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SAMPLEQUESTIONPAPER(ByCBSE)

Class-X(2017–18)Mathematics

GeneralInstructions:

1. Allquestionsarecompulsory.

2. Thequestionpaperconsistsof30questionsdividedintofoursectionsA,B,CandD.

3. SectionAcontains6questionsof1markeach.SectionBcontains6questionsof2marks

each.SectionCcontains10questionsof3markseach.SectionDcontains8questionsof4

markseach.

4. Thereisnooverallchoice.However,aninternalchoicehasbeenprovidedinfour

questionsof3markseachandthreequestionsof4markseach.Youhavetoattemptonly

oneofthealternativesinallsuchquestions.

5. Useofcalculatorsisnotpermitted.

SectionA

Questionnumbers1to6carry1markeach

1.Writewhethertherationalnumber willhaveaterminatingdecimalexpansionor

anor-terminatingrepeatingdecimalexpansion.

Ans.Nonterminatingrepeatingdecimalexpansion.

2.Findthevalue(s)ofk,ifthequadraticequation hasequalroots.

Ans.

3.FindtheeleventhtermfromthelasttermoftheAP:

27,23,19,...,–65.

Ans.

4.Findthecoordinatesofthepointony-axiswhichisnearesttothepoint(–2,5).

Ans.

5.Ingivenfigure, and Findtheratiooftheareaof

totheareaof

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Ans.

6.If findthevalueof

Ans.25

SectionB

Questionnumbers7to12carry2markseach.

7.Iftwopositiveintegerspandqarewrittenas areprime

numbers,thenverify:

Ans.

8.ThesumoffirstntermsofanAPisgivenby Findthesixteenthterm

oftheAP.

Ans.Sn=2n2+3n

S1=5=a1

S2=a1+a2=14 a2=9

d=a2–a1=4

a16=a1+15d=5+15(4)=65

9. Find the value(s) of k for which the pair of linear equations

haveinfinitelymanysolutions.

Ans.Forpairofequationskx+1y=k2and1x+ky=1

Wehave:

Forinfinitelymanysolutions,

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From(i)and(ii),k=1

10.If isthemid-pointofthelinesegmentjoiningthepoints(2,0)and

thenshowthattheline passesthroughthepoint

Ans.Since isthemid-pointofthelinesegmentjoiningthepoints

therefore,

Theline passesthroughthepoint

11.Aboxcontainscardsnumbered11to123.Acardisdrawnatrandomfromthebox.

Findtheprobabilitythatthenumberonthedrawncardis

(i)asquarenumber

(ii)amultipleof7

Ans.(i)P(squarenumber)

(ii)P(multipleof7)

12.Aboxcontains12ballsofwhichsomeareredincolour.If6moreredballsareput

intheboxandaballisdrawnatrandom,theprobabilityofdrawingaredballdoubles

thanwhatitwasbefore.Findthenumberofredballsinthebag.

Ans.Letnumberofredballsbe=x

If6moreredballsareadded:

Thenumberofredballs=x+6

Since,

Thereare3redballsinthebag.

SectionC

Questionnumbers13to22carry3markseach.

13.Showthatexactlyoneofthenumbers isdivisibleby3.

Ans.Letn=3k,3k+1or3k+2.

(i)Whenn=3k:

nisdivisibleby3.

n+2=3k+2 n+2isnotdivisibleby3.

n+4=3k+4=3(k+1)+1 n+4isnotdivisibleby3.

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(ii)Whenn=3k+1:

nisnotdivisibleby3.

n+2=(3k+1)+2=3k+3=3(k+1) n+2isdivisibleby3.

n+4=(3k+1)+4=3k+5=3(k+1)+2 n+4isnotdivisibleby3.

(iii)Whenn=3k+2:

nisnotdivisibleby3.

n+2=(3k+2)+2=3k+4=3(k+1)+1 n+2isnotdivisibleby3.

n+4=(3k+2)+4=3k+6=3(k+2) n+4isdivisibleby3.

Henceexactlyoneofthenumbersn,n+2orn+4isdivisibleby3.

14.Findallthezeroesofthepolynomial iftwoofitszeroes

are

Ans.Since arethetwozeroestherefore,

isafactorofgivenpolynomial.

Wedividethegivenpolynomialby

Forotherzeroes,

Zeroesofthegivenpolynomialare

15.Seventimesatwodigitnumberisequaltofourtimesthenumberobtainedby

reversingtheorderofitsdigits.Ifthedifferenceofthedigitsis3,determinethe

number.

Ans.Lettheten’sandtheunitsdigitbeyandxrespectively.

So,thenumberis10y+x.

Thenumberwhendigitsarereversedis10x+y.

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Now,7(10y+x)=4(10x+y) 2y=x…(i)

Alsox–y=3…(ii)

Solving(1)and(2),wegety=3andx=6.

Hencethenumberis36

16.Inwhatratiodoesthex-axisdividethelinesegmentjoiningthepoints

Findtheco-ordinatesofthepointofdivision.

OR

Thepoints formaparallelogram.Findthe

lengthofthealtitudeoftheparallelogramonthebaseAB.

Ans.Letx-axisdividesthelinesegmentjoining atthepointPinthe

ratio1:k.

Now,coordinatesofpointofdivision

SincePliesonx-axis,therefore

Hencetheratiois

Now,thecoordinatesofPare

OR

LettheheightofparallelogramtakingABasbasebeh.

NowAB

17.Ingivenfigure thenprovethat

OR

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InanequilateraltriangleABC,DisapointonthesideBCsuchthat Prove

that

Ans.

Since,

Also

And

OR

Construction:Draw

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18.Ingivenfigure aretwoparalleltangentstoacirclewithcentreO

andanothertangentABwithpointofcontactCintersecting atAand atB.

Provethat

Ans.JoinOC

In

OP=OC(radiiofsamecircle)

PA=CA(lengthoftwotangents)

AO=AO(Common)

(BySSScongruencycriterion)

Hence,

Similarly

Now,

19.Evaluate:

OR

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If thenevaluate:

Ans.

OR

20.IngivenfigureABPCisaquadrantofacircleofradius14cmandasemicircleis

drawnwithBCasdiameter.Findtheareaoftheshadedregion.

Ans.Weknow,AC=r

Requiredarea= +ar(semicircleonBCasdiameter)–ar(quadrantABPC

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21.Waterinacanal,6mwideand1.5mdeep,isflowingwithaspeedof10km/h.How

muchareawillitirrigatein30minutes,if8cmofstandingwaterisneeded?

OR

Aconeofmaximumsizeiscarvedoutfromacubeofedge14cm.Findthesurfacearea

oftheremainingsolidaftertheconeiscarvedout.

Ans.Lettheareathatcanbeirrigatedin30minutebeA

Waterflowingincanalin30minutes

Volumeofwaterflowingoutin30minutes

Volumeofwaterrequiredtoirrigatethefield

Equating(i)and(ii),weget

Or

Surfaceareaofremainingsolid whererandlaretheradiusandslant

heightofthecone.

22.Findthemodeofthefollowingdistributionofmarksobtainedbythestudentsinan

examination:

Marksobtained 0-20 20-40 40-60 60-80 80-100

Numberofstudents 15 18 21 29 17

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Giventhemeanoftheabovedistributionis53,usingempiricalrelationshipestimate

thevalueofitsmedian.

Ans.

So,themodemarksis68.

Empiricalrelationshipbetweenthethreemeasuresofcentraltendenciesis:

SectionD

Questionnumbers23to30carry4markseach.

23.Atraintravellingatauniformspeedfor360kmwouldhavetaken48minuteslessto

travelthesamedistanceifitsspeedwere5km/hourmore.Findtheoriginalspeedof

thetrain.

OR

Checkwhethertheequation hasrealrootsandifithas,findthemby

themethodofcompletingthesquare.Alsoverifythatrootsobtainedsatisfythegiven

equation.

Ans.Letoriginalspeedofthetrainbexkm/h.

Timetakenatoriginalspeed

Timetakenatincreasedspeed

Now,

OR

Discriminant

So,thegivenequationhastwodistinctrealroots

Multiplyingbothsidesby5.

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Verification:

Similarly,

24.AnAPconsistsof37terms.Thesumofthethreemiddlemosttermsis225andthe

sumofthelastthreetermsis429.FindtheAP.

Ans.LetthethreemiddlemosttermsoftheAPbea–d,a,a+d.

Wehave,(a–d)+a+(a+d)=225

Now,theAPis

a–18d,…,a–2d,a–d,a,a+d,a+2d,…,a+18d

Sumoflastthreeterms:

Now,firstterm

TheAPis3,7,11,…,147.

25.Showthatinarighttriangle,thesquareofthehypotenuseisequaltothesumofthe

squaresoftheothertwosides.

OR

Provethattheratiooftheareasoftwosimilartrianglesisequaltotheratioofthe

squaresoftheircorrespondingsides.

Ans.Given:ArighttriangleABCrightangledatB.

Toprove:

Construction:Draw

Proof:In

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Now, (correspondingsidesareproportional)

Similarly

Adding(1)and(2)

OR

Given:

Toprove:

Construction:Draw

In

Therefore,

But

Hence,

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26.DrawatriangleABCwithside Then,construct

atrianglewhosesidesare timesthecorrespondingsidesof

Ans.Draw inwhich andhence

Constructionofsimilartriangle asshownbelow:

27.Provethat

Ans.

28.Theanglesofdepressionofthetopandbottomofabuilding50metreshighas

observedfromthetopofatowerare respectively.Findtheheightofthe

towerandalsothehorizontaldistancebetweenthebuildingandthetower.

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Ans.

Now,

Now,

Heightoftower=TR=75m

Distancebetweenbuildingandtower

29.TwodairyownersAandBsellflavouredmilkfilledtocapacityinmugsofnegligible

thickness,whicharecylindricalinshapewitharaisedhemisphericalbottom.Themugs

are14cmhighandhavediameterof7cmasshowningivenfigure.BothAandBsell

flavouredmilkattherateof perlitre.ThedairyownerAusestheformula

tofindthevolumeofmilkinthemugandcharges forit.ThedairyownerB

isoftheviewthatthepriceofactualquantityofmilkshouldbecharged.What

accordingtohimshouldbethepriceofonemugofmilk?Whichvalueisexhibitedby

thedairyownerB?

Ans.Capacityofmug(actualquantityofmilk)

AmountdairyownerBshouldchargeforonemugofmilk

ValueexhibitedbydairyownerB:honesty(oranysimilarvalue)

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30.Thefollowingdistributionshowsthedailypocketallowanceofchildrenofalocality.

Themeanpocketallowanceis Findthemissingfrequencyk.

Dailypocketallowance(inRs.) 11-13 13-15 15-17 17-19 19-21 21-23 23-25

Numberofchildren 3 6 9 13 k 5 4

OR

Thefollowingfrequencydistributionshowsthedistance(inmetres)thrownby68

studentsinaJavelinthrowcompetition.

Distance(inm) 0-10 10-20 20-30 30-40 40-50 50-60 60-70

Numberofstudents 4 5 13 20 14 8 4

DrawalessthantypeOgiveforthegivendataandfindthemediandistancethrown

usingthiscurve.

Ans.

Dailypocketallowance

(inRs.)

Numberofchildren Mid-point

11-13 3 12 -3 -9

13–15 6 14 -2 -12

15-17 9 16 -1 -9

17–19 13 18 0 0

19–21 K 20 1 k

21–23 5 22 2 10

23-25 4 24 3 12

OR

Lessthan NumberofStudents

10 4

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20 9

30 22

40 42

50 56

60 64

70 68

MediandistanceisvalueofxthatcorrespondstoCumulativefrequency

Therefore,Mediandistance=36m