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SAMPLEQUESTIONPAPER01

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.Cantwonumbershave18astheirHCFand380astheirLCM?Givereason.

2.Findtherootoftheequation .

3.Determinewhether50cm,80cm,100cmcanbethesidesofarighttriangleornot.

4.Thelengthoftheshadowofamanisequaltotheheightofman.Theangleofelevationis

________.

5.Iftheperimeterandareaofacirclearenumericallyequal,thenfindtheradiusofthe

circle.

6.Ifthreecoinsaretossedsimultaneously,thenfindtheprobabilityofgettingatleasttwo

heads.

Section-B

7.Is7×6×5×4×3×2×1+5acompositenumber?Justifyyouranswer.

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8.The11thtermofanA.P.exceedsits4thtermby14.Findthecommondifference.

9.Findtherelationbetween‘x’and‘y’,ifthepoints ,(1,2)and(7,0)arecollinear.

10.Twotangentsmakinganangleof120°witheachotheraredrawntoacircleofradius6

cm,findthelengthofeachtangent.

11.Provethat

12.Aconeofheight20cmandradiusofbase5cmismadeupofmodellingclay.Achild

reshapesitintheformofasphere.Findthediameterofthesphere.

Section-C

13.UseEuclid’sdivisionlemmatoshowthatcubeofanypositiveintegeriseitheroftheform

9q,9q+1or9q+8forsomeinteger‘q’.

14.Obtainallotherzeroesofx4+5x3-2x2-40x-48iftwoofitszeroesare

15.Solvefor‘x’: ,x≠0.

16.Howmanytermsoftheseries54,51,48,…….betakensothattheirsumis513?Explainthe

doubleanswer.

Or

InanAPpth,qthandrthtermsarerespectivelya,bandc.Provethat

p(b-c)+q(c-a)+r(a-b)=0

17.ThepointA(3,y)isequidistantfromthepointsP(6,5)andQ(0,-3).Findthevalueofy.

Or

IfA(4,6),B(3,-2)andC(5,2)aretheverticesofΔABC,thenverifythefactthatamedianofa

triangleABCdividesitintotwotrianglesofequalareas.

18.If provethat

19.Anobserver1.5mtallis28.5mawayfromachimney.Theangleofelevationofthetopof

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thechimneyfromhereyesis45°.Whatistheheightofthechimney?

Or

Fromthetopofa7mhighbuilding,theangleofelevationofthetopofacabletoweris

60oandtheangleofdepressionofthefootofthetoweris30o.Findtheheightofthetower.

20.Aboyiscyclingsuchthatthewheelsofthecyclearemaking140revolutionsperminute.

Ifthediameterofthewheelis60cm,calculatethespeedinKmperhourinwhichtheboyis

cycling.

21.Thefollowingtableshowsthegaininweightby50childreninayear.Calculatemodal

gaininweight.

Gaininweight(inkg) 1-3 3-5 5-7 7-9 9-11 11-13

No.ofchildren 4 6 10 18 7 5

Or

ComputetheMedianforthegivendata

Class–interval 100-110 110-120 120-130 130-140 140-150 150-160

Frequency 6 35 48 72 100 4

22.Whatistheprobabilitythataleapyear,selectedatrandomwillcontain53Thursdays?

Section-D

23.Solvegraphicallythefollowingequations2x+3y=9;x–2y=1.Shadetheregionbounded

bythetwolinesandthexaxis.

Or

Checkgraphicallywhetherthepairofequationsx+y=8andx–2y=2isconsistent.Ifso,

solvethemgraphically.Alsofindthecoordinatesofthepointswherethetwolinesmeetthe

y-axis.

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24.AthiefawayfromaPoliceStationwithauniformspeed100m/minutes.Afteroneminute

aPolicemanrunsbehindthethieftocatchhim.Hegoesataspeedof100m/minuteinfirst

minuteandincreasesthespeed10m/minuteoneachsucceedingminute.Afterhowmany

minutesthePolicemancatchesthethief.

Nowanswerthesequestions:

(i)Whichmathematicalconceptisbeingusedtosolvetheaboveproblem?

(ii)Whichtraitofpersonalityofthepolicemanisshowed?

25.Intheadjoiningfigure,PQR,isarighttriangle,rightangledatQ.XandYarethepoints

onPQandQRsuchthatPX:XQ=1:2andQY:YR=2:1.Provethat9(PY2+XR2)=13PR2

Or

AquadrilateralABCDisdrawntocircumscribeacircle(fig-2).Provethat,AB+CD=AD+BC.

26.Provethatthelengthsoftwotangentsdrawnfromanexternalpointtoacircleareequal.

27.Constructanisoscelestrianglewhosebaseis7cmandaltitude5cmandthenconstruct

anothertrianglewhosesidesare timesthecorrespondingsidesoftheisoscelestriangle.

28.Supposeapersonisstandingonatowerofheight mandobservingacar

comingtowardsthetower.Heobservedthatangleofdepressionchangesfrom30°to45°,in

3seconds.Findthespeedofthecar.

29.Acontaineropensatthetopandmadeupofmetalsheetisintheformofafrustumofa

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coneofheight16cmwithdiametersofitslowerandupperendsas16cmand40cm

respectively.Findthecostofmetalsheetusedtomakethecontainer,ifitcostsRs.10per

100cm2.(Use =3.14)

30.Themeanofthefollowingfrequencydistributionis47.Findthevalueof‘p’.

Classes 0-20 20-40 40-60 60-80 80-100

Frequency 8 15 20 p 5

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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CBSESAMPLEPAPER01

CLASSXMATHEMATICS

MarkingScheme

1.No.BecauseHCFisalwaysafactorofLCMbuthere18isnotafactorof380.

2.16x2-27x-10=0

(16x+5)(x-2)=0

x= ,x=2

3.

Clearly,thesumofthesquaresofthelengthsoftwosidesisnotequaltothesquareofthe

lengthofthethirdside.Hence,givensidesdonotmakearighttrianglebecauseitdoesnot

satisfythepropertyofPythagorastheorem.

4.45°

5.PerimetreoftheCircle=AreaoftheCircle

6.Numberofpossibleoutcomes=8(HHH,HHT,HTH,HTT,THH,THT,TTH,TTT)

Numberoffavorableoutcomes(2head)=4

Soprobability=

7.Yes,5040+5=5045Ithasmorethantwofactors

8.LetthefirsttermofAPisaanddiscommondifferencethenAccordingtoQuestion

a11–a4=14;

=>a+10d–(a+3d)=14

=>a+10d-a-3d=14

=>7d=14

=>d=2

9. [ ]

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x=7–3y

10.

PT=PS(lengthoftangents)

60°

ΔOTPtan60°=

PT= cm

11.LHS=

=RHS

12.Volumeofcone=Volumeofsphere

r=5cm

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13.a=bq+r;b=3;r=0,1,2

a3=(3m)3

=9(3m3)

=9q

a3=(3m+1)3

=27m3+27m2+9m+1

=9q+1

a3=(3m+2)3

=27m3+54m2+18m+8

=9q+8

14.Twozeroesare

Therefore

15.

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=>2x2+2ax+bx+ab=0

=>2x(x+a)+b(x+a)=0

=>(x+a)(2x+b)=0

=>x=-a,

16.a=54,d=-3,sn=513

Sn=

513=

=>

n=18,19

Sincedisnegative.wegetdoubleanswerbecausesumof18termsand19termsiszero,as

fewtermsarepositiveandfewarenegative.

Or

A+(p-1)D=a..(i)

A+(q-1)D=b….(ii)

A+(r-1)D=c…….(iii)

(ii)-(iii)

b-c=(q-1)D–(r-1)D

Similarly,

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Adding(iv),(v)and(vi)

17.PA=QA17.ThepointA(3,y)isequidistantfromthepointsP(6,5)andQ(0,-3).Findthevalueofy.

18.

19.

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GiventheheightoftheobserverbeDE=1.5m

ThatisAB=1.5m

LetBC=hisheightofthechimney

HenceAC=(h–1.5)m

Givendistancebetweentheobserverandthechimneyis

AD=BE=28.5m

InrighttriangleDCA,θ=45°

tan45°=

∴h=28.5+1.5=30mThustheheightofthechimneyis30m.

20.Circumferenceofwheel=πd=60πcm

Distancecoveredin1revolution= km

Distancecoveredin140revolution= km

(Distancecoveredin1min)

Distancecoveredin1hr=

Speedofcycle=15.84km/hr

21.Modalclass=7-9

Mode-

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=

22.Thereare366daysinaleapyearthatcontain52weeksand2moredays.So,52

Thursdaysand2days.

These2dayscanbe:

(Mon,Tue},{Tue,Wed},{Wed,Thu},{Thu,Fri},{Fri,Sat},{Sat,Sun}and{Sun,Mon}(7cases).

Inordertohave53Thursdaysweshouldhaveeither{Thu,Fri}or{Wed,Thu}case.

No.ofsamplespaces=7.

No.ofeventthatgives53ThursdaysinaleapYear=2.

RequiredProbability=

23.

TableforEqn.1anditsline

TableforEqn.2anditsline

Solutionx=3andy=1

Shadedareaisrequiredsolution.

Or

x 0 4 8

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y=8–x 8 4 0

Threesolutionsforequation(i)aregiveninthetable:

Threesolutionsforequation(ii)aregiveninthetable:

x 0 2 8

–1 0 3

DrawingLineAC

DrawingLinePR

PlottingpointsA(0,8),B(4,4)andC(8,0)ongraphpaperthestraightlineACisobtainedas

graphoftheequation

(i)PlottingpointsP(0,–1),Q(2,0)andR(8,3)ongraphpaperthestraightlinePRisobtained

asgraphoftheequation

(ii).Fromthegraph,itisclearthatapointM(6,2)commontoboththelinesACandPR.

Sothepairofequationsisconsistentandthesolutionsoftheequationsarex=6andy=2.

FromthegraphitisseenthatthecoordinatesofthepointswherethelinesACandPRmeets

they-axisare(0,8)and(0,–1)respectively.

24.TimetakenbyThiefbeforebeingcaught=n+1

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DistancetravelledbyThief=100(n+1)

100(n+1)=

n=5minutes

i)ArithmeticProgression

ii)Responsibilityoftheirwork(duty)andhonesty

25.

Or

Since,thelengthsoftangentsdrawnfromanexternalpointtoacircleareequal.

AP=AS…(i)BP=BQ…(ii)

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CQ=CR…(iii)DR=DS…(iv)

Now,AB+CD

=AP+PB+CR+RD

=AS+BQ+CQ+DS

=(AS+DS)+(BQ+CQ)

=AD+BC

Henceproved.

26.Construction:DrawacirclewithcentreO.FromapointPoutsidethecircle,drawtwo

tangentsPandR.

ToProve:PQ=PR

Proof:InΔPOQandΔPOR

OQ=OR(radii)

PO=PO(commonside)

∠PQO=∠PRO(Rightangle)

ΔPOQ ΔPOR(ByRHSCongruencyrule)

Henceproved

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27.

28.

ΔPQBtan45°= ,x=15( ------------(1)

ΔPQAtan30°= x+y=15 ( ---------(2)

From(1)and(2)y=30m

SincethecarmovingfromAtoBin3seconds

Speed= =10m/sec

29.R=20cm,r=8cm,h=16cm

l= =20cm

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Totalsurfacearea=CSAoffrustum+areaofbase

=πl(R+r)+

=1959.36

Rateofmetalsheetused=Rs.10per100cm2

Costofmetalsheetused=1959.36× =Rs.195.94(Approximately)

30.

CI fi xi fixi

0-20 8 10 80

20-40 15 30 450

40-60 20 50 1000

60-80 p 70 70p

80-100 5 90 450

48+p 1980+70p