MaterialdownloadedfrommyCBSEguide.com. 1/18

CBSEClass-12Mathematics

SamplePaper(ByCBSE)

GeneralInstructions:

Allquestionsarecompulsory.

Thisquestionpapercontains29questions.

Question1-4inSectionAareveryshort-answertypequestionscarrying1markeach.

Questions5-12inSectionBareshort-answertypequestionscarrying2markseach.

Questions13-23inSectionCarelong-answer-Itypequestionscarrying4markseach.

Questions24-29inSectionDarelong-answer-IItypequestionscarrying6markseach.

SectionA

Questions1to4carry1markeach.

1.LetA={1,2,3,4}.LetRbetheequivalencerelationonA Adefinedby

(a,b)R(c,d)iffa+d=b+c.Findtheequivalenceclass[(1,3)].

2.IfA[aij]isamatrixoforder2 2,suchthat =-15andCijrepresentsthecofactorofaij,

thenfinda21C21a22C22

3.Giveanexampleofvectors and suchthat but ≠ .

4.Determinewhetherthebinaryoperation*onthesetNofnaturalnumbersdefinedbya*b

2abisassociativeornot.

SectionB

Questions5to12carry2markseach

5.If4sin-1cos-1x=π,thenfindthevalueofx.

6.Findtheinverseofthematrix Hence,findthematrixPsatisfyingthematrix

MaterialdownloadedfrommyCBSEguide.com. 2/18

equationP = .

7.Provethatif ≤x≤thencos-1x+cos-1

8.Findtheapproximatechangeinthevalueof ,whenxchangesfromx=2tox=2.002

9.Find

10.Verifythatax2+by2=1isasolutionofthedifferentialequationx(yy2+y12)=yy1

11.FindtheProjection(vector)of

12.IfAandBaretwoeventssuchthatP(A)=0.6,thenfindP(A/B).

SectionC

Questions13to23carry4markseach.

13.If

Thenfindthevalueof

14.Find‘a’and‘b’,ifthefunctiongivenbyf(x)=

OR

Determinethevaluesof‘a’and‘b’suchthatthefollowingfunctioniscontinuousatx=0:

MaterialdownloadedfrommyCBSEguide.com. 3/18

15.Ify=log thenprovethatx(x+1)2y2+(x+1)2y1=2.

16.Findtheequation(s)ofthetangent(s)tothecurvey=(x3-1)(x-2)3atthepointswherethe

curveintersectsthex–axis.

OR

Findtheintervalsinwhichthefunctionf(x)=-3log(1+x)+4log(2+x)- isstrictlyincreasing

orstrictlydecreasing.

17.Apersonwantstoplantsometreesinhiscommunitypark.Thelocalnurseryhasto

performthistask.Itchargesthecostofplantingtreesbythefollowingformula:

C(x)=x3-45x2+600x,WherexisthenumberoftreesandC(x)isthecostofplantingxtreesin

rupees.Thelocalauthorityhasimposedarestrictionthatitcanplant10to20treesinone

communityparkforafairdistribution.Forhowmanytreesshouldthepersonplacethe

ordersothathehastospendtheleastamount?

Howmuchistheleastamount?Usecalculustoanswerthesequestions.Whichvalueisbeing

exhibitedbytheperson?

18.Find

19.Findtheparticularsolutionofthedifferentialequation:

OR

Showthat(x-y)dy=(x+2y)dxisahomogenousdifferentialequation.Also,findthegeneral

solutionofthegivendifferentialequation.

MaterialdownloadedfrommyCBSEguide.com. 4/18

20.If arethreevectorssuchthat, thenprovethat andhence

showthat =0.

21.Findtheequationofthelinewhichintersectsthelinesandpassesthroughthepoint(1,1,

22.BagIcontains1white,2blackand3redballs;BagIIcontains2white,1blackand1red

balls;BagIIIcontains4white,3blackand2redballs.Abagischosenatrandomandtwo

ballsaredrawnfromitwithreplacement.Theyhappentobeonewhiteandonered.Whatis

theprobabilitythattheycamefromBagIII.

23.Fourbadorangesareaccidentallymixedwith16goodones.Findtheprobability

distributionofthenumberofbadorangeswhentwoorangesaredrawnatrandomfromthis

lot.Findthemeanandvarianceofthedistribution.

SectionD

Questions24to29carry6markseach.

24.Ifthefunctionf: bedefinedbyf(x)=2x-3andg: byg(x)=x3+5,thenfind

fogandshowthatfogisinvertible.Also,find(fog)-1,hencefind(fog)-1(9).

OR

Abinaryoperation*isdefinedontheset ofrealnumbersbya*b=

Ifatleastoneofaandbis0,thenprovethata*b=b*a.Checkwhether*iscommutative.Find

theidentityelementfor*,ifitexists.

25.If thenfindA-1andhencesolvethefollowingsystemof

equations:3x+4y+7z=14,2x-y+3z=4,x+2y-3z=0

OR

IfA= ,findtheinverseofAusingelementaryrowtransformationsandhence

MaterialdownloadedfrommyCBSEguide.com. 5/18

solvethefollowingmatrixequationXA=[101].

26.Usingintegration,findtheareainthefirstquadrantboundedbythecurve the

circle andthey-axis

27.Evaluatethefollowing:

OR

Evaluate asthelimitofasum.

28.fromtheline

measuredparalleltotheplane:x–y+2z–3=0.

29.Acompanyproducestwodifferentproducts.Oneofthemneeds1/4ofanhourof

assemblyworkperunit,1/8ofanhourinqualitycontrolworkandRs1.2inrawmaterials.

Theotherproductrequires1/3ofanhourofassemblyworkperunit,1/3ofanhourin

qualitycontrolworkandRs0.9inrawmaterials.Giventhecurrentavailabilityofstaffinthe

company,eachdaythereisatmostatotalof90hoursavailableforassemblyand80hours

forqualitycontrol.Thefirstproductdescribedhasamarketvalue(saleprice)ofRs9per

unitandthesecondproductdescribedhasamarketvalue(saleprice)ofRs8perunit.In

addition,themaximumamountofdailysalesforthefirstproductisestimatedtobe200

units,withouttherebeingamaximumlimitofdailysalesforthesecondproduct.

FormulateandsolvegraphicallytheLPPandfindthemaximumprofit.

MaterialdownloadedfrommyCBSEguide.com. 6/18

CBSEClass-12Mathematics

SamplePaper(ByCBSE)

Answer

1.

2.-15

3. (oranyothercorrectanswer)

4.(1*2)*3=22*3=212,1*(2*3)=1*26=264 .

Hence,*isnotassociative.

SectionB

5.

6.

7.Letscos-1x=θ.Then

ThegivenexpressiononLHS

8.

MaterialdownloadedfrommyCBSEguide.com. 7/18

9.

10.

Substitutingthisvalue,foraintheequation(1),weget,

11.

TherequiredProjection(vector)of

12.

13.

WhereCij=thecofactorofaijandaij=the(i,j)theelementofdeterminantD.

MaterialdownloadedfrommyCBSEguide.com. 8/18

Weknowthat

14.Since,fisdifferentiableat1,fiscontinuousat1Hence,

f(x)= (2x+1)=3

Asfiscontinuousat1,wehavea+b=3…(1)

=2a

Asfisdifferentiableat1,wehave2a=2,i.e.,a=1andb=2.

OR

f(0)=2.

Forthefunctiontobecontinuousat0,wemusthave

i.e.,wemust bmaybeanyrealnumberotherthan0.

15.

=

MaterialdownloadedfrommyCBSEguide.com. 9/18

(using(1))

Hence,proved.

16.Wheny=0,wehave(x-1)(x2+x+1)(x-2)=0,i.e.,x=1or2.

Therequiredequationsofthetangentsarey-0=-3(x-1)or,y=-3x+3andy-0=7(x-2)or,y=7x-14.

OR

Domainf=

In(-1,0),f’(x) Therefore,fisstrictlydecreasingin(-1,0].

In Therefore,fisstrictlyincreasingin .

17.WehaveC(x)=x3-45x2+600x,10≤x≤20.Forthetimebeingwemayassumethatthefunction

C(x)iscontinuousin[10,20].

C’(x)=3x2-90x+600=3(x-10)(x-20)

C’(x)=0ifx=10orx=20.

But,10,20 (10,20).

Therefore,themaximumortheminimumvaluewilloccuratthepoints.

C(10)=2500,C(20)=2000.

Hence,thepersonmustplacetheorderfor20treesandtheleastamounttobespent=Rs2000.

Value:Thepersoncaresforahealthyenvironmentdespitebeingeconomically

constrained.

18.

MaterialdownloadedfrommyCBSEguide.com. 10/18

Putt=-1,-1=2B,i.e.,B=-1/2.Put,t=1,1=4C,i.e.,C=1/4.

Putt=0,0=A+B+C,whichgives .

Thereforetherequiredintergral

19.Thegivendifferentialequationisyeydx=(y3+2xey)dy,y(0)=1

Or,

whichislinearinx.

I.F.=

MultiplyingbothsidesbytheI.F.andintegrating,weget,

(thegeneralsolution).

Whenx=0,y=1.0=-e-1+C Hence,therequiredparticularsolutionis

Thegivendifferentialequationis(x-y)dy=(x+2y)dxor,

Hence,homogeneous.

MaterialdownloadedfrommyCBSEguide.com. 11/18

Puty=vx Thenequationbecomes

Or,

Integrating,weget

Or,log(v2+v+1)-

OR,log(y2+xy+x2)-2 (thegeneralsolution).

20.

=0[Asthescalartripleproductofthreevectorsiszeroifanytwoofthemareequal.]

21.Generalpointonthefirstlineis

Generalpointonthefirstlineis

Directionratiosoftherequired

Therefore,thedirectionratiosoftherequiredlineare(6,20,34)or,(3,10,17).

Hence,therequiredequationofthelineis

22.LetE1=BagIischosen,E2=BagIIischosen,E2=BagIIIischosen,A=Thetwoballsdrawn

MaterialdownloadedfrommyCBSEguide.com. 12/18

fromthechosenbagarewhiteandred.

ByBayes’sTheorem,therequiredprobability

=

23.LetXdenotestherandomvariable.ThenX=0,1,2.

Xi Pi Xipi Xi2pi

0 60/95 0 0

1 32/95 32/95 32/95

Variance=

24. definedby

fog(x)=f(g(x))=f(x3+5)=2(x3+5)-3=2x3+7

Let

MaterialdownloadedfrommyCBSEguide.com. 13/18

Hence,fogisone-one.

Lety (Codomainfog).Hence,foreveryy (Codomainfog), such

thatfog

Hence,

Fogisonto.

Sincefogisbothone-oneandonto,itisinvertible.

(fog)-1: definedby(fog)-1(x)=

(fog)-1(9)=

OR

Leta,b suchthata=0,b≠0.

Thena*b=

Leta,b suchthata=0,b≠0.

Thena*b=a,b*a=

Leta,b suchthata=0,b=0.

Thena*b=a=0,b*a=b=0,

∴a*b=b*a.

Nowweneedtocheckwhether*iscommunicative.Onemorecaseisneededtobe

examined.Leta,b suchthata≠0,b≠0.Thena*b= anda*bmay

notbeequaltob*a,e.g.,(-1)*2=3,2*(-1)=1,hence,(-1)*2≠2*(-1).Thus*isnot

communicative.

MaterialdownloadedfrommyCBSEguide.com. 14/18

Theelement willbetheidentifyelementfor*ifa*e=e*a=aforalla .a*e=a

providede=0ande*a=aprovidede=0(As0*0=0and0*a= +a=afora≠0).Hence,0isthe

identifyelementfor*.

25.

.

Hence,A-1exist.LetCijrepresentthecofactorof(I,j)thelementofA.Then,

C11=-3,C12=26,

Thegivensystemofequationsisequivalenttothematrixequation

Hence,x=1,y=1,z=1

MaterialdownloadedfrommyCBSEguide.com. 15/18

26.

Solvingy= simultaneously,

(y=x2liesinquadrantI).

MaterialdownloadedfrommyCBSEguide.com. 16/18

Therequiredarea=theshadedarea

27.Thegivedefiniteintegral=I

=

Hence,fisodd.Therefore,

g(x)=

Hence,giseven.Thus

Hence,

MaterialdownloadedfrommyCBSEguide.com. 17/18

Letf(x)=3x2-2x+4.Thenthegivendefiniteintegral

= wherenh=4.

f(-2+rh)=3(-2+rh)2-2(-2+rh)+4=3r2h2-14rh+20

28.Thegeneralpointonthegivenline is

Thedirectionratiosofthelineparalleltotheplanex-y+2z-3=0intersectingthegivenlineand

passingthroughthepoint(-2,3,-4)are and + (-1)+

.

Thepointofintersectionis

Therequireddistance

MaterialdownloadedfrommyCBSEguide.com. 18/18

= unit.

28.Letx=thenumberofunitsofproduct1tobeproduceddaily

Y=thenumberofunitsofProducts2tobeproduceddaily

TomaximizeP=(9-1.2)x+(8-0.9)y=7.8x+7.1ysubjecttotheconstraints:

Atthepoint P

(0,0) 0

(200,120) 2412

(0,240) 1704

(200,0) 1560

(80,210) 2115

Themaximumprofit=Rs.2412.