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MaterialdownloadedfrommyCBSEguide.com. 1/16
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)
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