some important qestions foe board exam

1. MATHEMATICS CLASS XII
MOST IMPORTANT QUESTIONS (FOR 2015)
Relations & Functions (6 Marks)
1) Let N be the setof all natural Nos.& R be a relationonNxN definedby(a,b)R(c,d) suchthat
ad = bc. Prove that R isan equivalence relation
2) Let R be a relation onNxN definedby(a,b)R(c,d) : a+d = b+c. Prove that R isan equivalence relation.
3) Prove that the relationRon the setZ of all integersdefinedbyR={(a,b):a-bisdivisible byn} isanequivalence
relation.
4) A=R-{3}andB= R-{1}.Consider the functionF:A →B definedbyf(x)=
𝑥−2
𝑥−3
isf isone one and onto?Justify
5) Showthat the relationRin the setR of real numbersdefinedasR={ (a,b):a≤ b2
} isneitherreflexive nor
symmetricnortransitive.
6) Considerf:R+
→ [-5,∞) givenbyf(x)=9x2
+6x-5.Show thatf isinvertibleandfindinverseof f.
7) F: R→R, g: R→R such that f(x)=3x+1 and g(x)=4x-2.Findfog andshow that fog isone one and onto.
8) A binaryoperation*on Q-{-1} such that a*b= a+b+ab. Findthe identityelementonQ.Also findthe inverse of
an elementin Q-{-1}
9) Let A= N×N and * be the binaryoperatoron A definedby(a,b)*(c,d)=(a+c,b+d).show that* is commutative and
associative.Findthe identityelementfor* onA if any.
10) Let A = NxN & * be a B.O. on A definedby(a,b)*(c,d)=(ac,bd) forall a, b, c, d 𝜀N. Show that * is commutative &
Associative.
11) F, g : R → R be a function:
F(x) =
1
2𝑥
& g(x) = 3x + 1. Findgof & prove that gof isone one & onto.
12) Let f : N → R be a functiondefinedbyf(x) =4x2
+ 12x +15. Show that f : N → range (f) isinvertible.
INVERSE TRIGONOMETRY (4 Marks)
1. Write the following function in simplest form:
(i) tan−1 (
√1+x2−1
x
) (ii)cot−1 (
1
√x2− 1
) (iii) sin-1
√
x
1+ x
. (iv)sin−1 (
sin x+cos x
√2
)
2. Prove the following:
(i) tan-1
2 + tan -1
3 =
3π
4
(ii) cot-1
[
√1+ sin x
√1+sin x
+√1−sin x
−√1−sin x
]= x
2
(iii) tan-1[
√ 𝟏+𝐱 𝟐
√ 𝟏+𝐱 𝟐
+√ 𝟏−𝐱 𝟐
−√ 𝟏−𝐱 𝟐
]= π
4
+
1
2
cos−1 x2 (iv) tan-1
[
√1+cos x
√1+cos x
+√1−cos x
−√1−cos x
]= π
4
+
x
2
(v) tan- 163
16
= sin-1 5
13
+ cos-13
5
(vi) tan- 11
5
+ tan-11
7
+ tan-11
3
+ tan-11
8
=
π
4
.
3. Solve the following for x:
(i)tan−1 (
1 −x
1+x
) =
1
2
tan−1 x (ii). tan-1
2x + tan-1
3x =
π
4
(iii). tan−1 (
x− 1
x−2
) +tan−1 (
x+ 1
x+ 2
) =
π
4
(iv). 2 tan -1
(cos x) = tan -1
(2 cosec x) (v). sin -1
(1 –
x) – 2 sin -1
x =
π
2

2. 4. Prove : tan-1
(
π
4
+
1
2
cos−1 a
b
) + tan (
π
4
−
1
2
cos−1a
b
) =
2b
a
.
5. Prove that: tan -1
(
1
4
) + tan−1 (
2
9
) =
1
2
cos−1 (
3
5
) .
6. Prove that : 2tan-1
(
1
2
)+ tan-1
(
1
7
) = sin-1 31
25√2
.
7. .Prove that : cos[tan-1
{sin (cot-1
x)}] = √
1+𝑥2
2+𝑥2
.
8. Solve forx : 2 tan-1
( sinx) = tan-1
(2 secx),
9. Prove thatcot -1
7 + cot-1
8 + cot -1
18 = cot-1
3
MATRICES AND DETERMINANTS (1+4+4+4=13 Marks)
1 MARK QUESTIONS.
1. Find if [
cos 𝜃 −sin 𝜃
sin 𝜃 cos 𝜃
] isequal toidentifymatrix.
2. If matrix [
0 6 − 5𝑥
𝑥2 𝑥 + 3
] issymmetricfindx
3. (
3𝑥 − 2𝑦 5
𝑥 −2
) = (
3 5
−3 −2
) findy .
4. 𝐴 = [
2 3
5 7
] findA + AT
.
5. Findx if |
𝑥 1
3 𝑥
| = |
1 0
2 1
|
6. Findarea of triangle usingdeterminantsA (-3,5) B(3, -6) & C(7, 2)
7. For whatvalue of x, matrix [
3 − 2𝑥 𝑥 + 1
2 4
] is singular?
8. ‘A’is a square matrix of order4 : | 𝐴| = 1 find (i) |2𝐴| (ii) | 𝑎𝑑𝑗 𝐴| (iii) |−𝐴|
9. Findcofactor of a12 in |
2 −3 5
6 0 4
1 5 −7
|
10. |
𝑥 + 1 𝑥 − 1
𝑥 − 3 𝑥 + 2
| = |
4 −1
1 3
| findx.
11. A = [
2 3
5 −2
] write A-1
intermsof A
4 MARKS QUESTIONS
1. In a departmentstore,acustomerX purchases2 packetsof tea, 4 kg of rice & 5 dozenoranges.CustomerY
purchases1 packetof tea,5 kg rice and 24 oranges.Price on1 pack of teais Rs.54, 1 kg of rice isRs. 22 and that
of 1 dozenorangesisRs. 24. Use Matrix multiplicationmethodandcalculate eachindividualbill.
2. If A = [
3 2
2 1
] , VerifyA2
–4A – I = 0. Hence findA-1
.
3. If A = [
𝛼 1
0 𝛼
] , thenprove byP.M.I. thatAn
= [ 𝛼 𝑛 𝑛𝛼 𝑛−1
0 𝛼 𝑛
] .

3. 4. The departmentof educationruns120 collegesand15 universitieshavingastrengthof 300 lecturers,100
readersand50 professorsinthe universitiesand5000 lecturersand1000 readersincolleges.The monthly
salaryis Rs.9000 for professors,Rs.8000 for readersandRs. 5000 for lecturers.Findthe monthlysalarybill in
the collegesanduniversities.
5. Expressthe matrix [
6 2 −5
−2 −5 3
−3 3 −1
] as sum of symmetricandskew symmetricmatrix.
6. Findx if [x 4 -1] [
2 1 −1
1 0 0
2 2 4
] [
𝑥
4
−1
] = 0
7. 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
]
8. A manufacturerproducesthree productsx,y,z whichhe sellsintwomarkets.Annual salesare indicatedinthe
table :
Market Products
X Y Z
I
II
10,000
6,000
2,000
20,000
18,000
8,000
If unit sale price of x,y and z are Rs.2.50, Rs. 1.50 andRs. 1.00 respectively, findthe total revenueineach
market,usingmatrices.
9. Expressthe matrix B = [
2 −1 3
4 5 0
6 2 −1
] as sum of symmetricandskew symmetricmatrices.
10. Usingpropertiesof determinants,prove that
i) |
𝑥 + 4 2𝑥 2𝑥
2𝑥 𝑥 + 4 𝑥
2𝑥 2𝑥 𝑥 + 4
| = (5x + 4) (4 – x)2
ii) |
𝑎 − 𝑏 − 𝑐 2𝑎 2𝑎
2𝑏 𝑏 − 𝑐 − 𝑎 2𝑏
2𝑐 2𝑐 𝑐 − 𝑎 − 𝑏
| = (a+ b + c )3
iii) |
 2  + 
 2
 + 
 2  + 
| = (  - ) ( - ) (- ) ( +  + )
iv) |
1 1 + 𝑝 1 + 𝑝 + 𝑞
2 3 + 2𝑝 4 + 3𝑝 + 2𝑞
3 6 + 3𝑝 10 + 6𝑝 + 3𝑞
|= 1
v) |
𝑎2 𝑏𝑐 𝑎𝑐 + 𝑐2
𝑎2 + 𝑎𝑏 𝑏2 𝑎𝑐
𝑎𝑏 𝑏2 + 𝑏𝑐 𝑐2
| = 4a2
b2
c2
vi) |
𝑎2 + 1 𝑎𝑏 𝑎𝑐
𝑎𝑏 𝑏2 + 1 𝑏𝑐
𝑎𝑐 𝑏𝑐 𝑐2 + 1
| = (1 + a2
+ b2
+ c2
)

4. vii) |
1 𝑎 𝑎3
1 𝑏 𝑏3
1 𝑐 𝑐3
| = (a– b) (b – c) (c – a) (a + b + c)
viii) |
3𝑎 −𝑎 + 𝑏 −𝑎 + 𝑐
−𝑏 + 𝑎 3𝑏 −𝑏 + 𝑐
−𝑐 + 𝑎 −𝑐 + 𝑏 3𝑐
| = 3(a + b + c) ( ab + bc + ca)
11. For the matrix A = [
3 2
1 1
] , findthe numbersa & b: A2
+ aA + bI = 0. Hence findA-1
.
12. Findthe matrix A satisfyingthe matrix equation [
2 1
3 2
] A [
−3 2
5 −3
] = [
1 0
0 1
] .
13. Solve the equation: |
𝑥 + 𝑎 𝑥 𝑥
𝑥 𝑥 + 𝑎 𝑥
𝑥 𝑥 𝑥 + 𝑎
| = 0, a≠0
14. Usingpropertiesof determinantprove :- |
𝑥 𝑥2 1 + 𝑝𝑥3
𝑦 𝑦2 1 + 𝑝𝑦3
𝑧 𝑧2 1 + 𝑝𝑧3
| = (1 + pxyz)(x –y) (y – z) ( z – x)
15. A trust fundhasRs. 30,000 to investintwobonds.Firstpays5% interestperyear& secondpays7%. Using
matrix multiplication determine howtodivideRs.30,000 among twotypesof bondsif trust findmustobtain
annual interestof Rs.2000.
CONTINUITY AND DIFFERENTIABILITY (4+4=8 Marks)
1. Showthat the functiong(x) =| 𝑥 − 2| , x  R, iscontinuousbutnot differentiableatx = 2.
2. Differentiate log( x sin x
+ cot2
x) withrespecttox.
3. If y = log[ x + √𝑥2 + 𝑎2 ],showthat ( x2
+ a2
)
𝑑2 𝑦
𝑑𝑥2
+ x
𝑑𝑦
𝑑𝑥
= 0.
4. For what values of a and b the function defined is continuous at x = 1, f(x) = {
3𝑎𝑥 + 𝑏 𝑖𝑓 𝑥 < 1
11 𝑖𝑓 𝑥 = 1
5𝑎𝑥 − 2𝑏 𝑖𝑓 𝑥 > 1
5. . If √1 − 𝑥2 + √1 − 𝑦2 = a (x – y), Prove
𝑑𝑦
𝑑𝑥
= √
1−𝑦2
1−𝑥2
6. If x = a sint andy = a ( cos t + logtan
𝑡
2
), find
𝑑2 𝑦
𝑑𝑥2
.
7. 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.
8. If x = a(cos t + t sin t), y = b(sin t – t cos t), Prove that
𝑑𝑦2
𝑑𝑥2
=
𝑏 𝑠𝑒𝑐3 𝑡
𝑎2 𝑡
.
9. Differentiate tan-1 [
√1+ 𝑥2− √1− 𝑥2
√1+ 𝑥2+ √1− 𝑥2
] withrespecttocos-1
x2
.
10. y = ex
tan-1
x,thenshowthat (1 + x2
)
𝑑2 𝑦
𝑑𝑥2
- 2 (1 – x + x2
)
𝑑𝑦
𝑑𝑥
+ (1 – x2
) y = 0.
11. Verify lagrange’s Mean value theorem: (i) f(x) = x(x – 1) (x – 2) [ 0, ½] (ii) f(x) = x3
-5x2
– 3x [1, 3]

5. 12. 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
]
13. Differentiate sin-1
(
2𝑥
1+𝑥2
) w.r.t. tan-1
x .
14. If x = √ 𝑎sin−1 𝑡 , 𝑦 = √ 𝑎cos−1 𝑡 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
𝑑𝑦
𝑑𝑥
= −
𝑦
𝑥
.
15. If y =
sin−1 𝑥
√1−𝑥2
, show that (1-𝑥2)
𝑑2 𝑦
𝑑𝑥2
− 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦 = 0 .
16. Differentiate cos-1
{
1− 𝑥2
1+ 𝑥2
} withrespectof tan-1
{
3𝑥 − 𝑥3
1−3 𝑥2
} .
APPLICATION OF DERIVATIVES (4+6=10 Marks)
4 Marks Questions:
1. Show that the line
𝑥
𝑎
+
𝑦
𝑏
= 1 touchesthe curve y = be –x/a
at the pointwhere itcrossesthe y axis.
2. Separate the interval [0,
𝜋
2
]intosub – intervalsinwhichf(x) =sin4
x + cos4
x is increasingordecreasing.
3. The radiusof a spherical diamondis measuredas7 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. Showthat the curves4x = y2
and4xy = K cut at rightanglesif K2
= 512 .
5. Findthe intervalsinwhichthe functionf givenbe f(x) –sinx –cosx,0  x  2 is strictlyincreasingorstrictly
decreasing.
6. Findall pointsonthe curve y = 4x3
– 2x5
at whichthe tangentspassesthroughthe origin.
7. Findthe equationof normal tothe curve y = x3
+ 2x + 6 whichare parallel toline x+14y+4=0.
8. Showthat the curves y = aex
and y = be –x
cut at rightanglesif ab = 1.
9. Findthe intervalsinwhichthe function f f(x) = x3
+
1
𝑥3
, x ≠ 0 isincreasingordecreasing.
10. Prove that y =
4 𝑠𝑖𝑛 𝜃
(2+𝑐𝑜𝑠 𝜃)
– 𝜃 is an increasingfunctionof 𝜃 in[0,
𝜋
2
] .
11. Usingdifferentials,findthe approximatevalue of eachof following:
(i)(401)1/2
(ii) (26)1/3
(iii) (0.009)1/3
(iv)√25.3 (v)√36.6
6 Marks Questions:
1. Findthe area of the greatestrectangle thatcan be inscribedinanellipse
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1.
2. If the sumof the lengthsof the hypotenuse andaside of a righttriangle isgiven,show thatthe area of the
triangle ismaximumwhenthe angle betweenthemis

3
.
3. Prove that the radiusof right circularcylinderof greatestC.S.A whichcanbe inscribedinacone ishalf of that of
cone.

6. 4. A helicopterif flyingalongthe curve y= x2
+ 2. A soldierisplacedatthe point(3,2) . findthe nearestdistance
betweenthe soliderandthe helicopter.
5. Showthat the rightcircular cylinderof givensurface and maximumvolume issuchthatitsheightisequal to
diameterof base.
6. Prove that the volume of largestcone thatcan be inscribedinasphere of radiusR is8/27 timesvolume of
sphere.
7. Showthat the semi – vertical angle of cone of maximumvolume andof givenslantheightistan-1
√2.
8. A wire of length28 cm isto be cut into twopieces, one of the twopiecesisto made intoa square and other
intocircle.What shouldbe the lengthof twopiecessothatcombinedareaof square and circle isminimum?
9. Showthat the heightof cylinderof maximumvolume thatcanbe inscribedina sphere of radiusR is2R/√3 .
10. The sum of perimeterof circle andsquare isK. Prove thatthe sumof theirareasisleastwhen side of square is
double the radiusof circle.
11. Findthe value of x forwhich f(x) = [x(x – 2)]2
is an increasingfunction.Also,findthe pointsonthe curve,where
the tangentisparallel tox- axis.
12. Show that all the rectanglesinscribedinagivenfixedcircle,the square hasmaximumarea.
13. A wire of length36 cm iscut intotwo pieces,one of the piecesisturnedinthe formof a square and otherinthe
formof an equilateraltriangle.Findthe lengthof eachpiece sothatthe sumof the areas of the two be
minimum.
14. Prove that the surface areaof a solidcuboid,of square base andgivenvolume,isminimumwhenitisacube.
15. 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?
16. An openbox witha square base isto be made out of a givenquantityof cardboardof area c2
square units.Show
that the maximumvolume of the box is
𝑐3
6√3
cubicunits.
17. A windowisinthe formof a rectangle surmountedbyasemicircularopening.Total perimeterof windowis10m.
Findthe dimensionsof the windowtoadmitmaximumlightthroughwhole opening.
INTEGRALS (4+4+4=12 Marks)
1. Evaluate :- ∫
𝑥sin 𝑥 cos𝑥
𝑠𝑖𝑛4 𝑥+ 𝑐𝑜𝑠4 𝑥
𝜋
2
0 dx.
2. Evaluate :- ∫
𝑥2+ 1
( 𝑥−1)2 ( 𝑥+3)
dx.
3. Evaluate :- ∫ log( 1 + tan 𝑥) 𝑑𝑥
𝜋
4
0 , usingpropertiesof definite integrals.
4. Evaluate :- ∫
sin( 𝑥−𝑎)
sin( 𝑥+𝑎)
dx.

7. 5. Evaluate :- ∫
5𝑥 2
1+2𝑥+3𝑥2
dx.
6. Evaluate :- ∫( 2 sin 2𝑥 − cos 𝑥) √6 − 𝑐𝑜𝑠2 𝑥− 4 sin 𝑥 dx.
7. Evaluate :- ∫
2
( 1−𝑥)( 1+ 𝑥2 )
dx
8. Usingpropertiesof definite integrals,evaluate : ∫
𝑥 𝑑𝑥
25 𝑠𝑖𝑛2 𝑥+16 𝑐𝑜𝑠2 𝑥
𝜋
0 .
9. Evaluate :- ∫
𝑑𝑥
𝑠𝑖𝑛𝑥− sin2𝑥
dx
10. Evaluate :- ∫
𝑥2
𝑥2+ 3𝑥−3
𝑑𝑥
11. Evaluate :- ∫ 𝑒 𝑥 (
sin4𝑥−4
1−cos4𝑥
) dx
12. Evaluate :- ∫
𝑥2
( 𝑥−1)3 (𝑥+1)
dx
13. Show that ∫ (√ 𝑡𝑎𝑛 𝑥 + √ 𝑐𝑜𝑡 𝑥) 𝑑𝑥 = √2 𝜋
𝜋
2
0
14. Evaluate ∶ − ∫
𝑥 𝑑𝑥
4−𝑐𝑜𝑠2 𝑥
𝜋
0
15. Evaluate :- ∫ | 𝑥3 − 𝑥|
2
1 dx
16. Prove that :- ∫ sin−1 (√
𝑥
𝑎+𝑥
)
𝑎
0 𝑑𝑥 = −
𝑎
2
( 𝜋 − 2).
17. Evaluate :- ∫
1
𝑠𝑖𝑛𝑥 ( 5−4cos𝑥)
dx
APPLICATION OF INTEGRATION (6 MARKS)
1. Usingintegrationfindthe areaof the region{ (x,y) : x2
+ y2
 1  x +
𝑦
2
, x , y  R} .
2. Usingintegration,findthe areaof the regionenclosedbetweenthe twocirclesx2
+y2
= 4 and
(x – 2)2
+ y2
= 4.
3. Findthe area of the region{ (x,y) : y2
 6ax andx2
+ y2
 16a2
} usingmethodof integration.
4. Prove that the area betweentwoparabolasy2
4ax andx2
= 4ay is16 a2
/ 3 sq units.
5. Usingintegration,findthe areaof the followingregion. {( 𝑥, 𝑦):
𝑥2
9
+
𝑦2
4
≤ 1 ≤
𝑥
3
+
𝑦
2
}.
6. Findthe area of the region{(x,y) :x2
+ y2
≤ 4, 𝑥 + 𝑦 ≥ 2}.
7. Findthe area lyingabove x – axisand includedbetweenthe circle x2
+y2
= 8x andthe parabolay2
= 4x.
8. Findthe area of the regionincludedbetweenthe curve 4y = 3x2
and line 2y = 3x + 12
9. Sketchthe graph of y = | 𝑥 + 3| and Evaluate ∫ | 𝑥 + 3|
0
−6 dx .
10. Usingthe methodof integration,findthe areaof the regionboundedbythe lines3x – 2y + 1 = 0,
2x + 3y – 21 = 0 and x – 5y + 9 = 0.

8. DIFFERENTIAL EQUATIONS (1+1+6=8 MARKS)
1 MARK 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.
6 MARKS QUESTIONS.
1. Showthat the differential equation
xdy – ydx = √𝑥2 + 𝑦2 dx is homogeneousandsolve it.
2. Findthe particularsolutionof the differential equation:- cosx dy = sinx ( cos x – 2y) dx,giventhaty = 0, whenx
=
𝜋
3
.
3. 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.
4. 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
.
5. Solve the followingdifferentialequation:- (1+ y + x2
y) dx + (x + x3
)dy= 0, where y= 0 whenx = 1.
6. Solve the followingdifferentialequation: √1 + 𝑥2 + 𝑦2 + 𝑥2 𝑦2 + xy
𝑑𝑦
𝑑𝑥
= 0 .
7. Findthe particularsolutionof te differential equation:( xdy –ydx) y sin
𝑦
𝑥
= ( 𝑦𝑑𝑥 + 𝑥𝑑𝑦) 𝑥 cos
𝑦
𝑥
, giventhat y=
 whenx = 3.
8. 𝑆𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 ∶ − 𝑥𝑒
𝑦
𝑥 − 𝑦 sin
𝑦
𝑥
+ 𝑥
𝑑𝑦
𝑑𝑥
sin
𝑦
𝑥
= 0 . giventhaty = 0 where x = 1, i.e.,y(1) =
0
9. Solve the initial value problem: (x2
+ 1)
𝑑𝑦
𝑑𝑥
- 2xy = ( x4
+ 2x2
+ 1) cos x,y (0) = 0.
10. Solve :(x2
+ xy) dy= (x2
+ y2
) dx.
11. 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.
12. Solve the equation : (𝑥 𝑐𝑜𝑠
𝑦
𝑥
+ 𝑦 𝑠𝑖𝑛
𝑦
𝑥
) 𝑦 𝑑𝑥 = (𝑦 𝑠𝑖𝑛
𝑦
𝑥
− 𝑥 𝑐𝑜𝑠
𝑦
𝑥
) x dy.
13. Find the particular solution of the differential equation (1 + x3
)
𝑑𝑦
𝑑𝑥
+ 6x2
y = (1 + x2
), given that x = y = 1.

9. VECTORS AND THREE DIMENSIONAL GEOMETRY(1+1+1+4+4+6=17 MARKS)
1 MARK QUESTIONS.
1. If 𝑝⃗ is a unit vector and (𝑥⃗ − 𝑝⃗) ( 𝑥⃗ + 𝑝⃗) = 80 , then find | 𝑥⃗| .
2. Find the value of 𝜆 so that the vector 𝑎⃗ = 2 𝑖̂ + 𝜆𝑗̂ + 𝑘̂ and 𝑏⃗⃗ = 𝑖̂ − 2𝑗̂ +3 𝑘̂ are perpendicular to each
other.
3. If two vectors 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ are : | 𝑎⃗| = 2, | 𝑏⃗⃗| = 3 and 𝑎⃗. 𝑏⃗⃗ = 4, find | 𝑎⃗ − 𝑏⃗⃗|.
4. Find the angle between 𝑖̂ − 2𝑗̂ + 3𝑘̂ 𝑎𝑛𝑑 3𝑖̂ − 2𝑗̂ + 𝑘̂ .
5. 𝑎⃗ = 𝑖̂ + 2𝑗̂ − 𝑘̂, 𝑏⃗⃗ = 3𝑖̂ + 𝑗̂ − 5𝑘̂, find a unit vector in the direction 𝑎⃗ − 𝑏⃗⃗ .
6. Write Direction ratios of
𝑥−2
2
=
2𝑦−5
−3
= 𝑧 − 1 .
7. Find ‘ ’ when the projection of 𝑎⃗ = 𝛌𝑖̂ + 𝑗̂ + 4 𝑘̂ and 𝑏⃗⃗ = 2𝑖̂ + 6𝑗̂ + 3 𝑘̂ is 4 units.
8. Find the volume of the parallelopiped whose adjacent sides are represented by 𝑎⃗ , 𝑏⃗⃗ and 𝑐⃗ where
𝑎⃗ = 3𝑖̂ − 2𝑗̂ + 5𝑘̂ , 𝑏⃗⃗ = 2𝑖̂ + 2𝑗̂ − 𝑘̂ , 𝑐⃗= -4𝑖̂ + 3𝑗̂ + 2𝑘̂ .
9. Find the value of 𝜆 so that the vectors 𝑖̂ + 𝑗̂ + 𝑘̂ , 2𝑖̂ + 3𝑗̂ − 𝑘̂ , -𝑖̂ + 𝜆𝑗̂ + 2𝑘̂ are coplanar.
4 MARKS QUESTIONS.
1. Finda unitvectorperpendiculartothe plane of triangle ABCwhere the verticesare A (3, -1, 2), B ( 1, -1, -3) and
C ( 4, -3, 1).
2. Let 𝑎⃗ = 4𝑖̂ + 5𝑗̂ + 𝑘̂, 𝑏⃗⃗ = 𝑖̂ − 4𝑗̂ + 5𝑘̂ and 𝑐⃗ = 3𝑖̂ + 𝑗̂ − 𝑘̂. Find a vector 𝑑⃗ which is perpendicular to
both 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ 𝑎𝑛𝑑 : 𝑑⃗. 𝑐⃗ = 21.
3. Let 𝑎⃗, 𝑏⃗⃗ 𝑎𝑛𝑑𝑐⃗ be three vectors : | 𝑎⃗| = 3, | 𝑏⃗⃗| = 4, | 𝑐⃗| = 5 and each one of them being perpendicular to
the sum of other two, find | 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗| .
4. Usingvectors,findthe area of the triangle ABCwithverticesA (1,2, 3), B ( 2, -1, 4) and C ( 4, 5, -1) .
5. 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.
6. If 𝑎⃗ and𝑏⃗⃗ are unit vectors and 𝜃 is the angle between them, show that sin 𝜃/2 = ½ | 𝑎⃗ − 𝑏⃗⃗| .
7. If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is √3 .
8. The two adjacent side of a parallelogram are 2𝑖̂ − 4𝑗̂ − 5𝑘̂ and 𝑖̂ − 2𝑗̂ + 3𝑘̂ . Find the unit vector
parallel to its diagonal. Also, find its area.
9. The scalar product of the vector 𝑖̂ + 𝑗̂ + 𝑘̂ with a unit vector along the sum of vectors 2𝑖̂ + 4𝑗̂ − 5𝑘̂ and
𝛾𝑖̂ + 2𝑗̂ + 3𝑘̂ is equal to one, find the value of 𝛾 .
10. 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.

10. 11. 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) .
12. 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 .
13. 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.
14. Findthe shortestdistance betweenthe lines:
𝑟⃗ = 6𝑖̂ + 2𝑗̂ + 2𝑘̂ +  ( 𝑖̂ − 2𝑗̂ + 2𝑘̂ ) 𝑎𝑛𝑑 𝑟⃗ = −4𝑖̂− 𝑘̂ +  (3 𝑖̂ − 2𝑗̂ − 2𝑘̂ ) .
15. If any three vectors 𝑎⃗ , 𝑏⃗⃗ and 𝑐⃗ are coplanar,prove that the vectors 𝑎⃗ + 𝑏⃗⃗ , 𝑏⃗⃗ + 𝑐⃗ and 𝑐⃗ + 𝑎⃗ are also
coplanar.
16. If 𝑎⃗ , 𝑏⃗⃗ and 𝑐⃗ are mutuallyperpendicularvectorsof equal magnitudes,show thatthe vector 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ is
equallyinclinedto 𝑎⃗ 𝑏⃗⃗ 𝑎𝑛𝑑 𝑐⃗ .
17. Find the position vector of a point R which divides the line joining two points P and Q whose position
vectors are (2𝑎⃗ + 𝑏⃗⃗ ) & (𝑎⃗ − 3𝑏⃗⃗) respectively, externally in the ration 1 : 2. Also, show P is the mid
point of RQ .
18. . If 𝛼 = 3𝑖̂ + 4𝑗̂ + 5𝑘̂ and 𝛽 = 2𝑖̂ + 𝑗̂ − 4𝑘̂ , then express 𝛽⃗ in the form 𝛽⃗ = 𝛽⃗1 + 𝛽⃗2, where 𝛽⃗1 is parallel
to 𝛼⃗ and 𝛽⃗2 is perpendicular to 𝛼⃗ .
19. Find the angle between the line
𝑥+1
2
=
3𝑦+5
9
=
3−𝑧
−6
and the plane 10x + 2y – 11z = 3.
20. Find whether the lines 𝑟⃗ = ( 𝑖̂ − 𝑗̂ + 𝑘̂) + 𝜆(2𝑖̂ + 𝑗̂) and 𝑟⃗ = (2𝑖̂ − 𝑗̂) + 𝜇( 𝑖̂ + 𝑗̂ − 𝑘̂) intersect or not. If
intersecting, find their point of intersection.
6 MARKS QUESTIONS.
1. Show that the lines
𝑥+3
−3
=
𝑦−1
1
=
𝑧−5
5
,
𝑥+1
−1
=
𝑦−2
2
=
𝑧−5
5
are coplanar. Also find the equation of the plane
containing the lines.
2. Find the co-ordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane determined by
points A(1, 2, 3), B (2, 2, 1) and C (-1, 3, 6) .
3. Findthe equationof the line passingthroughthe point(-1,3, -2) and perpendiculartothe lines
𝑥
1
=
𝑦
2
=
𝑧
3
And
𝑥+2
−3
=
𝑦−1
2
=
𝑧+1
5
4. Findthe equationof the plane containingthe lines:
𝑟⃗ = 𝑖̂ + 𝑗̂ +  ( 𝑖̂ + 2 𝑗̂ − 𝑘̂ ) and 𝑟⃗ = 𝑖̂ + 𝑗̂ +  (− 𝑖̂ + 𝑗̂ − 2 𝑘̂ )
Findthe distance of thisplane fromoriginandalsofrom the point (1,1, 1).
5. Findthe coordinatesof the footof the perpendiculardrawnfromthe pointA (1, 8, 4) to the line joiningthe
pointB (0, -1, 3) and C ( 2, -3, -1).
6. Findthe distance of the point(-1, -5, -10) from the pointof intersectionof the line

11. 𝑟⃗ = (2 𝑖̂ − 𝑗̂ + 2 𝑘̂) +  (3 𝑖̂ + 4 𝑗̂ + 2 𝑘̂) and the plane 𝑟⃗. ( 𝑖̂− 𝑗̂ + 𝑘̂) = 5.
7. 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.
8. 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.
9. Findthe image of point(1, 6, 3) inthe line
𝑥
1
=
𝑦−1
2
=
𝑧−2
3
.
10. 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.
11. Findwhetherthe lines 𝑟⃗ = ( 𝑖̂ − 𝑗̂ − 𝑘̂ ) +  ( 𝑖̂ + 𝑗̂ ) and 𝑟⃗ = ( 2𝑖̂ − 𝑗̂ ) +  ( 𝑖̂ + 𝑗̂ − 𝑘̂ ) intersectornot.If
intersecting,findtheirpointof intersection.
12. Findthe distance betweenthe pointP(5,9) and the plane determinedbythe pointsA(3, -1,2), B(5, 2, 4) and C( -
1, -1, 6).
13. Find the equation of plane through the points (1, 2, 3) & (0, -1, 0) and parallel to the line
𝑟⃗ .(2 𝑖̂+ 3𝑗̂ + 4𝑘̂) + 5 = 0 .
14. Prove that the image of the point (3, -2, 1) in the plane 3x-y+4z = 2 lie on plane x+y+z+4 = 0.
15. Findthe vectorequationof the plane throughthe points( 2, 1, -1) and ( -1, 3, 4) andperpendiculartothe plane
x – 2y + 4z = 10.
16. 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.
17. 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 whichisperpendiculartothe plane
𝑟⃗ .(5𝑖̂ + 3𝑗̂ − 6 𝑘̂ ) + 8 = 0 .
PROBABILITY(4+6=10MARKS)
4 MARKS QUESTIONS.
1. Probabilityof solvingspecificproblembyX& Y are
1
2
and
1
3
. If bothtry to solve the problem, findthe probability
that:
(i) Problemissolved. (ii) Exactlyone of themsolvesthe problem.
2. Three ballsare drawn withoutreplacementfromabag containing5 white and4 greenballs.Findthe probability
distributionof numberof greenballs.
3. A die isthrownagainand againuntil three sixesare obtained.Findthe probabilityof obtaining3rd
six in6th
throwof die.

12. 4. A speakstruthin60% of the casesand B in 90% of the cases.In whatpercentage of casesare theylikelyto
contradicteach otherinstatingthe same fact?
5. Findthe meannumberof headsinthree tossesof a faircoin.
6. Bag I contains3 redand 4 blackballsandBags II contains4 red and 5 black balls.One ball istransferredfrom
Bag I to bag II and thentwoballsare drawn at random( withoutreplacement) fromBagII.The ballsso
drawnare foundto be bothred incolour.Findthe probabilitythatthe transferredball isred.
7. In a hockey match,both teamA andB scoredsame numberof goalsup to the endof the game,so to decide the
winner,the refereeaskedboththe captainstothrow a die alternatelyanddecidedthatthe team, whose captain
getsa six first,will be declaredthe winner. If the captainof teamA wasaskedto start, findtheirrespectively
probabilitiesof winningthe matchandstate whetherthe decisionof the referee wasfairornot.
8. In answeringaquestiononaMCQ testwith4 choicesperquestion,astudentknowsthe answer,guessesor
copiesthe answer.Let½ be the probabilitythathe knowsthe answer,¼ be the probabilitythathe guessesand
¼ that he copiesit.Assumingthatastudent,whocopiesthe answer,will be correctwiththe probability¾,what
isthe probabilitythatthe studentknowsthe answer,giventhathe answereditcorrectly?
Arjundoesnotknowthe answertoone of the questioninthe test.The evaluationprocesshasnegative marking.
Whichvalue wouldArjunvioletif he resortstounfairmeans?How wouldanact like the above hamperhis
character developmentinthe comingyears?
9. The probabilitythata studententeringauniversitywill graduate is0.4.findthe probabilitythatoutof 3
studentsof the university:
a)None will graduate, b)Onlyone willgraduate, c) All will graduate.
10. How manytimesmusta man tossa faircoin,so that the probabilityof having atleastone headismore than
80% ?
11. On a multiple choice examinationwiththree possible answer (outof whichonlyone iscorrect) for each of the
five questions,whatisthe probabilitythatacandidate wouldgetfouror more correct answerjustby guessing?
12. A familyhas2 children.Findthe probabilitythatbothare boys,if itis knownthat
(i) At leastone of the childrenisa boy (ii) the elderchildisaboy.
13. An experimentsucceedtwice oftenasitfails.Findthe probabilitythatinthe nextsix trailsthere willbe atleast4
successes.
14. From a lotof 10 bulbs,whichinclude 3defectives,asample of 2 bulbsisdrawn at random.Findthe probability
distributionof the numberof defective bulbs.
6 Marks Questions:
1. Findthe probabilitydistributionof numberof doubletsinthree throwsof apairof dice.Alsofindmean&
variance.

13. 2. 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.
3. 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 twooptions,the patientselectedatrandomsuffersaheartattack. Findthe probabilitythatthe patient
followedacourse of meditationandyoga.Interpretthe resultandstate whichof the above statedmethodsis
more beneficialforthe patient.
4. An insurance companyinsured2000 cyclist,4000 scooterdriversand6000 motorbike drivers.The probabilityof
an accidentinvolvingacyclist,scooterdriveranda motorbike driverare 0.01, 0.03 and 0.15 respectively.One of
the insuredpersonsmeetswithanaccident.Whatisthe probabilitythathe isa scooterdriver?Whichmode of
transportwouldyousuggestto a studentandwhy?
5. Two bagsA and B contain4 white and3 blackballsand2 white and2 black ballsrespectively.FrombagA,two
ballsare drawn at randomand thentransferredtobag B. A ball isthendrawnfrombag B and isfoundto be a
blackball.What isthe probabilitythatthe transferredballswere 1white and1 black?
6. A biaseddie istwice aslikelytoshowanevennumberasan oddnumber.The die isrolledthree times.If
occurrence if an evennumberisconsideredasuccess,thenwritthe probabilitydistributionof numberof
successes.Alsofindthe meannumberof success.
7. Three bags containballsasshowninthe table below :
Bag Number of White Balls Number of Black
balls
Number of Red
balls
I 1 2 3
II 2 1 1
III 4 3 2
A bag is chosen at random and two balls are drawn from it. They happen to be white and red.
What is the probability that they come from the III Bag?
8. There are three coins. One is two headed coin (having head on both faces),another is a biased coin that comes up
tail 25% of the times and third is an unbiased coin. One of the three coins is chosen at random and tossed, it
shows heads, what is the probability that it was the two headed coin?
LINEAR PROGRAMMING (6 MARKS)
1. If a young man rides his motorcycle at 25 km/h, he has to spend Rs. 2/km on petrol. If he rides at a faster
speed of 40 km/h, the petrol cost increase at Rs. 5/km. He has Rs. 100 to spend on petrol and wishes to find
what is the maximum distance he can travel within one hour? Solve it graphically. Write two methods to save
petrol in daily life.

14. 2. An aeroplane cancarry a maximumof 200 passengers.A profitof Rs.1000 is made on eachexecutive class
ticketanda profitof Rs. 600 is made on eacheconomyclassticket.The airline reservesatleast20 seatsfor
executiveclass.However,atleast4 timesasmany passengersprefertotravel byeconomyclassthanby the
executiveclass.Determine howmanyticketsof eachtype mustbe soldinorderto maximise the profitfor
the airline.Whatisthe maximumprofit?Doyoufeel thatair travel issafernow than inoldendays?Discuss
briefly.
3. Kelloggisnewcereal formedof amixture of branand rice that containsat least88 grams of proteinandat
least36 milligramsof iron.Knowingthatbrancontains80 grams of proteinand40 milligramsof ironper
kilogram,andthat rice contains100 grams of proteinand30 milligramsof ironperkilogram, findthe
minimumcostof producingthiscereal if brancosts Rs. 15 perkilogramandrice costsRs. 14 per kilogram.
4. A manufacturerproduces pizzaandcakes.Ittakes1 hour of work onmachine.A and 3 hourson machine B
to 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 shouldbe producedeachdayso as to maximize hisprofitsif he operateshismachinesforatthe most
12 hours a day?Form the above as linearprogrammingproblemandsolveitgraphically.Whypizzaand
cakesare not goodfor health?
5. A toycompanymanufacturestwotypesof dollsA and B. markettestsand available resourceshave
indicatedthatthe combinedproductionlevel shouldnotexceed1200 dollsperweekandthe demandfor
dollsof type B isat mosthalf of thatfor dollsof type A.further,the productionlevelof dollsof type A can
exceedthree timesthe productionof dollsof othertype byatmost 600 units.If the company makesprofit
of Rs.12 and Rs 16 per doll respectivelyondollsA andB, how many of each shouldbe producedweeklyin
orderto maximise the profit?Whoisinterestedincolourful dolls?
6. A retiredpersonhasRs.70,000 to investandtwotypesof bondsare available inthe marketforinvestment.
Firsttype of bondsyieldsanannual income of 8% on the amountinvestedandthe secondtype of bond
yields10%per annum.Aspee norms,he has to investaminimumof Rs.10,000 in the firsttype andnot
more than Rs. 30,000 inthe secondtype.How shouldhe planhisinvestment,soastoget maximumreturn,
afterone year of investment?
7. One kindof cake requires300 g of flourand15g of fat,anotherkindof cake requires150g of flourand30g
of fat.Findthe maximumnumberof cakeswhichcanbe made from7.5kg of flourand 600g of fat,assuming
that there isno shortage of the otheringredientsusedinmakingthe cakes.Make itasan LPPand solve it
graphically.