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SAMPLEQUESTIONPAPER04

Class-X(2017–18)

Mathematics

Timeallowed:3HoursMax.Marks:80

GeneralInstructions:

(i)Allquestionsarecompulsory.

(ii)Thequestionpaperconsistsof30questionsdividedintofoursectionsA,B,CandD.

(iii)SectionAcontains6questionsof1markeach.SectionBcontains6questionsof2marks

each.SectionCcontains10questionsof3markseach.SectionDcontains8questionsof4

markseach.

(iv)Thereisnooverallchoice.However,aninternalchoicehasbeenprovidedinfour

questionsof3markseachandthreequestionsof4markseach.Youhavetoattemptonlyone

ofthealternativesinallsuchquestions.

(v)Useofcalculatorsisnotpermitted.

SECTION-A

1.Findthezeroesofthequadraticpolynomial .

2.Writethesumofthefirstfifteennaturalnumbers.

3.Findthecoordinateofthemid-pointofthelinesegmentjoiningthepointswhose

coordinatesare and .

4.Infig-1,centreofthecircleisO.FromoutsidepointP,twotangentsPAandPBaredrawn

totouchthecircle.GivenÐAPB=60o.FindÐAOB.

5.Aladderoflength4m.makesanangleof45owiththegroundwhenplacedagainstan

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electricpost.Determinethedistancebetweenthefeetoftheladderandtheelectricpost.

6.CaptainofIndiancricketteamtossestwodifferentcoins,oneof₹1andotherof₹2simultaneously.Whatistheprobabilitythathegetsatleastonehead?

SECTION-B

7.Findthediscriminantoftheequation andhencefindthenatureits

roots.

8.Ifthe10thtermofanA.P.is47andits1sttermis2,findthesumofitsfirst15terms.

9.Findarelationbetweenxandysuchthatthepoint(x,y)isequidistantfromthepoints(7,

1)and(3,5).

10.Infig-2, and .Provethat, isisosceles.

11.Twoconcentriccirclesareofradius5cmand3cm.Findthelengthofthechordofthe

largercirclewhichtouchesthesmallercircle.

12.Theheightandbasediameterofasolidcylinderare210mand24cmrespectively.Find

thevolumeofthecylinder.

SECTION-C

13.Thetaxichargesinacityconsistofafixedchargetogetherwiththechargeforthe

distancecovered.Foradistanceof10km,thechargepaidis₹200andforajourneyof15km,thechargepaidis₹275.Whatarethefixedchargesandthechargeperkm?Howmuchapersonhastopayfortravellingadistanceof25km?

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14.Showthat isirrational.

15.Showthatanypositiveoddintegerisoftheform4q+1or4q+3,whereqissomeinteger.

16.The4thtermofanA.P.isequaltothreetimesthe1stterm&the7thtermexceedstwice

the3rdtermby1.Findthe1sttermandcommondifference.

Or

ThefirstandthelasttermsofanAPare17and350respectively.Ifthecommondifferenceis

9,howmanytermsarethereandwhatistheirsum?

17.IfA(-5,7),B(-4,-5),C(-1,-6)andD(4,5)aretheverticesofaquadrilateral,findthearea.

Or

Findthecoordinatesofthepointswhichdividetheline-segmentjoiningthepoints(-4,0)and

(0,6)infourequalparts.

18.Provethatthesumofthesquaresofthesidesofarhombusisequaltosumofthesquares

onitsdiagonals

19.Provethat,thelengthsoftangentsdrawnfromanexternalpointtoacircleareequal.

Or

Infig-3,PQandRSaretwoparalleltangentstoacirclewithcentreOandanothertangentEF

withpointofcontactCintersectingPQatEandRSatF.Provethat∠EOF=90°.

20.Evaluate:

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21.Aboxcontains90discswhicharenumberedfrom1to90.Ifonediscisdrawnatrandom

fromthebox,findtheprobabilitythatitbearsaperfectsquarenumber.

Or

Twohorsesareconsideredforarace.Theprobabilityofselectionofthefirsthorseis1/4and

thatofsecondis1/3.Whatistheprobabilitythat:

(a)bothofthemwillbeselected.

(b)onlyoneofthemwillbeselected.

(c)noneofthemwillbeselected.

22.Aboxcontains5redmarbles,8whitemarblesand4greenmarbles.Onemarbleistaken

outoftheboxatrandom.Whatistheprobabilitythatthemarbletakenoutwillbe

i)whiteii)notgreen?

SECTION-D

23.Twowatertapstogethercanfillatankin hours.Thetapoflargerdiametertakes10

hrslessthanthesmalleronetofillthetankseparately.Findthetimeinwhicheachtapcan

separatelyfillthetank.

Or

Atrain,travellingatauniformspeedfor360km,wouldhavetaken48minuteslesstotravel

thesamedistanceifitsspeedwere5km/hrmore.Findtheoriginalspeedofthetrain.

24.Findthesumofallthree-digitnumberswhichleavestheremainder3whendividedby5.

25.ConstructatrianglesimilartotriangleABCwithitssideequalto ofthecorresponding

sidesofthetriangleABC.

Or

Constructanisoscelestrianglewhosebaseis7cmandaltitude5cmandthenconstruct

anothertrianglewhosesidesare timesthecorrespondingsidesoftheisoscelestriangle.

26.Provethat: .

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27.Infig-3, of ABCisarightangleandAB=3units,AC=4units.Semicirclesare

drawnonAB,ACandBCasdiameters.Alsoacirclecircumscribingthe ABCisdrawn.Find

theareaoftheshadedregion.

28.Twoshipsaresailingintheseaoneithersideofalight-house.Theanglesofdepressionof

twoshipsasobservedfromthetopofthelight-houseare60oand45orespectively.Ifthe

distancebetweentheshipsis meters,findtheheightofthelight-house.

29.Apersonconnectsapipeofinternaldiameter20mfromacanalintoaemptycylindrical

tankinhisfieldwhichis10mindiameterand2mdeep.Iftherateofflowofwaterthrough

thepipeis6km/hr,thenafterhowmuchtimetheflowofwatershouldbestoppedtoavoid

overflowofwaterfromthetank?Whatvaluewehavefromtheproblem?

30.Themedianofthefollowingfrequencydistributionis35.Findthevalueofx.

ClassInterval 0-10 10-20 20-30 30-40 40-50 50-60 60-70

Frequency 2 3 5 6 x 3 2

Or

Computethemodeforthefollowingfrequencydistribution.

Sizeofitems: 0-4 4-8 8-12 12-16 16-20 20-40 24-28 28-32 32-36 36-40

Frequency: 5 7 9 17 12 10 6 3 1 0

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CBSESAMPLEPAPER04

CLASSXMathematics

Solution

SECTION-A

1.Wehave

=(x+5)(x+2)

So,thezeroesof are-5and-2.

2.Thesumofthefirstfifteennaturalnumbers=

3.Coordinatesofthemid-pointare

4.

5.

6.Possibleoutcomesare(H,H),(H,T),(T,H),(T,T)whichareequallylikely.

Outcomesfavorabletotheeventare(H,H),(H,T),(T,H).HencetherequiredProbability=

SECTION-B

7.Here, .

Discriminant

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Hencethegivenquadraticequationhastwoequalrealroots.

8.LetthecommondifferenceoftheA.Pbed.

Given,thefirstterm(a)=2andthe10thterm( )=47

As,nthterm,

So,

i.e.,9d=47-2

i.e.,

As,thesumofthefirstn-termsofanAP,

Therefore,thesumofthefirst15-termsoftheAP

9.SincethepointP(x,y)isequidistantfromthepointsA(7,1)andB(3,5)

So,PA=PBi.e.,

i.e.,

i.e.,

i.e.,

i.e., ,Thisistherequiredrelqtion.

10.

Hence, (correspondingangles).....(i)

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............(ii)

Therefore,PQ=PR[sidesoppositetoequalangles]

Hence, isisosceles.

11.LetCisthecentreoftwoconcentriccirclesofradii5cmand3cm.

LetAB,achordofthebiggercircletouchesthesmallercircleatM.

CM=3cm,CA=5cm

[radiusthroughpointofcontactis totangent]

[byPythagorasTheorem]

i.e.,

Therefore,AB=2AM=2*4=8cm.

[Linesegment,drawnfromcentreofacircleperpendiculartoanychord,bisectsthechord]

Hence,thelengthofthechordis8cm.

12.LetV,handrdenotethevolume,heightandbaseradiusofthecylinderrespectively.

Here,h=210m,

Then,

Hencethevolumeofthecylinderis

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SECTION-C

13.Letthefixedcharge=₹xandthechargeperkm=₹y

Accordingtothefirstcondition,x+10y=200…(i)

Accordingtothesecondcondition,x+15y=275…(ii)

Nowsubtracting(i)from(ii),weget,

5y=75

i.e.,y=15

From(i),x+10*15=200(puttingvalueofy]

i.e.,x=200-150

i.e.,x=50

Hence,thefixedcharge=50andthechargeperkm=₹15

Nowfortravellingadistanceof25km,

thepersonhastopay=₹(50+25*15)=₹425.

14.letusassumethat, isrational.

i.e.,letfortwocoprimenumbersaandb(a,bareintegers,b 0),

i.e.,

Since, isrationalso, isrational.

Thisisacontradiction,because isirrational.

Thiscontradictionisonlyforourincorrectassumption,

that, isrational.

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Hence,theconclusionisthat, isirrational.

15.AccordingtoEuclid’sdivisionlemma,

Givenpositiveintegersaandb,

thereexistuniqueintegersqandrsatisfyinga=bq+r,0≤r<b.

Letusstartwithtakinga,whereaisapositiveoddinteger.

Weapplythedivisionalgorithmwithaandb=4.

Since,0≤r<4,thepossibleremaindersare0,1,2and3.

i.e.,acanbe4qor4q+1or4q+2,or4q+3,whereqisthequotient.

However,sinceaisodd,acannotbe4qor4q+2(sincetheyarebothdivisibleby2)

Therefore,anyoddintegerisoftheform4q+1or4q+3.

16.Let,a,dand bethefirstterm,commondifferenceandthe termoftheAP.

As,

Accordingtothefirstcondition,

i.e.,a+(4-1)d=3a

i.e.,3d=2a….(i)

Also,accordingtothe2ndcondition,

i.e.,a+(7-1)d-[a+(3-1)d]=1

i.e.,a+6d-a-2d=1

i.e.,4d=1

i.e.,

Nowfrom(i), i.e.,

Hencethefirstterm= andcommondifference=

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17.ByjoiningAtoC,weget and

As,areaofatrianglewithverticesatpoints

is

…(i)

So,usingformula(i)

Sotheareaof is .

Again,usingformula(i),

So,theareaofquadrilateralABCD=

18.LetABCDisrhombus.

ItsdiagonalsACandBDintersectatpointO.

Toprovethat,

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Sincediagonalsofarhombusbisectseachotherperpendicularly,

so,AO=OC= ;BO=OD=

andACisperpendiculartoBDatpointO.

and areallrightangledtriangle.

UsingPythagorasTheorem,

From ,

From ,

From ,

From ,

Adding(i),(ii),(iii),(iv),weget,

[sinceAO=COandBO=DO]

i.e.,

Henceproved

19.LetfromanexternalpointPtwotangentsPQandPR

aredrawntothecirclewithcentreatO.

ThetangentstouchthecircleatpointsQandR.

Wehavetoprovethat,PQ=PR.

LetusdrawthelinesegmentsOQ,ORandOP.

NowinDPOQandDPOR,

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[each=90oasOQandORareradii

throughpointsofcontacts]

OQ=OR[radiiofsamecircle]

OP=OP[Commonside]

Therefore, [RHS]

Hence,PQ=PR[correspondingpartofcongruenttriangles]

20.Weknow,

,

,

=

21.Onediscisdrawnatrandomfromtheboxmeansthatallthediscsareequallylikelytobe

drawn.

LettheeventofdrawingonediscbearingaperfectsquarenumberbeE.

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Giventhattotalnumberofdiscsinthebox=90

Therefore,totalnumberofallpossibleoutcomes=90

1,4,9,16,25,36,49,64,81areperfectsquarenumbersbetween1to90.1Therefore,

outcomesfavorabletotheeventE=9

So,

22.Onemarbleistakenoutoftheboxatrandommeans,

allthemarblesareequallylikelytobetakenout.Therefore,thetotalnumberofpossible

outcomes=5+8+4=17

LettheeventoftakingoutofonewhitemarblebeW

andalsolettheeventoftakingoutofonegreenmarblebeG

ThennumberofoutcomesfavorabletotheeventW=8

Therefore, [answerof(i)]

AgainthenumberofoutcomesfavourabletotheeventG=4

So,

Therefore,P(notgreen)=1-P(G) [answerof(ii)]

SECTION-D

23.Letthetapofsmallerdiametercanfillthetankseparatelyinxhrs

Thenthetapofbiggerdiametercanfillthetankseparatelyin(x-10)hrs.

Accordingtothequestion,

twowatertapstogethercanfillatankin hoursi.e.,in hrs.

Then,in1hrsmallerdiametertapcanfill partofthetank,

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in1hrbiggerdiametertapcanfill partofthetank,

and1hrtwotapstogethercanfill partofthetank,

So,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

Now,usingthequadraticformula,weget,

Since ,so,x=25

Therefore,tapofsmallerdiametercanfillthetankseparatelyin25hrs,andthetapofbigger

diametercanfillthetankseparatelyin15hrs.

24.Thefirstthree-digitnumberwhichleavesremainder3whendividedby5is103.[as103=

5*20+3]

Lastthree-digitnumberwhichleavesremainder3whendividedby5is998.[as998=5*199

+3]

Nowthethree-digitnumberswhichleavesremainder3whendividedby5are103,108,111,

…,998,whichformanAP.

Firstterm,a=103,commondifference,d=108-103=5

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Let,998bethe term( )oftheseries.

As,

So,

i.e.,

i.e.,

Therefore,998isthe180thtermoftheseries.

Nowsumofthetermsoftheseries

=

=99090

25.Stepsofconstruction:

DrawanyrayBXmakinganacuteanglewithBConthesideoppositetothevertexA.

Locate4(thegreaterof3and4in )points onBXsothat

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JoinB4CanddrawalinethroughB3(the3rdpoint,3beingsmallerof3and4in )

paralleltoB4CtointersectBCatC′.

DrawalinethroughC′paralleltothelineCAtointersectBAatA′.

Thus, A′BC′istherequiredtriangle

26.

Henceproved.

27.In ABC, ,AB=3units,AC=4units

So,byPythagorasTheorem,

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Areaofthesemi-circleonBC(asdiameter)

i.e.,areaofBECB=

Areaofthesemi-circleonAC(asdiameter)

i.e.,areaofADCA=

Areaofthesemi-circleonAB(asdiameter)

i.e.,areaofAFBA=

Alsofromthefig.(areaofAHBA+areaofAGCA)

=areaofsemi-circleBHGCB-areaof ABC

= -

= -

=

AreaofAFBHA+areaofAGCDA

=areaofAFBA+areaofADCA-(areaofAHBA+areaofAGCA)

= + - squnits.

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Hence,areaoftheshadedpart

i.e.,areaofareaofBECB+areaofAGCDA+areaofAFBHA

= + + - sq.units

=+ + -sq.units

=

28.Lethmetersbetheheightofthelight-houseAB.

AlsolettwoshipsbeatCandD(infig)

Byquestion,CD= meters.

Angleofdepressionoftwoshipsasobserved

fromthetop(A)ofthelight-houseare60oand45orespectively.

Inthefig. ,

Therefore,

(alternateangle)

Nowin ,

Again,in ,

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[by(i)]

Hence,theheightofthelight-houseis meters

29.Radiusofthecylindricaltank= ,itsdepthis2m.

Internalradiusofthepipe=

Sovolumeofthetank=

Flowofwaterthroughthepipe=3km/hr

Volumeofwaterwillflowthroughthepipeperminute

=

Therefore,timetakentofillthetank

So,flowofwatershouldbestoppedafter1hr40mintoavoidoverflowfromthetank.

Value:Waterismostimportantsubstanceintheearthforlives(human,animals,plants)so

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weshoulduseourwaterwiselyandnotwasteit.

30.

ClassInterval Frequency CumulativeFrequency

0-10 2 2

10-20 3 5

20-30 5 10

30-40 6 16

40-50 x 16+x

50-60 3 19+x

60-70 2 21+x

Givenmedian=35.Thisliesintheclass30-40

So,l=30,f=6,cf=thecumulativefrequencyoftheclasspreceding30-40=10

Weknow,

Here,

Hencevalueofxis5.