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CBSEClass–XIIMathematics

SamplePaper-02

Timeallowed:3hours(MM:100)

GeneralInstructions:

(i)Allquestionsarecompulsory.

(ii)Thisquestionpapercontains29questions.

(iii)Question1-4inSectionAareveryshort-answertypequestionscarrying1markeach.

(iv)Question5-12inSectionBareshort-answertypequestionscarrying2markseach.

(v)Question13-23inSectionCarelong-answer-Itypequestionscarrying4markseach.

(vi)Question24-29inSectionDarelong-answer-IItypequestionscarrying6markseach.

SectionA

1.Giveanexampleofarelationwhichissymmetricbutnotreflexiveandtransitive.

Ans.LetA={1,2,3,4}

LetR={(1,2),(2,1)}

2.Isthemeasureof5secondsisscalarorvector?

Ans.Scalar.

3.Whatisthedomainof ?

Ans.[-1,1]

4.Showthat

Ans.

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SectionB

5.Findtheprincipalvalueof .

Ans.Let

6.If and .Findrelationgivenby .

Ans.

ATQ.

7.VerifyRolle’sTheoremforthefunctionf(x)=x2+2x–8,x [-4,2]

Ans.Thefunction

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Continuousin[-4,2]anddifferentiablein(-4,2)

Also

HencealltheconditionofallRolle’sTheorem,isverified

TheirexistavalueC

Suchthat (c)=0

(c)=2c+2

0=2C+2

C=-1

8.Find

Ans.

9.

Ans.Put

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10.Verifythatthefunctionisasolutionofthecorrespondingdiffeq.

Ans.

Henceproved.

11.Findtheanglebetweenvectors

Ans.

12.Adiethrownthreetimes.EventsAandBaredefinedasbelow.

A:4onthethirdthrow

B:6onthefirstand5onthesecondthrow.

FindtheprobabilityofAgiventhatBhasalreadyoccurred.

Ans.Totalsamplespace=216

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SectionC

13.Solve:

Ans.

14.Differentiate

Ans.

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15.Usedifferentiationtoapproximate

Ans.Let

Let

Then

Putthevalueofdyinequation(1)

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16.Findaunitvectorperpendiculartoeachofthevector

Ans.

Unitvector=

17.IfAandBareindependenteventssuchthatP(AUB)=0.6,P(A)=0.2.FindP(B)

Ans.SinceAandBareindependentevents

18.ProbabilityofsolvingspecificproblemindependentlybyAandBare and

respectivelyofbothtrytosolvetheproblemindependently,findtheprobabilitythat

(i)theproblemissolved

(ii)Exactlyoneofthemsolvestheproblem.

Ans.E1:Asolvestheproblem

E2:Bsolvestheproblem

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P(E1)= andP(E2)=

(i)P(theproblemissolved)

=1–P(theproblemisnotsolved)

(ii)PExactlyoneoftheirsolvestheproblem

19.Solve

Ans.

Solutionis:

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20.If andIistheidentitymatrixoforder2,showthat

Ans.

21.If .

Ans.

Differentiatingbothsidesw.r.tx,weget

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22.Foranytwovectors

Ans.

23.Findthevectorequationoftheplanepassingthroughtheintersectionoftheplanes

Ans.

Equationofplane:

SectionD

24.LetA=R–{3}andB=R–{1}.Considerthefunction :A Bdefinedby

Is one-oneandonto?Justifyyouranswer.

Ans.A=R–{3}andB=R–{1}and

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Let A,then and

Now,for

isone-onefunction.

Now,

=

Therefore, isanontofunction.

25.Integrate

Ans.

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26.Findtheshortestdistancebetweenthelines

Ans.

27.Usingintegrationfindtheareaoftheregion

Ans.Given:

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Tofindthepointsofintersectionofthecircle andtheparabola

,wewillsubstitute .

Therefore,thepointsofintersectionare(0,0),(a,a)and(a,−a).

Now,

Areaoftheshadedregion=I

AreaofIfromx=0tox=a

Letx−a=tforthefirstpartoftheintegral

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Areaoftheshadedregion squareunits

28.Thesumofthreenumbersis6.Ifwemultiplythethirdnumberby3andaddsecond

numbertoitweget11.Byaddingfirstandthirdnumberswegetdoubleofthesecond

number.Representthefollowinginformationmathematicallyandsolveusingmatrices.

Ans.SupposethefactoryproducesxunitsofmachineAandyunitsofmachineB.

Then,ProfitZ=10,500x+9000y

Themathematicalformulationoftheproblemisasfollows:

MaxZ=10,500x+9000y

s.t10x+20y 480,x+2y 48(metalconstraint)

15x+10y 400,3x+2y 80(paintingconstraint)

x 0,y 0

Wegraphtheaboveinequalities.Thefeasibleregionisasshowninthefigure.Weobserve

thefeasibleregionisboundedandthecornerpointsareO,B,EandC.Theco-ordinatesofthe

cornerpointsare(0,0),(0,24),(16,16),(80/3,0).

CornerPoint Z=10,500x+9000y

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(0,0) 0

(0,24) 2,16,000

(16,16) 3,12,000

(80/3,0) 2,80,000

Thusprofitismaximizedbyproducing16unitseachofmachineAandB.

29.AfactorymanufacturestwotypesofmachinesAandB.Eachtypeismadeofcertain

metal.Thefactoryhasonly480kgsofthismetalavailableinaday.Tomanufacture

machineA,10kgsofmetalisrequiredand20kgsisrequiredforB.MachineAandB

require15and10minutestobepainted.Paintingdepartmentcanuseonly400minutes

inaday.Thefactoryearnsprofitof10,500onmachineAand9000onmachineB.State

asalinearprogrammingproblemandmaximizestheprofit.

Ans.Thepointofintersectionofthetwocurves:

Rejectingy=-8,weget .

Shadedarea=Requiredarea=Ar(OAB)+Ar(OBC)=2Ar(OAB)