.Assumingthat
the events,‘A comingintime’and‘B comingintime’are independent,findthe probabilityof onlyone of them
comingto the school intime.
264. Findthe meannumberof headsinthree tossesof a faircoin.

15. 265. Usingpropertiesof determinants,showthat |
3𝑎 −𝑎 + 𝑏 −𝑎 + 𝑐
𝑎 − 𝑏 3𝑏 𝑐 − 𝑏
𝑎 − 𝑐 𝑏 − 𝑐 3𝑐
| = 3 (a + b + c)(ab + bc + ca).
266. Showthat the function: f(x) =cot -1
( sinx + cos x) isa strictlydecreasingfunctioninthe interval (0,
𝜋
4
) .
267. Findthe intervalsin whichthe functionf givenbyf(x) =x3
+
1
𝑥3
, 𝑥 ≠ 0 is (i) increasing (ii) decreasing.
268. Evaluate :- ∫
𝑠𝑖𝑛2 𝑥
sin𝑥+𝑐𝑜𝑠 𝑥
𝜋
2
0 dx .
269. If 𝑎⃗ , 𝑏⃗⃗ and 𝑐⃗ are mutuallyperpendicularvectorsof equal magnitudes,show thatthe vector 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ is
equallyinclinedto 𝑎⃗ 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ .
270. A manufacturerproducespizzaandcakes.Ittakes1 hour of work onmachine.A and 3 hourson machine Bto
produce a packetof pizza.Ittakes3 hours onmachine A and 1 hour onmachine B to produce a packetof cakes.
He earnsa profitof Rs. 17.50 perpacket onpizzaand Rs. 7 perpacketof cake.How manypacketsof each should
be producedeachday so as to maximize hisprofitsif he operateshismachinesforatthe most 12 hoursa day?
271. A helicopterif flyingalongthe curve y= x2
+ 2. A soldierisplacedatthe point(3,2) . findthe nearestdistance
betweenthe soliderandthe helicopter.
272. Findthe area of the smallerregionboundedbythe ellipse
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1 and the line
𝑥
𝑎
+
𝑦
𝑏
= 1 .
273. Solve the systemof the followingequations:
2
𝑥
+
3
𝑦
+
10
𝑧
= 4 ,
4
𝑥
−
6
𝑦
+
5
𝑧
= 1,
6
𝑥
+
9
𝑦
-
20
𝑧
= 2
274. Findthe equationof the plane whichcontainsthe twoparallel lines :
𝑥−3
3
=
𝑦+4
2
=
𝑧−1
1
and
𝑥+1
3
=
𝑦−2
2
=
𝑧
1
.
275. Findthe equationof the plane passingthroughthe line of intersectionof the planes 𝑟⃗ = ( 𝑖̂ + 3𝑗̂) - 6 = 0 and 𝑟⃗
= (3 𝑖̂ − 𝑗̂ − 4 𝑘̂) = 0, whose perpendiculardistance fromoriginisunity.
276. Prove that 2 tan-1
(
1
5
) + sec−1 (
5√2
7
) + 2 tan−1 (
1
8
) =
𝜋
4
.
277. Solve forx : tan -1
( x - 1) + tan-1
x + tan-1
(x + 1) = tan-1
3x .
278. If x = sint , y = sinpt,prove that ( 1 − 𝑦2)
𝑑2 𝑦
𝑑𝑥2
= x
𝑑𝑦
𝑑𝑥
+ p2
y= 0.
279. Evaluate :- ∫ √
1+𝑥
𝑥
dx
280. Usingpropertiesof determinants,prove the following:- |
−𝑦𝑧 𝑦2 + 𝑦𝑧 𝑧2 + 𝑦𝑧
𝑥2 + 𝑥𝑧 −𝑥𝑧 𝑧2 + 𝑥𝑧
𝑥2 + 𝑥𝑦 𝑦2 + 𝑥𝑦 −𝑥𝑦
| = ( xy + yz + zx)2
.
281. Evaluate :- ∫ 𝑥 tan−1 𝑥 dx .
282. Evaluate :- ∫
𝑑𝑥
𝑠𝑖𝑛𝑥− sin2𝑥
dx

16. 283. If 𝑎⃗ , 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ are three unitvectorssuchthat 𝑎⃗ . 𝑏⃗⃗ = 𝑎⃗ 𝑐⃗ = 0 andangle between 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ is
𝜋
6
,prove that
𝑎⃗ =  2( 𝑏⃗⃗  𝑐⃗) .
284. ∫
𝑥 sin𝑥 cos𝑥
𝑠𝑖𝑛 4 𝑥+ 𝑐𝑜𝑠4 𝑥
𝜋
2
0 dx
285. Prove that the radiusof the rightcircularcylinderof greatestcurvedsurface areawhichcan be inscribedina
givencone ishalf of that of the cone.
286. If the lengthof three sidesof trapezium, otherthanthe base,are equal to10cm, thenfindthe area of the
trapezium whenitismaximum.
287. Findthe distance of the point(-1, -5, -10) from the pointof intersectionof the line 𝑟⃗ = (2 𝑖̂ − 𝑗̂ + 2 𝑘̂) + 
(3 𝑖̂ + 4 𝑗̂ + 2 𝑘̂) and the plane 𝑟⃗. ( 𝑖̂− 𝑗̂ + 𝑘̂) = 5.
288. Showthat the lines:- 𝑟⃗ = ( 𝑖̂ + 𝑗̂ − 𝑘̂ ) +  (3 𝑖̂ − 𝑗̂ ) and 𝑟⃗ = ( 4𝑖̂ − 𝑘̂ ) +  (2 𝑖̂+ 3 𝑘̂ ) are coplanar.Also,
findthe equationof the plane containingboththeselines.
289. A toycompanymanufacturestwotypesof dollsA and B. markettestsand available resourceshave indicated
that the combinedproductionlevelshouldnotexceed1200 dollsperweekandthe demandfordollsof type B is
at most half of that for dollsof type A.further,the productionlevel of dollsof type A canexceedthree timesthe
productionof dollsof othertype byat most 600 units.If the companymakesprofitof Rs.12 andRs 16 perdoll
respectivelyondollsA andB,how manyof eachshouldbe producedweeklyinordertomaximise the profit?
290. Differentiate cos-1
{
1− 𝑥2
1+ 𝑥2
} withrespectof tan-1
{
3𝑥 − 𝑥3
1−3 𝑥2
} .
291. If y = xx
,prove that
𝑑2 𝑦
𝑑𝑥2
-
1
𝑦
(
𝑑𝑦
𝑑𝑥
)
2
−
𝑦
𝑥
= 0
292. Showthat the function: f(x) =cot-1
( sinx + cos x) is a strictlydecreasingfunctioninthe interval (0,
𝜋
4
) .
293. Findthe pointsonthe curve x2
+ y2
– 2x – 3 = 0 at whichthe tangentsare parallel tox – axis.
294. Solve forx : tan-1
(x + 1) + tan-1
(x – 1) = tan-1 8
13
.
295. Showthat the relationRin te setA = { x : x  Z, 0  x  12 } givenbyR = {(a,b) : | 𝑎 − 𝑏| is even} isanequivalence
relation.Findthe setof elementsrelatedto1.
296. Showthat the cone of the greatestvolume whichcanbe inscribedinagivensphere hasanaltitude equal to2/3
of the diameterof the sphere.
297. A windowhasthe shape of a rectangle surmountedbyanequilateraltriangle.If the perimeterof the window is
12 m, findthe dimensionsof the rectangle thatwill produce the largestareaof the window.
298. Findthe area boundedbythe curves(x – 1)2
+ y2
= 1 and x2
+ y2
= 1.
299. A card from a pack of 52 playingcardsislost.From the remainingcardsof the pack three cards are drawnat
random( withoutreplacement) andare foundtobe all spades.Findthe probabilityof the lostcardbeingspade.
300. Findthe coordinatesof the footof the perpendiculardrawnfromthe pointA (1, 8, 4) to the line joiningthe
pointB (0, -1, 3) and C ( 2, -3, -1)