1

Maths Book

Part 2

By Abhishek Jain

2

Topics 1. Simple interest and compound

interest 2. Probability

3. Data Interpretation 4. Mixtures & Alligation

5. Algebra 6. Geometry & mensuration

7. Triginometry & Height and distance

3

Simple&Compoundinterest

Simpleinterest:-Interest:TheamountofmoneythatyoupaytoborrowmoneyortheamountofmoneythatyouearnonadepositAnnualInterestRate:Thepercentofinterestthatyoupayformoneyborrowed,orearnformoneydepositedGeneralInformationS.I=!"#

$%%

1.Principle:Themoneyborrowedorlentoutforacertainperiodiscalledtheprincipalorthesum(P).2.Interest:Theborrowerpaysacertainamountfortheuseofthismoneyiscalledinterest(S.I).3.Time:TheborrowingisforaspecifiedperiodcalledTime(T).4.Rateofinterest:Thespecifiedtermisexpressedaspercentoftheprincipaliscalledrateofinterest(R%).5.Amount:ThesumoftheprincipalandtheinterestiscalledtheamountorFuturevalue.𝐴𝑚𝑜𝑢𝑛𝑡 = 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 + 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝑃 + !"#

$%%= 𝑃 1 + "#

$%%

Compoundinterest:-Wheninterestchargedoveraperiodoftimeisaddedupintheprincipal,theinterestsochargedonthisnewprincipaliscalledcompoundinterest.IfP=sumorPrincipaln=timeinyearsR=ratepercentperannum

Then,amount=P 1 + "$%%

9

(i)Wheninterestiscompoundedhalf-yearly,

Amount=P 1 +:;$%%

<9

(ii)Wheninterestiscompoundedquarterly,

Amount=P 1 +:=$%%

>9

4

(iii)Wheninterestiscompoundedannuallybuttimeinfraction,say2$?years.

Amount=P 1 + "$%%

<× 1 +

"AB$%%

$

(iv)Whenratesaredifferentfordifferentyears,sayR1%,R2%,andR3%for1st,2ndand3rdyearrespectivelythen,Amount=𝑃 1 + "$

$%%1 + "<

$%%1 + "C

$%%

(v)PresentworthofRs.Xduenyearshenceisgivenby:Presentworth=

D

$E :AFF

G

VeryImportantFormulae’sThedifferencebetweenthesimpleinterestandcompoundinterestfor2year(orterms)isgivenbytheformula

Difference=𝑃 "$%%

<

Thedifferencebetweenthesimpleinterestandcompoundinterestfor3year(orterms)isgivenbytheformula

Difference=𝑃 "; "EC%%$%%H

ConceptofEqualInstallmentsinCompoundinterest

P=𝑿

𝟏E 𝒓𝟏𝟎𝟎

𝒏 +𝑿

𝟏E 𝒓𝟏𝟎𝟎

(𝒏O𝟏) + ⋯……… . . + 𝑿

𝟏E 𝒓𝟏𝟎𝟎

P=PrincipalX=installmentR=rateN=numberofyearsEx.1.RobinlendsRs.9toRahulontheconditionthattheloanisrepayablein10monthsin10equalinstallmentsofRe.1each.Findtherateofinterestperannum.Sol.LettherateofinterestpermonthberTotalamountrepaid=Rs.10Interest=Re.1T$%%

(9+8+7+6+5+4+3+2+1)=1

Sor=$%%>?

Hence,therateofinterestperannum=($%%>?

)12=26<C%

5

Ex.2.AmilkmanborrowedRs.2500fromtwomoneylenders.Foroneloan,hepaid5%p.a.andfortheotherhepaid7%p.a.ThetotalinterestpaidfortwoyearswasRs.265.Howmuchdidheborrowat5%andhowmuchat7%?Sol.LettheloanbeRs.xat5%p.aandRs(2500–x)at7%p.a.

Totalinterestfor2years=[V×?×<E(<?%%WV)×X×<]

$%%

Givensimpleinterestforthetotalamount=Rs.265[V×?×<E(<?%%WV)×X×<]

$%%=265

Solving,wegetx=2125Amountborrowedat5%=Rs.2125Amountborrowedat7%=Rs.375Ex.3.ThreepersonsAmar,AkbarandAnthonyinvesteddifferentamountsinafixeddepositschemeforoneyearattherateof12%perannumandearnedatotalinterestofRs.3240attheendoftheyear.IftheamountinvestedbyAkbarisRs.5000morethantheamountinvestedbyAmarandtheamountinvestedbyAnthonyisRs.2000morethantheamountinvestedbyAkbar,whatistheamountinvestedbyAkbar?Sol.P=Amar+Akbar+Anthony=x+x+5000+x+7000=3x+12000

C<>%×$%%<

=3x+12000⇒27000-12000=3x⇒15000=3x⇒x=5000Akbar=5000+5000=Rs.10000Ex.4.VankatlaltakesmoneyfromtheEmployeesCooperativeSocietyatlowerrateofinterestandinvestsinascheme,whichgiveshimacompoundinterestof20%,compoundedannually.Findtheleastnumberofcompleteyearsafterwhichhissumwillbemorethandouble.Sol.LetP=Rs.100After4years,theamountwillbe100x1.2x1.2x1.2x1.2=Rs.207.3,whichismorethanthedoubleofRs.100Ex.5.RaminvestedacertainamountofmoneyandearnedacompoundinterestofRs.420inthesecondyearandRs.462inthethirdyear.AtwhatrateofinterestdidRaminvest?Sol.C.I.inthirdyear–C.I.insecondyear=462–420=42Thus,Rs.42istheinterestonRs.420.i.e.10%of420Hence,Rate=10%

6

Ex.6.Asumisinvestedatcompoundinterestpayableannually.TheinterestintwosuccessiveyearsstartingfromthefirstyearisRs.500andRs.540respectively.Findthesum.

Sol.Rate%=>%×$%%?%%×$

=8%?%%×$%%[×$

=Rs.6250Ex.7.SeemaclosedheraccountinaninvestmentoptionschemebywithdrawingRs.10000.Oneyearago,shehadwithdrawnRs.6000,twoyearsago,shehadwithdrawnRs.5000andthreeyearsago,shehadnotwithdrawnanymoney.Howmuchmoneyhadshedeposited(approx)atthetimeofopeningtheaccountfouryearsago,giventhatannualsimpleinterestis10%?Sol.Letxbethemoneyshedepositedatthetimeofopeningtheaccount.So,after1year(i.e.3yearsago)itwouldamountto1.1x.Sincenomoneywaswithdrawnatthispoint,after2years(i.e.2yearsago)itwouldamountto1.2x.Atthispoint,seemawithdrawsRs.5,000.Hence,herprincipalforthenextyear=(1.2x–5000)Nextyear,sheearns10%interestonthis,whichwillamountto1.1(1.2x–5000)Atthispoint,shewithdrawsRs.6,000.Hence,herprincipalforthenextyearwouldbe(1.32x–11500).Sheearns10%interestonthis,whichamountsto1.1(1.32x–11500)=(1.452x–12650)Butthisisequalto10000.Hence,x=15600Ex.8.Thesimpleinterestonasumofmoneyfor2yearsisRs.100andthecompoundinterestonthesamesumatthesamerateforthesametimeisRs.104.Findtherateofinterest.Sol.Simpleinterestfor2years=Rs.100Compoundinterestfor2years=Rs.104Simpleinterestfor1styearissameascompoundinterestfor1styear.Simpleinterestforbothyearsissame.So,S.I.for1year=Rs.50Compoundinterestforfirstyear=Rs.50Compoundinterestfor2ndyear=Rs.54ThismeansRs.4extra.So,whatpercentis4of50.or >

?%×100 = 8%

Ex.9.RaghavobtainsRs.1100afterlendingoutRs.xat5%perannumfor2yearsandobtainsRs.1800afterlendingoutRs.yat10%perannumfor2years.Findthevalueofx+y.Sol.HereCaseI:x+0.1x=1100CaseII:y+0.2y=1800Solvingtheabovetwoequations,wegetx+y=2500.

7

Ex.10.Thecompoundinterestandsimpleinterestoncertainsumfor2yearsarerespectivelyRs.41andRs.40.Findthesumandratepercent.Sol.LetthesumbeRs.P,rateofinterestber%AnnualS.I.=Rs.2020×1×

T$%%

=1⇒r=5%

P×1×?$%%

=20⇒P=Rs.400Ex.11.Mr.ThomasinvestedanamountofRs.13,900dividedintwodifferentschemesAandBatthesimpleinterestrateof14%p.a.and11%p.a.respectively.Ifthetotalamountofsimpleinterestearnedin2yearsbeRs.3508,whatwastheamountinvestedinSchemeB?Sol.LetthesuminvestedinSchemeAbeRs.xandthatinSchemeBbeRs.(13900-x).ThenD×$>×<

$%%+ ($C_%%WD)×$$×<

$%%=3508

⇒28x-22x=350800-(13900x22)⇒6x=45000⇒x=7500.So,suminvestedinSchemeB=Rs.(13900-7500)=Rs.6400.Ex.12.ReenatookaloanofRs.1200withsimpleinterestforasmanyyearsastherateofinterest.IfshepaidRs.432asinterestattheendoftheloanperiod,whatwastherateofinterest?Sol.Letrate=R%andtime=Ryears.Then$<%%×"×"

$%%=432

⇒12R2=432⇒R2=36⇒R=6.Ex.13.Anautomobilefinancierclaimstobelendingmoneyatsimpleinterest,butheincludestheinteresteverysixmonthsforcalculatingtheprincipal.Ifheischarginganinterestof10%,theeffectiverateofinterestbecomes:Sol.LetthesumbeRs.100.Then,SIFor6months=Rs.$%%×$%×$

$%%×<=Rs.5

SIForlast6months=Rs.$%?×$%×$

$%%×<=Rs.5.25

So,amountattheendof1year=Rs.(100+5+5.25)=Rs.110.25Effectiverate=(110.25-100)=10.25%Orshortcut=5+5+?×?

$%%=10.25%

8

Ex.14.AsumofRs.725islentinthebeginningofayearatacertainrateofinterest.After8months,asumofRs.362.50moreislentbutattheratetwicetheformer.Attheendoftheyear,Rs.33.50isearnedasinterestfromboththeloans.Whatwastheoriginalrateofinterest?Sol.LettheoriginalratebeR%.Then,newrate=(2R)%.Note:Here,originalrateisfor1year(s);thenewrateisforonly4monthsi.e.$

Cyear(s).

SoX<?×"×$

$%%+C`<.?×<"×$

$%%×C=33.5

Or2175+725R=33.5×100×3(2175+725)R=33.50x100x3(2175+725)R=10050(2900)R=10050R=10050/2900=3.46Sooriginalrate=3.46%Ex.15.AcertainamountearnssimpleinterestofRs.1750after7years.Hadtheinterestbeen2%more,howmuchmoreinterestwouldithaveearned?Sol.WeneedtoknowtheS.I.,principalandtimetofindtherate.Sincetheprincipalisnotgiven,sodataisinadequate.Ex.16.Thereis60%increaseinanamountin6yearsatsimpleinterest.WhatwillbethecompoundinterestofRs.12,000after3yearsatthesamerate?Sol.LetP=Rs.100.Then,S.I.Rs.60andT=6years.SoR=

$%%×`%$%%×`

=10%p.a.Now,P=Rs.12000.T=3yearsandR=10%p.a.

SoCI=Rs.12000 1 + $%$%%

C− 1 =Rs.3972

Ex.17.AlbertinvestedanamountofRs.8000inafixeddepositschemefor2yearsatcompoundinterestrate5p.c.p.a.HowmuchamountwillAlbertgetonmaturityofthefixeddeposit?Sol.

Amount=Rs.8000 1 + ?$%%

<=8000× <$

<%× <$<%=Rs.8820

Ex.18.Theeffectiveannualrateofinterestcorrespondingtoanominalrateof6%perannumpayablehalf-yearlyis:Sol.AmountofRs.100for1year

Whencompoundedhalfyearly=100 1 + C$%%

<=106.09

9

AmountofRs.100for1yearwhencompoundedhalf-yearly

=Rs.

100x

1+3

2

=Rs.106.09100

orEffectiverate=(106.09-100)%=6.09%Ex.19.Simpleinterestonacertainsumofmoneyfor3yearsat8%perannumishalfthecompoundinterestonRs.4000for2yearsat10%perannum.Thesumplacedonsimpleinterestis:Sol.

CI=Rs.Rs.4000 1 + $%$%%

<− 1 =Rs.840orsum=Rs.><%×$%%

C×[=Rs.1750

Ex.20.RiteshdepositedRs.8,000inStateBankofIndia.Hehadtopay10%ofthefirstyear’sinterestashiscollegefees,afterwhichRs.900oftheinterestamountwasleft.Findtheratepercentatwhichinterestwaspaid.Sol.Lettherate%bex.

S.I.=[%%%×D×b

$%%=[%%%D

$%%(t=1year)

10%ofinterestpaid⇒90%ofinterestleft=Rs.900(given)Socompleteinterest(100%)=Rs.1000

1000=[%%%D$%%

⇒x=

$%%[=12.5%

Ex.21.Asumwasputatsimpleinterestatacertainratefor4years.Haditbeenputat5%higherrate,itwouldhavefetchedRs.200more.Findthesum.Sol.Rs.200moreinterestisonaccountof5%higherrate.

200=c×?×>$%%

Ex.22.AsumofmoneyamountstoRs.767in3yearsandRs.806in4yearsatsimpleinterest.Findthesum.Sol.Interestfor1year=806–767=39Interestfor3year=117ThereforePrincipal=767–117=Rs.650Ex.23.Amanborrowsmoneyatsimpleinterestatarateof5%perannumandlendsitoutat4%perannumcompoundedhalfyearly.Findhisnetgainorlossattheendof2yearsasapercentageofsumborrowedbyhim.Sol.LetPrincipalbeRs.100AtSI,amount=Rs.110AtCI,amount=100×(1.02)4=108.24Thereforeloss%=1.76

10

Exercise

1.AcertainsumofmoneyatsimpleinterestamounttoRs.1040in3yearsandtoRs.1360in7years.Thenthatsumisa.Rs.800b.Rs.850c.Rs.820d.Rs.780e.Noneofthese2.OutofasumofRs.625,apartwaslentat5%andtheotherat10%simpleinterest.Iftheinterestonthefirstpartaftertwoyearsisequaltotheinterestonthesecondpartafterfouryears,thenthesecondsumisequaltoa.Rs.120b.Rs.122c.Rs.125d.Rs.100e.Noneofthese3.x,yandzarethreesumsofmoneysuchthatyisthesimpleinterestonxandzisthesimpleinterestonyforthesametimeandsamerateofinterest.Then,wehavea.y2=zb.x2=yzc.z2=xyd.y2=xze.Noneofthese4.AsumofRs.2668amountstoRs.4669in5yrattherateofsimpleinterest.Findtheratepercenta.10%b.35%c.20%d.12%e.Noneofthese5.Whatwillbesimpleinterestfor1yrand4monthsonasumofRs.25800attherateof14%perannum?a.Rs.4816b.Rs.4810c.Rs.4916d.Rs.4618e.Noneofthese6.Atwhatrateofannualsimpleinterest,acertainsumwillamounttofourtimesin15yr?a.10%b.15%c.20%d.12%e.Noneofthese7.Asumbecomesitsdoublein10yr.Findtheannualrateofsimpleinteresta.10%b.15%c.20%d.12%e.Noneofthese8.Atsimpleinterest,asumbecomes3timesin20yr.Findthetimeinwhichthesumwillbedoubleatthesamerateofinterest?a.10yrb.15yrc.20yrd.12yre.Noneofthese9.Acertainsumbecomes3foldat4%annualrateofinterest.Atwhatrate,itwillbecome6fold?a.10%b.15%c.20%d.12%e.Noneofthese10.AtasimpleinterestasumamountstoRs.1012in2.5yrandbecomesRs.1057.54in4yr.Whatistherateofinterest?a.2%b.3%c.5%d.4%e.Noneofthese11.FindthedifferenceinamountandprincipalforRs.4000attherateof5%annualinterestin4yra.Rs.800b.Rs.850c.Rs.820d.Rs.780e.Noneofthese

11

12.RakeshlentoutRs.8750at7%annualinterest.Findthesimpleinterestin3yra.Rs.1850b.Rs.1836.5c.Rs.1837.5d.Rs.1838e.Noneofthese13.PriyadepositsanamountofRs.65800toobtainasimpleinterestattherateof14%perannumfor4yr.WhattotalamountwillPriyagetattheendof4yr?a.Rs.102486b.Rs.102648c.Rs.103648d.Rs.102864e.Noneofthese14.Asumwaslentoutforacertaintimeatsimpleinterest.ThesumamountstoRs.400at10%annualinterestrate.Whenthesumwaslentoutat4%annualinterestrate,itamountstoRs.200.Findthesuma.Rs.<%%

Cb.Rs.$%%

Cc.Rs.<%%

Xd.Rs.>%%

Ce.Noneofthese

15.Asumwasinvestedfor4yratacertainrateofsimpleinterest,ifithadbeeninvestedat2%moreannualrateofinterest,whenRs.56morewouldhavebeenobtained.Whatisthesum?a.Rs.800b.Rs.720c.Rs.750d.Rs.700e.Noneofthese16.<

Cpartofmysumislentoutat3%,$

`partislentourat6%andremainingpartislentoutat

12%.Allthethreepartsarelentoutatsimpleinterest.IftheannualincomeisRs.25,whatisthesum?a.Rs.600b.Rs.550c.Rs.650d.Rs.520e.Noneofthese17.AsumofRs.1521islentoutintwopartsinsuchawaythattheinterestononepartat10%for5yrisequaltothatofanotherpartat8%for10yr.Whatwillbethetwopartsofsum?a.Rs.800andRs.885b.Rs.Rs.936andRs.585c.Rs.Rs.926andRs.585d.Rs.Rs.936andRs.595e.Noneofthese18.HarshamakesafixeddepositofRs.20000inBankofIndiaforaperiodof3yr.Iftherateofinterestbe13%slperannumchargedhalf-yearly,whatamountwillhegetafter42months?a.Rs.29000b.Rs.29200c.Rs.29100d.Rs.28100e.Noneofthese19.ThedifferenceofsimpleinterestfromtwobanksforRs.1000in2yrisRs.20.Findthedifferenceinratesofinteresta.2%b.3%c.5%d.4%e.Noneofthese20.SureshborrowedRs.800at6%andNareshborrowedRs.600at10%.Afterhowmuchtime,willtheybothhaveequaldebts?a.16<

Cyrb.16$

Cyrc.15<

Cyrd.15$

Cyre.Noneofthese

21.RajulentRs.400toAjayfor2yrandRs.100toManojfor4yrandreceivedfrombothRs60ascollectiveinterest.Findtherateofinterest,simpleinterestbeingcalculateda.2%b.3%c.5%d.4%e.Noneofthese

12

22.Rameshinvestedanamountthatis10%ofRs.10000atsimpleinterest.After3yr,theamountbecomesRs.2500.Findoutthe4timesofactualinterestratea.200%b.300%c.500%d.400%e.Noneofthese23.OnwhatsumofmoneywilltheS.Ifor4yearsat8%p.a.ishalfoftheS.IonRs.400for2yearsat10%p.a.?a.Rs.100b.Rs.120c.Rs.125d.Rs.144e.Noneofthese24.Anilborrowedcertainmoneyattherateof6%perannumforthefirst2yr,attherateof9%perannumforthenext3yrandattherateof14%perannumfor4yr.IfhepaysatotalinterestofRs.22800attheendof9yr,howmuchmoneydidheborrow?a.Rs.22000b.Rs.24000c.Rs.48000d.Rs.30000e.Noneofthese25.AsumofRs.500amountstoRs.650in3yratsimpleinterest.Iftheinterestrateisincreasedby3%,itwouldamounttohowmuch?a.Rs.690b.Rs.680c.Rs.685d.Rs.695e.Noneofthese26.Neetaborrowedsomemoneyattherateof6%perannumforthefirst3yr,attherateof9%perannumforthenext5yearandattherateof13%perannumfortheperiodbeyond8yr.IfshepaysatotalinterestofRs.8160attheendof11yr,howmuchmoneydidsheborrow?a.Rs.8000b.Rs.6000c.Rs.4000d.Rs.10000e.Noneofthese27.ReenahadRs.10000withheroutofthismoneyshelentsomemoneytoAkshayfor2yrat15%simpleinterest.ShelentremainingmoneytoBrijeshforanequalnumberofyearsattherateof18%.After2yr,ReenafoundthatAkshayhadgivenherRs.360moreasinterestascomparedtoBrijesh.TheamountofmoneywhichReenahadlenttoBrijeshmustbea.Rs.8000b.Rs.6000c.Rs.4000d.Rs.10000e.Noneofthese28.AsumofRs.800amountstoRs.956in3yratsimpleinterest.Iftheinterestrateisincreasedby3%,itwouldamounttohowmuch?a.Rs.1020b.Rs.1025c.Rs.1030d.Rs.1015e.Noneofthese29.Whatwillbetheratioofsimpleinterestearnedbycertainamountatthesamerateofinterestfor12yrandfor18yr?a.3:2b.2:3c.4:5d.5:4e.Noneofthese30.AsumofRs.1550waslentpartlyat5%andpartlyat8%perannumsimpleinterest.Thetotalinterestreceivedafter4yrwasRs.400.Theratioofthemoneylentat5%tothatlentat8%?a.16:15b.15:16c.17:9d.9:17e.Noneofthese31.TheannualpaymentofRs.160in5yrat5%perannumsimpleinterestwilldischargeadebtofa.Rs.800b.Rs.600c.Rs.400d.Rs.880e.Noneofthese

13

32.Rameshlentout40%ofacertainsumattheannualrateof15%,helent50%oftheremainingattheannualrateof10%andtherestamountwaslentoutat18%,perannum.Findtheannualrateonwholesuma.14%b.13%c.14.4%d.datainadequatee.Noneofthese33.AprivatefinancecompanyAclaimstobelendingmoneyatsimpleinterest.Butthecompanyincludestheinterestevery6monthsforcalculatingprincipal.IfcompanyAischarginganinterestof10%,theeffectiverateofinterestafter1yrbecomesa.10.5%b.10.25%c.10.75%d.11%e.Noneofthese34.RajnishinvestedcertainsuminthreedifferentschemesP,QandRwiththeratesofinterest10%perannum,12%perannumand15%perannum,respectively.Ifthetotalinterestaccruedin1yrwasRs.3200andtheamountinvestedinschemeRwas150%oftheamountinvestedinschemePand240%oftheamountinvestedinschemeQ.WhatwastheamountinvestedinschemeQ?a.Rs.8000b.Rs.6000c.Rs.4000d.Rs.5000e.Noneofthese35.Thesimpleinterestonasumofmoneyat9%perannumfor5yrishalfthesum,isa.Rs.80b.Rs.60c.Rs.40d.datainadequatee.Noneofthese36.AanchalborrowedRs.500at3%perannumSlandRs.600at4.5%perannumSlontheagreementthatthewholesumwillbereturnedonlywhenthetotalinterestbecomesRs.252.Thenumberofyearsafterwhichtheborrowedsumistobereturned,isa.2yrb.3yrc.5yrd.6yre.Noneofthese37.Thesimpleinterestonasumofmoneyat8%perannumfor6yrishalfthesum.Whatisthesum?a.Rs.800b.Rs.600c.Rs.400d.datainadequatee.Noneofthese38.Theratesofsimpleinterestintwobanksxandyareintheratioof10:8.Ranjiwantstodeposithistotalsavingsintwobanksinsuchawaythatshereceivesequalhalf-yearlyinterestfromboth.Sheshoulddepositthesavingsinbanksxandyintheratioofa.3:2b.2:3c.4:5d.5:4e.Noneofthese39.Asumbecomes6foldat5%perannum.Atwhatrate,thesumbecomes12fold?a.10.5%b.10.25%c.10.75%d.11%e.Noneofthese40.Asumofmoneybecomes9timesin20yr.Findthe10timesofrateofinteresta.200%b.300%c.500%d.400%e.Noneofthese41.ThesimpleinterestonasumofmoneywillbeRs.200after5yr.Inthenext5yr,principleistripled.Whatwillbethetotalinterestattheendofthe10thyr?a.Rs.800b.Rs.600c.Rs.400d.900e.Noneofthese

14

42.WhatmustbetheprincipalthatamountstoRs.720in2years6monthsas5%perannumsimpleinterest?a.Rs.800b.Rs.640c.Rs.700d.840e.Noneofthese43.Acertainsumgivenonsimpleinterestbecamedoublein20yrs.Inhowmanyyearswillitbefourtimes?a.20yrb.30yrc.50yrd.60yre.Noneofthese44.FindoutthecapitalrequiredtoearnamonthlyinterestofRs.600permonthas6%simpleinteresta.Rs.1,80,000b.Rs.1,60,000c.Rs.1,40,000d.Rs.1,20,000e.Noneofthese45.AmanderiveshisincomefromaninvestmentofRs.2,000atacertainrateofinterestandRs.1,600at2%higher.Thewholeinterestin3yrsisRs.960.Findtherateofinteresta.8%b.6%c.5%d.4%e.Noneofthese46.AsumofRs.1,550waslentpartlyat5%andpartlyat8%simpleinterest.Thetotalinterestreceivedafter3yrswasRs.300.Theratioofmoneylentat5%to8%isa.13:12b.12:13c.16:15d.15:16e.Noneofthese47.Rs.793isdividedintothreepartsassuchthattheiramountafter2,3,and4yrsmaybeequaltherateofinterestbeing5%.Findratiobetweentheseparts

a.<$$%

: $$$?

: $$<%

b.$$$%

: $$$?

: $$<%

c.C$$%

: $$$?

: $$<%

d.C$$%

: <$$?

: $$<%

e.Noneofthese48.IftheC.Ionacertainsumfor3yearsat20%p.a.isRs.728,whatisthesuminvested?a.1500b.800c.1200d.1000e.Noneofthese49.AmanbuysahouseandpaysRs.8,000cashandRs.9,600at5yearscreditat4%perannumsimpleinterest.Findthecashpriceofthehousea.Rs.18000b.Rs.16000c.Rs.14000d.Rs.12000e.Noneofthese50.FindthesimpleinterestonRs.600from3rdMarchto15thMayofayearat6%p.a.a.Rs.7.2b.Rs.7c.Rs.6.5d.6e.Noneofthese51.AsumofRs.2,600islentintwopartssothattheinterestonthefirstpartforaperiodof3yearsat5%maybeequaltotheinterestonthesecondpartfor6yearsat4%.Thesecondpartisequaltoa.Rs.1250b.Rs.1200c.Rs.1000d.Rs.1100e.Noneofthese52.Asumdoublesin20yearsatsimpleinterest.Howmuchistherate?a.8%b.6%c.5%d.4%e.Noneofthese

15

53.AsumofRs.5984becomesRs.8976in6yearsatSI.Whatistherate?a.8$

C%p.a.b.8<

C%p.a.c.7$

C%p.a.d.8$

?%p.a.e.Noneofthese

54.Thesimpleinterestonacertainprincipal@4%p.afor5yearsisRs.800.Howmuchistheprincipalamount?a.Rs.8000b.Rs.6000c.Rs.4000d.Rs.2000e.Noneofthese55.AcertainsumofmoneybecomesRs.1250inaspanof5yearsatsimpleinterestandfurthertoRs.1700inthespanof8years.Atthesamerate,whatwoulditamounttoattheendof12years?a.Rs.2100b.Rs.2500c.Rs.2300d.Rs.2000e.Noneofthese56.Thesimpleinterestearnedonacertainsumofmoneyfor10yearsattherateof5%p.a.washalfthesum.Howmuchisthesum?a.Rs.3000b.Rs.5000c.Rs.4000d.datainadequatee.Noneofthese57.Rs.800becomesRs.956in3yearsatcertainsimplerateofinterest.Iftherateofinterestisincreasedby4%,whatamountwillRs.800becomein3years?a.Rs.1050b.Rs.1052c.Rs.1060d.Rs.1048e.Noneofthese58.Srinivasaninveststwoequalamountsintwobanksgiving10%and12%rateofinterestrespectively.AttheendofyeartheinterestearnedisRs.1650.Findthesuminvestedineacha.Rs.7000b.Rs.7200c.Rs.7500d.Rs.5000e.Noneofthese59.ThesimpleinterestonasumofmoneywillbeRs.600after10years.Iftheprincipalistrebledafter5years,whatwillbetotalinterestattheendofthetenthyear?a.Rs.1000b.Rs.1200c.Rs.1400d.Rs.800e.Noneofthese60.ApersoninvestsRs.5000at5%p.a.simpleinterestsforacertainperiodandearnsRs.750.IfheearnsRs.720onRs.6000inthesametimeperiodwhatistherateofinterest?a.8%b.6%c.5%d.4%e.Noneofthese61.TherateofinterestatwhichanamountofRs.1800oncompoundinterestbecomesRs.1984.50in2yearisa.8%b.6%c.5%d.4%e.Noneofthese62.WhichistheprincipalamountwhichearnsRs.132ascompoundinterestforthesecondyearat10%perannum?a.Rs.1000b.Rs.1200c.Rs.1400d.Rs.800e.Noneofthese63.TheamountofRs.7500atcompoundinterestat4%perannumfor2years,isa.Rs.8112b.Rs.8110c.Rs.8100d.Rs.8200e.Noneofthese

16

64.Thedifferenceincompoundinterestandsimpleinterestonacertainamountat10%perannumattheendofthethirdyearisRs.620.Whatistheprincipalamount?a.Rs.30000b.Rs.50000c.Rs.40000d.Rs.20000e.Noneofthese65.Tofindoutthetotalcompoundinterestaccruedonasumofmoneyafter5years,whichofthefollowinginformationgiveninthestatementsPandQwillbesufficient?P:ThesumwasRs.20,000Q:Thetotalamountofsimpleinterestonsamesumatsamerateofinterestandforthesametimeis2000a.onlyPneededb.onlyQneededc.BothPandQareneededd.answercannotbeansweredevenbyusingbothPandQe.Noneofthese66.Thedifferencebetweencompoundinterestandthesimpleinterestearnedonasumofmoneyattheendof4yearsisRs.256.40.Tofindoutthesum,whichofthefollowinginformationgiveninthestatementsPandQis/arenecessary?P:Amountofsimpleinterestaccruedafter4years500Q:Rateofinterestis10%perannuma.onlyPneededb.onlyQneededc.BothPandQareneededd.answercannotbeansweredevenbyusingbothPandQe.Noneofthese67.Tofindoutthetotalcompoundinterestaccruedonasumofmoneyafter5years,whichofthefollowinginformationgiveninthestatements.AandBis/aresufficient?P:Therateofinterestwas6%perannumQ:Thetotalsimpleinterestonthesameamountafter5yearsatthesameratewillbeRs.600a.onlyPneededb.onlyQneededc.BothPandQareneededd.answercannotbeansweredevenbyusingbothPandQe.Noneofthese68.AsumofmoneyinvestedatcompoundinterestamountstoRs.800in3yearsandRs.840in4years.Whatistherateofinterestforperannum?a.8%b.6%c.5%d.4%e.Noneofthese69.AmanborrowedRs.800at10%perannumsimpleinterestandimmediatelylentthewholesumat10%perannumcompoundinterest.Whatdoeshegainattheendof2years?a.Rs.8b.Rs.6c.Rs.5d.Rs.4e.Noneofthese

17

70.Thecompoundinterestonacertainsumat5%perannumfor2yearsisRs.328.Thesimpleinterestforthatsumatthesamerateandforthesameperiodwillbea.Rs.300b.Rs.320c.Rs.340d.Rs.360e.Noneofthese71.AsumofmoneyatcompoundinterestamountstoRs.578.40in2yearsandtoRs.614.55in3years.Therateofinterestperannumisa.6C

>%b.5$

>%c.6$

>%d.6?

>%e.Noneofthese

72.Acertainamountofmoneyisinvestedatthesimpleinterestof15%perannum.Ifithadbeeninvestedatcompoundinterest,anextrainterestofRs.450wouldhavebeenobtainedinthesecondyear.Whatmustbetheamountinvested?a.Rs.30000b.Rs.50000c.Rs.40000d.Rs.20000e.Noneofthese73.WhatisthedifferencebetweentheCIandSIonasumofRs.1600at5%p.a.forperiodof2years?a.Rs.8b.Rs.6c.Rs.5d.Rs.4e.Noneofthese74.WhatisthedifferencebetweenthecompoundinterestandthesimpleinterestonacapitalofRs.16,000attherateof15%perannumforaperiodof2years?a.Rs.480b.Rs.360c.Rs.350d.Rs.450e.Noneofthese75.AtwhatrateofinterestperannumwouldthedifferencebetweenthecompoundinterestandthesimpleinterestattheendoftwoyearsonthecapitalofRs.60000beRs.1944?a.18%b.16%c.15%d.14%e.Noneofthese76.Thedifferencebetweencompoundinterestandsimpleinterestonanamountforaperiodof1$

<yearsis62rupees.Rateofinterestis20%perannuminbothcasesandincaseof

compoundinteresttheinterestisbeingcompoundedhalfyearly.Whatistheamount?a.Rs.3000b.Rs.5000c.Rs.4000d.Rs.2000e.Noneofthese77.Theincomeofacompanyincreases20%perannum.IfitsincomeisRs.26,64,000intheyear1999,whatwasitsincomeintheyear1997?a.Rs.1850000b.Rs.1800000c.Rs.1852000d.Rs.1950000e.Noneofthese78.Thedifferencebetweenthesimpleinterestonacertainsumattherateof10%perannumfor2yearsancompoundinterestwhichiscompoundedevery6monthsisRs.124.05.Whatistheprincipalsum?a.Rs.7000b.Rs.8000c.Rs.9000d.Rs.10000e.Noneofthese79.Thedifferencebetweenthecompoundinterestandsimpleinterestonacertainsumat5%for2yearsisRs.1.50.Thesumisa.Rs.300b.Rs.500c.Rs.400d.Rs.600e.Noneofthese

18

80.Thedifferentbetweenthecompoundinterestandsimpleinterestonacertainsumofmoneyfor2yearsat10%perannumisRs.15.Findthesumofmoneya.Rs.1300b.Rs.1500c.Rs.1400d.Rs.1600e.Noneofthese81.Exchangerateofdollarvsrupeeincreasesattherateof10%permonth.IfthecurrentrateisRs.40perdollar,whatwillbetherateattheendofamonth?a.Rs.45b.Rs.44c.Rs.40d.Rs.41e.Noneofthese82.BalanborrowedRs.1,000at10percentperannumsimpleinterest.Heimmediatelylentthewholesumat10percentperannumcompoundinterest.Attheendof2years,hewouldgaina.Rs.30b.Rs.50c.Rs.10d.Rs.20e.Noneofthese83.AcertainamountearnssimpleinterestofRs.1750after7years.Hadtheinterestbeen2%more,howmuchmoreinterestwouldithaveearned?a.Rs.135b.Rs.125c.Rs.124d.cannotbedeterminede.Noneofthese84.HowmuchwillRs.25000amounttoin2yearsatcompoundinterest,iftheratesforthesuccessiveyearsbe4and5percentperyear?a.Rs.27500b.Rs.27300c.Rs.27400d.Rs.27600e.Noneofthese85.WhatisthedifferencebetweencompoundinterestandsimpleinterestforthesumofRs.2000overa2yearperiod,ifthecompoundinterestiscalculatedat20%andsimpleinterestiscalculatedat23%?a.Rs.45b.Rs.44c.Rs.40d.Rs.41e.Noneofthese86.FindthecompoundinterestofRs.1000attherateof20%perannumfor18monthswheninterestiscompoundedhalf-yearlya.Rs.331b.Rs.325c.Rs.330d.Rs.320e.Noneofthese87.Atwhatpercentageperannum,willRs.10,000amounttoRs.17280inthreeyears?(Compoundinterestbeingreckoned)a.20%b.30%c.50%d.40%e.Noneofthese88.VinaydepositedRs.8000inICICIBank,whichpayshim12%interestperannumcompoundedquarterly.Whatistheamountthathereceivesafter15months?a.Rs.9750b.Rs.9730c.Rs.9740d.Rs.9760e.Noneofthese89.RanjeetmakesadepositofRs.50,000inthePunjabNationalBankforaperiodof2$

<years.

Iftherateofinterestis12%perannumcompoundedhalf-yearly,findthematurityvalueofthemoneydepositedbyhim(approx.)a.Rs.66900b.Rs.66800c.Rs.66911d.Rs.66800e.Noneofthese90.VinodmakesadepositofRs.1,00,000intheSyndicateBankforaperiodof2years.Iftherateofinterestbe12%perannumcompoundedhalf-yearly,whatamounthewillgetafter2

19

years(approx.)a.Rs.126000b.Rs.125000c.Rs.126250d.Rs.126247e.Noneofthese91.AsumofmoneyisborrowedandpaidbackintwoequalannualinstalmentsofRs.882,allowing5%compoundinterest.Thesumborrowedwasa.Rs.1540b.Rs.1740c.Rs.1640d.Rs.1660e.Noneofthese92.Ifthedifferencebetweenthesimpleinterestandcompoundinterestonsomeprincipalamountat20%perannumfor3yearsisRs.48,thentheprincipalamountmustbea.Rs.275b.Rs.350c.Rs.375d.Rs.325e.Noneofthese93.Asumofmoneydoublesitselfin5years.Inhowmanyyearswillitbecomefourfold(ifinterestiscompounded)?a.8yrb.9yrc.6yrd.10yre.Noneofthese94.Ifthecompoundinterestonacertainsumat10%perannumfor2yearsisRs.21.Whatcouldbethesimpleinterest?a.Rs.20b.Rs.30c.Rs.40d.Rs.10e.Noneofthese95.Ifthecompoundinterestonacertainsumofmoneyfor2yearsat10%isRs.25200,findthesimpleinterestatthesamerateforthesametimea.Rs.25000b.Rs.24600c.Rs.24000d.Rs.23000e.Noneofthese96.ThedifferencebetweenCIandSIonasumofmoneyfor3yearsat5%perannumisRs.61.Findthesuma.Rs.7000b.Rs.8000c.Rs.9000d.Rs.10000e.Noneofthese97.Asumofmoneydoublesitselfatcompoundinterestin15years.Inhowmanyyearsitwillbecomeeighttimes?a.40b.35c.45d.50e.Noneofthese98.AsumofRs.400amountstoRs.441in2years.Whatwillitamounttoiftherateofinterestisincreasedby5%?a.Rs.480b.Rs.484c.Rs.450d.Rs.425e.Noneofthese99.Ifthedifferencebetweenthecompoundinterest,compoundedeverysixmonths,andthesimpleinterestonacertainsumofmoneyattherateof12%perannumforoneyearisRs.36,thesumisa.Rs.7000b.Rs.8000c.Rs.9000d.Rs.10000e.Noneofthese100.AbuilderborrowsRs.2550tobepaidbackwithcompoundinterestattherateof4%perannumbytheendof2yearsintwoequalyearlyinstalments.Howmuchwilleachinstalmentbe?a.Rs.1350b.Rs.1340c.Rs.1352d.Rs.1360e.Noneofthese

20

Answers1.A 2.C 3.D 4.B 5.A 6.C 7.A 8.A 9.A 10.B11.A 12.C 13.B 14.A 15.D 16.E 17.B 18.C 19.E 20.A21.C 22.A 23.C 24.B 25.D 26.A 27.C 28.E 29.B 30.A31.D 32.D 33.B 34.D 35.D 36.D 37.D 38.C 39.D 40.D41.A 42.B 43.D 44.D 45.A 46.C 47.B 48.D 49.B 50.A51.C 52.C 53.A 54.C 55.C 56.D 57.B 58.C 59.B 60.D61.C 62.B 63.A 64.D 65.C 66.B 67.D 68.C 69.A 70.B71.C 72.D 73.D 74.B 75.A 76.D 77.A 78.B 79.D 80.B81.B 82.C 83.D 84.B 85.C 86.A 87.A 88.E 89.C 90.D91.C 92.C 93.D 94.A 95.C 96.B 97.C 98.B 99.D 100.C

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Permutation&combinationThefactorialfunction(symbol:!)meanstomultiplyaseriesofdescendingnaturalnumbers.Examples:

4!=4×3×2×1=247!=7×6×5×4×3×2×1=50401!=1

PERMUTATIONSApermutationis"are-arrangementofelementsofaset".So,whatdoesthismean?ItmeansapermutationisONLYinterestedinre-arrangingtheelementsofthesetAnyduplicationofthecollectedelementsindifferentordersisfine.Apermutationthereforetendstobealargenumber.Example:Takingthe4letters,ABCD,writedownallthepermutationsof3oftheseleters:ABCBACCABDABACBBCACBADBAABDBADCADDACADBBDACDADCAACDBCDCBDDBCADCBDCCDBDCBorthereare24permutations.Inotherwords,justtakingeachletterandcollectingthemintosetsof3fromthe4andwritingthemout,gives24variations.Done.Here,ifyoulike,theordermatters,sinceABCisdifferenttoACBanddifferenttoBCAanddifferenttoCABetc.Permutationsseetheseasalldifferentanswers.COMBINATIONSAcombinationis"oneormoreelementsselectedfromasetwithoutregardtotheorder"The"withoutregard"meansthatthecollectionmattersratherthanorderincombinations,sointheaboveexample,thefactweABC,ACB,BAC,BCA,CAB,CBA...forcombinations,theseareall1combinationoflettersA,BandC.So,questionsconcerningpickingateamof5peoplefromasquadof11...youwouldneedcombinations,sinceitishaving"Bert,Ernie,Fred,BillandBob"thatmatters,notthefactthatyouhavesomanydifferentpermutationsofthese5people.

22

Example:Takingthe4letters,ABCD,writedownallthecombinationsof3oftheseletters:ABC,ABD,ACD,BCDortherearejust4combinations.Youcannotpickanyother3lettersfromABCD,thatisnotpartoftheabove4combinations.Itisenlighteningtoseethelettermissingineach:inorderwehave"noD","noC","noB"andfinally"noA"...thissometimeshelpsyouto"see"allthepossibleanswers.Tocalculatecombinations,itisa2-stageprocess:1.Youcalculatealltheequivalentpermutationsfirst.2.Youthencorrectthislistbycuttingoutanyduplicates.Asyoucanseefromthis,combinationsareasubsetofPermutations.MathematicsofPermutationsTofindthenumberofpermutationsofrelementsfromasetofn,theformulais:

P(n,r)=9!9WT !

So,theaboveexamplewouldbeMathematicsofCombinationsTofindthenumberofcombinationsofkelementsfromasetofn,theformulais:

C(n,r)=9!

T!× 9WT !

Ex.1.Inhowmanywayscanthelettersoftheword“PREUNIVERSITY”bearranged?

Solution:Numberofways=𝟏𝟑!

𝟐!×𝟐!×𝟐!

Becausethereare13lettersinwhichthereare2R’s,2’Eand2I’s.Ex.2.Howmanyeightletterwordscanbeformedfromthelettersoftheword“Courtesy”,beginningwithCandendingwithY?Sol.ThefirstandthelastplacearefixedwithCandYrespectively.Theremaining6letterscanbearrangedin6!Ways.Therefore,thetotalnumberofways=720.Ex.3.Outof8gentlemenand5ladies,acommitteeof5istobeformed.Findthenumberofwaysinwhichthiscanbedonesoastoincludeatleast3ladies.Sol.Thepossiblecombinationsfulfillingthegivenconditionsare:3ladiesand2gentlemen,4ladiesand1gentlemen,andallthe5ladies.a)3ladiesand2gentlemencanbeselectedin:⁵C₃X⁸C₂=280waysb)4ladiesand1gentlemancanbeselectedin:⁵C₄X⁸C₁=40waysc)5ladiescanbeselectedin1waysThetotalnumberofwaysthecommitteecanbeformed=321ways

23

Ex.4.Howmanydifferent7digittelephonenumberscanbeformedfrom0,1,2,3,4,5,6,7,8,9?Sol.Since0cannotbethefirstdigitinthe7-digittelephonenumber,thereare9differentpossibilitiesforthefirstposition.Theremaining6digitscanbeselectedfromthe10digitsin10⁶ways.Thetotalnumberofways=10⁶x9(Rememberthatintelephonenumbers,digitscanberepeated.Hence,foreverysingledigit,exceptforthefirstposition,thereare10differentpossibilities)Ex.5.From3mangoes,4applesand2oranges,howmanyselectionsoffruitscanbemade,takingatleastoneofeachkind?Sol.Youmaybetemptedtoadopttheapproachwetookinthepreviousquestion.However,thereisnoconditiongivenregardingthemaximumnumberoffruitsinselection.Hence,wehavetotryadifferenttechnique:Numberofwaysinwhichmangoescanbeselected=2³.Butthisalsoincludesthecasewhereallthreemangoesarenotselected.Hence,numberofwaysinwhichatleastonemangoesisselected=(2³-1)=7Similarly,numberofapplescanbeselectedin(2⁴–1)ways=15numberoforangescanbeselectedin(2²–1)ways=3Thetotalnumberofwaysis15x3x7=315ways(wehavetomultiplyandnotsimplyadd)Ex.6.Inhowmanydifferentwayscanthelettersoftheword'LEADING'bearrangedinsuchawaythatthevowelsalwayscometogether?Sol.Theword'LEADING'has7differentletters.WhenthevowelsEAIarealwaystogether,theycanbesupposedtoformoneletter.Then,wehavetoarrangethelettersLNDG(EAI).Now,5(4+1=5)letterscanbearrangedin5!=120ways.Thevowels(EAI)canbearrangedamongthemselvesin3!=6ways.

Requirednumberofways=(120x6)=720.Ex.7.Inhowmanydifferentwayscanthelettersoftheword'CORPORATION'bearrangedsothatthevowelsalwayscometogether?Sol.Intheword'CORPORATION',wetreatthevowelsOOAIOasoneletter.Thus,wehaveCRPRTN(OOAIO).Thishas7(6+1)lettersofwhichRoccurs2timesandtherestaredifferent.Numberofwaysarrangingtheseletters=7!/2!=2520Now,5vowelsinwhichOoccurs3timesandtherestaredifferent,canbearrangedIn5!/3!=20ways.SoRequirednumberofways=(2520x20)=50400.

24

Ex.8.Inhowmanywayscanthelettersoftheword'LEADER'bearranged?Sol.Theword'LEADER'contains6letters,namely1L,2E,1A,1Dand1R.Requirednumber=`!

<!=720/2=360

Ex.9.Howmany3-digitnumberscanbeformedfromthedigits2,3,5,6,7and9,whicharedivisibleby5andnoneofthedigitsisrepeated?Sol.Sinceeachdesirednumberisdivisibleby5,sowemusthave5attheunitplace.So,thereis1wayofdoingit.Thetensplacecannowbefilledbyanyoftheremaining5digits(2,3,6,7,9).So,thereare5waysoffillingthetensplace.Thehundredsplacecannowbefilledbyanyoftheremaining4digits.So,thereare4waysoffillingit.Requirednumberofnumbers=(1x5x4)=20Ex.10.Howmanymultiplesof5aretherefrom10to95?Sol.Asyouknow,multiplesof5areintegershaving0or5inthedigittotheextremeright(i.e.theunit’splace).Thefirstdigitfromtherightcanbechosenin2ways.Theseconddigitcanbeanyoneof1,2,3,4,5,6,7,8,9i.e.Thereare9choicesfortheseconddigit.Thus,thereare2×9=18multiplesof5from10to95.Ex.11.Inacity,thebusroutenumbersconsistofanaturalnumberlessthan100,followedbyoneofthelettersA,B,C,D,EandF.Howmanydifferentbusroutesarepossible?Sol.Thenumbercanbeanyoneofthenaturalnumbersfrom1to99.Thereare99choicesforthenumber.Thelettercanbechosenin6ways.Numberofpossiblebusroutesare99×6=594Ex.12.Thereare3questionsinaquestionpaper.Ifthequestionshave4,3and2solutionsrespectively,findthetotalnumberofsolutions.Sol.Herequestion1has4solutions,question2has3solutionsandquestion3has2solutions.Bythemultiplication(counting)rule,Totalnumberofsolutions=4×3×2=24Ex.13.ConsiderthewordROTOR.Whicheverwayyoureadit,fromlefttorightorfromrighttoleft,yougetthesameword.Suchawordisknownaspalindrome.Findthemaximumpossiblenumberof5-letterpalindromes.Sol.Thefirstletterfromtherightcanbechosenin26waysbecausethereare26alphabets.Havingchosenthis,thesecondlettercanbechosenin26ways.Thefirsttwoletterscanbechosenin26×26=676waysHavingchosenthefirsttwoletters,thethirdlettercanbechosenin26ways.Allthethreeletterscanbechosenin676×26=17576ways.Itimpliesthatthemaximumpossiblenumberoffiveletterpalindromesis17576becausethefourthletteristhesameasthesecondletterandthefifthletteristhesameasthe

25

firstletter.Ex.14.Howmany3-digitnumberscanbeformedwiththedigits1,4,7,8and9ifthedigitsarenotrepeated?Sol.Threedigitnumberswillhaveunit’s,ten’sandhundred’splace.Outof5givendigitsanyonecantaketheunit’splace.Thiscanbedonein5ways-------(i)Afterfillingtheunit’splace,anyofthefourremainingdigitscantaketheten’splace.Thiscanbedonein4ways-------(ii)Afterfillinginten’splace,hundred’splacecanbefilledfromanyofthethreeremainingdigits.Thiscanbedonein3ways-------(iii)Bycountingprinciple,thenumberof3digitnumbers=5×4×3=60Ex.15.SupposeyoucantravelfromaplaceAtoaplaceBby3buses,fromplaceBtoplaceCby4buses,fromplaceCtoplaceDby2busesandfromplaceDtoplaceEby3buses.InhowmanywayscanyoutravelfromAtoE?Sol.ThebusfromAtoBcanbeselectedin3ways.ThebusfromBtoCcanbeselectedin4ways.ThebusfromCtoDcanbeselectedin2ways.ThebusfromDtoEcanbeselectedin3ways.So,bytheGeneralCountingPrinciple,onecantravelfromAtoEin3×4×2×3=72Ex.16.SupposeyouwanttoarrangeyourEnglish,Hindi,Mathematics,History,GeographyandSciencebooksonashelf.Inhowmanywayscanyoudoit?Sol.Wehavetoarrange6books.Thenumberofpermutationsofnobjectsisn!=n.(n–1).(n–2)...2.1Heren=6andtherefore,numberofpermutationsis6.5.4.3.2.1=720Ex.17.Suppose7studentsarestayinginahallinahostelandtheyareallotted7beds.Amongthem,ParvindoesnotwantabednexttoAnjubecauseAnjusnores.Then,inhowmanywayscanyouallotthebeds?Sol.Letthebedsbenumbered1to7.Case1:SupposeAnjuisallottedbednumber1.Then,Parvincannotbeallottedbednumber2.SoParvincanbeallottedabedin5ways.AfterallottingabedtoParvin,theremaining5studentscanbeallottedbedsin5!ways.So,inthiscasethebedscanbeallottedin5×5!=600ways.Case2:Anjuisallottedbednumber7.Then,Parvincannotbeallottedbednumber6AsinCase1,thebedscanbeallottedin600ways.Case3:Anjuisallottedoneofthebedsnumbered2,3,4,5or6ParvincannotbeallottedthebedsontherighthandsideandlefthandsideofAnju’sbed.Forexample,ifAnjuisallottedbednumber2,bedsnumbered1or3cannotbeallottedtoParvin.Therefore,Parvincanbeallottedabedin4waysinallthesecases.

26

AfterallottingabedtoParvin,theother5canbeallottedabedin5!ways.Therefore,ineachofthesecases,thebedscanbeallotted4×5!=480ways.soThebedscanbeallottedin:2×600+5×480=1200+2400=3600waysEx.18.Inhowmanywayscanananimaltrainerarrange5lionsand4tigersinarowsothatnotwolionsaretogether?Sol.Theyhavetobearrangedinthefollowingway:|L|T|L|T|L|T|L|T|L|The5lionsshouldbearrangedinthe5placesmarked‘L’.Thiscanbedonein5!ways.The4tigersshouldbeinthe4placesmarked‘T’.Thiscanbedonein4!ways.Therefore,thelionsandthetigerscanbearrangedin5!×4!=2880waysEx.19.Thereare4booksonfairytales,5novelsand3plays.Inhowmanywayscanyouarrangethesesothatbooksonfairytalesaretogether,novelsaretogetherandplaysaretogetherandintheorder,booksonfairytales,novelsandplays.Sol.Thereare4booksonfairytalesandtheyhavetobeputtogether.Theycanbearrangedin4!ways.Similarly,thereare5novels.Theycanbearrangedin5!ways.Andthereare3plays.Theycanbearrangedin3!ways.So,bythecountingprincipleallofthemtogethercanbearrangedin4!×5!×3!=17280waysEx.20.Inhowmanywayscan4girlsand5boysbearrangedinarowsothatallthefourgirlsaretogether?Sol.Let4girlsbeoneunitandnowthereare6unitsinall.Theycanbearrangedin6!ways.Ineachofthesearrangements4girlscanbearrangedin4!ways.orTotalnumberofarrangementsinwhichgirlsarealwaystogether=6!×4!=720×24=17280Ex.21.Howmanyarrangementsofthelettersoftheword‘BENGALI’canbemade(i)Ifthevowelsarenevertogether.(ii)Ifthevowelsaretooccupyonlyoddplaces.Sol.Thereare7lettersintheword‘Bengali;ofthese3arevowelsand4consonants.(i)Consideringvowelsa,e,iasoneletter,wecanarrange4+1lettersin5!waysineachofwhichvowelsaretogether.These3vowelscanbearrangedamongthemselvesin3!ways.orTotalnumberofwords=5!×3!or120×6=720Sotherearetotalof720waysinwhichvowelsareALWAYSTOGEGHER.Now,Sincetherearenorepeatedletters,thetotalnumberofwaysinwhichthelettersoftheword‘BENGALI’canbearranged:

27

or7!=5040So,Totalno.ofarrangementsinwhichvowelsarenevertogether:orALLthearrangementspossible–arrangementsinwhichvowelsareALWAYSTOGETHERor5040–720=4320(ii)Thereare4oddplacesand3evenplaces.3vowelscanoccupy4oddplacesinP(4,3)waysand4constantscanbearrangedinP(4,4)itmeansNumberofwords=P(4,3)×P(4,4)=4!×4!=576Ex.22.12pointslieonacircle.Howmanycyclicquadrilateralscanbedrawnbyusingthesepoints?Sol.Foranysetof4pointswegetacyclicquadrilateral.Numberofwaysofchoosing4pointsoutof12pointsisC(12,4)=495Therefore,wecandraw495quadrilaterals.

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Exercise1.Whatisthevalueof18C16?a.140b.135c.145d.150e.Noneofthese2.HowmanysixletteredwordsstartingwiththeletterTcanbemadefromallthelettersofthewordTRAVEL?a.120b.135c.130d.150e.Noneofthese3.Thenumberof2digitevennumbersformedfromthedigits1,2,3,4,5and6ifrepetitionofdigitsisnotalloweda.12b.13c.14d.15e.Noneofthese4.Howmanydifferentwordscanbeformedbytakinganythreelettersformtheword‘CHEMISTRY’?a.512b.502c.504d.500e.Noneofthese5.6studentsappearinanexamination.Inhowmanywayscantheresultbeannounced?a.32b.64c.128d.36e.Noneofthese6.Inhowmanyways4boysand3girlscanbeseatedinarowsothatboysandgirlsarealternate?a.132b.164c.128d.144e.Noneofthese7.Whatwillbethesumofallthefourdigitnumbersthatcanbemadewiththedigits0,1,2and4?(Repetitionofdigitsinanumberisnotallowed)a.45100b.45102c.45108d.44100e.Noneofthese8.Inhowmanydifferentways3differentringscanbeworninfivefingersofahand?a.210b.140c.60d.280e.Noneofthese9.Inhowmanydifferentwayscanthelettersofthefollowingbearrangedsothatthevowelsmayoccupyonlytheoddposition“DETECT”a.45b.36c.120d.60e.Noneofthese10.Howmanyfourdigitnumberscanbemadewiththedigits0,1,2and7sothatatleastoneofthedigitsisrepeatedineverynumber?a.144b.164c.158d.174e.Noneofthese11.Atotalof66gameswereplayedinatournamentwhereeachplayerplayedoneagainsttherest.Thenumbersofplayersarea.10b.12c.11d.13e.Noneofthese

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12.Inhowmanywayscanyouselect2oddnumbersand2evennumbersoutofthefirst128wholenumbers?a.64C2х63C2b.64C2х64C2c.63C2х63C2d.65C2х63C2e.Noneofthese13.Thereare13couples,5singlemalesand7singlefemalesinaparty.Everymaleshakehandwitheveryfemaleoncebutnooneshakeshandwithhiswife.Howmanyhandshakestookplaceintheparty?a.340b.345c.347d.346e.Noneofthese14.Howmanynumbersaretherebetween100and1000suchthateverydigitiseither4or5?a.10b.8c.9d.11e.Noneofthese15.FindthetotalnumberofwordsthatcanbemadebyusingallthelettersfromthewordMACHINEusingonlyoncea.5!b.6!c.7!/2!d.7!e.Noneofthese16.Howmanyfivedigitnumberscanbeformedusingthedigits0,1,2,3,4and5whicharedivisibleby5,withoutrepeatingthedigits?a.216b.120c.96d.288e.Noneofthese17.Howmanyoddintegersfrom0to8000(inclusive)havedistinctdigits?a.1700b.1740c.1736d.1750e.Noneofthese18.Thereare5boysand6girls.Acommitteeof4istobeselectedsothatitmustconsistatleastoneboyandatleastonegirl?a.310b.312c.300d.320e.Noneofthese19.Apalindromeisanumberthatreadsthesamelefttorightasitdoesfromrighttoleft,suchas131.Howmany6-digitspalindromesaretherewhichareeven?a.200b.400c.300d.500e.Noneofthese20.HowmanydifferentwordscanbemadefromthewordEDUCATIONsothatallthevowelsarealwaystogether?(Donotbotheraboutanymeaninglesswords)a.14400b.12100c.16900d.14200e.Noneofthese21.Fivedistinctpairsofshoesaredisplaced.Inhowmanydifferentwayscanthreeshoesbeselectedcontainingamatchedpair?a.20b.40c.30d.50e.Noneofthese22.Fromaclassof12students5aretobechosenforanexcursion.But3veryclosefriendsdecideamongthemselvesthateitherallthreeofthemwillgoornoneofthemwillgo.Inhowmanywayscantheexcursionpartybechosen?a.144b.165c.169d.162e.Noneofthese

30

23.Howmanyevennumbersoffourdigitscanbeformedwiththedigits1,2,3,4,5,6(repetitionsofdigitsareallowed)?a.640b.642c.648d.646e.Noneofthese24.ThelettersofthewordLABOURarepermutedinallpossiblewaysandthewordsthusformedarearrangedasinadictionary.WhatistherankofthewordsLABOUR?a.242b.240c.246d.244e.Noneofthese25.Apersonwantstoselect2toysforthischild.Onevarietyoftoyshas9modelsandanothervarietyhas6models.Inhowmanywayscanheselectthe2toysonefromeachofthevariety?a.44b.54c.52d.48e.Noneofthese26.Findthevalueof9C5a.120b.117c.130d.126e.Noneofthese27.If(28)C(2r):(24)C[2(r-4)]=225:11,findra.8b.5c.2d.9e.Noneofthese28.Howmanymotorvehicleregistrationnumbersof4digitscanbeformedwiththedigits0,1,2,3,4,5(nodigitbeingrepeated)?a.200b.400c.300d.500e.Noneofthese29.Inhowmanywayscanthelettersoftheword‘POSSESS’bearrangedsothatthefourS’sareinalternatepositionsonly?a.8b.7c.6d.9e.Noneofthese30.Inhowmanywavescanacommitteeof3menand2womenbeformedoutofatotalof4menand4women?a.24b.25c.26d.23e.Noneofthese31.7C2+5C1+8C3-7C5-5C4-8C5a.2b.0c.4d.5e.Noneofthese32.Thetotalnumberofpermutationoftheword‘KOLKATA’willbea.1242b.1240c.1260d.1244e.Noneofthese33.Whatisthenumberofdiagonalsofaregularpolygonwith10sides?a.45b.35c.36d.55e.Noneofthese34.Asixfaceddie,aneightfaceddieandatenfaceddiearethrowntogether.Whatistheprobablenumberofoutcomes?a.240b.460c.480d.500e.Noneofthese

31

35.Inaboxthereare5distinctwhiteand6distinctblackballs.Apersonhastopickuptwoballsfromtheboxsuchthatthereisoneeachofboththecolours.Inhowmanywayshecanpicktheballs?a.20b.40c.30d.50e.Noneofthese36.Howmanytrianglescanbeformedbyjoiningtheverticesofaheptagon?a.45b.35c.36d.55e.Noneofthese37.Inaculturalfestival,sixprogrammersaretobestaged,threeonadayfortwodays.Inhowmanywaystheprogrammerscouldbearranged?a.720b.640c.680d.780e.Noneofthese38.Alltheoddnumbersfrom1to9arewrittenineverypossibleorder.Howmanynumberscanbeformedifrepetitionisnorallowed?a.240b.120c.480d.360e.Noneofthese39.Howmanynumberslyingbetween3000and4000andmadewiththedigits3,4,5,6,7and8aredivisibleby5?Repetitionsarenotallowed?a.24b.12c.48d.36e.Noneofthese40.Inameetingbetweentwocountrieseachcountryhas12delegates.Allthedelegatesofonecountryshakehandswithalldelegatesoftheothercountry.Findthenumberofhandshakespossible?a.144b.165c.169d.162e.Noneofthese41.Inthepreviousproblemifallthedelegatesshakehandwitheachotherirrespectiveofthecountrytheybelongtothentotalnumberofhandshakesisa.270b.265c.288d.276e.Noneofthese42.FivepersonsA,B,C,D,EoccupyseatsinarowsuchthatAandBsitnexttoeachother.Inhowmanypossiblewayscanthesefivepeoplesit?a.24b.12c.48d.36e.Noneofthese43. Thereare5boysand6girls.Acommitteeof4istobeselectedsothatitmustconsistatleastoneboyandatleastonegirl?a.310b.120c.480d.360e.Noneofthese44.Findthenumberofdiagnolsinhexagon?a.8b.7c.6d.9e.Noneofthese45.Howmanynumbersdivisibleby2andlyingbetween50000to70000canbeformed,fromthedigits3,4,5,6,7,8,9nodigitbeingrepeatedinanynumber?a.240b.120c.480d.360e.Noneofthese

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46.Howmanydiagonalsarethereinann-sidedpolygon(n›3)?a.n(n-3)/2b.n(n-2)/2c.n(n-3)/6d.(n-1)(n-3)/2e.Noneofthese47.Aclassinaschoolhas40students.Threestudentsofthisclassaretobeselectedasclassmonitor,gamesinchargeandlibrarian.Inhowmanywayscantheybeselected,ifastudentcanholdonlyonepositionatatime?a.59820b.59280c.59082d.59028e.Noneofthese48.Sherrybuys7novelsfromabookfair.Merrybuys8novelsfromthefair,noneofwhichiscommonwiththoseboughtbysherry.Theydecidetoexchangetheirbooksoneforone.Inhowmanywayscantheyexchangetheirbooksforthefirsttime?a.56b.54c.52d.58e.Noneofthese49.Howmany6-digitevennumberscanbeformedfromthedigits1,2,3,4,5,6and7sothatthedigitsshouldnotrepeatandthesecondlastdigitiseven?a.480b.360c.720d.960e.Noneofthese50.Outof7consonants4vowels,howmanywordsof3consonantsand2vowelscanbeformed?a.14400b.22500c.25600d.25200e.Noneofthese51.Howmany3-letterswordswithorwithoutmeaning,canbeformedoutofthelettersoftheword,'LOGARITHMS',ifrepetitionoflettersisnotallowed?a.720b.120c.1440d.240e.Noneofthese52.Apolygonhas54diagonals.Findthenumberofsidesa.14b.16c.15d.12e.Noneofthese53.Fourdicearerolledsimultaneously.Whatisthenumberofpossibleoutcomesinwhichatleastoneofthedieshows5?a.1296b.671c.625d.125e.Noneofthese54.ThefigurebelowshowsthenetworkconnectingcitiesA,B,C,D,EandF.Thenarrowsindicatepermissibledirectionoftravel.WhatisthenumberofdistinctpathsfromAtoF?

a.8b.9c.10d.12e.Noneofthese

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55.Theproductofanyrconsecutivepositiveintegersisdivisiblebywhichofthefollowing?a.r!b.(r-1)!c.(r+1)!d.both(a)and(b)e.Noneofthese56.Inhowmanywayscan3childreninafamilyhavealldifferentbirthdays?a.366х365х364b.365х364х363c.366х365х363d.365х365х364e.Noneofthese57.Inhowmanywayscanaleapyearhave53Sundays?a.3b.4c.2d.1e.Noneofthese58.Inhowmanywayscan5lettersbepostedin3postboxes,ifanynumberofletterscanbepostedinallofthethreepostboxes?a.125b.243c.25d.81e.Noneofthese59.Ontheir10thweddinganniversaryaBengalicouplebought10differentsweetsandthendistributeditbetweentwooftheirfamilyfriendssuchthatbothofthemgotfivesweetseach.Findthenumberofdifferentwaysinwhichthisdistributioncanbedonea.252b.25c.100d.125e.Noneofthese60.InthecountryofUtopiathelanguagecontainsonly4alphabets.FindthemaximumnumberofwordsthatcanbethereintheUtopiandictionaryifnoalphabetcanberepeatedinaworda.68b.69c.64d.60e.Noneofthese61.Acompanycouldadvertiseaboutitsnewproductin4magazines,3newspapersand2televisionchannels.Butinalatermoveitdecidedtogiveadvertisementsinonly2ofthemagazines,1ofthenewspapersand1oftheTVchannels.Inhowmanywayscantheyadvertisetheirproduct?a.48b.36c.73d.96e.Noneofthese62.Thefirst5oddnaturalnumbersarewrittenineverypossibleorder.Howmanynumberscanbeformedifnorepetitionisallowed?a.4!b.5!c.6!d.7!e.Noneofthese63.Inhowmanydifferentwaysonecanwear3differentringsinfingersofonehand?a.210b.60c.120d.180e.Noneofthese64.Inastaircasethereare3steps.Apersoncanjumponestep,twostepsorthreesteps.Inhowmanywayscanhereachthetop?a.4b.5c.6d.7e.Noneofthese65.Howmanywayscan5prizesbegivenawayto4boys,ifeachboyiseligibleforalltheprizes?a.54b.53c.45d.44e.Noneofthese

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66.Aftergroupdiscussionandinterview6candidateswereselectedforadmissioninacollege.Butunfortunatelythenumberofseatsleftis2.So,itwasleftwiththeprincipaltoselect2candidatesoutofthem.Inhowmanywayscanheselect2candidates?a.14b.15c.16d.17e.Noneofthese67.Inanexamination10questionsaretobeansweredchoosingatleast4fromeachofpartAandpartB.Ifthereare6questionsinpartAand7inpartB,inhowmanywayscan10questionsbeanswered?a.248b.236c.273d.266e.Noneofthese68.AfteraconditioningcampofIndianCricketteam,thefinalteamof11playersoutofatotalof15playersistobedecidedsuchthattwoplayersSouravandSachinisalwayschosen.Findthetotalnumberofwaysthefinalteamcanbeselecteda.714b.715c.716d.717e.Noneofthese

Answers1.E 2.A 3.D 4.C 5.B 6.D 7.C 8.A 9.B 10.D11.B 12.A 13.C 14.B 15.D 16.A 17.C 18.A 19.B 20.A21.B 22.D 23.C 24.A 25.B 26.D 27.E 28.C 29.C 30.A31.B 32.C 33.B 34.C 35.C 36.B 37.A 38.B 39.B 40.A41.D 42.C 43.A 44.D 45.D 46.A 47.B 48.A 49.C 50.D51.A 52.D 53.B 54.C 55.D 56.A 57.C 58.B 59.A 60.C61.B 62.B 63.A 64.A 65.C 66.B 67.D 68.B

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ProbabilityConceptofProbability:Probabilitydealswiththeanalysisofrandomphenomena.Itisawayofassigningeveryeventavaluebetweenzeroandone,withtherequirementthattheeventmadeupofallpossibleresultsisassignedavalueofone.Experiment:Anoperationwhichcanproducesomewell-definedoutcomesiscalledanexperimentRandomExperiment:Anexperimentinwhichallpossibleoutcomesareknowandtheexactoutputcannotbepredictedinadvance,iscalledarandomexperiment.Examples:

i. Rollinganunbiaseddice.ii. Tossingafaircoin.iii. Drawingacardfromapackofwell-shuffledcards.iv. Pickingupaballofcertaincolourfromabagcontainingballsofdifferentcolours.

Details:

i. Whenwethrowacoin,theneitheraHead(H)oraTail(T)appears.ii. Adiceisasolidcube,having6faces,marked1,2,3,4,5,6respectively.Whenwethrow

adie,theoutcomeisthenumberthatappearsonitsupperface.iii. Apackofcardshas52cards.

Ithas13cardsofeachsuit,nameSpades,Clubs,HeartsandDiamonds.Cardsofspadesandclubsareblackcards.Cardsofheartsanddiamondsareredcards.Thereare4honoursofeachunit.ThereareKings,QueensandJacks.Theseareallcalledfacecards.SampleSpace:Whenweperformanexperiment,thenthesetSofallpossibleoutcomesiscalledthesamplespace.Examples:

1. Intossingacoin,S={H,T}2. Iftwocoinsaretossed,theS={HH,HT,TH,TT}.3. Inrollingadice,wehave,S={1,2,3,4,5,6}.

Event:Anysubsetofasamplespaceiscalledanevent.

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ProbabilityofOccurrenceofanEvent:LetSbethesampleandletEbeanevent.Then,P(E)=n(E)/n(S)Ex.1.Ticketsnumbered1to20aremixedupandthenaticketisdrawnatrandom.Whatistheprobabilitythattheticketdrawnhasanumberwhichisamultipleof3or5?Sol.Here,S={1,2,3,4,....,19,20}.LetE=eventofgettingamultipleof3or5={3,6,9,12,15,18,5,10,20}.P(E)=n(E)/n(S)=9/20Ex.2.Abagcontains2red,3greenand2blueballs.Twoballsaredrawnatrandom.Whatistheprobabilitythatnoneoftheballsdrawnisblue?Sol.Totalnumberofballs=(2+3+2)=7.LetSbethesamplespace.Thenn(S)=numberofwaysofdrawing2ballsoutof7=7C2=21LetE=Eventofdrawing2balls,noneofwhichisblue.Thenn(E)=numberofwaysofdrawing2ballsoutof(2+3)balls=5C2=10SoP(E)=10/21Ex.3.Inabox,thereare8red,7blueand6greenballs.Oneballispickeduprandomly.Whatistheprobabilitythatitisneitherrednorgreen?Sol.Totalnumberofballs=(8+7+6)=21.LetE =eventthattheballdrawnisneitherrednorgreen

=eventthattheballdrawnisblue.

Son(E)=7P(E)=n(E)/n(S)=7/21=1/3Ex.4.Whatistheprobabilityofgettingasum9fromtwothrowsofadice?Sol.Intwothrowsofadie,n(S)=(6x6)=36.LetE=eventofgettingasum={(3,6),(4,5),(5,4),(6,3)}.P(E)=n(E)/n(S)=4/36=1/9Ex.5.Threeunbiasedcoinsaretossed.Whatistheprobabilityofgettingatmosttwoheads?Sol.HereS={TTT,TTH,THT,HTT,THH,HTH,HHT,HHH}LetE=eventofgettingatmosttwoheads.ThenE={TTT,TTH,THT,HTT,THH,HTH,HHT}.P(E)=n(E)/n(S)=7/8

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Ex.6.Twodicearethrownsimultaneously.Whatistheprobabilityofgettingtwonumberswhoseproductiseven?Sol.Inasimultaneousthrowoftwodice,wehaven(S)=(6x6)=36.ThenE={(1,2),(1,4),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,2),(3,4),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,2),(5,4),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}P(E)=n(E)/n(S)=27/36=3/4Ex.7.Acardisdrawnfromapackof52cards.Theprobabilityofgettingaqueenofcluborakingofheartis:Sol.Here,n(S)=52.LetE=eventofgettingaqueenofcluborakingofheart.Then,n(E)=2.P(E)=n(E)/n(S)=2/52=1/26Ex.8.Abagcontains4white,5redand6blueballs.Threeballsaredrawnatrandomfromthebag.Theprobabilitythatallofthemarered,is:Sol.LetSbethesamplespace.Thenn(S)=numberofwaysofdrawing3ballsoutof15=15C3=

$?×$>×$CC×<×$

= 455LetE=eventofgettingallthe3redballs.Son(E)=5C3=5C2=10orP(E)=10/455=2/91Ex.9.Twocardsaredrawntogetherfromapackof52cards.Theprobabilitythatoneisaspadeandoneisaheart,is:Sol.LetSbethesamplespace.Thenn(S)=52C2=

?<×?$<×$

= 1326LetE=eventofgetting1spadeand1heart.Son(E)=numberofwaysofchoosing1spadeoutof13&1heartoutof13=13C1×13C1=13×13=169HenceP(E)=n(E)/n(S)=169/1326=13/102Ex.10.Onecardisdrawnatrandomfromapackof52cards.Whatistheprobabilitythatthecarddrawnisafacecard(Jack,QueenandKingonly)?Sol.Clearly,thereare52cards,outofwhichthereare12facecards.P(gettingafacecard)=12/52=3/13

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Ex.11.Abagcontains6blackand8whiteballs.Oneballisdrawnatrandom.Whatistheprobabilitythattheballdrawniswhite?Sol.Letnumberofballs=(6+8)=14.Numberofwhiteballs=8.P(drawingawhiteball)=8/14=4/7Ex.12.Inaclass,thereare15boysand10girls.Threestudentsareselectedatrandom.Theprobabilitythat1girland2boysareselected,is:Sol.LetSbethesamplespaceandEbetheeventofselecting1girland2boys.Thenn(S)=numberofwaysofselceting3studentsoutof25=25C3=

<?×<>×<CC×<×$

= 2300n(E)=10C1×15C2=1050sop(E)=n(E)/n(S)=1050/2300=21/46Ex.13.Inalottery,thereare10prizesand25blanks.Alotteryisdrawnatrandom.Whatistheprobabilityofgettingaprize?Sol.p(gettingaprize)=n(E)/n(S)=10/35=2/7Ex.14.Fromapackof52cards,twocardsaredrawntogetheratrandom.Whatistheprobabilityofboththecardsbeingkings?Sol.LetSbethesamplespace.n(S)=52C2=

?<×?$<×$

= 1326letE=eventofgetting2kingsoutof4son(E)=4C2=

>×C<×$

= 6sop(E)=n(E)/n(S)=5/1326=1/221Ex.15.Twodicearetossed.Theprobabilitythatthetotalscoreisaprimenumberis:Sol.Clearly,n(S)=(6x6)=36.LetE=Eventthatthesumisaprimenumber.ThenE=={(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5)}Son(E)=15sop(E)=n(E)/n(S)=15/36=5/12

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Exercise1.Fromapackof52cards,twocardsaredrawnatrandom.Whatistheprobabilitythateitherbothareredorbotharequeensisa.55/221b.52/221c.52/243d.56/221e.Noneofthese2.Aboxcontains20ballsbearingnumbers1,2,3…..20.Aballisdrawnatrandomfromthebox.Whatistheprobabilitythatthenumberontheballisdivisibleby2or3?a.9/20b.11/20c.17/20d.13/20e.Noneofthese3.Abagcontains2red,5blueand7greenballs.Aballisdrawnatrandom.Whatistheprobabilitythatitiseitherablueoragreenball?a.5/7b.5/14c.6/7d.3/7e.Noneofthese4.Fromapackofcardstwocardsaredrawnatrandom.Whatistheprobabilitythateitherbotharekingsorbotharequeens?a.3/221b.4/221c.5/221d.2/221e.Noneofthese5.Thereare8blueand4whiteballsinabag.Aballisdrawnatrandom.Withoutreplacingitanotherballisdrawn.Findtheprobabilitythatboththeballsdrawnarebluea.12/35b.14/33c.15/33d.17/33e.Noneofthese6.Thereare5greenand6blackballsinabag.Aballisdrawnatrandomwithreplacement.Findtheprobabilitythatboththeballsdrawnareblacka.36/121b.14/121c.15/121d.12/121e.Noneofthese7.Onecardisdrawnatrandomfromapackofcards.Findtheprobabilitythatthecarddrawniseitherakingoraqueen?a.3/26b.13/51c.2/13d.1/26e.Noneofthese8.Alotof12bulbscontains4defectivebulbs.Threebulbsaredrawnatrandomfromthelot,oneaftertheother.Theprobabilitythatallthreearenon-defectiveisa.11/56b.11/28c.15/56d.17/28e.Noneofthese9.100studentsappearedfortwoexaminations,60passedthefirst,50passedthesecondand30passedboth.Theprobabilitythatastudentselectedatrandomhasfailedinbothexaminationsisa.1/5b.2/5c.3/5d.4/5e.Noneofthese10.6boysand6girlssitinarowrandomly,findtheprobabilitythatallthe‘6’girlssittogether.a.3/132b.1/132c.5/132d.7/132e.Noneofthese11.Twodicearethrownsimultaneously.Whatistheprobabilityofgettingtwonumberswhoseproductiseven?a.1/5b.3/4c.1/4d.4/15e.Noneofthese

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12.Ina6faceddiewithnumbers1to6inscribed.Findtheprobabilityofgettinga2digitnumberwhichisprimeonadding2successivethrowsofthediea.2/5b.2/6c.5/6d.3/5e.Noneofthese13.Outof20consecutivepositiveintegers,twoarechosenatrandom.Theprobabilitythattheirsumisoddisa.11/19b.10/17c.13/19d.10/19e.Noneofthese14.Whentwodicearerolled,whatistheprobabilitythatthesumofthenumbersappearedonthemis11?a.1/18b.2/19c.3/17d.4/5e.Noneofthese15.Theprobabilityofdrawingaredcardfromadeckofplayingcards,isa.1/13b.2/13c.1/26d.1/2e.Noneofthese16.Fivecoinsaretossedatatime.Thentheprobabilityofobtainingatleastonetailisa.15/32b.31/32c.1/2d.15/16e.Noneofthese17.Ifa4digitnumberisformedatrandomusingthedigits1,3,5,7,9withoutrepetition,thentheprobabilitythatitisdivisibleby5isa.1/5b.2/5c.3/5d.4/5e.Noneofthese18.Theprobabilityofgettingacompositenumberwhenasix-facedunbiaseddieistossed,isa.1/5b.2/5c.2/3d.1/3e.Noneofthese19.Twounbiaseddicearethrownsimultaneously.Theprobabilityofgettingthesumdivisibleby3isa.2/3b.1/3c.5/12d.5/6e.Noneofthese20.Anumberisselectedatrandomfromtheset{1,2,3…..50}.Theprobabilitythatnisaprimeisa.1/5b.2/5c.3/10d.7/10e.Noneofthese21.Ifthreeunbiasedcoinsaretossedsimultaneously,thentheprobabilityofexactlytwoheadsisa.3/8b.5/8c.7/8d.1/2e.Noneofthese22.LetEbethesetofallintegerswith1intheirunitplace.Theprobabilitythatanumberchosen{2,3,4….50}isanelementofEisa.2/13b.2/7c.4/49d.5/49e.Noneofthese23.Anumberxischosenatrandomfrom{1,2,….10}.Theprobabilitythatxsatisfiestheequation(x-3)(x-6)(x-10)=0isa.1/5b.2/5c.3/10d.7/10e.Noneofthese

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24.Apersongetsasmanyrupeesasthenumberhegetswhenanunbiasedsix-faceddieisthrown.IftwosuchdicearethrowntheprobabilityofgettingRs.10isa.1/12b.2/11c.3/13d.7/12e.Noneofthese25.Abagcontains3redballs,4whiteballsand7blackballs.Theprobabilityofdrawingaredorablackballisa.5/14b.5/7c.5/49d.25/49e.Noneofthese26.Ifadieisthrown,thentheprobabilityofgettinganevennumberoranumbergreaterthan3,isa.2/3b.2/7c.2/5d.3/5e.Noneofthese27.Iftwodicearethrownsimultaneously,thentheprobabilityofhaving6onfirstdieandanynumberotherthan6onotherdieisa.25/36b.5/6c.5/36d.7/36e.Noneofthese28.Inasinglethrowoftwodice,findtheprobabilityofgettingatotalof3or5a.1/6b.2/5c.7/10d.3/10e.Noneofthese29.Aboxcontains36ticketsnumbered1to36,oneticketdrawnatrandom.Findtheprobabilitythatthenumberontheticketiseitherdivisibleby3orisaperfectsquarea.5/9b.1/9c.2/9d.4/9e.Noneofthese30.ThelettersB,G,I,N,Rarerearrangedtoformtheword‘BRING’.Finditsprobabilitya.7/120b.1/120c.11/120d.119/120e.Noneofthese31.Aboxofelectronicdiodecontains120standardand80sub-standardones.Twodiodesaretakenatrandom.Whatistheprobabilitythatoneisstandardandtheotherissub-standard?a.97/199b.99/199c.96/199d.95/199e.Noneofthese32.Aboxcontains20white,30black,40blueand30redballs.Computetheprobabilitythatoneoftheballsextractedatrandomfromtheboxturnsouttobewhite,blackorreda.2/3b.2/7c.2/5d.3/5e.Noneofthese33.Ifonerollsafair-sideddietwice,whatistheprobabilitythatthediewilllandonthesamenumberonboththeoccasions?a.1/6b.2/5c.7/10d.3/10e.Noneofthese34.Ifsevencoinsaretossed,whatistheprobabilityofobtainingatleast2heads?a.1/16b.9/16c.7/16d.15/16e.Noneofthese35.Abagcontains4red,5blueand6blackballs.Ifoneballisdrawnatrandom,findtheprobabilitythatitisnotablackballa.2/3b.2/7c.2/5d.3/5e.Noneofthese

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36.Acoinistossedtwice.Findtheprobabilityofgettingnoheada.3/4b.1/4c.4/5d.1/5e.Noneofthese37.Threeunbiasedcoinsaretossed.Findtheprobabilityofgettingatleasttwoheadsa.2/5b.2/7c.1/2d.3/5e.Noneofthese38.Iftwodicearethrowntogether,whatistheprobabilityofthesumbeinglessthan6,onthesidefacingup?a.3/18b.5/18c.1/18d.3/9e.Noneofthese39.Fromapackof52cards,whatistheprobabilityofpickingaspadecardifonlyonecardisdrawn?a.3/4b.1/4c.4/5d.1/5e.Noneofthese40.Ifabagcontains5yellowand6redballsand2ballsaredrawnouttogether,whataretheoddsagainstallballsbeingred?a.7/11b.6/11c.8/11d.9/11e.Noneofthese41.Whatistheprobabilitythatarandomlychosen2digitnumberisdivisibleby5?a.3/4b.1/4c.4/5d.1/5e.Noneofthese42.Thereare16candidatesforthreeopeningsinacompany.Iffiveofthemarewomen,whatistheprobabilitythatexactlyonewomanwouldbechosenforthejob?a.54/221b.52/225c.55/221d.55/224e.Noneofthese43.Aschoolhas4labsand10classrooms.Ifastudentcangotoanyoftheserooms,whatistheprobabilitythatthestudentsenteredaparticularroom?a.3/14b.1/14c.4/15d.1/15e.Noneofthese44.IfthelettersofthewordEnglisharerearranged,whatistheprobabilitythatawordwillstartwithavowel?a.2/7b.3/7c.5/7d.1/7e.Noneofthese45.Reshmadecidedtogetmarriedandwantedtoinform3outofher5friendsfirst,beforeconveyingtoeveryoneelse.IfSushmaisoneamongher5friends,whatistheprobabilitythatSushmawillbeamongthefirst3tohearaboutReshma’swedding?a.3/8b.5/8c.1/8d.3/5e.Noneofthese46.Whenfourcoinsaretossedsimultaneously,whatistheprobabilityofgettingexactlytwoheads?a.3/8b.5/8c.1/8d.3/5e.Noneofthese47.Fromapackofcards,if4randomcardsaredrawn,whatistheprobabilitythateachcardisfromadifferentsuit?a.132/52C2b.13/52C1c.133/52C3d.134/52C4e.Noneofthese

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48.Findtheprobabilityofgettinganumberlessthanfivewhenadieisrolled?a.2/3b.2/7c.2/5d.3/5e.Noneofthese49.Twodicearethrown,findtheprobabilityofgettinganeventotalonthem?a.2/3b.1/3c.1/2d.3/5e.Noneofthese50.Threedicearethrown,findtheprobabilityofgettingasumgreaterthan17ontheirfaces?a.1/216b.1/108c.1/54d.5/216e.Noneofthese51.Ifthreecoinsaretossed,findtheprobabilityoftheeventshowingexactlyoneTailonthem?a.3/8b.5/8c.1/8d.3/5e.Noneofthese52.Twocoinsaretossed.Findtheprobabilitythattwotailsresult,giventhatthereisatleastonetail?a.1/3b.2/3c.1/6d.1/2e.Noneofthese53.Onecardisdrawnatrandomfromapackof52cards.Whatistheprobabilitythatthecarddrawnisafacecard?a.3/13b.2/13c.1/13d.5/13e.Noneofthese54.FindtheprobabilitythatvowelselectedatrandomfromanEnglishbookisa‘0’a.3/4b.1/4c.4/5d.1/5e.Noneofthese55.Inasinglethrowwithtwodice,findtheprobabilityofthrowinga10?a.1/13b.2/13c.1/16d.1/12e.Noneofthese56.When4coinsaretossedsimultaneouslyfindtheprobabilityofgettingexactlyonetail?a.1/4b.5/8c.1/8d.3/4e.Noneofthese57.Twodicearethrownsimultaneously.Findtheprobabilitythatthesumofthenumbersisatmost4?a.1/3b.2/3c.1/6d.1/2e.Noneofthese58.Findtheprobabilitythatanon-leapyearshouldhave52Tuesdays?a.2/3b.2/7c.6/7d.3/5e.Noneofthese59.Asingleletterisselectedatrandomfromtheword‘CONICS’.Findtheprobabilitythatitisavowel?a.1/3b.2/3c.1/6d.1/2e.Noneofthese60.If3dicearerolled,findthetotalnumberofexhaustiveoutcomes?a.53b.73c.43d.63e.Noneofthese61.Iffourstudentsarechosenatrandom,findtheprobabilitythatnotwoofthemwereborn

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onthesamedayoftheweek?a.124/343b.120/343c.125/343d.130/343e.Noneofthese62.Anintegerischosenfrom3to17whatistheprobabilitythatitisprime?a.3/4b.1/4c.2/5d.1/5e.Noneofthese63.Acardisdrawnatrandomfromadeckofcards.Findtheprobabilityofgettingthe3ofdiamond?a.1/13b.2/13c.1/26d.1/52e.Noneofthese64.Findtheprobabilityofgettinganoddnumberinthrowofanunbiaseddie?a.1/3b.2/3c.1/6d.1/2e.Noneofthese65.Whatistheprobability,thataleapyearselectedatrandomwillcontain53Sundays?a.5/7b.1/7c.3/7d.2/7e.Noneofthese66.Findtheprobabilityofgettinga‘king’ora‘Queen’inasingledrawfromawellshuffledpackofplayingcards?a.1/13b.2/13c.1/16d.1/12e.NoneoftheseDirections(67-69):Onecardisdrawnfromapackof52cards,eachcardbeingequallylikelytobedrawn67.Findtheprobabilitythatthecarddrawnisreda.1/3b.2/3c.1/6d.1/2e.Noneofthese68.Findtheprobabilitythatthecarddrawnisakinga.1/13b.2/13c.1/16d.1/12e.Noneofthese69.Findtheprobabilitythatthecarddrawnisredandakinga.1/26b.2/13c.1/16d.1/12e.Noneofthese

Answers1.A 2.D 3.C 4.D 5.B 6.A 7.C 8.B 9.A 10.B11.B 12.E 13.D 14.A 15.D 16.B 17.A 18.D 19.B 20.C21.A 22.C 23.C 24.A 25.B 26.A 27.C 28.A 29.D 30.B31.C 32.A 33.A 34.D 35.D 36.B 37.C 38.B 39.B 40.C41.D 42.D 43.B 44.A 45.D 46.A 47.D 48.A 49.C 50.A51.A 52.A 53.A 54.D 55.D 56.A 57.C 58.C 59.A 60.D61.B 62.C 63.D 64.D 65.D 66.B 67.D 68.A 69.A

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DatainterpretationGraphsWhiletablesexpressactualnumbers,graphsareadiagrammaticrepresentationofdata.Theybringouttherelationshipbetweendatamoreclearlythannumbersinatable.Forexample,apie-chartcanbringoutclearlythepercentagethatAnilAmbaniownsofRelianceCommunicationsLtdandthefactthatheisthelargestshareholder,whileatablewouldrequireyoutoactuallycalculatethepercentageofeachshareholder'sownershiptofindoutthelargestshareholder.Graphsarefarbettertounderstandchangesinvariables-whetheraparticularvaluehasrisenorfallenoverthepastfewyearsandhenceanalyzethetrends.PieChartsTheyderivetheirnamefromitsshape,likethatofapiedividedintovariousportions.Theyalwaysrepresentdataintheformofapercentageofthetotal,withthetotalpercentagebeing100.Insuchachart,thelengthofthearc(andthereforetheangleeachsectorsubtendsatthecentre)isproportionaltothequantityitrepresents.Suchchartsareoftenusedinthecorporateworldandinnewspapers.Sinceacirclecomprises360degrees,eachpercentofapie-chartisequalto360dividedby100,or3.6degrees.Thisfactwillbeimportantforthecalculationsyouareexpectedtoperform.BarGraphsBargraphsrepresentdataintheformofcolumnsorbars.Bargraphscanbehorizontalorvertical.Thelengthofthebarisproportionaltothedatavaluerepresentedbyit.LineGraphsLinegraphrepresentsdataintheformofstraightlinesthatconnectvariousdatavalues.Bothlinegraphsandbargraphsareusedtoconveysamethingsandhencecanbeusedinter-changeably.Forexample,alinegraphcanbegeneratedbyjoiningthetipofthebargraph.

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DirectionforQ1-Q5:Followingtableshowsthescoresofstudentsin8BANKPO(sectionwise).AssumeallBANKPOEXAMcontain3sections,eachsectionshave50questions.Thetotalofthescoresin3sectionsiscomparedwith“cutoff”andifhistotalscoreishigherthanthe“cutoff”,itissaidthathehasclearedthe“cutoff”. English Quant Rasoning Cut-off TotalscoreEXAM1 23 12 19 56 54EXAM2 24 17 18 62 59EXAM3 29 11 23 60 63EXAM4 21 14 21 57 56EXAM5 19 13 25 59 57EXAM6 26 17 21 60 64EXAM7 25 19 23 58 67EXAM8 27 20 21 65 68Q1.InhowmanyBANKPOEXAMSdidheclearthecutoff?Solution:asclearfromthetable,thereare4BANKEXAMSinwhichtotalmarksaregreaterthancutoffsoin4heclearthecutoff.Soansweris4Q2.InwhichBANKEXAMSdidhescorethemaximumtotalmarks?Solution:asclearfromthetable,inBANKEXAMS,hegotmaximumscorei.e.68soanswerisBANKEXAM8Q3.Whatishisimproving,ifimprovementisdefinedas=(maximumscore–minimumscore)?Solution:Hismaximumscoreis68andminimumscoreis54sotheimprovementis(68-54)=14soansweris14Q4.Inwhichsectionhasheshownthemaximumimprovementamongthese8BANKEXAMS?Solution:ImprovementinEnglish=29-19=10ImprovementinQuant=20-11=9Improvementinreasoning=25-18=7 HencemaximumimprovementisinEnglishQ5.Ifthesection-wisecut-offinallBANKEXAMSis15forQuantand20eachforEnglishandReasoning,theninhowmanyBANKEXAMSdidheclearallthecut-offs?Solution:Thereareonly3BANKEXAMS,inwhichsectionwisecutoffhecleartheseareBANKEXAMS6,7,8DirectionsforQ6-Q8:Studythegraphcarefullyandanswerthedatainterpretationsquestionsthatfollow.Thegraphshowstheimportsandexportsofcottoninrupeescroresfrom1990-91to1994-94.

47

Q6.Whatistheratioofthenumberofyearshavingaboveaverageexportstothosehavingbelowaverageexportsinthegivenperiod?Solution:Averageexports=638+1226+1661+1538+1305/5=1273.6Years1990-91&1991-92haveexportslessthanaverageexportwhileothershavemorethanthat.Sorequiredratiois3:2.Q7.Inwhichyearwasthegapinimportsandexportstheleast?Solution:Thegapinimportsandexportsfortheyear1990-91is186,1991-92is212,1992-93is524,1993-94is25and1994-95is353.Soyear1993-94hastheminimumgap.Alsoit’scanbedeterminedbylookingatthegraph.Q8.Theimportsin1994-95wereapproximatelyhowmuchpercentmorethantheimportsof1990-91?Solution:Importin1994-95is1,658crore.Importin1990-91is824croreExcess=1658=1658–824=834croretherefore,Excesspercentage=834/824X100=101.2%.DirectionsforQ9toQ13.Studythefollowinglinegraphwhichgivesthenumberofstudentswhojoinedandlefttheschoolinthebeginningofyearforsixyears,from1996to2001.InitialStrengthofschoolin1995=3000.

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Q9.Thestrengthofschoolincreased/decreasedfrom1997to1998byapproximatelywhatpercent?Solution:Importantdatanotedfromthegivengraph:In1996:Numberofstudentsleft=250andnumberofstudentsjoined=350.In1997:Numberofstudentsleft=450andnumberofstudentsjoined=300.In1998:Numberofstudentsleft=400andnumberofstudentsjoined=450.In1999:Numberofstudentsleft=350andnumberofstudentsjoined=500.In2000:Numberofstudentsleft=450andnumberofstudentsjoined=400.In2001:Numberofstudentsleft=450andnumberofstudentsjoined=550Therefore,thenumbersofstudentsstudyingintheschool(i.e.,strengthoftheschool)invariousyears:In1995=3000(given).In1996=3000-250+350=3100.In1997=3100-450+300=2950.In1998=2950-400+450=3000.In1999=3000-350+500=3150. In2000=3150-450+400=3100.In2001=3100-450+550=3200.Percentageincreaseinthestrengthoftheschoolfrom1997to1998= C%%%W<_?%

C<%%×100 %=1.7%

Q10.Thenumberofstudentsstudyingintheschoolduring1999was?Solution:Ascalculatedabove,thenumberofstudentsstudyingintheschoolduring1999=3150.Q11.Duringwhichofthefollowingpairsofyears,thestrengthoftheschoolwassame?Solution:Ascalculatedabove,intheyears1996and2000thestrengthoftheschoolwassamei.e.,3100.

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Q12.Thenumberofstudentsstudyingintheschoolin1998waswhatpercentofthenumberofstudentsstudyingintheschoolin2001?Solution:Requiredpercentage= C%%%

C<%%×100 %=93.75%

Q13.Amongthegivenyears,thelargestnumberofstudentsjoinedtheschoolintheyear?Solution:Ascalculatedabove,thelargestnumberofstudents(i.e.,550)joinedtheschoolintheyear2001.DirectionforQ14-Q18:Thecircle-graphgivenhereshowsthespendingsofacountryonvarioussportsduringaparticularyear.Studythegraphcarefullyandanswerthequestionsgivenbelowit.

Q14.WhatpercentoftotalspendingisspentonTennis?Solution:PercentageofmoneyspendonTennis=45/360*100=12.5%Q15.HowmuchpercentmoreisspentonHockeythanthatonGolf?Solution:LetthetotalspendingsonsportsbeRs.x.Then,AmountspentonGolf= C`

C`%×𝑥%=x/10%

Amountspentonhockey= `CC`%

×𝑥%=7x/40%Difference=Rs.XD

>%− D

$%=Rs.CD

>%

Requiredpercentage=Rs.nAFon=F×100=75%

Q16.HowmuchpercentlessisspentonFootballthanthatonCricket?Solution:LetthetotalspendingsonsportsbeRs.x.Then,AmountspentonCricket=Rs. [$

C`%×𝑥%=_D

>%

AmountspentonFootball=Rs. ?>C`%

×𝑥%=CD<%

Difference=Rs._D>%−CD

<%=CD

>%

Requiredpercentage=Rs.Hn=Fpn=F×100=33.33%

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Q17.IfthetotalamountspentonsportsduringtheyearbeRs.1,80,00,000,theamountspentonBasketballexceedsonTennisby:Solution:AmountspentonBasketballexceedsthatonTennisby=?%W>?

C`%×1800000

=Rs.250000Q18.IfthetotalamountspentonsportsduringtheyearwasRs.2crores,theamountspentonCricketandHockeytogetherwas:Solution:AmountspentonCricketandHockeytogether=[$E`C

C`%×2=Rs.8000000

Q19.Ifthereareatotalof200items,inastationaryshop,findthedifferenceinthenumberofpencilstopens

Solution:Totalno.ofitems=200,No.ofpens= $`

$%%×200=32

No.ofpencils= C[$%%

×200=76So,Difference=76-32=4420.Abagcontainsballsof4colorsasshowninthefigure,ifaballispickedatrandomwhichcoloredballhasthemaximumprobabilitytobepicked.

Solution:As,clearlyshowninfigure,blackareaismaximumsoprobabilitytobepickedanyballwillbemaximumforblack21.Thegivenpie-chart,showsthepercentageofchildreninterestedineachfield,findthenumberofstudentswhoareinterestedinmusicfromagroupof120students.

Solution:%ofstudentsinterestedinmusic=30%Totalnumberofstudents=120Numberofstudentsinterestedinmusic=30%of120=36

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Directions(Q22-):Studythefollowingtablecarefullyandanswerthequestionthatfollows:PRODUCTIONOFMACHINES(inunits)

Q22.Theproductionofmachinesin1985isapproximatelyhowmanytimesthatoftheproductionin1983?Solution:Youshould,firstofall,writedownthetotalproductionofmachinesforeachofthegivenyears.430/390=1.1.Q23.InwhichofthefollowingyearsdidthetypeVmachinesregisterthehighestproductionascomparedtothetotalproductionforthatyear?Solution:In1980,theratioworksoutto60/217=0.27.Q24.InwhichofthefollowingyearswasthereamaximumincreaseintheproductionoftypeIIImachinesoverthepreviousyear?Solution:In1983,Increase=70–53=17.Q25.Whatwastheapproximate%increaseintheproductionofalltypesofmachinestogether,fromtheyear1981totheyear1984?Solution:Therequired%growthrate=(420–280)/280=50%

TYPE 1980 1981 1982 1983 1984 1985I 60 80 75 85 90 94II 28 40 40 60 65 71III 39 45 53 70 75 78IV 30 60 120 90 100 92V 60 55 52 85 90 95

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Exercise

DirectionforQ1-Q3

1.Howmanyboysattendedthe1995convention?a.358 b.390 c.407 d.540 e.7162.Whichyeardidthesamenumberofboysandgirlsattendtheconference?a.1995 b.1996 c.1997 d.1998 e.None3.Whichtwoyearsdidtheleastnumberofboysattendtheconvention?a.1995and1996 b.1995and1998 c.1996and1997d.1997and1994 e.1997and1998DirectionforQ4-Q5

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4.Whichvacationdestinationismostcommonforthestudents?a.Beach b.HistoricalSites c.Cruises d.Mountains e.Other5.If500studentsattendWashingtonMiddleSchool,howmanyaregoingtothemountainsforvacation?a.25 b.60 c.75 d.100 e.125DirectionforQ6-Q8

6.Whichtwoyearsweretheleastnumberoftiressold?a.1998and1999 b.1998and2000 c.1998and2001d.1999and2000 e.2000and20017.Whichyeardidthestoresell1/3moretiresthantheyearbefore?a.1998 b.1999 c.2000d.2001 e.Thisdidnotoccurduringthe4yearspan.8.Whatwastheaveragenumberoftiressoldbythestorefrom1998to2001?a.9,000 b.9,375 c.9,545 d.9,770 e.9,995

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DirectionforQ9-Q11

9.WhatwastheaveragenumberofbabiesthatDr.Jonesdeliveredeachyearfrom1995to1998?a.35 b.40 c.45 d.50 e.5510.HowmanybabiesdidDr.Jonesdeliverin1998?a.25 b.35 c.45 d.55 e.6511.IfDr.Jonesdelivered85babiesin1999,howmanyrattleswouldrepresentthisnumber?a.6½ b.7 c.7½D.8 e.8½DirectionforQ12-14

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12.IfXYZAutoCompanysold23,000vehiclesin1999,howmanywereSUV's?a.2,990 b.3,030 c.3,450 d.4,760 e.4,77513.If7,650trucksweresoldin1999,howmanytotalvehiclesweresoldin1999byXYZAutoCompany?a.35,000 b.40,000 c.45,000 d.50,000 e.55,00014.If3,7502-doorsedansweresoldin1999,thenhowmany4-doorsedansweresoldin1999byXYZAutoCompany?a.3,578 b.4,950 c.5,120 d.5,670 e.5,000DirectionforQ15-Q17.

15.Howmuchdidtheinfantgaininthefirstmonthoflife?a.0.5pounds b.1.2pounds c.1.5pounds d.0.25poundse.0.75pounds16.WhatwastheaverageweightoftheinfantfromApriltoOctober,roundedtothenearestounce?a.10pounds b.10.5pounds c.10.9poundsd.11.5pounds e.11.9pounds17.Betweenwhichtwomonthsdidtheinfantgainthemostweight?a.AprilandMay b.JuneandJuly c.JulyandAugustd.AugustandSeptember e.SeptemberandOctoberDirections(Q18-Q22):Studythefollowinginformationandanswerthefollowingquestion:Mr.Mohanhastodecidewhetherornottotestabatchof1000crank–shaftsbeforesendingthemtothebuyeroverseas.Incasehedecidedtotest,hehastwooptions:(1)Usetest1(2)Usetest2.Test1costsRs.2percrankshaft.Howeverthistestisnotperfect.Itsends20%ofthebadonestothebuyerasgood.Test2willcostRs.3percrank–shaft.Itindeedbringsout

56

allthebadones.AdefectivecrankshaftidentifiedbeforesendingcanbecorrectedatacostofRs.25percrank–shift.Alldefectivecrank–shaftsareidentifiedandthebuyer'sendandapenaltyofRs.50perdefectivecrankshafthastobepaidbyMr.Mohanasperthecontractsigned.Note:OnceadefectivepieceisidentifiedbyMr.Mohan,thenecessarilysendsitforcorrection.Mr.Mohanhasappointedyou,abuddingmanagementconsultant,toadvisehim.Youhavetoadvisehimpurelyonacommercialbasis18.Mr.Mohanshouldnottestiftheno.ofdefectivecrank–shaftsinthelotis:a.lessthan100b.morethan200c.between120&190d.indeterminatee.Noneofthese19.Iftheno.ofdefectivecrank–shaftsinthelotisbetween200and400,Mohan:a.shoulduseeithertest1ortest2b.shouldusetest1onlyc.shouldusetest2onlyd.cannotdecidee.Noneofthese20.Ifthereare200defectivecrank–shaftsinthelot,Mohan:a.shoulduseeithertest1ortest2b.shouldusetest1ornotuseanytestc.shouldusetest2ornotuseanytestd.cannotdecidee.Noneofthese21.IfMohanistoldthatthelothas160defectivecrank–shafts,heshould:a.usetest1onlyb.usetest2onlyc.donotestingd.eitherusetest1ordonoteste.Noneofthese22.Thetotalcostoftest1whenthenumberofdefectivepiecesis600is:a.Rs.16680b.Rs.20000c.Rs.30000d.Rs.22600e.NoneoftheseDirections(Q.23–27):Studythefollowinggraphandanswerthequestionsthatfollow:No.ofstudents(inthousands)whooptedforthreedifferentspecializationsduringthegivenfiveyearsinauniversity

57

23.Outofthetotalnumberofstudentswhooptedforthegiventhreesubjects,intheyear2009,38%weregirls.HowmanyboysoptedforMathematicsinthesameyear?a.3100b.3200c.3000d.Cannotbedeterminede.Noneofthese24.Ifthetotalnumberofstudentsintheuniversityintheyear2007was455030,thetotalnumberofstudentswhooptedforthegiventhreesubjectswasapproximatelywhatpercentofthestudents?a.19b.9c.12d.5e.2325.WhatisthetotalnumberofstudentswhooptedforHindiandMathematicsintheyears2006,2007and2009together?a.97000b.93000c.85000d.96000e.Noneofthese26.ThetotalnumberofstudentswhooptedforMathematicsintheyears2005and2008togetherisapproximatelywhatpercentofthetotalnumberofstudentswhooptedforallthreesubjectsinthesameyear?a.38b.28c.42d.32e.4827.WhatistheratioofthenumberofstudentswhooptedforEnglishintheyear2006and2008togethertothenumberofstudentswhooptedforHindiintheyear2005and2009together?a.11:5b.12:7c.11:7d.12:5e.Noneofthese

58

DirectionsforQ28–32:Studythefollowinglinegraphwhichgivesthenumberofstudentswhojoinedandlefttheschoolinthebeginningofyearforsixyears,from1996to2001.InitialStrengthofschoolin1995=3000

28.Thestrengthofschoolincreased/decreasedfrom1997to1998byapproximatelywhatpercent?a.10%b.1.7%c.3%d.5%e.Noneofthese29.Thenumberofstudentsstudyingintheschoolduring1999was?a.3150b.3000c.3100d.3050e.Noneofthese30.Duringwhichofthefollowingpairsofyears,thestrengthoftheschoolwassame?a.1999&2000b.2000&2001c.1996&2000d.1998&2000e.Noneofthese31.Thenumberofstudentsstudyingintheschoolin1998waswhatpercentofthenumberofstudentsstudyingintheschoolin2001?a.93%b.93.75%c.94%d.92%e.Noneofthese32.Amongthegivenyears,thelargestnumberofstudentsjoinedtheschoolintheyear?a.1997b.1998c.1999d.2001e.2000Directions(33to35):-Studythebarchartandanswerthequestionbasedonit.ProductionofFertilizersbyaCompany(in1000tonnes)OvertheYears

59

33.Whatwasthepercentagedeclineintheproductionoffertilizersfrom1997to1998?a.33.33%b.25%c.20%d.50%e.Noneofthese34.Inhowmanyofthegivenyearswastheproductionoffertilizersmorethantheaverageproductionofthegivenyears?a.1b.2c.3d.4e.Noneofthese35.Theaverageproductionof1996and1997wasexactlyequaltotheaverageproductionofwhichofthefollowingpairsofyears?a.2000and2001b.1999and2000c.1998and2000d.1995and2001e.NoneoftheseDirections(36to38):-Thefollowingpie-chartshowsthepercentagedistributionoftheexpenditureincurredinpublishingabook.Studythepie-chartandtheanswerthequestionsbasedonit.VariousExpenditures(inpercentage)IncurredinPublishingaBook

36.WhatisthecentralangleofthesectorcorrespondingtotheexpenditureincurredonRoyalty?a.15°b.24°c.54°d.48°e.Noneofthese37.Royaltyonthebookislessthantheprintingcostby:a.25%b.20%c.30%d.15%e.Noneofthese

60

38.Ifforacertainquantityofbooks,thepublisherhastopayRs.30,600asprintingcost,thenwhatwillbeamountofroyaltytobepaidforthesebooks?a.Rs.19,450 b.Rs.21,200c.Rs.22,950d.Rs.26,150e.NoneoftheseDirectionsfor(Q39to41):RefertothefollowingBar-chartandanswerthequestionsthatfollow:

39.Whatistheaveragevalueofthecontractsecuredduringtheyearsshowninthediagram?a.Rs.103.48croreb.Rs.105crorec.Rs.100crored.Rs.125.2croree.Noneofthese40.Comparedtotheperformancein1985(i.e.takingitasthebase),whatcanyousayabouttheperformancesintheyears’84,’85,’86,’87,’88respectively,inpercentageterms?a.150,100,211,216,97b.100,67,141,144,65c.150,100,200,215,100d.120,100,220,230,68e.Noneofthese41.Whichistheyearinwhichthehighestpercentagedeclineisseeninthevalueofcontractsecuredcomparedtotheprecedingyear?a.1985b.1988c.1984d.1986e.Noneofthese

61

Directionsforthequestionsfrom42to46:Thefollowingtablegivesthenationalincomeandthepopulationofacountryfortheyears1984–85to1989–90.Foreachothefollowingquestionschoosethebestalternative:

42.Theincreaseinthepercapitaincomecomparedtothepreviousyearislowestfortheyear:a.1985-86b.1986-87c.1987-88d.1989-90e.Noneofthese43.Thepercapitaincomeishighestfortheyear:a.1984-85b.1985-86c.1987-88d.1989-90e.Noneofthese44.Thedifferencebetweenthepercentageincreaseinpercapitaincomeandthepercentageincreaseinthepopulationcomparedtothepreviousyearishighestfortheyear:a.1985-86b.1986-87c.1987-88d.1988-89e.Noneofthese45.Therateofincreaseinpopulationwaslowestintheyear:a.1985-86b.1987-88c.1989-90d.datainadequatee.Noneofthese46.Increaseinthepercapitaincomecomparedtothepreviousyearamongtheyearsgivenbelowwashighestfortheyear:a.1985-86b.1986-87

62

c.1987-88d.1989-90e.NoneoftheseDirection(Q47–51):Thegraphbelowshowstheendofthemonthmarketvaluesof4sharesfortheperiodfromJanuarytoJune.Answerthefollowingquestionsbasedonthisgraph.

47.Whichshareshowedthegreatestpercentageincreaseinmarketvalueinanymonthduringtheentireperiod?a.Ab.Bc.Cd.De.Noneofthese48.Inwhichmonthwasthegreatestabsolutechangeinmarketvalueforanysharerecorded?a.Marchb.Aprilc.Mayd.Junee.Noneofthese49.Inwhichmonthwasthegreatestpercentageincreaseinmarketvalueforanysharerecorded?a.Februaryb.Marchc.Aprild.Maye.Noneofthese50.Anindividualwishestosell1shareofCand1shareofDtobuy1shareofAattheendofa

63

month.Atwhichmonth-endwouldtheindividual’slossfromthisdecision,duetosharevaluechanges,bethemost?a.Februaryb.Marchc.Aprild.Junee.Noneofthese51.Anindividualdecidestosell1shareofCand1shareofDtobuy1shareofAattheendofthemonth.Whatcanbetheindividual’sgreatestgainfromthisdecision,duetosharevaluechanges?a.5b.10c.15d.datainadequatee.NoneoftheseDirection(Q52–56):Answerthequestionsbasedonthefollowingtable.

52.Themaximumpercentagedecreaseinmarketshareisa.60%b.50%c.33.3%d.20%e.Noneofthese53.Thecityinwhichminimumnumberofproductsincreasedtheirmarketsharesin1993-94isa.Mumbaib.Delhic.Kolkatad.Chennai

64

e.Noneofthese54.Themarketsharesofwhichproductsdidnotdecreasedbetween1993-94inanycity?a.HDb.COc.BNd.datainadequatee.Noneofthese55.Thenumberofproductswhichhad100%marketshareinfourmetropolitancitiesisa.0b.1c.2d.3e.Noneofthese56.Thenumberofproductswhichdoubledtheirmarketsharesinoneormorecitiesisa.0b.1c.2d.3e.NoneoftheseDirectionfor(Q57to60):Analyzethetableandanswerthequestionscarefully

65

57.Whatwasthetotalnumberofengineeringstudentsin1989–90?a.28500b.4400c.4200d.42000e.Noneofthese58.ThegrowthrateinstudentsofGovt.Engg.CollegescomparedtothatofPrivateEngg.Collegesbetween1988–89and1989–90isa.moreb.lessc.equald.3/2e.Cannotbedetermined59.ThetotalnumberofEngg.Studentsin1991–92,assuminga10%reductioninthenumberoverthepreviousyear,isa.5700b.57000c.44800d.noneofthesee.Cannotbedetermined60.In1990–91,whatpercentofEngg.StudentswerestudyingatIIT’s?a.16b.15c.14d.12e.NoneoftheseDirectionfor61to64:Refertothepie-chartgivenbelow:

66

61.WhatfractionofGhoshbabu’sweightconsistsofmuscularandskinprotein?a.1/13b.1/30c.1/20d.Cannotbedeterminede.Noneofthese62.Ratioofdistributionofproteininmuscletothedistributionofproteininskinisa.3:1b.3:10c.1:3d.31/2:1e.Noneofthese63.WhatpercentofGhoshBabu’sbodyweightismadeupofskin?a.0.15b.10c.1.2d.Cannotbedeterminede.Noneofthese64.Intermsoftotalbodyweight,theportionofmaterialotherthanwaterandproteinisclosesttoa.3/20b.1/15c.85/100d.1/20e.Noneofthese

Answers

1.A 2.A 3.A 4.A 5.B 6.B 7.B 8.B 9.C 10.D11.E 12.A 13.C 14.E 15.A 16.A 17.D 18.A 19.C 20.A21.A 22.B 23.D 24.B 25.E 26.D 27.A 28.B 29.A 30.C31.B 32.D 33.B 34.D 35.D 36.C 37.A 38.C 39.A 40.A41.B 42.B 43.D 44.B 45.C 46.E 47.D 48.A 49.A 50.A51.B 52.C 53.B 54.C 55.A 56.B 57.E 58.B 59.D 60.C61.C 62.E 63.D 64.A

67

Mixture&AlligationMixture:Mixingoftwoormorethantwotypeofquantitiesgivesusamixure.Quantitiesoftheseelementscanbeexpressedaspercentageorratio.i.e.Percentage(20%ofsugarinwater)Fraction(Asolutionofsugarandwatersuchthatsugar:water=1:4)Alligation:Alligationisarulewhichisusedtosolvetheproblemsrelatedtomixtureanditsingredient.Itistherulethatenablesustofindtheratioinwhichtwoormoreingredientsatthegivenpricemustbemixedtoproduceamixtureofdesiredprice.AlligationRuleWhentwoelementsaremixedtomakeamixtureandoneoftheelementsischeaperandotheroneiscostlierthen,

𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑐ℎ𝑒𝑎𝑝𝑒𝑟𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑐𝑜𝑠𝑡𝑙𝑖𝑒𝑟 =

𝐶. 𝑃𝑜𝑓𝑐𝑜𝑠𝑡𝑙𝑖𝑒𝑟 − 𝑚𝑒𝑎𝑛𝑜𝑓𝑝𝑟𝑖𝑐𝑒𝑀𝑒𝑎𝑛𝑝𝑟𝑖𝑐𝑒 − 𝐶𝑃𝑜𝑓𝑐ℎ𝑒𝑎𝑝𝑒𝑟

HereMeanPriceisCPofmixtureperunitquantity.Aboverulecanbewrittenas,

Then,CheaperQuantity:CostlierQuantity=(D–M):(M–C)ExampleInwhatproportionmustsugaratRs40perkgbemixedwithsugaratRs60perkgsothatthemixturebeRs55perkg?Sol:Here,CPofCheaper(C)=40,

68

CPofCostlier(D)=60andMeanPrice(M)=55Sofromtheruleofalligationwecansaythat

Cheapersugar/costliersugar=5/15ProportionofCheaperSugarandCostlierSugaris1:3Mixtureofmorethantwoelements.Thismethodisabittrickyinitiallybutifyoupracticeitthenitbecomesquiteeasy.Ifthemixtureisofmorethantwoingredients,thenwritethepricesofeachingredientbelowoneanotherinascendingorder.Writethemeanpricetotheleftofthelist.Nowmakecouplesofpricesinsuchawaythatonepriceofthecoupleisbelowmeanpriceandanotherpriceofthecoupleisabovemeanprice.Nowfindthedifferencebetweeneachpriceandmeanpriceandwriteitoppositetothepricelinkedtoit.Thisdifferenceisrequiredanswer.Don’tworryifyoudon’tunderstandaboveparagraph.Trytounderstandaboveparagraphwiththeexamplesgivenbelow.ExampleHowmustashopownermix4typesofriceworthRs95,Rs60,Rs90andRs50perkgsothathecanmakethemixtureofthesesugarsworthRs80perkg?Sol:Herethepricesofsugarsare95,60,90and50.Andthemeanpriceis80.Nowreadtheaboveparagraphandtheimagegivenbelowtounderstandthismethod.

Sotheproportionofsugaris

69

50:60:90:95=15:10:20:30or50:60:90:95=3:2:4:6ExampleInwhatratiomustapersonmixthreekindofteaeachofwhichhasapriceof70,80and120rupeesperkg,insuchawaythatthemixturecostshim100rupeesperkg?Sol:Herethepricesofteaare70,80and120Andmeanpriceis100,so

Sotheproportionofteais70:80:120=20:20:50or70:80:120=2:2:5SomeShortcutFormulasRule1IfndifferentvesselsofequalsizearefilledwiththemixtureofPandQintheratiop1:q1,p2:q2,……,pn:qnandcontentofallthesevesselsaremixedinonelargevessel,then

wxy9zb{|}!wxy9bzb{|}w

= ~A

~A��AE ~;~;��;

E...…….. ~G~G��G

�A~A��A

E �;~;��;

E...…….. �G~G��G

ExampleThreeequalbucketscontainingthemixtureofmilkandwateraremixedintoabiggerbucket.Iftheproportionofmilkandwaterintheglassesare3:1,2:3and4:2thenfindtheproportionofmilkandwaterinthebiggerbucket.Sol:Let’ssayPstandsformilkandQstandsforwater,So,p1:q1=3:1p2:q2=2:3p3:q3=4:2

𝑄𝑢𝑎𝑛𝑖𝑡𝑦𝑜𝑓𝑃𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑄 =

𝑃$𝑃$ + 𝑄$

+ 𝑃<𝑃< + 𝑄<

+. . . …… . . 𝑃9𝑃9 + 𝑄9

𝑄$𝑃$ + 𝑄$

+ 𝑄<𝑃< + 𝑄<

+. . . …… . . 𝑄9𝑃9 + 𝑄9

70

wxy9zb{|}!wxy9bzb{|}w

= H

H�AE;

;�HE=

=�;A

H�AEH

;�HE;

=�;=109/71

Soinbiggerbucket,Milk:Water=109:71Rule2Ifndifferentvesselsofsizesx1,x2,…,xnarefilledwiththemixtureofPandQintheratiop1:q1,p2:q2,……,pn:qnandcontentofallthesevesselsaremixedinonelargevessel,then

𝑄𝑢𝑎𝑛𝑖𝑡𝑦𝑜𝑓𝑃𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑄 =

𝑃$𝑋$𝑃$ + 𝑄$

+ 𝑃<𝑋<𝑃< + 𝑄<

+. . . …… . . 𝑃9𝑋9𝑃9 + 𝑄9

𝑄$𝑋$𝑃$ + 𝑄$

+ 𝑄<𝑋<𝑃< + 𝑄<

+. . . …… . . 𝑄9𝑋9𝑃9 + 𝑄9

ExampleThreebucketsofsize2liter,4literand5litercontainingthemixtureofmilkandwateraremixedintoabiggerbucket.Iftheproportionofmilkandwaterintheglassesare3:1,2:3and4:2thenfindtheproportionofmilkandwaterinthebiggerbucket.Sol:Let’ssayPstandsformilkandQstandsforwater,So,p1:q1=3:1,x1=2p2:q2=2:3,x2=4p3:q3=4:2x3=5,so

𝑄𝑢𝑎𝑛𝑖𝑡𝑦𝑜𝑓𝑃𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑄 =

𝑃$𝑋$𝑃$ + 𝑄$

+ 𝑃<𝑋<𝑃< + 𝑄<

+. . . …… . . 𝑃9𝑋9𝑃9 + 𝑄9

𝑄$𝑋$𝑃$ + 𝑄$

+ 𝑄<𝑋<𝑃< + 𝑄<

+. . . …… . . 𝑄9𝑋9𝑃9 + 𝑄9

wxy9zb{|}!wxy9bzb{|}w

= H×;H�AE

;×=;�HE

=×B=�;

A×;H�AE

H×=;�HE

;×B=�;

=109/71

𝑄𝑢𝑎𝑛𝑖𝑡𝑦𝑜𝑓𝑀𝑖𝑙𝑘𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑤𝑎𝑡𝑒𝑟 = 193/137

Soinbiggerbucket,Milk:Water=193:137

71

Rule3:RemovalandReplacementIfavesselcontains“x”litresofliquidAandif“y”litresbewithdrawnandreplacedbyliquidB,thenif“y”litresofthemixturebewithdrawnandreplacedbyliquidB,andtheoperationisrepeated‘n’timesinall,then:

𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑙𝑖𝑞𝑢𝑖𝑑𝐴𝑎𝑓𝑡𝑒𝑟𝑛𝑡ℎ𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑞𝑢𝑎𝑛𝑖𝑡𝑦𝑜𝑓𝑙𝑖𝑞𝑢𝑖𝑑𝑜𝑓𝐴 = 1 −

𝑦𝑥

9

ExampleAcontaineriscontaining80literofwine.8literofwinewastakenoutfromthiscontainerandreplacedbywater.Thisprocesswasfurtherrepeatedtwotimes.Howmuchwineisthereinthecontainernow?Sol:Herex=80,y=8andn=3,so

𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦𝑜𝑓𝑙𝑖𝑞𝑢𝑖𝑑𝐴𝑎𝑓𝑡𝑒𝑟3𝑟𝑑𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛80 = 1 −

880

C

= 58.32𝑙𝑖𝑡𝑟𝑒𝑠Quantityofwineafter3rdoperation=58.32liters.Rule4:pgramofingredientsolutionhasa%ingredientinit.Toincreasetheingredientcontenttob%inthesolution?Quantityofingredientneedtobeadded=c(�Wy)

$%%W�

Example125literofmixtureofmilkandwatercontains25%ofwater.Howmuchwatermustbeaddedtoittomakewater30%inthenewmixture?Sol:Let’ssayp=125,b=30,a=25SofromtheequationQuantityofingredientneedtobeadded=$<?(C%W<?)

$%%WC%

Quantityofwaterneedtobeadded=8.92liter.

72

Ex.1.HowmanykgofRs.33akgcoffeehastobemixedwithRs.27akgcoffeetomakeamixtureof60kgworthRs.32akg?Sol. Bytheruleofallegation

Totalweightoftwotypeofcoffeeis=60kgTotalweightofcoffeeofRs.33akgis=?

`×60=50kg

Ex.2.Twovesselscontainmilkandwatermixedintheratioof2:3and3:4.Findtheratioinwhichthesetwoaretobemixedtogetanewmixtureinwhichtheratioofmilktowateris7:10Sol.Byalligationrule

Requiredratio= <

$$_∶ $[?=10:7.

Ex.3.Inacocktailtherewasamixtureofalcoholandjuice.Thepriceofthemixtureof30ml.ofalcoholand90ml.ofjuiceisRs.120.Thepriceofthealcoholincreasesby20%andthepriceofjuiceincreasesby30%.NowthesamecocktailcostRs.150.Whataretheoldpricesof100ml.ofalcoholand100mlofjuice?Sol.Letpriceof1mlofalcoholisRs.xandpriceof1mlofjuiceisRs.y.Therefore,30x+90y=120orx+3y=4...............(i)Aftertheincreaseinthepriceofalcoholby20%andofjuiceby30%,thenewpricesofcocktailis36x+117y=150...............(ii)Now,multiply(i)by36&subtracting(i)from(ii),wegety=2/3,putthisvalueofyin(i),wegetx=2Now,priceof1mlofalcoholisRs.2andpriceof1mlofjuiceisRs.2/3Therefore,priceof100mlofalcoholand100mlofjuiceisRs.200&Rs.66.67Ex.4.TheaveragemonthlysalaryoflabourersandsupervisorsinafactoryisRs.1,250permonth,whereastheaveragemonthlysalaryofallthe6supervisorsisRs.2,450.IftheaveragemonthlysalaryofthelabourersisRs.950,findthenumberoflabourers.

73

Sol.

ByAlligationSo,theratioof�xc�T�z�|T�

�y�|xT��=300/1200=$

>

Becausethereare6supervisors,Numberoflaboures=4 6=24.

Ex.5.TwoliquidsAandBareintheratio5:1incontainerXand1:3incontainerY.InwhatratioshouldthecontentsofthetwocontainersbemixedsoastoobtainamixtureofAandBintheratio1:1?Sol.

Wecansolvethisbyalligation.Butwhilewealligate,wehavetobecarefulthatithastobedonewithrespecttoanyoneofthetwoliquids,viz,eitherAorB.Wecanverifythatinbothcases,wegetthesameresult.WithrespecttoliquidA:TheproportionofAinthefirstvesselis?`andthatinthesecondvesselis$

>,andwefinallyrequire$

<partofA.Nowsolvewiththehelp

ofallegationEx.6.Adishonestmilkmanmixes20litresofwaterwith80litresofmilk.Aftersellingone-fourthofthismixture,headdswatertoreplenishthequantitythathehassold.Whatisthecurrentproportionofwatertomilk?Sol.100litresofmixturehave80lmilkand20lwater.When25litresofmixtureistakenoutthenitcontains20lmilkand5lwater.Somilkleft=60litresandwaterleft=15litresIf25litresofwaterisaddedthenratioofwatertomilk=40:60=2:3Ex.7.Agoldbiscuitof11kgcontainedonly82%goldandtherestcopper.Ifitwasalloyedwithanothergoldbiscuitandtheresultingalloyhasamassof28kgandcontained90%ofgold,findthepercentageofgoldinthesecondbiscuit.Sol.Weightof1stbiscuit=11kg%ageofgoldin1stbiscuit=82%Weightof2ndbiscuit=28–11=17kgLetthe%ageofgoldin2ndbiscuit=x%ByAlligation

74

Ratio= $�b�z��xzb

<9��z��xzb= $$

$X

$$$X= DW_%

[

17x–1530=8817x=1618x=95.17Ex.8.Therearetwoalloysofgold,silverandplatinum.Thefirstalloyisknowntocontain40percentofplatinumandthesecondalloy26percentofsilver.Thepercentageofgoldisthesameinbothalloys.Havingalloyed150kgofthefirstalloyand250kgofthesecond,wegetanewalloythatcontains30percentofgold.Howmanykilogramofplatinumisthereinthenewalloy?Sol.Sincethepercentageofgoldinbothalloysisthesame,anymixtureofthetwowillcontainthesamepercentageconcentrationofgold.Hence,wegetFirstalloy=Gold:Silver:Platinum=30:30:40Andsecondalloy=Gold:Silver:Platinum=30:26:44So,theweightofplatinuminthefirstalloy=150 >%

$%%=60kgAndtheweightofplatinumin

thesecondalloy=250 >>$%%

=110kgSo,thetotalweightofplatinuminthemixture=60+110kg=170kgEx.9.Avesselcontains40litresofmilkandamilkmandelivers10litrestothefirsthouseandaddsanequalquantityofwater.Hedoesexactlythesameatthesecondandthirdhouse.Whatistheratioofmilkandwater,whenhehasfinisheddeliveringatthethirdhouse?Sol.Totalamountofmilk=40litresAmountofmilkreplacedbywater=10litres

Now,accordingtothequestion:�z��T��yz9z9�y}b�TC|c�Tybz|9�z9zbzy��xy9bzb{|}�z��

= 1 − $%>%

C=<X

`>

Itmeanstheratioofmilktowater=27:37Ex.10.ThreecontainersA,BandCarehavingmixturesofmilkandwaterintheratioof1:5,3:5and5:7respectively.Thecapacitiesofthecontainersareintheratio5:4:5.Ifthecontentsofallthethreecontainersaremixedtogether,findtheratioofmilktowaterintheresultingmixture.Sol.Assumethatthereare500,400and500litresrespectivelyinthe3containers.Thenwehave,83.33,150and208.33litresofmilkineachofthethreecontainers.Thus,thetotalmilkis441.66litres.Hence,theamountofwaterinthemixtureis

75

1400–441.66=958.33litres.Hence,theratioofmilktowateris441.66:958.33=53:115.Ex.11.TwoboxesAandBwerefilledwithcoffeeandchicorymixedinAintheratioof5:3andinBintheratioof7:3.Whatquantitymustbetakenfromeachtoformamixturewhichshallcontain6kgofcoffeeand3kgofchicory(inkgrespectively)?Sol.SupposeXkgaretakenfromA;then(9–X)kgaretakenfromB.5/8ofthemixtureinAand7/10ofthemixtureinBiscoffee.5X/8+7/10(9–X)=6,HenceX=44kgmustbetakenfromAand5kgfromBEx.12.Kalicharangetssomecoinsmadeofanalloyofgoldandsilver.Thealloywithaweightof100gmcontains20%ofgold.Whatpieceofanothergold-silveralloycontaining60%ofsilvermustbealloyedwiththefirstpieceofalloyinordertoobtainanewalloywiththe32%ofgold?Sol.Goldisthefirstalloy=20%Goldisthesecondalloy=40%Byruledalligation:

Theratioofthetwoalloys=2:3Thequantityofsecondalloys=100×C

<=150gm.

Ex.13.Atoyweighing24gramsisanalloyoftwometalsandisworthRs.174,butiftheweightsofmetalsinalloybeinterchanged,thetoywouldbeworthRs.162.IfthepriceofonemetalbeRs.8pergram,findthepriceoftheothermetalinthealloyusedtomakethetoy.Sol.Let’ssayonemetalisxgrams.Sotheotheroneis[24–x]grams.Nowaccordingtothequestion8x+(24–x)y=1748(24–x)+xy=1628x+24y–xy=174192–8x+xy=16224y+192=33624y=144y=6.Rs.6pergram.

76

Ex.14.Johnstolesomewhiskyfromabottlewhichhisfatherhadkeptawayinthecupboard.Thisbottleofwhiskycontained50%alcohol.Johnreplacedwhathehaddrunkbyanotherbrandofwhiskywhichcontainedonly25%alcohol.Thewhiskyinthebottlenowhasonly35%alcohol.HowmuchofthebottledidJohnsteal?Sol.

��z��{�zb�?%%y��|�|���z��{�zb�<?%y��|�|�

= $%$?= <

C

Thesetwospiritsmustbemixedintheratio2:3inordertogetthewhiskyof35%alcohol.Thismeansthatonly2/5oftheoriginalbottleofwhiskyisleftandsoJohnhasstolen3/5ofthebottle.Ex.15. Alumpoftwometalsweighing18gramsisworthRs.74butiftheirweightsbeinterchanged,itwouldbeworthRs.60.10.IfthepriceofthegoldbeRs.7.20pergram,findtheweightoftheothermetalinthemixture.Sol.Lettheweightofthegoldbexgm&Weightoftheothermetal=(18–x)gmLetthepriceoftheothermetalbeRs.y/gm.Now,accordingtothequestion7.20x+y(18–x)=747.20(18–x)+yx=60.10Solvingthesetwoequationssimultaneously,x=10,y=0.25Quantityoftheothermetal=18–10=8gramsEx.16.If20litersofwaterisaddedtoatubalreadyfilledwith120litersofdilutedmilk,theconcentrationofmilkinthetubwillbe80%.Whatistheconcentrationofmilkinthetubbeforeaddingwatertothetub?Sol.$<%�E<%×%

$>%=80

120k=80×140k=<[%

C=93.33%.

Ex.17.Theratioofmilktowaterinamixtureis2:3andtheratioofwatertomilkinanothermixtureis3:4.If70litresofeachtypeofmilkisaddedto100litresofpuremilk,whatwillbetheratioofmilktowaternow?Sol.70litresoffirsttype

77

<?×70=28litresofmilk

70litresofsecondtype=>X×70=40litresofmilk

100litresofpuremilk=100litresofmilkTotaloutof240litres,thereis168litresofmilk.So,theremaining72litresiswater.So,requiredratio=$`[

X<= X

C

Ex.18.Analloycontainszincandtininratio3:4.Anotheralloycontainszincandsilverinratio4:3.Ifboththealloysaremeltedandmixedinequalratio,whatwillbetheratioofzincandtininthenewalloy?Sol.7kgofthefirstalloycontains3kgofzincand4kgofTin.7kgofthesecondalloycontains4kgofzincand3kgofsilver,Ifboththesealloysareadded,wewillget7kgofzincand4kgoftin.So,7:4.Ex.19.Analloycontainszinc,copperandIronintheratio4:3:2.If20kgofcopperisaddedto90kgofthisalloy,whatwillbetheratioofzincandcopperinthealloy?Sol.Initially,Zinc=40kgCopper=30kgIron=20kgIf20kgofcopperisadded,thetotalquantityofcopperwillbe50kg,andtheratioofzincandcopper=4:5.Ex.20.From200litersof80%concentratedmilk,30litersistakenoutandwaterisadded.ThenewconcentrationofmilkinthemixtureisSol.If30litersistakenoutfrom200litersitwillbe170ltsoutofthis;Themilkpercentis80%.Sothevolumeofmilkintheremainingsolution=170× [%

$%%=136lts

Percentageofmilkinthefinalsolution=$C`<%%

×100=68.1Ex.21.InwhatproportionmustwheatatRs.1.80perkgbemixedwithwheatatRs.2.50perkgtogetamixtureworthRs.2.0perkg?Sol.ApplytheRuleofAlligation:⇒requiredratio=5:2

78

Ex.22.20litresofamixturecontains40%alcoholandtherestiswater.If5litresofwaterisadded,thepercentageofalcoholinthenewmixturewillbeSol.Totalamountofmixture=20lt&amountofAlcohol=40%=8ltNow,AccordingtotheQuestion,Newmixture=20+5=25lits.Percentageofalcohol=(8/25)×100=32%Ex.23.TwovesselsA,Bcontainmilkandwatermixedintheratioof5:2and8:5respectively.Findtheratioinwhichthesemixturesaretobemixedtogetanewmixturecontainingmilkandwaterintheratio9:4.Sol.Ratioofmilk&waterinAis5:2&inBis8:5.Thenewmixtureshouldhavemilk&waterintheratio9:4.RatioofmilkinAis?

X&inBis [

$C.Theresultingmixtureshouldcontainmilk&

waterintheratio9:4.Henceratioofmilkshouldbe9/13Nowbyapplyingtheruleofalligationwegettheratio $

$C× _$

<= X

<

HenceA&Bshouldbemixedintheratio7:2sothattheresultingmixturecontainMilk&waterintheratio9:4.

Ex.24.McGrathearnedaprofitofRs.300byselling100kgofamixtureofAandBtypesofrice,atatotalpriceofRs.1,100.WhatwastheproportionofAandBtypesofriceinthemixtureifthecostpricesofAandBareRs.10andRs.5perkgrespectively?Sol.TotalamountwhichMcgrathgetaftersellingthe100kgofrice=Rs.1,100&Profit=Rs.300∴C.P.of100kg=1,100-300=Rs.800C.P.of1kg=Rs.8Now,ApplytheRuleofAlligation:

𝑄y𝑄�

= 8 − 510 − 8 =

32

79

Exercise

1.InwhatproportionmustagrocermixwheatatRs.2.04perkgandRs.2.88perkgsoastomakeamixtureofworthRs.2.57perkg?a.4:3b.3:4c.2:3d.3:2e.Noneofthese2.Amixtureofcertainquantityofmilkwith8Lofwaterisworth45paiseperlitre.Ifpuremilkbeworth54paiseperlitre,howmuchmilkisthereinthemixture?a.50Lb.20Lc.30Ld.60Le.Noneofthese3.Atraderhas50kgofpulses,partofwhichhesellsat8%profitandrestat18%profit.Hegains14%onthewhole.Whatisthequantitysoldat18%profit?a.50kgb.20kgc.30kgd.40kge.Noneofthese4.Amerchanthas2000kgofrice,partofwhichhesellsat36%profitandtherestat16%profit.Hegains28%onthewhole.Findthequantitysoldat16%a.500kgb.900kgc.800kgd.400kge.Noneofthese5.Abutlerstolewinefromabuttofsherrywhichcontained80%ofspiritandhereplaceditbywinecontainingonly32%spirit.Thenthebuttwasof48%strengthonly.Howmuchofthebuttdidhesteal?a.2/3b.3/2c.1/3d.4/5e.Noneofthese6.TwovesselsPandQcontainwineandwaterintheratiosof5:2and8:5,respectively.Findtheratioinwhichthesemixturesaretobemixedtogetanewmixturecontainingwineandwaterintheratioof9:4a.4:3b.7:2c.2:7d.3:4e.Noneofthese7.Acontainerisfilledwithliquid,6partofwhicharewaterand10partmilk.Howmuchofthemixturemustbedrawnoffandreplacedwithwatersothatthemixturemaybehalfwaterandhalfmilk?a.2/3b.3/2c.1/5d.4/5e.Noneofthese8.Abutlerstolewinefromabuttofsherrywhichcontains15%ofspiritandhereplacedwhathehadstolenbywinecontaining6%ofspirit.Thebuttwasthen9%strongonly.Howmuchofthebuttdidhestole?a.2/3b.3/2c.1/5d.4/5e.Noneofthese9.Ravina’ssavingsandexpenditureareintheratioof2:3.Herincomeincreasesby10%.Herexpenditurealsoincreasesby12%.Byhowmanypercentdoeshersavingincrease?a.6%b.7%c.8%d.6%e.Noneofthese10.Inazoo,therearelionsandparrots.Ifcounted,thereare100headsand290legs.Howmanyparrotsarethere?a.45b.50c.55d.60e.Noneofthese

80

11.4Laredrawnfromacontainerfullofmilkandisthenfilledwithwater.Thisoperationisperformedthreemoretimes.Theratioofthequantityofmilkleftinthecontainerandthatofwateris16:65.Howmuchmilkdidthecontainerholdinitially?a.11Lb.13Lc.14Ld.20Le.Noneofthese12.AmixtureworthRs.3.25akgisformedbymixingtwotypesofflour,onecostingRs.3.10perkgwhiletheotherRs.3.60perkg.Inwhatproportionmusttheyhavebeenmixed?a.4:3b.7:3c.2:5d.3:4e.Noneofthese13.Inwhatproportionmustwaterbemixedwithmilksoastogain20%bysellingthemixtureatthecostpriceofthemilk?(Assumethatwaterisfreelyavailable)a.1:5b.5:1c.2:3d.3:2e.Noneofthese14.Prabodhbought30kgofriceattherateofRs.8.50perkgand20kgofriceattherateofRs.9.00perkg.Hemixedthetwo.Atwhatprice(approx)perkgshouldhesellthemixtureinordertoget20%profita.Rs.10.25b.Rs.10c.Rs.10.5d.11e.Noneofthese15.Prabhupurchased30kgofriceattherateofRs.17.50perkgandanother30kgriceatacertainrate.HemixedthetwoandsoldtheentirequantityattherateofRs.18.60perkgandmade20percentoverallprofit.Atwhatpriceperkgdidhepurchasethelotofanother30kgrice?a.Rs.13.25b.Rs.13c.Rs.13.5d.12e.Noneofthese16.Inwhatproportionmustwaterbemixedwithmilksoastogain50%bysellingthemixtureatthecostpriceofthemilk?(Assumethatwaterisfreelyavailable)a.1:2b.2:1c.2:3d.3:2e.Noneofthese17.Amixturecontainsspiritandwaterintheratio3:2.Ifitcontains3litresmorespiritthanwater,thequantityofspiritinthemixtureisa.10Lb.13Lc.14Ld.9Le.Noneofthese18.Avesselcontains50litresmilk.Themilkmandelivers10litrestothefirsthouseandaddsanequalquantityofwater.Hedoesexactlythesameatthesecondandthirdhouse.Whatistheratioofmilkandwaterwhenhehasfinisheddeliveringatthethirdhouse?a.61:64b.64:61c.23:29d.29:23e.Noneofthese19.Severallitresofacidweredrawnofffroma54litresvesselfullofacidandanequalamountofwaterisadded.Againthesamevolumeofthemixturewasdrawnoffandreplacedbywater.Asaresult,thevesselcontained24litresofpureacid.Howmuchacidwasdrawnoffinitially?a.10Lb.13Lc.18Ld.9Le.Noneofthese20.Amixtureofacertainquantityofmilkwith16litresofwaterisworth90paisaperlitre.Ifpuremilkbeworth108paiseperlitre,howmuchmilkinthereisthemixture?a.100Lb.90Lc.80Ld.110Le.Noneofthese

81

21.Amixtureofacertainquantityofmilkwith32litresofwaterisworthRs.1.50perlitre.IfpuremilkbeworthRs.4.50perlitre,howmuchmilkisthereinthemixture?a.10Lb.16Lc.18Ld.9Le.Noneofthese22.Theamountofwheat@Rs.610perquintalwhichshouldbeaddedto126quintalsofwheatcostingRs.285perquintalsothat20%maybegainedbysellingthemixtureatRs.480perquintalwillbea.50kgb.20kgc.30kgd.40kge.Noneofthese23.Threevesselscontainequalmixturesofmilk&waterintheratio6:1,5:2,&3:1respectively.Ifallthesolutionsaremixedtogether,theratioofmilktowaterinthefinalmixturewillbea.19:65b.65:19c.23:29d.29:23e.Noneofthese24.TomixspiritsworthRs.8,Rs.6andRs.3pergallonformakingamixtureworthRs.5pergallon,howmuchofeachquantityistobetaken?a.2:1:2b.1:2:2c.2:2:1d.1:1:2e.Noneofthese25.Twoequalcontainersarefilledwithamixtureofwaterandalcohol.Oneofthemcontainsthreetimesasmuchalcoholastheother.Themixturesinthetwocontainersarethenmixedanditisfoundthattheratioofwatertoalcoholis3:2.Findtheratioofwatertoalcoholineachoftheoriginalcontainersa.2:3,1:4b.3:2.4:1c.2:3,4:1d.3:2,1:4e.Noneofthese26.A100litresolutionofmilkandwatercontainswaterandmilkintheratio1:4.10%ofthesolutionisremovedandreplacedbymilktwice,insuccession.Whatwouldbethequantityofwaterintheresultantsolution?a.16Lb.16.2Lc.15Ld.17Le.Noneofthese27.Acontainercontains240litresofwine.80litresistakenoutofthecontainereverydayandanequalquantityofwaterisputintoit.Findthequantityofthewinethatremainsinthecontainerattheendofthefourthdaya.47.4Lb.46Lc.48Ld.47Le.Noneofthese28.AcancontainsamixtureoftwoliquidsAandBintheratio7:5.When9litresofmixturearedrawnoffandcanisfilledwithB,theratioofAandBbecomes7:9.HowmanylitresofliquidAwascontainedbythecaninitially?a.16Lb.20Lc.21Ld.25Le.Noneofthese29. Analloyofaluminumandleadcontains37%aluminumbyweight.Theweightofaluminumwhichmustbeaddedto400poundsofthisalloytomakethepercentageofaluminum70isa.440poundsb.400poundsc.370pounds d.380poundse.Noneofthese30.Amixtureof20kgsofspiritandwatercontains10%water.Howmuchwatermustbeaddedtothismixturetoraisethepercentageofwaterto25%?

82

a.4kgb.5kgc.7kgd.3kge.Noneofthese31.InwhatratiomusttwokindsofteaworthRs.18andRs.28perkgbemixedsothatbysellingthemixtureatRs.32perkgtheremaybeagainof20%?a.2:15b.15:2c.13:2d.2:13e.Noneofthese32.Agrocermixes26kgofteawhichcostsRs.20akgwith30kgofteawhichcostsRs.36akgandsellsthemixtureatRs.30akg.Hisprofitpercentisa.4%b.6%c.5%d.4.5%e.Noneofthese33.TeaworthRs.126perkgandRs.135perkgaremixedwithathirdvarietyintheratio1:1:2.IfthemixtureisworthRs.153perkg,thepriceofthethirdvarietyperkgwilla.Rs.175.5b.Rs.177.5c.Rs.176.5d.Rs.174.5e.Noneofthese34.Nikitabought30kgofwheatattherateofRs.4.75perkg,40kgofwheatattherateofRs.4.25perkgandmixedthem.ShesoldthemixtureattherateofRs.4.45perkg.Hertotalprofitorlossinthetransactionwasa.Rs.1profitb.Rs.1lossc.Rs.2profitd.Rs.2losse.Noneofthese35.AsumofRs.41wasdividedamong50boysandgirls.Eachboygets90psandeachgirlgets65ps.Findthenumberofboys.a.15b.35c.16d.34e.Noneofthese36.Acontainercontains80litresofwine.8litresistakenoutofthecontainereverydayandanequalquantityofwaterisputintoit.Findthequantityofthewinethatremainsinthecontainerattheendofthe2nddaya.64.8Lb.64Lc.48Ld.60Le.Noneofthese37.Avesselcontains50litresmilk.Themilkmandelivers5litrestothefirsthouseandaddsanequalquantityofwater.Hedoesexactlythesameatthesecondandthirdhouse.Whatistheratioofmilkandwaterwhenhehasfinisheddeliveringatthethirdhouse?a.271:729b.729:271c.64:61d.61:64e.Noneofthese38. VesselsAandBcontainmilkandwaterintheratio4:5and5:1respectively.InwhatproportionshouldquantitiesbetakenfromthevesselsAandBrespectivelytoformamixtureinwhichmilk:waterisintheratio5:4?a.5:3b.3:5c.5:2d.2:5e.Noneofthese39.Twoequaltanksfilledwithmixtureofmilkandroohafzaintheproportionof2:1and1:1respectivelywereemptiedintoathirdtank.Whatistheproportionofroohafzaandmilkinthethirdtank?a.7:3b.3:7c.7:5d.5:7e.Noneofthese40.Twovesselscontainmilkandwatermixedintheratioof2:3and3:4.Findtheratioinwhichthesetwoaretobemixedtogetanewmixtureinwhichtheratioofmilktowateris

83

7:10a.7:10b.10:7c.3:7d.7:3e.Noneofthese41.ApersonhasachemicalofRs.50perlitre.InwhatratioshouldwaterbemixedinthatchemicalsothataftersellingthemixtureatRs.40perlitrehemaygetaprofitof50%?a.7:8b.15:2c.13:2d.8:7e.Noneofthese42.Amerchanthas1000kgofsugar,partofwhichhesellsat8%profitandtherestat18%profit.Hegains14%onthewhole.Thequantitysoldat18%profitis:a.400kgb.360kgc.600kgd.640kge.Noneofthese43.Alumpoftwometalsweighting18gramsisworthRs.87butiftheirweightsbeinterchanged,itwouldbeworthRs.78.60.IfthepriceofonemetalbeRs.6.70pergram,findtheweightoftheothermetalinthemixturea.8gb.6gc.4gd.10ge.Noneofthese44.Intwoalloys,copperandzincarerelatedintheratiosof4:1and1:3.10kgof1stalloy,16kgof2ndalloyandsomeofpurecopperaremeltedtogether.Analloywasobtainedinwhichtheratioofcoppertozincwas3:2.Findtheweightofthenewalloya.32kgb.35kgc.30kgd.40kge.Noneofthese45.Freshfruitcontains72%wateranddryfruitcontained20%water.Howmuchdryfruitfrom100kgoffreshfruitcanbeobtained?a.32kgb.35kgc.30kgd.40kge.Noneofthese46.Amixturecontainsacidandwaterintheratioof1:4andanothercontainsthemintheratio4:5.Ifwewantathirdmixturefromtheaboveoneswitharatio2:7,theproportioninwhichthetwovarietiesshouldbemixedisa.2:15b.15:2c.10:1d.1:10e.Noneofthese47.InwhatratiomustagrocermixtwovarietiesofteaworthRs.60akgandRs.65akgsothatbysellingthemixtureatRs.68.20akghemaygain10%?a.5:4b.4:5c.2:3d.3:2e.Noneofthese48.AcertainproductCismadeoftwoingredientsAandBintheproportionof2:5.ThepriceofAisthreetimesthatofB.TheoverallcostofCisRs.5.20perkgincludinglabourchargesof80paisaperkg.FindthecostofBperkg?a.Rs.2.80b.Rs.1.80c.Rs.3.80d.Rs.0.80e.Noneofthese49.TwovesselsAandBcontainmixtureofspiritandwater.Amixtureof3partsfromAand2partsfromBisfoundtocontain29%ofspiritandamixtureof1partfromAand9partsfromBisfoundtocontain34%ofspirit.FindthepercentageofspiritinAandBa.32,25b.35,25c.30,40d.40,30e.Noneofthese50.Twolumpscomposedofgold,silverandcoppertogetherweight20kg,onelumpcontains

84

gold75%andsilver31.25gramsperkg.Theothercontainsgold85%andsilver30gramsperkg.Thetotalquantityofsilverintwolumpsis617.5grams.Ifthetwolumpsaremeltedandformedintoone,whatpercentofgoldwillitcontain?a.82%b.80%c.74%d.78%e.Noneofthese51.Threevesselswhosecapacitiesareas5:3:2arecompletelyfilledwithmilkmixedwithwater.Theratioofmilkandwaterinthemixtureofvesselsareas3:2,2:1and3:1respectively.Findthepercentageofwaterinthenewmixtureobtainedwhen1/3rdoffirst,1/2ofsecondand2/3rdofthethirdvesselsistakenoutandmixedtogethera.25%b.33.33%c.20%d.30%e.Noneofthese52.HowmanykilogramofsugarcostingRs.9perkgmustbemixedwith27kgofsugarcostingRs.7perkgsothattheremaybeagainof10%bysellingthemixtureatRs.9.24perkg?a.63kgb.62kgc.60kgd.64kge.Noneofthese53.Whatwillbetheratioofpetrolandkeroseneinthefinalsolutionformedbymixingpetrolandkerosenethatarepresentinthreevesselsintheratio4:1,5:2and6:1respectively?a.22:83b.83:22c.10:1d.datainadequatee.Noneofthese54.AmixtureworthRs.3.25akgisformedbymixingtwotypesofflour,onecostingRs.3.10perkgwhiletheotherRs.3.60perkg.Inwhatproportionmusttheyhavebeenmixed?a.2:15b.15:2c.7:3d.3:7e.Noneofthese55.10gallonsaredrawnfromacaskfullofwine.Itisthenfilledwithwater.10gallonsofthemixturearedrawnandthecaskisagainfilledwithwater.Thequantityofwinenowleftinthecasktothatofthewaterinitis16:9.Howmuchdoesthecaskhold?a.50gallonsb.20gallonsc.30gallonsd.40gallonse.Noneofthese56.Theratioofkerosenetopetrolin100kgsofadulteratedpetrolnormallyusedbythreewheelers7:25.Theamountofkerosenetobeaddedto100kgsofadulteratedpetroltomaketheratio9:25isa.6.5kgb.6.25kgc.7kgd.7.25kge.Noneofthese57.Inwhatproportionmustwaterbemixedwithmilktogain20%bysellingitatthecostpricea.2:15b.15:2c.1:5d.5:1e.Noneofthese58.ThecontentsinthebeakersAandBare80cm3ofsugarand80cm3ofsandrespectively.Now20cm3ofsugaristakenoutfromAandputintoB.Afterthroughmixing,20cm3ofthemixtureistakenoutfromBandputintoA.FindthepercentageofsandinthemixtureinAa.25%b.33.33%c.20%d.30%e.Noneofthese59.Avesselisfullofamixtureofkeroseneandpetrolinwhichthereis18%kerosene.Eightlitresaredrawnoffandthenthevesselisfilledwithpetrol.Ifthekeroseneisnow15%,howmuchdoesthevesselhold?a.24Lb.48Lc.36Ld.42Le.Noneofthese

85

60.Twosolutionsof90%and97%purityaremixedresultingin21litresofmixtureof94%purity.Howmuchisthequantityofthefirstsolutionintheresultingmixture?a.10Lb.16Lc.18Ld.9Le.Noneofthese61.Alokbought25kgand35kgofacommodity@Rs.6perkgandRs.7perkgrespectively.Hemixedthetwoquantitiesandsoldthemixture@Rs.6.75perkg.Howmuchistheoverallgain/losstohim?a.10gainb.16lossc.18gaind.9losse.Noneofthese62.Sureshpurchased20kgofteaofonevarietyatRs.30perkgand30kgofteaofanothervarietyatRs.25perkg.Howmuchisthetotalprofit?a.Rs.20b.Rs.25c.Rs.30d.datainadequatee.Noneofthese63.HowmanylitresofRs.16perlitrewineshouldamerchantaddwith25litresofRs.20perlitrewine,sothatheearns25%bysellingthemixtureatRs.22?a.35Lb.32.5Lc.35Ld.37.5Le.Noneofthese64.Twovesselscontainmixturesofmilkandwater.Onecontains80%milkandtheothercontains60%milk.Theproportioninwhichtheyshouldbemixedtogetliquidwith75%milkand25%waterisa.3:1b.1:3c.2:3d.3:2e.Noneofthese65.Goldis19timesasheavyaswaterandcopperis9timesasheavyaswater.Inwhatratioshouldthesebemixedtogetanalloy15timesasheavyaswater?a.3:1b.1:3c.2:3d.3:2e.Noneofthese66.Twokindsofteawereprepared.Inthefirst,30gramsofsugarwasmixedwith190gramsoftea.Inthesecond,40gramsofsugarwasmixedwith270gramsoftea.Whichteawouldbesweeter?Ifthetwokindsofteaaremixedtogether,determinethepercentageofteainthemixturea.Ist,86><

?Cb.2nd,86><

?Cc.Ist,85><

?Cd.2nd,85><

?Ce.Noneofthese

67.Amixtureiscomposedof4partsofbrandyand1partofwater,1kgofwaterisadded,andtheresultingmixture3timesasmuchbrandyaswater.Findthequantityofbrandyintheoriginalmixturea.10kgb.12kgc.16kgd.8kge.Noneofthese68.Adishonestmilkmanmixed1litreofwaterforevery3litresofmilkandthusmadeup36litresofmilk.Ifhenowadds15litresofmilktothemixture,findtheratioofmilkandwaterinthenewmixturea.3:14b.14:3c.12:3d.3:12e.Noneofthese69.AdishonestmilkmanpurchasedmilkatRs.10perlitresandmixed5litresofwaterinit.BysellingthemixtureattherateofRs.10perlitreheearnsaprofitof25%.Thequantityofthemixturethathehadwas?

86

a.40Lb.20Lc.30Ld.10Le.Noneofthese70.Acisterncontains50litresofwater.5litresofwateristakenoutofitandreplacedbywine.Theprocessisrepeatedagain.Findtheproportionofwineandwaterintheresultingmixturea.19:81b.81:19c.81:100d.100:81e.Noneofthese71.Acontainerhasacapacityof20gallonsandisfullofspirit.4gallonsofspiritisdrawnoutandthecontainerisagainfilledwithwater.Thisprocessisrepeated5times.Findouthowmuchspiritisleftintheresultingmixturefinally?a.6C>`

C>Cgallonsb.6C>`

`<?gallonsc.5C>`

`<?gallonsd.6C>C

`<?gallonse.Noneofthese

72.Avesselisfullofrefinedoil.1/4oftherefinedoilistakenoutandthevesselisfilledwithmustardoil.Iftheprocessisrepeated4timesand10litresofrefinedoilisfinallyleftinthevessel,whatisthecapacityofthevessel?a.<?`%

[%litresb.<??_

[$litresc.<?`%

[$litresd.<>`%

[$litrese.Noneofthese

73.InwhatratioshouldtwoqualitiesofcoffeepowderhavingtheratesofRs.47perkgandRs.32perkgbemixedinordertogetamixturethatwouldhavearateofRs.37perkg?a.3:1b.1:3c.2:1d.1:2e.Noneofthese74.AsumofRs.36.90ismadeupof90coinsthatareeither20paisacoinsor50paisacoins.Findouthowmany20paisacoinsarethereinthetotalamount?a.25b.20c.27d.26e.Noneofthese75.Adiscountgrocerprofessestosellpurebutteratcostprice,buthemixesitwithadulteratedfatandtherebygains25%.Findthepercentageofadulteratedfatinthemixturesassumingthatadulteratedfatisfreelyavailablea.25%b.20%c.27%d.26%e.Noneofthese76.Athiefstealsfourgallonsofliquidsoapkeptinatraincompartment’sbathroomfromacontainerthatisfullofliquidsoap.Hethenfillsitwithwatertoavoiddetection.Unabletoresistthetemptationhesteals4gallonsofthemixtureagain,andfillsitwithwater.Whentheliquidsoapischeckedatastationitisfoundthattheratiooftheliquidsoapnowleftinthecontainertothatofthewaterinitis36:13.Whatwastheinitialamountoftheliquidsoapinthecontainerifitisknownthattheliquidsoapisneitherusednoraugmentedbyanybodyelseduringtheentireperiod?a.25gallonsb.20gallonsc.28gallonsd.26gallonse.Noneofthese77.Amixtureof70litresofalcoholandwatercontains10%ofwater.Howmuchwatermustbeaddedtotheabovemixturetomakethewater12.5%oftheresultingmixture?a.2Lb.4Lc.3Ld.1Le.Noneofthese78.Amixtureof20litresofbrandyandwatercontains10%water.Howmuchwatershouldbeaddedtoittoincreasethepercentageofwaterto25%?

87

a.2Lb.4Lc.3Ld.1Le.Noneofthese79.AmerchantpurchasedtwoqualitiesofpulsesattherateofRs.200perquintalandRs.260perquintal.In52quintalsofthesecondquality,howmuchpulseofthefirstqualityshouldbemixedsothatbysellingtheresultingmixtureatRs.300perquintal,hegainsaprofitof25%?a.20qb.22qc.24qd.26qe.Noneofthese80.AmanbuysmilkatRs.8.5perlitreanddilutesitwithwater.Hesellsthemixtureatthesamerateandthusgains11.11%.Findthequantityofwatermixedbyhimineverylitreofmilka.0.111Lb.1Lc.2Ld.0.5Le.Noneofthese81.Therearetwomixturesofhoneyandwater,thequantityofhoneyinthembeing25%and75%ofthemixture.If2gallonsofthefirstaremixedwith3gallonsofthesecond,whatwillbetheratioofhoneytowaterinthenewmixture?a.9:11b.11:9c.2:3d.3:2e.Noneofthese82.Therearetwokindsofalloysoftinandcopper.Thefirstalloycontainstinandcoppersuchthat93:33%ofitistin.Inthesecondalloythereis86.66%tin.Whatweightofthefirstalloyshouldbemixedwith25kgofthesecondalloysoastomakecontaining90%oftin?a.25kgb.20kgc.27kgd.26kge.Noneofthese83.Twocontainersofequalcapacityarefullofamixtureofoilandwater.Inthefirst,theratioofoiltowateris4:7andintheseconditis7:11.Nowboththemixturesaremixedinabiggercontainer.Whatistheresultingratioofoiltowater?a.149:247b.247:149c.243:144d.144:243e.Noneofthese84.Twovesselscontainspiritandwatermixedrespectivelyintheratioof1:3and3:5.Findtheratioinwhichthesearetobemixedtogetanewmixtureinwhichtheratioofspirittowateris1:2a.3:1b.1:3c.2:1d.1:2e.Noneofthese85.ApersonpurchasedacupboardandacotforRs.18,000.Hesoldthecupboardataprofitof20%andthecotataprofitof30%.Ifhistotalprofitwas25%,findthecostpriceofthecupboarda.Rs.9000b.Rs.15000c.Rs.3000d.Rs.5000e.Noneofthese

88

Answers

1.E 2.E 3.C 4.C 5.A 6.B 7.C 8.A 9.B 10.C11.E 12.B 13.A 14.C 15.C 16.A 17.D 18.B 19.C 20.C21.B 22.E 23.B 24.D 25.C 26.B 27.A 28.C 29.A 30.A31.D 32.C 33.A 34.B 35.D 36.A 37.B 38.C 39.C 40.B41.D 42.C 43.A 44.B 45.B 46.C 47.D 48.A 49.B 50.D51.B 52.A 53.D 54.C 55.A 56.B 57.C 58.C 59.B 60.D61.A 62.D 63.D 64.A 65.D 66.A 67.B 68.B 69.B 70.A71.B 72.C 73.D 74.C 75.B 76.C 77.A 78.B 79.D 80.A81.B 82.A 83.A 84.D 85.A

89

Simplification(Algebra)

IMPORTANTCONCEPTSI.'BODMAS'Rule:Thisruledepictsthecorrectsequenceinwhichtheoperationsaretobe executed,soastofindoutthevalueofagivenexpression. Here,'B'standsfor'bracket','O'for'of','D'for'division'and'M'for'multiplication','A' for'addition'and'S'for'subtraction'. Thus,insimplifyinganexpression,firstofallthebracketsmustberemoved,strictlyin theorder(),{}and[].Afterremovingthebrackets,wemustusethefollowingoperationsstrictlyintheorder:(1)of(2)division(3)multiplication(4)addition(5)subtraction.II.Modulusofarealnumber:Modulusofarealnumberaisdefinedas |a|={a,ifa>0 -a,ifa<0 Thus,|5|=5and|-5|=-(-5)=5.III.Virnaculum(orbar):WhenanexpressioncontainsVirnaculum,beforeapplyingthe 'BODMAS'rule,wesimplifytheexpressionundertheVirnaculum.SURDSANDINDICESIMPORTANTFACTSANDFORMULAE1.LAWSOFINDICES:(i)amxan=am+n(ii)am–an=am-n(iii)(am)n=amn(iv)𝑎�G = 𝑎 �G (v)(ab)n=anbn

(vi) y�

9=y

G

�G

(vii)a0=12.SURDS:Letabearationalnumberandnbeapositiveintegersuchthata1/n= 𝑥G isirrational.Then 𝑥G iscalledasurdofordern

Ex.1.𝟔𝟒𝟏𝟐𝟏W

𝟗𝟔𝟒

𝟖𝟏𝟏E

𝟑𝟖

[Bank2000]

90

Sol.

�=A;AW

p�=

�AAE

H�= `>

;W _×$<$$<$×`>

× [×$$[;E C×$$

= `>;W C×C×$$×$$[×[×$$×$$

× [×$$`>ECC

= `>;WCC;[×[×$$×$$

× [×$$`>ECC

= `>ECC (`>WCC)[×[×$$×$$

× [×$$`>ECC

=64 + 3388 =

3188

Ex.2.If𝟐𝒏 = 𝟒𝟓E𝟒𝟓E𝟒𝟓

𝟑𝟓E𝟑𝟓× 𝟔𝟓E𝟔𝟓E𝟔𝟓E𝟔𝟓

𝟐𝟓E𝟐𝟓E𝟐𝟓andn>0,thenthevalueofn2is[SSC2000]

Sol.>BE>BE>B

CBECB× `BE`BE`BE`B

<BE<BE<B= C(>B)

<(CB)× >(`B)C(<B)

>B

CB× <×`B

<B= >B

CB× `B

<=

(>;×>H)`B

CB×<== >H×`B

CB

>H×`×`×`×`×`C×C×C×C×C

4C×2? = 2`×2? = 2$$29 = 2$$So,n=11andn2=121

Ex.3.Simplify𝒂𝟏𝟐E𝒂O

𝟏𝟐

𝟏W𝒂+ (𝟏W𝒂

O𝟏𝟐)𝟏E 𝒂

[Bank2000]Sol.

yA;EyO

A;

$Wy+

$WyOA;

$E y= y

A;EyO

A;

$EyA; $Wy

A;+

$WyOA;

$EyA;

1 − 𝑎 = 1< − 𝑎A;<= 1 + 𝑎

A; 1 − 𝑎W

A;

yA;EyO

A; E $WyO

A; $WyO

A;

$EyA; $Wy

A;

yA;EyO

A;E$WyO

A;Wy

A;E$

$Wy

= <$Wy

Ex.4.Whichofthefollowingequationsareequivalent?[Bank2004]

(i)(𝟏𝟐𝑴 + 𝟑

𝟐𝑵)𝟐(ii)𝟒

𝟗N2+𝟏

𝟒M2+𝟐

𝟑MN(iii) (iv)

Sol.

91

$<𝑀 + C

<𝑁

<= >

_𝑁< + $

>𝑀< + <

C𝑀𝑁

So,(i)and(ii)aresame.$>

$`_𝑁< + 𝑀< + [

C𝑀𝑁

$>𝑀 + >

C𝑁

<----(iv)

Options(i),(ii)and(iv)areequal.Ex.5.Inaclassof63children,thechildrenareseatedinrowsandcolumnssuchthattherearetwochildrenineachcolumnthatthenumberofchildrenseatedineachrow.Howmanychildrenarethereineachrow?[Asst.Grade2005]Sol.Letthenumberofchildreninrow=xThenumberofchildreninthecolumn=(x+2)Accordingtothequestion,X(x+2)=63x2+2x–63=0x2+9x–7x–63=0x(x–9)–7(x–9)=0(x–9)(x–7)=0x=9or-7(Negativevalueisnotpossible)Thereare9childrenineachrow.Ex.6.If 𝒙 − 𝒚 = 𝟑and 𝒙 + 𝒚 = 𝟏𝟗,then 𝒙𝒚 =?[SSC2003]Sol. 𝑥 − 𝑦 = 3-----eq1𝑥 + 𝑦 = 19-----eq2

Addingboththeequations,2 𝒙 = 𝟐𝟐𝑥 = <<

<= 11

Subtractingboththeequations,-2 𝑦 = −16𝑦 = 8

So, 𝑥𝑦 = 11×8 = 88Ex.7.2 C

D×𝑦 $

<= 7 C

>[Bank2002]

Sol.Takingthequotients2,yand72y=7andy=3Substitutethevalueofy,2 C

D×3 $

<= 7 C

>

2 CD=

XH=CA;= 2 C

$>

92

So,comparingtheequations,x=14andy=3Ex.8.AmanhasRs.480inthedenominationsofone-rupeenotes,five-rupeenotesandten-rupeenotes.Thenumberofnotesofeachdenominationisequal.Whatisthetotalnumberofnotesthathehas?Sol.Letnumberofnotesofeachdenominationbex.Thenx+5x+10x=48016x=480x=30.Hence,totalnumberofnotes=3x=90.Ex.9.TherearetwoexaminationsroomsAandB.If10studentsaresentfromAtoB,thenthenumberofstudentsineachroomisthesame.If20candidatesaresentfromBtoA,thenthenumberofstudentsinAisdoublethenumberofstudentsinB.ThenumberofstudentsinroomAis:Sol.LetthenumberofstudentsinroomsAandBbexandyrespectively.Then,x-10=y+10x-y=20....(i)andx+20=2(y-20)x-2y=-60....(ii)Solving(i)and(ii)weget:x=100,y=80.TherequiredanswerA=100.Ex.10.a-b=3anda2+b2=29,findthevalueofab.Sol.2ab=(a2+b2)-(a-b)2=29-9=20ab=10.Ex.11.Inaregularweek,thereare5workingdaysandforeachday,theworkinghoursare8.AmangetsRs.2.40perhourforregularworkandRs.3.20perhoursforovertime.IfheearnsRs.432in4weeks,thenhowmanyhoursdoesheworkfor?Sol.Supposethemanworksovertimeforxhours.Now,workinghoursin4weeks=(5x8x4)=160.

160x2.40+xx3.20=4323.20x=432-384=48x=15.Hence,totalhoursofwork=(160+15)=175Ex.12.Amanhassomehensandcows.Ifthenumberofheadsbe48andthenumberoffeetequals140,thenthenumberofhenswillbe:Sol.Letthenumberofhensbexandthenumberofcowsbey.Then,x+y=48....(i)and2x+4y=140 x+2y=70....(ii)Solving(i)and(ii)weget:x=26,y=22.Therequiredanswer=26.

93

Ex.13.Thepriceof2sareesand4shirtsisRs.1600.Withthesamemoneyonecanbuy1sareeand6shirts.Ifonewantstobuy12shirts,howmuchshallhehavetopay?Sol.LetthepriceofasareeandashirtbeRs.xandRs.yrespectively.Then,2x+4y=1600....(i)andx+6y=1600....(ii)Divideequation(i)by2,wegetthebelowequation.x+2y=800.---(iii)Nowsubtract(iii)from(ii)x+6y=1600(-)x+2y=800----------------4y=800----------------Therefore,y=200.Nowapplyvalueofyin(iii)x+2x200=800x+400=800Thereforex=400Solving(i)and(ii)wegetx=400,y=200.Costof12shirts=Rs.(12x200)=Rs.2400.Ex.14.4/15of5/7ofanumberisgreaterthan4/9of2/5ofthesamenumberby8.Whatishalfofthatnumber?Sol.Letthenumberbex.then >

$?of?

Xofx–>

_of<

?ofx=8or >

<$x– [

>?x=8

or( ><$− [

>?)x=8or`%W?`

C$?x=8or >

C$?x=8

orx=[×C$?>

=630orx/2=315Hencerequirednumber=315.Ex.15.Therationinacampof500menisenoughtolastfor8weeks.Howlongwilltherationlastiftherewere400men?Sol.500menwilleattherationin8weeks.1manwilleattherationin8 500weeks=4000weeks.400menwilleattherationin4000/400=10weeks.Ex.16.If2x+3y+z=55,x–y=4andy–x+z=12,thenwhatarethevaluesofx,yandz?Sol.Thegivenequationsare:2x+3y+z=55…(i);x+z–y=4…(ii);y–x+z=12…(iii)Subtracting(ii)from(i),weget:x+4y=51…(iv)Subtracting(iii)from(i),weget:3x+2y=43…(v)Multiplying(v)by2andsubtracting(iv)fromit,weget:5x=35orx=7.Puttingx=7in(iv),weget:4y=44ory=11.Puttingx=7,y=11in(i),weget:z=8.

94

Ex.17.If4n+4n-1=20,thenthevalueofnnisSol.4x+4x-1=204x=42X=2xX=22=4Ex.18.Ifx+y=1,findthevalueofx3+y3+3xy.Sol.(x+y)3=x3+y3+3xy(x+y)puttingvalue

(1)3=x3+y3+3xy(1)x3+y3+3xy=1.

Ex.19.Divide17intotwopartssothatthedifferencebetweensquaresofnumbersis119.Sol. LetonenumberbexAndsecondnumber=17–xAccordingtothegivencondition(x)2–(17–x)2=119x2–(289+x2–34x)=119x2–289–x2+34x–119=034x–408=034x=408x=>%[

C>=12

Onenumber=12Secondnumber=12–7=5Ex.20.Theproductoftwoconsecutiveoddnumbersis575.Findthesumofthenumbers.Sol. Letfirstoddnumber=nSecondoddnumber=n+2Accordingtocondition,n(n+2)=575n2+2n–575=0n2+25n–23n–575=0n(n+25)–23(n+25)=0(n+25)(n–23)=0n=–25Rejected1stoddnumber=23Secondoddnumber=25.Sum=23+25=4Ex.21.Inagroupofchildren,everychildgivesagifttoeveryother.Ifthenumberofgiftsis272,findthenumberofchildren.Sol.Letnbethenumberofchildrenn(n–1)=272n2–n–272=0n2–17n+16n–272=0n(n–17)+16(n–17)=0

95

(n+16)(n–17)=0n=–16,17Numberofchildren=17Ex.22.Inacricketmatch,WarnetookonewicketmorethantwicethenumberofwicketstakenbyBrettLee.Iftheproductofthenumberofwicketstakenbythesetwois10,findthenumberofwicketstakenbyeach.Sol.LetthenumberofwicketstakenbyBrettLeebexNumberofwicketstakenbyWarne=2x+1Accordingtothequestion,x(2x+1)=102x2+x=102x2+x–10=02x2+5x–4x–10=0x(2x+5)–2(2x+5)=0Eitherx–2=0 x=2Or2x+5=0 x=W?

<(rejected)

NumberofwicketstakenbyBrettLee=2andnumberofwicketstakebyWarne=2(2)+1=5Ex.23.AandBhaveacertainnumberofstamps.AsaidtoB,“Ifyougivemeoneofyourstamps,weshallhaveequalnumberofstamps.”Breplied,“Ifyougivemeoneofyourstamps,Ishallhavetwiceasmanyasyouwillbeleftwith”.FindthetotalnumbersofstampsAandBhave.Sol.LetthenumberofstampsAhadbexThenumberofstampsBhad=yThen,x+1=y–1…(i)Andy+1=2(x–1)…(ii)Solving(i)and(ii),wegetx=5Andy=7Hence,thenumberofstampsAandBhaveis5+7=12.Ex.24.Anumberconsistsoftwodigits,thesumofthedigitsbeing10.If18issubtractedfromthenumber,thedigitsarereversed.Findthenumber.Sol. Letthedigitatunit’splacebeyandthedigitatten’splacebexOriginalnumber=10x+y

Whendigitsarereversedthenumberbecomes=10y+xAccordingtotheconditions,x+y=10…(i)10x+y–18=10y+x…(ii)Equation(ii)reducesto9x–9y=18x–y=2…(iii)Adding(i)&(iii),weget2x=12x=6

96

Putx=6inequation(iii);y=4Originalnumber=64Ex.25.Thedifferenceoftwodigitsofanumberis3.If4timesthenumberisequalto7timesthenumberobtainedbyreversingthedigits,findtheoriginalnumber.Sol.Letthedigitatunit’splacebeyAndthedigitatten’splacebexOriginalnumber=10x+yNumberobtainedbyreversingthedigits=10y+xAccordingtothecondition,x–y=3(i)4(10x+y)=7(10y+x)40x+4y=70y+7x33x=66yx=``

CC𝑦

x=2yPutx=2yin(i)weget2y–y=3y=3Puty=3ineqn(i);x=6Numberis63Ex.26.Abagcontains89coinsof50paisaand25paisa.IfthetotalworthofthesecoinsisRs28.50,findthenumberof25paisacoinsand50paisacoins.Sol.Letnumberof50paisacoinsbexAndnumberof25paisacoinsbeyx+y=89…(i)50x+25y=28502x+y=114…(ii)Solving(i)and(ii),wegetx=35Putvalueofxinequation(i)y=89–25y=64Ex.27.Solvethefollowingequationforx.2x2–14x+14=–10Sol.2x2–14x+14=–102x2–14x+24=02x2–8x–6x+24=02x(x–4)–6(x–4)=0(2x–6)(x–4)=0x=3,4Ex.28.Janehas5dollarsmorethanTom.Ifbothofthemhaveatotalof13dollars,findtheamountwithTom.Sol.LetJ–numberofdollarswithJane

97

T–numberofdollarswithTomJ=T+5andJ+T=13(T+5)+T=132T+5=132T=13–52T=8T=4Ex.29.Findx3+y3+z3–3xyz,ifx+y+z=9andxy+yz+zx=11.Sol.x3+y3+z3–3xyz=(x+y+z)(x2+y2+z2–xy–yz–zx)=(x+y+z)[(x+y+z)2–3(xy+yz+zx)]=9[81–3(11)]=9×[81–33]=9×48=432.Ex.30.Simplify:b–[b–(a+b)–{b–(b–a+b)+2a}]Sol.Givenexpression=b–[b–(a+b)–{b–(b–a+b)+2a}]=b–[b–a–b–{b–2b+a+2a}]=b–[–a–{b–2b+a+2a}]=b–[–a–{–b+3a}]=b–[–a+b–3a]=b–[–4a+b]=b+4a–b=4a.

Ex.31.Evaluate: (𝟐𝟒𝟖 +√(𝟓𝟏 + √(𝟏𝟔𝟗))).

Sol.Givenexpression= (248 +√(51 + 13)= (248 +√64)= (248 + 8)

= (256)=16.Ex.32.If𝒂 + 𝟏

𝒂W𝟐=4,thenthevalueof(𝒂 − 𝟐)𝟐 + 𝟏

(𝒂W𝟐)𝟐

Sol.weneedsquareof(a-2)and1/(a-2)directsquareofbothsidewillnotgivetheresult,weneed(𝑎 − 2) + $

yW<togetthedesireresult,𝑎 + $

yW<=4

so,(𝑎 − 2) + $yW<

=2andnowsquarebothsides,(𝑎 − 2)< + $

(yW<);+2=4

or(𝑎 − 2)< + $(yW<);

=2

Ex.33.Ifxy(x+y)=1Thenthevalueof 𝟏

𝒙𝟑𝒚𝟑− 𝒙𝟑 − 𝒚𝟑

Sol.𝑥𝑦 𝑥 + 𝑦 = 1or 𝑥 + 𝑦 = $D{

98

Sincewehavetoget $DH{H

− 𝑥C − 𝑦Cwhichisintermofx³andY³

so,wehavetotakecubeonboththesides

(𝑥 + 𝑦)C = 1

𝑥C𝑦C

X3+y3+3xy(x+y)= $DH{H

or3xy(x+y)= $DH{H

− 𝑥C − 𝑦C

or,3x1= $DH{H

− 𝑥C − 𝑦C(since𝑥𝑦 𝑥 + 𝑦 = 1)

so, $DH{H

− 𝑥C − 𝑦C = 3(answer)Ex.34.Ifa3−b3−c3=0,thenthevalueofa9−b9−c9−3a3b3c3is,Sol.wheneveryoufindcubeofthreevariablesand3abcor3xyzalwaysthinkabouttheformula,when,x+y+z=0then,x3+y3+z3–3xyz=0herex=a3,y=-b3,z=-c3therefore,a9–b9–c9–3a3b3c3=0Ex.35.If(x+7954×7956)beasquarenumber,thenthevalueof'x'isSol.Theproblemis,(x+7954×7956)wehavetoconvertinasquare,anysumissquareofanumberifwegetitintheabovetwoformsherefocuson7954x7956,theycanbecomeasquareformulax2+y2+2xyorx2+y2-2xyif,eitheryouwrite,7954=7956-2or7956=7954+2let’stake7956=7954+2then,(x+7954×7956)=x+7954x(7954+2)=x+79542+2x1x7954clearlyifyouputx=1,itwillgiveaperfectsquare,1+79542+2x1x7954(1+7954)2therefore,x=1Ex.36.If𝒙

𝒂= 𝟏

𝒂− 𝟏

𝒙,thenthevalueofx–x2is

Sol.youhavetofindx–x2andgiven,Dy= $

y− $

D

youneedxandx²,commondenominatorashouldbeononeside,therefore,Dy− $

y= − $

Dor,DW$

y= − $

D

crossmultiplication,x2–x=-aORx–x2=a

99

Exercise

1.Whensimplified,theproduct 1 −$<

1 −$C

1 −$>… . . 1 − $

9gives

a.$9b. $

9E$c.<

9d. $

9E<e.Noneofthese

2.IfD

{= C

>,thenfindthevalueof{WD

{ED

a.C>b.C

<c.$

Cd.<

?e.Noneofthese

3.Atotalof324coinsof20paiseand25paisemakeasumofRs.71.Thenumberof25paisecoinsisa.125b.123c.124d.130e.Noneofthese4.IfD

{=<

$thevalueofDE{

DW{is

a.2b.1c.3d.4e.Noneofthese5.IfaxbxcmeansyE�

�forallnumbersexcept0,then(axbxc)xaxbisequalto

a.yE�Ey���

b.yE�Wy���

c.yW�Ey���

d.yE�Ey�y�

e.Noneofthese6.Ifa=4.965,b=2.343,c=2.622,findthevalueofa3-b3-c3-3abca.2b.1c.3d.4e.Noneofthese7.Thevalueof$<%+ $

C%+ $

><+ $

?`+ $

X<+ $

_%+ $

$$%+ $

$C<is

a.$Xb.$

[c.<

?d.$

`e.Noneofthese

8.Thesquarerootof %.C><×%.`[>

%.%%%C><×%.%%%$X$is

a.4000b.40000c.400000d.4000000e.Noneofthese9.Simplify:$$

;W?;×<?×<

a.7$`b.7 C

$%c.7 $

$%d.6 $

$%e.Noneofthese

10.Thefluidcontainedinabucketcanfillfourlargebottlesorsevensmallbottles.Afulllargebottleisusedtofillandemptysmallbottle.Whatfractionofthefluidisleftoverinthelargebottlewhenthesmalloneis-full?a.$

Xb.$

[c.C

Xd.$

`e.Noneofthese

11.AmanearnsRs.20onthefirstdayandspendsRs.15onthenextday.HeagainearnsRs.20onthethirddayandspendsRs.15onthefourthday.Ifhecontinuestosavelikethis,howsoonwillhehaveRs.60inhand?

100

a.on17thdayb.on16thdayc.on15thdayd.on18thdaye.Noneofthese12.Afires5shotstoB’s3butAkillsonly1birdinthe3shotswhileBkillsonly1birdin2shots.WhenBhasmissed27times,Ahaskilledhowmanybirds?a.20birdsb.10birdsc.30birdsd.40birdse.Noneofthese13.Afterdistributingthesweetsequallyamong25children,8sweetsremain.Hadthenumberofchildrenbeen28,22sweetswouldhavebeenleftafterequallydistributing.Whatwasthetotalnumberofsweets?a.350b.356c.250d.358e.Noneofthese14.SanketearnstwiceasmuchinthemonthofMarchasineachoftheothermonthsoftheyear.WhatpartofhisentireannualearningwasearnedinMarch?a. <

$Cb. C

$Xc. >

$Cd. <

$Xe.Noneofthese

15.Thevalueof

C <CE `

− > C`E <

+ `<E C

a.2b.0c.3d.4e.Noneofthese16. %.%?;E%.>$;E%.%XC;

%.%%?;E%.%>$;E%.%%XC;is

a.200b.100c.300d.400e.Noneofthese17.Thevalueof(%.CCXE%.$<`)

;W(%.CCXW%.$<`);

%.CCX×%.$<`is

a.2b.1c.3d.4e.Noneofthese18.Thesimplifiedvalueof 900 − 0.09 − 0.000009isa.29.697b.29.197c.29.597d.29.797e.Noneofthese

19.Thesimplifiedvalueof 5 + 11 + 19 + 29 + 49

a.2b.1c.3d.4e.Noneofthese20.Thevalueof99 _?

__×99is

a.9986b.9896c.9976d.9996e.Noneofthese21. $

CW [− $

[W X+ $

XW `− $

`W ?+ $

?W<

a.2b.1c.3d.4e.Noneofthese

101

22.%.$<<<?E%.%<X%.<?W%.$?E%.%_

isequaltoa.0.2b.0.1c.0.7d.0.8e.Noneofthese23.Thesumoftheseries(1+0.6+0.06+.0006+.00006+…)isa.1 <

Cb.2 <

Cc.1 $

Cd.2 $

Ce.Noneofthese

24. %.%%_×%.%C`×%.%$`×%.%[%.%%<×%.%%%[×%.%%%<

isequalto

a.26b.18c.36d.42e.Noneofthese25. >.>$×%.$`

<.$×$.`×%.<$issimplifiedto

a.2b.1c.3d.4e.Noneofthese26.<?`×<?`W$>>×$>>

$$<isequalto

a.200b.100c.300d.400e.Noneofthese

27. CEDE CWDCEDW CWD

=2thenxisequalto

a.$C<b. <

$Cc. ?

$<d.$<

?e.Noneofthese

28.Simplifiedformof 𝑥WHB

BWBH

?

is

a.xb.(x+1)c.(x–1)d.x3e.Noneofthese29.4/15of5/7ofanumberisgreaterthan4/9of2/5ofthesamenumberby8.Whatishalfofthatnumber?a.320b.315c.420d.350e.Noneofthese

30.Simplify: C`

`× H=|}

;p÷

o=|}

;o

?

a.20b.18c.13d.14e.Noneofthese31.If1.5a=0.04b,then�Wy

�Eyisequalto

a.XC

XXb.XC

X[c.X<

XXd.X$

XXe.Noneofthese

32. %.$

;W%.%$;

%.%%%$+ 1 isequalto

a.200b.100c.130d.140e.Noneofthese33. C.%`HW$._[H

C.%`;EC.%`×$._[E$._[;isequalto

102

a.1.8b.1.5c.1.3d.1.08e.Noneofthese34. ?.`<>HE>.CX`H

?.`<>×?.`<>W ?.`<>×>.CX` E>.CX`×>.CX`?

a.20b.10c.13d.14e.Noneofthese35.Giventhat3.718= $

%.<`[_then $

%.%%%CX$[isequalto

a.2679b.2789c.2689d.2537e.Noneofthese36.(53×87+159×21+106×25)isequaltoa.10500b.10550c.10400d.10600e.Noneofthese37. $

$.>+ >

>.X+ $

X.$+ $

$%.$C+ $

$C.$`isequalto

a. ?$`b. <

$Cc. $

$<d. ?

$<e.Noneofthese

38. >

$?𝑜𝑓 ?

[×6 + 15 − 10is

a.2b.1c.3d.4e.Noneofthese39.Whatis1/6thof3?a.0.5b.1.5c.0.33333d.1.333e.Noneofthese40.Multiply0.932by100a.90.1b.92.4c.93.2d.89e.Noneofthese41.Divide0.045by100a.0.45b.0.045c.0.0045d.0.00045e.Noneofthese42.If2x=5and3y=8,then>D

_{isequalto

a. ?$`b. <

$Cc. $

$<d. ?

$<e.Noneofthese

43.Thesumoffirst50positiveintegersis1275.Whatisthesumoftheintegersfrom51to100?a.3770b.3789c.3775d.3540e.Noneofthese44. $

<− $

C+ $

C− $

>+ $

>+ $

<isequalto

a.2b.1c.3d.4e.Noneofthese

45. $.?<.?

<isequalto

a.0.45b.0.36c.0.49d.0.16e.Noneofthese46.Find3+0.3+0.03+0.003a.3.5b.3.666c.3.33333d.3.333e.Noneofthese

103

47.Simplify15.876-(2.49+4.056)÷$

<

a.18<?b.18<

Cc.18$

?d.18$

Ce.Noneofthese

48.Simplify%.>[÷%.$<E%.%>×<?

%.%?

a.200b.100c.130d.140e.Noneofthese49.Simplify<C

$_× $_$X×85

a.115b.100c.125d.120e.Noneofthese50.FindPintheexpression,if !

$E A

A� ~AO~

= 1

a.2b.1c.3d.4e.Noneofthese

51.SimplifyH;÷

A;×

H;

H;÷

A;|}

H;÷ $

[

a.20b.18c.13d.24e.Noneofthese

52.Findthevalueof(CECECEC)÷C?E?E?E?÷?

a.C

>b.?

>c.$

>d.X

>e.Noneofthese

53.Findthevalueof?E?×$_W$?WX

$C×$CW$?`

a.2b.1c.3d.4e.Noneofthese54.Simplify1÷[1+1÷{1+1÷(1÷1)}]a.$

>b.<

Cc.$

Cd.<

?e.Noneofthese

55.Simplify$>;B+ $

<=p

a. X

$$b. X

$Cc.X

_d. X

$?e.Noneofthese

56.If217×15=3255,then2.17×0.15isa.1.3255b.0.3255c.0.03255d.0.003255e.Noneofthese57.24.315×256.2×0.00019isthesameasa.243.15×2.562×0.019b.243.15×2.562c.2430.15×2.562×0.019d.243.15×2.562×0.00019e.Noneofthese

104

58.If13+23+33+…..+93=2025,thenfindthevalueof(0.1)3+(0.2)3+…..+(0.9)3a.2.0025b.2.025c.20.025d.2.00025e.Noneofthese59.If12+22+32+….+102=385,thenfindthevalueof(0.11)2+(0.22)2+….+(0.99)2a.4.6585b.4.06585c.4.006585d.4.0006585e.Noneofthese60.Ifx#y=x+y,thenfindthevalueof(3#4)#3a.20b.10c.30d.40e.Noneofthese61.Ifa+bbedefinedbytherelationy

;

�,whereb≠0.Findthevalueof(24+16)+4

a.322b.314c.236d.324e.Noneofthese62.If $

%.`<?=1.6.Findthevalueof $

%.%%%`<?

a.150b.170c.160d.180e.Noneofthese63.Ifx=2yandy=2z/3,whatisthevalueofzintermsofx?a.CD

?b.CD

>c.?D

>d.CD

Xe.Noneofthese

64.%.[C

�

%.[Coisapproximatelyequalto

a.2b.1c.3d.4e.Noneofthese65.Ifxispositiveand6 −𝑥< = $?

$`, 𝑡ℎ𝑒𝑛 𝑥 =?

a.C>b.C

<c.$

Cd.<

?e.Noneofthese

66.Findthevalueof$?

$E ?<

− $W ?<

a.2b.1c.3d.4e.Noneofthese67.$[

`%(0.1254)isequalto

a.0.3762b.0.03762c.0.3765d.0.003762e.Noneofthese68.Simplify[1-2(3-4)-1]-1a.4b.3c.1d.2e.Noneofthese69.Simplify $

<+ $

<C>− $

<X[− C

>

a.<XC<b.<?

CCc.<X

CCd.<?

C<e.Noneofthese

105

70.Simplify1 + $<E A

H�AH

a.CC<?b.C?

<Cc.CC

<Cd.C?

<?e.Noneofthese

71.Findx,if6 − 5 − 𝑥 − 2 − C

<= 3

a.C>b.C

<c.$

Cd.<

?e.Noneofthese

72.Simplify0.00175÷0.025÷0.07a.2b.1c.3d.4e.Noneofthese73.Whatleastfractionshouldbeaddedto $

<×C+ $

C×>+ $

>×?+ ⋯+ $

<$×<<,sothattheresultis

equalto1a.C

>b.C

<c.$

Cd.<

?e.Noneofthese

Answers

1.A 2.E 3.C 4.C 5.A 6.E 7.D 8.D 9.C 10.C11.A 12.C 13.D 14.A 15.B 16.B 17.D 18.A 19.C 20.B21.E 22.D 23.A 24.C 25.B 26.D 27.D 28.A 29.B 30.B31.A 32.B 33.D 34.B 35.C 36.D 37.A 38.E 39.B 40.C41.D 42.D 43.C 44.B 45.B 46.D 47.B 48.B 49.A 50.B51.B 52.C 53.E 54.B 55.A 56.B 57.E 58.B 59.A 60.B61.D 62.C 63.B 64.B 65.B 66.B 67.B 68.C 69.A 70.C71.E 72.B 73.E

106

GeometryAngles:-Anangleisafigureformedbytworayswithacommoninitialpoint,sayO.ThispointiscalledthevertexTypesofAngles:-1)Arightangleisanangleof900.e.g.AngleAOB

2)Ifanangleislessthan900,itiscalledacute.3)Ifanangleisgreaterthan900butlessthan1800,itiscalledobtuse.4)Ifanangleisof1800,itiscalledastraightangle,ananglegreaterthan1800butlessthan3600iscalledareflexangle.5)Twoangleswhosesumis1800arecalledsupplementaryangles,eachoneisasupplementoftheother.6)Twoangleswhosesumis900arecalledcomplementaryangles,eachoneisacomplementoftheother.7)Twoadjacentangleswhosesumis1800aretheanglesofalinearpair.AnglesandIntersectinglines:Whentwolinesintersect,twopairsofverticallyoppositeanglesareformed.Verticallyoppositeanglesareequal.Thus,∠𝑐&∠𝑑areequal.∠a&∠bareequal.Also,sumofalltheanglesatapoint=3600i.e.

AnglesandParallelLines:Ifatransversal(cuttingline)cutstwoparallellinescorrespondinganglesareequali.e.∠a=∠e,∠d=∠f,∠b=∠h,∠c=∠g.Alternateanglesareequal

107

i.e.∠c=∠f,∠d=∠e.Interioranglesonthesamesideofthetransversalaresupplementary,i.e.∠c+∠e=∠d+∠f=180°

Triangles:-Atriangleisapolygonwiththreeverticesandthreesides(edges).Ithasthreeinternalangles.Whenweaddalltheinternalanglestogether,wewilldefinitelyget180°.1.Acutetriangle:Thetrianglewhichhasallacuteangles(i.elessthan900).2.Obtusetriangle:Thetrianglewhichhasoneobtuseangle(i.egreaterthat900)3.Rightangletriangle:Thetrianglewhichhasonerightanglei.e900.4.Scalenetriangle:Ascalenetriangleisatrianglethathasnoequalsides5.Isoscelestriangle:Anisoscelestriangleisatrianglethathastwoequalsides.6.Equilateraltriangle:Thetriangleofwhichallsidesareequalisknownasequilateraltriangle&allanglesareequalto600.PropertiesofTriangles1.Sumoftheanglesofatriangleare180°(anglesofatrianglearesupplementary)

2.Theexteriorangleofatriangleisequaltothesumoftheinterioroppositeangles.

108

Here,∠1=∠2+∠33.Anglesoppositetotwoequalsidesofatriangleareequal&viceversa.

Here,∠1=∠24.Iftwosidesofatriangleareunequalthenthegreaterangleisoppositetogreaterside&viceversa

5.Twotrianglesarecongruentiftwoangles&includedsideofonetriangleisequaltothecorrespondingtwoangles&includedsideoftheothertriangles.

∠1=∠3&∠2=∠4,BC=EF

or∆ABC≅ ∆DEF

6.Iftwosides&includedangleofatriangleareequaltocorrespondingtwosides&includedangleofanothertrianglethenthetwotrianglesarecongruent.

109

AB=DE,AC=DF,∠1=∠2∆ABC≅ ∆DEF7.Iftwoangles&nonincludedsideofonetriangleareequaltocorrespondingtwoangles&nonincludedsideofanothertrianglethenthetwotrianglesarecongruent.

8.Ifthreesidesofatriangleareequaltothreesidesofanothertriangleeachtoeachthenthetrianglesarecongruent

AB=DE,AC=DF,BC=EF9.Trianglesonthesamebase&betweenthesameparallelsareequalinArea

Here,areaofΔABC=areaofΔBDCifADisparalleltoBC10.Tworighttrianglesarecongruentifthehypotenuse&onesideofonetrianglearerespectivelyequaltohypotenuse&onesideoftheothertriangle.

110

AB=DE,AC=DFor∆ABC≅ ∆DEF

11.Sumofanytwosidesofatriangleisgreaterthanthethird.SimilarTriangles1.Ifalineisdrawnparalleltoonesideofatriangletheothertwosidesaredividedproportionally&viceversa

ifDE||BCso¸¹

¹º= ¸»

»¼

ifso¸¹

¹º= ¸»

»¼thenDE||BC

2.Iftwotrianglesareequiangular(i.e.thecorrespondinganglesareequal)thenthetrianglesaresimilar&hencetheirsidesareproportional

1 4,∠ =∠ 2 5,∠ =∠ 3 6,∠ =∠

111

𝐴𝐵𝐷𝐸 =

𝐵𝐶𝐸𝐹 =

𝐴𝐶𝐷𝐹

3.Ifthecorrespondingsidesoftwotrianglesareproportionalthenthetrianglesareequiangular/similar.4.Ratioofareasoftwosimilartrianglesisequaltotheratioofthesquaresoftwocorrespondingsides.5.Inarighttrianglethesquareofthehypotenuseisequaltothesumofthesquaresontheothertwosides

6.Inthegivenfigure(obtuseangledtriangle)

𝐴𝐶<=𝐴𝐵< +𝐵𝐶<+2AB.BDif∠B>90°7.Inthefigure(Acuteanglestriangle)

𝐴𝐶<=𝐴𝐵<+𝐵𝐷<-2BC.BDif∠B<90°

112

Areaoftriangle1)Whenlengthsofthesidesaregiven:-

Area= 𝑠 𝑠 − 𝑎 𝑠 − 𝑏 (𝑠 − 𝑐)where,semiperimeter(s)=yE�E�

<

2)Whenlengthsofthebaseandaltitude(height)aregiven:-

Area= 𝑏ℎ<

$ 3)Whenlengthsoftwosidesandtheincludedanglearegiven:-

Area= 𝑎𝑏 sin 𝜃<

$

113

4)ForEquilateralTriangle:-

Area=C>𝑎<

5)ForIsoscelesTriangle

Area=�>× 4𝑎< −𝑏<

6)WhenthreemedianaregivenTheareaofatrianglecanbeexpressedintermsofthemediansby:A=>

C 𝑆(𝑆 − 𝑀$)(𝑆 − 𝑀<)(𝑆 − 𝑀C)

Where,

S=ÉA�É;�ÉH

<

114

ApolloniusTheorem

IfADisthemedian,then:AB²+AC²=2(AD²+BD²)AngleBisectorTheorem

IfADistheanglebisectorforangleA,then:-¸ºº¹= ¸¼

¼¹

Inradiusandcircumradiusoftriangle#Inradius

IncaseofEqualilateraltriangle=𝒔𝒊𝒅𝒆𝟐 𝟑

Incaseofrightangletriangle=

𝒔𝒖𝒎𝒐𝒇𝒑𝒓𝒆𝒑𝒆𝒏𝒅𝒊𝒄𝒖𝒍𝒂𝒓𝒔𝒊𝒅𝒆𝒔W𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆𝟐

Incaseofothertriangles= 𝟐×𝑨𝒓𝒆𝒂

𝒔𝒖𝒎𝒐𝒇𝒂𝒍𝒍𝒔𝒊𝒅𝒆𝒔

115

#Circumradius

Incaseofequilateraltriangle=𝒔𝒊𝒅𝒆𝟑

Incaseofrightangletriangle=𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

𝟐

Incaseofothertriangles=

𝒑𝒓𝒐𝒅𝒖𝒄𝒕𝒐𝒇𝒔𝒊𝒅𝒆𝒔𝟒×𝒂𝒓𝒆𝒂

116

Circles:-

Circleillustrationwithcircumference(C),diameter(D),radius(R),andcentreororigin(O)

• Arc:anyconnectedpartofthecircle.• Centre:thepointequidistantfromthepointsonthecircle.• Chord:alinesegmentwhoseendpointslieonthecircle.• Circularsector:aregionboundedbytworadiiandanarclyingbetweentheradii.• Circularsegment:aregion,notcontainingthecentre,boundedbyachordandanarc

lyingbetweenthechord'sendpoints.• Circumference:thelengthofonecircuitalongthecircle.• Diameter:alinesegmentwhoseendpointslieonthecircleandwhichpassesthrough

thecentre;orthelengthofsuchalinesegment,whichisthelargestdistancebetweenanytwopointsonthecircle.Itisaspecialcaseofachord,namelythelongestchord,anditistwicetheradius.

• Radius:alinesegmentjoiningthecentreofthecircletoanypointonthecircleitself;orthelengthofsuchasegment,whichishalfadiameter.

• Secant:anextendedchord,acoplanarstraightlinecuttingthecircleattwopoints.• Semicircle:aregionboundedbyadiameterandanarclyingbetweenthediameter's

endpoints.Itisaspecialcaseofacircularsegment,namelythelargestone.• Tangent:astraightlinethattouchestheboundaryofcircleatasinglepoint.

117

Importantformulas

Circumference=2𝜋𝑟

Area=𝜋𝑟²

Areaofsemicircle=ÚT²<

Circumferenceofsemicircle=𝜋𝑟 + 2𝑟

Lengthofarc(𝒍)=2𝜋𝑟 ÛC`%°

Areaofsector=𝜋𝑟< ÛC`%°

Propertiesofcircle

1.Theperpendicularfromthecentreofacircletoachordbisectsthechord.

Oiscentre,ABischord&OPisperpendiculartoAB,AP=PB2.Perpendicularbisectorsoftwochordsofacirclepassesthroughitscentre(i.e.intersectatcentre).3.Iftwochordsofacircledrawnfromthesamepointareequalthenthelinebisectingtheanglebetweenthempassesthroughcentre(oristhediameter)

118

AB=BC,∠1=∠2thenBPisdiameter,centreliesonBP4.Equalchordsofacircleareequidistantfromthecentre&viceversa.

AB=CDorOP=OQ(Oisthecentreofthecircle)5.Anglesinthesamesegmentofacircleareequal

∠1=∠2(beinginthesamesegment)6.Angleinasemicircleisarightangle

ABisthediameter∠1=∠2=9007.Anglewhichanarcsubtendsatthecentreisdoubletheanglesubtendedbythesamearcatanyotherpartofthecircumference.

119

ArcABsubtends∠1atcentre&∠2&∠3atcircumference 1 2 2 2 3⇒∠ = ∠ = ∠ 8.Fromtheabovewecometoknowthat∠2=∠3henceanglesinthesamesegmentofacircleareequal.9.Equalchordsofacirclesubtendequalanglesatthecentre

ChordAB=ChordCD(Oisthecentreofthecircle)10.Oppositeanglesofacyclicquadrilateralaresupplementary

ABCDiscyclicquadrilateralthen∠1+∠2=∠3+∠4=180011.Atangenttoacircleisperpendiculartotheradiusthroughthepointofcontact

ABistangenttothecircleatPwithcentreO

120

12.Thelengthoftwotangentsfromanexternalpointareequal

OA=OB13.Iftwochordsofacircleintersectinsideoroutsidethecirclewhenproduced,therectangleformedbythetwosegmentsofonechordisequalinareatotherectangleformedbythetwosegmentsoftheother.

CirclewithcentreOchordsAB&CDintersectatP(fig.1inside&fig.2outside)

ð PA.PB=PC.PD&ifPTistangentthenPT2=PA.PB14.Ifachordisdrawnthroughthepointofcontactofatangenttoacircle,thentheangleswhichthischordmakeswiththegiventangentareequalrespectivelytotheanglesformedinthecorrespondingalternatesegments.

PQistangenttocirclethroughpt.A&ABisachord&∠1=∠2&∠3=∠4

121

CommonTangentToAPairOfCircles

Commontangentsarelinesorsegmentsthataretangenttomorethanonecircleatthesametime.Thepossibilityofcommontangentsiscloselylinkedtothemutualpositionofcircles.1.Iftwocirclestouchinside,thetwointernaltangentsvanishandthetwoexternalonesbecomeasingletangent.

2.Iftwocirclesintersect,thecommontangentisreplacedbyacommonsecant,whencethereareonlytwoexternaltangents.

3.Iftwocirclestoucheachotheroutside,thetwointernaltangentscoincideinacommontangent,thustherearethreecommontangents.

4.Iftwocirclesareseparate,therearefourcommontangents,twoinsideandtwooutside.

Lengthofcommontangent.Ifr1,r2aretheradiioftwocirclesanddisdistancebetweentheircenters,then

(i)thelengthofadirectcommontangent= 𝑑< − (𝑟$< − 𝑟<<)(ii)thelengthofatransversecommontangent= 𝑑< − (𝑟$< + 𝑟<<)Lengthofthetangentoftwocircleswhichtoucheachotheratanexternalpointwithradiusr1&r2,thenthelengthofdirectcommontangent= 𝟒𝒓𝟏𝒓𝟐

122

Quadrilaterals:-

Quadrilateral Shape Properties ImportantResultsSquare

1.Allsidesareequal2.allanglesare90°3. Diagonals are equaland Bisect each other at90°

Area=a²=$<×D²

(D=diagonal)Perimeter=4aDiagonal= 2𝑎

Rectangle

1.Oppositesidesareequalandparallel2.Allanglesare90°3.Diagonalsareequalandbisecteachother

Area=𝑙×𝑏Perimeter=2𝑙 + 2𝑏Diagonal= 𝑙< + 𝑏<

Rhombus

1.Allsidesareequalandoppositesidesareparallel2.Oppositeanglesareequal3.Diagonalsarenotequal4.Diagonalsbisecteachotherat90°

Area=$<×𝑑$𝑑<

Perimeter=sumofallsides

side²=2 2

1 2d d2 2

⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Parallelogram

1.oppositesidesareparallelandequal2.Oppositeanglesareequal3.Diagonalsofparallelogrambisecteachother

Area=base×heightOrb×hPerimeter=sumofallsides

Trapezium

1.onlyonepairofoppositesidesareparallel2.Thediagonalscutthequadrilateralintofourtrianglesofwhichoneoppositepairaresimilar3.Thediagonalscuteachotherinmutuallythesameratio(thisratioisthesameasthatbetweenthelengthsoftheparallelsides).

Area=𝟏𝟐(sumofparallelsides)×

(height)

Iflengthsofonediagonalandtwooffsetsaregivenofanyquadrilateral

123

Area=$<𝑑(ℎ$ +ℎ<)

Iflengthsoftwodiagonalsandtheincludedanglearegivenofanyquadrilateral

Area=$<𝑑$𝑑< sin 𝜃

Solids:-

124

Figure Shape Volume CSA/LSA Totalsurfacearea

Cube 𝑎C 4𝑎< 6𝑎²

Cuboids

𝑙𝑏ℎ 2 𝑙ℎ + 𝑏ℎ 2(𝑙ℎ + 𝑏ℎ + 𝑙𝑏)

Cylinder

𝜋𝑟²ℎ 2𝜋𝑟ℎ 2𝜋𝑟 ℎ + 𝑟

Cone 13𝜋𝑟

<ℎ 𝜋𝑟𝑙𝑙 = ℎ< + 𝑟<

𝜋𝑟(𝑙 + 𝑟)

Sphere 43𝜋𝑟

C 4πr< 4πr<

Hemisphere

23𝜋𝑟

C 2𝜋𝑟< 3𝜋𝑟<

125

Rightprism

𝑎𝑟𝑒𝑎𝑜𝑓𝑏𝑎𝑠𝑒×ℎ𝑒𝑖𝑔ℎ𝑡

𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑒𝑜𝑓𝑏𝑎𝑠𝑒×ℎ𝑒𝑖𝑔ℎ𝑡

𝐿𝑆𝐴 + 2×𝑎𝑟𝑒𝑎𝑜𝑓𝑏𝑎𝑠𝑒

Rightpyramid

13𝑎𝑟𝑒𝑎𝑜𝑓𝑏𝑎𝑠𝑒

×ℎ𝑒𝑖𝑔ℎ𝑡

12𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑒𝑜𝑓𝑏𝑎𝑠𝑒

×𝑠𝑙𝑎𝑛𝑡ℎ𝑒𝑖𝑔ℎ𝑡

𝐿𝑆𝐴 + 𝑎𝑟𝑒𝑎𝑜𝑓𝑏𝑎𝑠𝑒

126

Polygon:-Aclosedplanefiguremadeupofseverallinesegmentsthatarejoinedtogether.Thesidesdonotcrosseachother.Exactlytwosidesmeetateveryvertex.

Side-oneofthelinesegmentsthatmakeupthepolygon.Vertex-pointwheretwosidesmeet.Twoormoreofthesepointsarecalledvertices.Diagonal-alineconnectingtwoverticesthatisn'taside.InteriorAngle-Angleformedbytwoadjacentsidesinsidethepolygon.ExteriorAngle-Angleformedbytwoadjacentsidesoutsidethepolygon.Sumofinteriorangleandexteriorangleofanypolygonisequalto180°TypesofPolygonsRegular-allanglesareequalandallsidesarethesamelength.Regularpolygonsarebothequiangularandequilateral.Equiangular-allanglesareequal.Equilateral-allsidesarethesamelength.

Convex-astraightlinedrawnthroughaconvexpolygoncrossesatmosttwosides.Everyinteriorangleislessthan180°.

Concave-youcandrawatleastonestraightlinethroughaconcavepolygonthatcrossesmorethantwosides.Atleastoneinteriorangleismorethan180°.

127

Numberofsides Name Numberofsides NameN N–Gon 6 Hexagon3 Triangle 7 Heptagon4 Quadrilateral 8 Octagon5 Pentagon 10 Decagon

Exteriorangleofanypolygon=C`%9

Where𝑛 =Numberofsides

Interiorangleofanypolygon= 180 − C`%9

𝑤ℎ𝑒𝑟𝑒𝑛 =NumberofsidesNumberofdiagonals𝑛(𝑛 − 3)

2

𝑛 =NumberofsidesNumberoftrianglesinanypolygon𝑛 𝑛 − 4 𝑛 − 5

6

𝑛 =Numberofsides

128

Example1:-In the given figure, if AB || CD, find the measure of angle BGF.

Solution:-ItisgivenAB||CD70°=30°+∠ECD∠ECD=40°∠ECD+∠CEF=140°+40°=180°So,EF||CD∠EFG=∠FGBExample2:-In Δ ABC, ∠A = 40o. D is a point on AC such that BD = BC. If ∠ABD = 30o, find ∠ABC. Solution:-

∠BCD=∠BDC=θ(ΔDBCisisosceles)110°=𝜃 + ∅(exteriorangle)2𝜃 + ∅=180°(ΔCDB>Anglesumproperty)or𝜃 + ∅ + 𝜃=180°orθ=180°–110°=70°∅=110°–70°=40°∠ABC=30°+∅=30°+40°=70°.Example3:-Inthefollowingfigure,OAbisects∠A,∠ABO=∠OCAand∠BOC=100o.Findthemeasureof∠AOB.

129

Solution:-

IntriangleAOBandAOC,twoanglesarecorrespondinglyequal.⇒∠AOB=∠AOC=zNow,∠AOB+∠AOC+∠BOC=360°z+z+100o=360o

orz=130o

Example4:-A1,A2,A3,…,A15areequallyspacedpointsontheboundaryofacircle.WhatisthesizeofangleA1A3A7?

Solution:-SincethepointsA1,A2,A3,…,A15areevenlyspaced,theygenerateequalanglesatO,eachofmeasureC`%°

$?=24°

Now,joinA1,A3andA7toO,thecentreofthecircle,asshown.Thus,

130

thesizeofangleA1A3A7=(7-1)x24o=144oExample5:-Inthegivenfigure,∠A=36o,∠C=30o,∠D=45oand∠CED=xo.Findx.

Solution:-ΔABC;∠CBA=180°–(30°+36°)=180°–66°=114°∠EBD=180°–∠EBA=180°–114°=66°x°=66°+45°=111°(exteriorangle)Example6:-SupposethatAB=AC=CDandAD=BD.Whichofthefollowingoptionsisthemeasureof∠ABCindegrees?

Solution:-IntriangleABC,AB=AC,∠ABC=∠ACB=xIntriangleACD,AC=CD,∠ADC=∠CAD=yIntriangleABD,AD=BD,∠A=∠B=xSo,∠BAC=x–yweget3x-y=180°andfromtriangleACDweget,2x+y=180°solving,∠ABC=x=72°

131

Example7:-Inthegivenfigure,ifAB=NBandAC=CM,findthevaluesofx,yandz.

Solution:-

∠BNA=∠BAN=z40°=2zorz=20°y=180°–(50°+40°)=180°–90°=90°∠CMA=∠CAM=x50°=2xX=25°Example8:-Inthetrianglebelow,anglePORis60o.AssumethatthelinesegmentsOQandQRhavethesamelength,andlinesegmentsOPandPQhavethesamelength.FindtheangleQRO.

Solution:-

132

InΔOPQ,OP=PQ∠PQO=∠POQ=ySimilarlyinΔOQR,OQ=QR∠QRO=∠QOR=xy=2x(sumofinterioranglesisequaltoexteriorangle)Also,y+x=60o(Given∠POR=60o)3x=60oorx=20oExample9:-InΔABC,DE||BCanditdividesthetriangleintotwoequalparts.TheratioofareaΔADEtoareaΔABCis

Solution:-Area∆ADE=Areaquad.∆DBEC⇒Area∆ABC=2Area∆ADE

Area∆ADE/Area∆ABC=2Example10:-IntriangleAEF,CDisparalleltoEF,AD=DF,CD=4andDF=3units.WhatisthelengthofEF?

Solution:-InΔADCandΔAFE∠A=∠A{Common}AD=DF{given}∠ADC=∠AFE{correspondinganglesbetweentwo||lines}HenceΔABCissimilartoΔAFE.

Now,¸¹¸à=¼¹

ȈandweknowAF=AD+DF=3+3=6units

133

EF=`×>C

=8units Example11:- Inthegivenfigure,DF||AG,DE||AB,AB=15,CD=8,AD=a,DE=10,FG=bandCG=6.Theratioa:bisequalto

Solution:-

In ΔABC and ΔDEC

[Ey$?

= [$%

(By similar triangles).

⇒ a = 4 In ΔACG and ΔDCF [Ey[

= ``W�

(By similar triangle)

⇒ b = 2 ∴ a : b = 4 : 2 = 2 : 1 Example 12:- In the given figure, EF || CD, DE || BC, AF = 2FD and DE = 4 units. What is the measure of BC?

Solution:-ΔAFE~ΔADCAF/AD=AE/AC

134

⇒AF/(AF+FD)=AE/AC⇒2FD/3FD=AE/AC⇒2/3=AE/ACΔADE~ΔABC⇒DE/BC=AE/AC=2/3⇒BC=C

<×DE=C

<×4=6units

Example13:-GivenΔABCwithrightangleCandasecondrighttriangleABDsuchthatbothtrianglessharethesamehypotenuse.IfBC=1,AC=bandAD=2,thenBD=

Solution:-

InΔABCsincetherightangleisC,thehypotenuseisAB.SinceBC=1andAC=b,wegetthehypotenuseABas 𝑏< + 1 .SincethisisthehypotenuseoftheΔABDaswell,andsinceAD=2,wegetBD 𝑏< + 1 − 2<= 𝑏< − 3.

Example14:-MediansQXandRYofΔPQRareperpendicular,QX=8andRY=12.TheareaofΔPQRis

Solution:-ThemediansmeetatG.thenRG=(2/3)RY=8andtheareaof∆QRXis(1/2)xQXxRG=(1/2)x8x8=32.SinceQXisamedian,trianglesPQXandXQRhavethesamearea.HencetheareaofΔPQR=64.Example15:-Inthegivenfigure,therearethreecirclesofradius2cmtouchingeachother.ThetriangleABCisanequilateraltriangleofside4cm.Thelengthofarcs(MP+PN+NM)is

135

Solution:-Asthecirclesarecongruent(havingsameradius) AndangleA=angleB=angleC=60°each Arc(MP)=arc(PN)=arc(MN)= Û

C`%×2𝜋𝑟

Lengthsofarcs(MP+PN+NM)=3× `%C`%

×2𝜋𝑟=2πcmExample16:-Inthegivendiagram,ACBisanarcofacircleandCDistheperpendicularbisectorofthechordAB.IfAD=3andCD=9,theareaoftheentirecircleis

Solution:-Here,ACBisanisoscelesΔwithAC=CB= 90inscribedinacircle.

ADCisarightangle= 𝐴𝐷< +𝐷𝐶<=AC3< +9<=AC=BC

Now,inanyΔ,R=y��

>á(R=circumradius)

Here,a=b=AC=ABR= y;�

>×â= >y;W�; _%

>×_%W`;=90/18=5

R=5unitsandArea=π(5)²=25πsq.unitsExample17:-ABCisatriangleinwhich∠A=60o,AC=4cmandAB=3cm.IfBDistheperpendicularfromBtothesideAC,whatisthelengthofBD?Solution:-Area∆ABC=½×AC×AB×sinA=½×base×altitude∴½×3×4×sin60=½×AC×BD

136

i.e.12× C<=4×BD

⇒BD=1.5 3cm.Example18:-AC:AD::2:5andareaoftriangleAODis8 5squareunits.FindCDintermsofr,whereABistangenttothecirclewithcentreO.

Solution:-AC=2𝑥,AD=5𝑥

CD=AD–AC=3𝑥

AB2=AC×AD=2𝑥×5𝑥=10𝑥2

AB= 10𝑥AreaofΔAOD=½×AB×OB

8 5=$<× 10𝑥×r

⇒𝑥=[ <T

CD=3𝑥=3×[ <T=<> <

T

Example19:-In the given figure, O is the center of the circle and AE is a diameter. If AB = BC and ∠BFC = 25°, find the value of ∠ABC.

Solution:-anglessubtendedbyequalchordsareequal∠AFB=25°=∠BFCincyclicquadrilateralABCF∠AFC=25°+25°=50°

137

∠ABC=180°-50°=130°(becausesumofoppositeanglesofacyclicquadrilateralis1800)

Example20:-Atangenttoacircleofradius3cmfromanexternalpointPisoflength4cm.FindtheshortestdistanceofPfromapointonthecircumferenceofthecircle.Solution:-

In triangle OAP, OP2 = 32 + 42 or OP = 5 cm Also OP = OM + MP So, MP = 5 – 3 = 2 cm. Example21:-PMandPNaretangentstothecirclewithcentreQandradius7cm.ThelengthofPMis7cm.WhatisthelengthofPQ?

Solution:-PQ2=PM2+MQ2(ΔPQMisrightangledtriangle)=72+72=49+49orPQ=7√2cmExample22:-IfAB=30,PM=8andDC=16andPisthecentreofcirclethenfindthelengthofPN?

Solution:-AnyperpendicularfromcentretochordbisectthecordSo,AM=15,PM=8IntriangleAPM,AP2=PM2+AM2

So,AP=17=CP

138

IntriangleCPN,CP=17,CN=8So,byPythagorastheoremwewillhavePN=15Example23:-IntriangleABC,∠ABC=90o,DliesonACsuchthatBCDisanequilateraltriangle.IfBChaslength1,findthelengthofAB.Solution:-SinceΔBCDisanequilateraltriangle,themeasureof∠BCD=60oand∠ABD=30o.ABDisanisoscelestrianglewithAD=BD=1andAC=2Now,byPythagorastheorem,AB= 3Example24:-InatriangleABC,AB=6,BC=8andAC=10.AperpendicularfromBmeetsthesideACatD.AcircleisdrawnwithradiusBDandcentreB.IfthecirclecutsABandBCatPandQ,respectively,AP:QCisequaltoSolution:-

∆ABCisrightangledatBas(10,8,6)isaPythagoreantripletLetBP=BD=BQ=x(radiusofthesamecircle)AD=yForrightangledΔBDCx2+(10–y)2=8262–y2+(10–y)2=8236–y2=64–(100–20y+y2)36–y2=64–100+20y–y2y=C`

$%

BD2=36– C`$%

<

BD=4.8Hence,AP=6–4.8=1.2=`

?

QC=8–4.8=3.2AP:QC=1.2:3.2=3:8Example25:-InacyclicquadrilateralABCD,ABisparalleltoCDandCD=2ABthenAD:BCisequalto

139

Solution:-

Example26:-Inthefigure,ABisadiameterofthecircle,TDisatangent.If∠AHD=36o,∠CDTis______.

Solution:-∠ADB=90o(∠insemi-circle)∠DAH=180o–90o–36o=54o∠DAH=∠CDK=54o(∠inalt.segment)⇒∠CDT=180o-54o(adj.∠sonstraightline)=126oExample27:-Inthegivenfigure,PQ=RQ.∠RQP=72o,PCandQCaretangentstothecirclewithcentreO.Calculate∠PCQ.

140

Solution:-

Angle2=angle4=54o

Angle1=angle3=54oAnglesinalternatesegmentthereforeanglePCQ=72o Example28:-GiventwoconcentriccircleswithcentreP,ACisachordofthelargercircletangenttothesmalleratB.If|BC|=2units,findtheareainbetweentwocirclesinsquareunits.Solution:-

LetR,rbetheradiiofbigger&smallercirclesrespectively.InΔABPR2=r2+(AB)2…………(1)AsBC=2ACisachordPBisperpendiculartoit&mustbisectthechord.AB=BC=2From(1)R2–r2=4Areainbetweentwocircles=πR2–πr2=π(R2–r2)=4πExample29:-Inthegivenfigure,ABCQisaquadrantofacirclewithradius14cm.WithACasdiameter,asemicircleisdrawn.Findtheareaoftheshadedportion.

141

Solution:-Requiredarea=AreaAQCPA=AreaACPA–AreaACQA=Areaofsemi-circlewithACasdiameter–(AreaABCQA–AreaofΔABC)=Areaofsemi-circlewithACasdiameter–areaofaquadrantofacirclewithABasradius+AreaofΔABCNowAC= 14< + 14<=14 2cmAreaofsemi-circlewithACasdiameter=$

<𝜋(AC/2)2=$

<𝜋(7 2)2=49𝜋

(AreaABCQA–AreaofΔABC)=$>𝜋142−$

<×14×14=49𝜋−98

Requiredarea=49𝜋 − (49𝜋−98)=98cm2Example30:-Thefigureshowsthelengthsofthesidesofanequiangularpolygon.Whatistheareaofthepolygon?

Solution:-Asshowninthefigure.

Therearetwosquaresofside1,onerectanglewithdimensionsof3x1andfourrightangledisoscelestriangles.Therefore,Area=2(1x1)+(3x1)+4($

<x1x1)=7.

Example31:-PQisthediameterofacirclewithradius5units.ThelengthofRSis6units.WhatistheareaofthetrapeziumPQRS?

Solution:-

142

RT=3,QT=4.Area=$

<(6+10)×4=32

Example32:-Asolidconsistsofacircularcylinderfittedwitharightcircularconeplacedonthetop.Theheightoftheconeis9cm.Iftotalvolumeofthesolidis3timesthevolumeofcone,findtheheightofcircularcylinder.Solution:-

LetrbetheradiusofconeandcylinderHeightofcone=9cmVolumeofcone=$

Cπr2h=$

Cπr2×9=$

<volumeofcylinder=$

<πr2H=

Soheighth=6cmExample33:-Acylindricalcontainerwhosediameteris12cmandheightis15cm,isfilledwithice-cream.Thewholeice-creamisdistributedto10childreninequalconeswithhemisphericaltops.Iftheheightoftheconicalportionistwicethediameterofitsbase,findthediameteroftheice-creamcone.Solution:-Volumeofcylindricalcontainer=10×volumeofeachcone

143

⇒πr2h=10× $C𝜋𝑟$<ℎ$ +

<C𝜋𝑟$C

⇒π×62×15=$%ÚC[r12h1×2r13]

=$%Ú

C[4r13+2r13][ h1=2d1=2.2r1=4r1]

⇒`

;×$?×C$%

=6r13⇒r1=3Diameterofice-creamcone=6cmExample34:-ThecostofwhitewashingwallsofaroomisRs.56.Howmuchwillitcosttowhitewashanotherroomwhichistwiceinlength.Breadthandheightofthepreviousroom?Solution:Areaofwalls=2h(l+b)Areaofwallsforthebiggerroom=2(2h)(2l+2b)=4×2h(l+b)Thecostwillalsobe4timesthecostofsmallerroom.Totalcost=4×56=Rs.224.Example35:-Ametallicrightcircularconeofheight9cmandbaseradius7cmismeltedintoacuboidwithtwosidesas11cmand6cm.Whatisthethirdsideofthecuboid?Solution:-Letthethirdsideofcuboid=xcmNow,accordingtothequestion:1/3πr2h=l×b×h⇒66x=462⇒x=7cmExample36:-Asolidisintheformofacylinderwithhemisphericalends.Thetotalheightofthesolidis19cmandthediameterofthecylinderis7cm.Thesurfaceareaofthesolidis?Solution:-Radiusofthehemisphere=Radiusofthecylinder=HeightofthehemisphereHeightofcylinder=Totalheight-2x(radiusofthehemisphere).Heightofcylinder=12cmradiusofHemisphere=3.5cmSurfacearea=Surfaceareaofcylinder+2x(Surfaceareaofhemisphere)=(2πrh+2πr2)+2x(2πr3)=(2xπx3.5+2xπx3.5x3.5)+2x(2xπx3.5x3.5x3.5)=418Example37:-ABCDisarectanglewithAD=1.DPFandCQFaretwoequalarcsdrawnwithAandBascentersrespectively.EisthemidpointofCD.AnotherarcwithEascentretouchesthetwoarcsDPFandCQFatPandQrespectively.Whatistheareaoftheshadedportion?

144

Solution:-AsAFandBFarebothradiiofequalarcs,AF=BF=AD=1.

AB=2Areaoftherectangle=2.

NotethatBEisthediagonalofsquareFBCE,andtherefore,BE= 2EQ=BE–BQ= 2-1( BQ=BC=1)AreaofsemicirclewithcentreE=$

<π( 2-1)2.

Also,thetotalareaofthetwosegmentswithcentresAandB=2×$>×π×12=Ú

<

Areaofshadedregion=2-Ú

<[1+( 2-1)2]=2-Ú

<[2-2 2+2]=2-π(2- 2)

Example38:-Fiveequalsquaresareplacedsidebysidetomakeasinglerectanglewhoseperimeteris372cm.Whatistheareaofeachsquare?Solution:-Letxbethelengthofthesideofeachsquare.Arectangleformedbyplacingfivesquaressidebysidehasthelengthequalto5xandbreadthequaltox.Hence,theperimeteroftherectangleis12x.Therefore,12x=372cm.i.e.x=31cm.Hence,areaofeachofthesquare=x2=312=961cm2Example39:-Ifallthesidesofatriangleareincreasedby200%,whatwillbethepercentageincreaseinarea?Solution:-Newsides=3xoldsidesofΔ

Newarea=9xoldareaHence,theincrease=800%Example40:-Twocirclesofradius8cmeach,cutorthogonallyasshowninthefigure.Findthelengthoftheoutlineofthefigure(i.e.ABCDA).

145

Solution:-∠PCQ=90o∠CPQ=∠CQP=45o∠CPA=∠CQA=90oItmeans,¼thofperimeterofboththecirclesisoverlapping.orArcABCDA=2π(8)[1–¼]×2=24π.

Example41:-Arectangularsheetofpaperis30cm×20cm.ItisfoldedinsuchawaythatcornerBfallsonACatB'andPQ,alongwhichitisfoldedisparalleltoACasshowninthefigure.FindtheareaofAPQCDA.

Solution:-

NotethatPBQB'isakite.Since,BP=PB'andBQ=QB',BO=OB'and∠BOP=∠B'OP=90°AndBO=OB'andPQ||ACAP=PB=X=½AB=10andBQ=QC=Y=½BC=15AreaoftrianglePBQ=½×10×15=75cm2=>AreaofAPQCDA=600–75=525cmExample42:-Thefigureshowsthelengthsofarectangleininches.Howmuchistheshadedareainsquarein

146

Solution:-Theshadedregionistheareaofthe8by12rectangle,minus[theareaoftherighttrianglewithbase8andheight9]plus[theareaoftherighttrianglewithbase2andheight3]=3.SotheshadedareaA=8(12)–$

<(8)(9)–$

<(2)(3)=96–36–3=57.

Example43:-Onesideofarhombusis10cmandoneofitsdiagonalsis12cm.TheareaoftherhombusisSolution:-

AB2=AE2+EB2orEB2=AB2–AE2=100–36=64EB=8Areaofrhombus=2×AreaofΔABC

Example44:-Thediameterofawheelis63cm.Thedistancecoveredbythewheelin100revolutionsisSolution:-Circumference=2×<<

X× `C

<=198cm.

Thedistancetraveledinonerevolution=198cm.Hence,thedistancetraveledbythewheelin100revolutions=198×100=198mExample45:-Asphericalballofradius4cmistobedividedintoeightequalpartsbycuttingitalongtheaxisasshown.Findthesurfaceareaofeachpieceincm2.

147

Solution:-Theballafterbeingcutwillhaveeightparts,eachofthesamevolumeandsurfacearea.Thefigurewillbesomewhatlikethefigure(1)ifseenfromthetopbeforecutting.Aftercuttingitlookssomethinglikethefigure(2).

Now,Thesurfaceareaofeachpiece=Area(ACBD)+2(AreaofCODBC).Thedarkenedsurfaceisnothing,butarcABfromsideglance,whichmeansitssurfaceareaisoneeighththeareaofthesphere,thatis,$

[×4πr2=$

<πr2.Now,CODBCcanbeseenasasemicirclewithradius4

cm.Therefore,2(AreaCODB)=2×$<×πr2=πr2

⇒surfaceareaofeachpiece=$<×πr2+πr2= C

<×π×42=24π

Example46:-Fourcircularcoins,eachofradius1.4cmareplacedflatonatable;suchthattheircentersarethecornersofasquareandthateachcointouchestwooftheothers.Findthearealyingvacantbetweentheirrims.Solution:-

Arealyingbetweencoins=Areaofsquare–4×areaof1sector

=(2.8cm)2–4×_%°C`%

×𝜋× 1.4 <=(1.4)2[4–π]=(1.4)2×(4–3.14)=1.68cm2

Example47:-AparallelogramPTRUisdrawnwithinanotherparallelogramPQRS,suchthatTRbisects∠SRQand∠RTQ=500.Findthe∠TQR

148

Solution:-PU||RT,∠UPT=∠RTQ=50o(Correspondingangles)∠TRU=50o(Oppositeangleofparallelogram)RTbisectsangleURQ.∠TRQ=50o∠TQR=80o(SRQPisparallelogram)Example48:-Ifthesumofanglesofapolygonis1080,thenthenumberofsidesofthepolygonwillbe_________?Solution:-1080=180(n–2)ORn=8Example49:-Inthefiguregiven,ifx=120oandy=100o,thenz=?

Solution:-x+y+a=360°.120°+100°+a=360°.a=140°.z+a=180°.z+140°=180°.z=40°.Example50:-Ahallis15mlongand12mbroad.Ifthesumoftheareasofthefloorandtheceilingisequaltothesumoftheareasoffourwalls,thevolumeofthehallis:Solution:-2(15+12)xh=2(15x12)ORh=180/27m=20/3mOrvolume=15x12x20/3=1200m3Example51:-Ahollowironpipeis21cmlonganditsexternaldiameteris8cm.Ifthethicknessofthepipeis1cmandironweighs8g/cm3,thentheweightofthepipeis:Solution:-Externalradius=4cm,Internalradius=3cm.Volumeoftheiron=<<

X× 4< − 3< ×21=462cm3

149

SoweightoftheironWeightofiron=(462x8)gm=3696gm=3.696kg.Example52:-50mentookadipinawatertank40mlongand20mbroadonareligiousday.Iftheaveragedisplacementofwaterbyamanis4m3,thentheriseinthewaterlevelinthetankwillbe:Solution:-Totalvolumeofwaterdisplaced=(4x50)m3=200m3.Orriseinwaterlevel=200/40×20=0.25mor25cmExample53:-Thecurvedsurfaceareaofacylindricalpillaris264m2anditsvolumeis924m3.Findtheratioofitsdiametertoitsheight.Solution:-ÚT;�<ÚT�

= _<><`>

sor=7m&2𝜋𝑟ℎ=264Soh=6mSoreuiredratio=2r/h=14/6=7/3Example54:-Alargecubeisformedfromthematerialobtainedbymeltingthreesmallercubesof3,4and5cmside.Whatistheratioofthetotalsurfaceareasofthesmallercubesandthelargecube? Solution:-Volumeofthelargecube=(33+43+53)=216cm3.Lettheedgeofthelargecubebea.So,a3=216soa=6cm.Sorequiredratio=6(32+42+52)/6×62=50/36=25/18Example55:-Ahorizontalpipeforcarryingfloodwaterhasdiameter1m.Whenwaterinitis10cmdeep,thewidthofwatersurfaceABisequalto

Solution:-

150

LetObethecentreofcircle.InΔOAD502=402+x22500–1600=x2x2=900x=30AB=2x=60cm.Example56:-Theradiusofarightcircularcylinderisincreasedby25%.Bywhatpercentshoulditsheightbechangedsothatitsvolumeremainsthesame?Solution:-LetradiusberandheightbehVolumeofcylinder=r2hAsradiusisincreasedby25%Hence,thenewradius=1.25randletheightbeH.Asvolumeremainsthesame,so𝜋r2h=𝜋(1.25)2HH/h=1/1.252=(4/5)2=0.82=0.64æW��=0.64-1=-0.36

æW��x100=-36%

So,theheightshouldbedecreasedby36%.Example57:-Inthefigure,AB||CDandPQintersectsthematPandQrespectively.

IfPRandQRarethebisectorsofTypeequationhere. APQand PQCrespectively,findout PRQSolution:-Since AB || CD

151

Now, ∠1 + ∠2 + ∠4 + ∠5 = 180o … (1) Also ∠2 + ∠3 + ∠4 = 180o … (2) ∠1 = ∠2 and ∠4 = ∠5 [ given that PR & QR are bisectors] Now from equation (1)

Now from equation (1) 2∠2 + 2∠4 = 180o

∠2 + ∠4 = 90o From equation (2)

90 + ∠3 = 180o

or∠3 = 90o Example58:-Giventhat AOB=88ointhefigure,findthemeasureof OCB,whereOisthecentreofthecircle.

Solution:-

Asshowninthefig,JoinBC&AB∠AOB=880(given)∠BOC=180–880=920(angleoflinearpair)

Now OBCisanisoscelestriangleOB=OC=radius∠OCB=∠OBC=x

Nowx+x+92=18002x=880 x=440

Example59:-Inthefigure,TAisatangenttothecircle.

152

If∠BAT=120o,findthemeasureof∠ACBand∠CAT.Solution:-As∠TAC=∠ABC(anglesinthealternatesegment)Now∠TAB=1200Or∠TAC+ CAB=1200Or∠ABC+∠CAB=1200...(from(i))Nowin∆ABC,∠ABC+∠CAB+∠BCA=1800120o+∠BCA=1800Hence∠BCA=600Now∠BCA+∠TCA=1800∠TCA=1800–600=1200Nowin∆TAC,∠TAC+∠TCA+∠ATC=1800∠TAC=1800–1200-250=350∠ABC=350Example60:-Acylindricalcisternofradius42cmispartlyfilledwithwater.Ifarectangularblock24cm×21cm×11cmiswhollyimmersedinthewater,byhowmuchwillthewaterlevelrise?Solution:-Volumeofcylindricalcolumnofincreasedheight=volumeofrectangularblock⇒<<

X×42×42×h=24×21×11

orh=1cmExample61:-Fourhorsesaretetheredat4cornersofasquarefieldofside70metressothattheyjustcannotreachoneanother.Thearealeftungrazedbythehorsesinsq.m.is:Solution:Thelengthoftheropeinwhichthehorsestiedshouldbeequaltohalfofthesideofthesquareplotsothattheyjustcannotreachoneanother.Therefore,thelengthoftheropeis35m(70/2).Theareacoveredbyeachhorseshouldbeequaltotheareaofsectorwithradiusof70/2=35m(lengthoftherope).Totalareacoveredbythefourhorses=4×areaofsectorofradius35metres

=Arealeftungrazedbythehorses=Areaofsquarefield-Areacoveredbyfourhorses=702-(22/7)*35*35=4900-3850=1050sq.m.

153

Example62:-Acircustenthasacylindricalbasesurmountedbyaconicalroof.Theradiusofthecylindricalpartis30manditsheightis9m.Thetotalheightofthetentis24m.Findtheamountofairavailableperperson,if15personsareseatedinthetentSolution:-Radiusofcylindricalpart=30mHeightofcylinder=9mTotalheightoftent=24mHeightofconicalpart=24–9=15mVolumeofair=volumeofcylinder+volumeofcone.

=πr2h+$

Cπr2H

=πr2 ℎ +$C𝐻

=<<X

Example63:-Ifthesumoftheinterioranglesofaregularpolygonmeasuresupto1440degrees,howmanysidesdoesthepolygonhave?Solution:WeknowthatthesumofanexteriorangleandaninteriorangleofaPolygon=180o.Wealsoknowthatsumofalltheexterioranglesofapolygon=360o.TheQuestionstatesthatthesumofallinterioranglesofthegivenpolygon=1440o.Therefore,sumofalltheinteriorandexterioranglesofthepolygon=1440+360=1800.Ifthereare'n'sidestothispolygon,thenthesumofalltheexteriorandinteriorangles=180*n=1800.Therefore,n=10.Example64:-Theheightofacylinderis14cmanditscurvedsurfaceis264sq.cm.TheradiusofitsbaseisSolution:-Curvedsurfaceareaofacylinder=2πrhgivenh=14cmandcurvedsurfacearea=264cm22πrh=264cm2

Solvingwegetr=3cmExample65:-Ateapotisintheshapeofhemisphericalbase,aconicallidandacylindricalbody.Theoverallheightoftheteapotis20cm.Thebasediameteris3cmandthecylindricalpartis8cmlong.Whatisthevolumeoftheteapot?

154

Solution:-

Volumeofteapot=volumeofhemisphere+volumeofcylinder+volumeofcone=<

Cπr3+πr2h+$

Cπr2h=219π/8cubic.Cm

Example66:-Acylinderandaconehavethesameheightandthesameradiusofthebase.TheratiobetweenthevolumesofthecylinderandtheconeisSolution:-RatioofthevolumeofthecylinderandtheconeisÚT

;�AHÚT

;�=3:1

Example67:-Agoatistiedtoonecornerofasquareplotofside12mbyarope7mlong.Findtheareaitcannotgraze.Solution:-

Areathegoatcan’tgrazeistheshadedareashownwhichisequaltoareaofsquareminusareaofquadrantofcirclewithradiusequaltolengthofropei.e.7m.Areaitcan’tgraze=(12)2– _%

C`%× <<

X×7×7

144–38.5=105.5sq.mExample68:-:Ifaregularhexagonisinscribedinacircleofradiusr,thenitsperimeterisSolution:

155

Inthecirclewitheachsideofhexagonwewillget6congruentequilateraltrianglesHenceSidesofhexagon=radiusofcircle SoSide=rOr,Perimeter=6r.Example69:-Thediagonalsofarhombusare64cmand48cm.TheheightoftherhombusisSolution:Areaofrhombus=$

<×d1×d2

Areaofrhombus=$<×64×48=1536cm2

AB= 𝑂𝐴< + 𝑂𝐵<= 32< + 24<=40cmAreaofrhombus=base×heightor1536=40×hHence,h=$?C`

>%=38.4cm.

Example70:-Theareaofatrapeziumis384cm2.Ifitsparallelsidesareintheratioof3:5andtheperpendiculardistancebetweenthemis12cm.thenthesmalleroftheparallelsidesisSolution:Lettheparallelsidesbe3xand5xcmrespectivelyThen,Area=$

<(3x+5x)12=384

Or,8x=64x=8Hencetheparallelsidesare24cmand40cm.Example71:-Thebaseandtheheightofarightangledtrianglearethesameasthoseofatrapezium.Iftheotherparallellineofthetrapeziumis1.5timesofthefirstparallelline,thenwhatwillbetheratiooftheareasofthetriangleandthetrapezium?Solution:Letxisthebaseandhistheheightofthetrapezium.Areaoftriangle=$

<xh

Areaoftrapezium=$<𝑥 + C

<𝑥 ×h=?

>𝑥ℎ

Requiredratio=$<𝑥ℎ:?

>𝑥ℎ=2:5.

Example72:-SRisadirectcommontangenttothecirclesofradiiof8cmand3cmrespectively,theircentresbeing13cmapart.ifSandRaretherespectivepointsofcontactsthenfindthelengthofSR

Solution:

UsetheformulaD=√d2-(R-r)2

D=√169-25orD=√144orD=12cmisthelengthofSR.

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Exercise

1.Anairjetflies10milessouth,then4mileseast,then7milesnorthandthen8mileswestwhereitfinallylanded.Findtheshortestdistancefromthestartingpointofthejourneyandthepointwhereitfinallyends.(a)7miles (b)6miles (c)8miles (d)5miles2.Theareaofacircleis154cm²,whichisequaltotheareaofarectanglewithonesideequivalenttotheradiusofthecircle.Findtheothersideoftherectangle.(a)22/7cm (b)11cm (c)22cm (d)11/7cm3.Findthesupplementofangle75°.(a)105° (b)90°(c)15° (d)125°4.Findtheanglewhosesupplement&thriceitscomplementareintheratioof5:6.(a)60° (b)30° (c)90° (d)120°5.Findthelargeranglemadebythehandsoftheclockat8:00.(a)120° (b)180° (c)240° (d)200°6.Theperimeterofarectangleis220meters,andthedifferencebetweenlengthandbreadthis30meters.Findtheareaoftherectangle.(a)2524m² (b)3200m² (c)2400m² (d)2800m²7.Findthediagonalofasquarewhosesideisof8m.(a)8 2m (b)16m (c)8m (d)18 2m8.Asolidmetalcylinderhavingaradiusof5cmandheightof18cmismelteddownandrecastasaconehavingradiusof3cm.Findtheheightofthecone.(a)150cm (b)100cm (c)120cm (d)125cm9.Given:RadiusofcircleAis2.5unitsandradiusofcircleBistwicetheradiusofA.ColumnA ColumnBAreaofcircleA CircumferenceofcircleB(a)ThequantityincolumnAisgreater.(b)ThequantityincolumnBisgreater.(c)Boththequantitiesareequal.(d)Therelationshipcannotbedetermined.10.Calculatetheareaofthesquarehavingperimeterequaltotheareaofarectangleas44cm².(a)120cm² (b)142cm² (c)121cm² (d)144cm²11.Calculatetheareaofarectanglewithlengthas(1-a)andbreadthas(1+a).

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(a)a² (b)1/a² (c)1+a² (d)1-a²12.Calculatebothsidesofarectangle,giventheperimeterandareaoftherectangleas24mand36m²respecively.(a)10m,2m (b)12m,3m (c)6m,6m (d)18m,2m13.FindtheareaoftriangleABC,whereABisthediameterofacircle.Cliesontheperimeterofthatcircleatadistanceof5unitsfromAand12unitsfromB.(a)32units (b)35units (c)30units (d)31units14.Anisoscelesrighttrianglehashypotenuseof16inches.Findthelengthofotherside.(a)6inches (b)8 2inches(c)7 2inches (d)6 2inches15.Findthenumberofrevolutionsmadebythewheelperkilometer,with14cmradius.(a)App.1000revolutions(b)App.1245revolutions(c)App.1136revolutions(d)App.1263revolutions16.Iftheradiusofthecircleistripled,theareaismultipliedby:(a)8 (b)2 (c)4 (d)remainsunchanged17.Thelengthofawirefencearoundacirculargardenis44meters.Whatisthearea(insq.meters)ofthe2metersconcretepathlaidinsidethefence?(a)24𝜋m² (b)25𝜋m² (c)32𝜋m² (d)33𝜋m²18.Theareaofarhombusis154sq.m.Ifoneofitsdiagonalsis22m,findthelengthoftheotherdiagonal.(a)20m (b)22m (c)14m (d)27m19.Arectangularparkwithlengthandbreadthof11mand22m,issurroundedbyapathof3mwide.Findtheareaofthepath.(a)100m² (b)108m² (c)200m² (d)234m²20.AlinesegmentABis32mlong.ApointCislocatedonABsuchthatAC:CBis5:3.FindthelengthofCB.(a)10m (b)12m (c)20m (d)22m21.Iftheanglesofaquadrilateralareintheratioof3:4:5:6.Calculatethesmallestangle.(a)60° (b)80° (c)100° (d)120°22.Thebaseofrightangletriangleis'b'units.Iftheareaofthetriangleis'a'units,findtheheightofthetriangle.(a)2abunits (b)2a/bunits (c)2abunits (d)Cannotbedetermined

158

23.Theperimeteroftherectangleis28cmandthebreadthis6timesthelength.Findtheareaofrectangle.(a)20cm² (b)28cm² (c)14cm² (d)24cm²24.Thesidesofatriangleareintheratio5:6:7.Ifitsperimeteris36cm.Findthelongestsideofthetriangle.(a)10cm (b)14cm (c)12cm (d)16cm25.Arectanglehasanareaof36cmandperimeterof30cm.Findthelargersideofit.(a)15cm (b)18cm (c)10cm (d)12cm26.Theareasoftwocirclesare4:1,findtheratioofthecircumferencesofthecircles:(a)4:1 (b)1:2 (c)1:4 (d)2:127.Abuffaloistiedtothegroundwitharope.Whatshouldbethelengthoftherope,sothatthebuffalocangrazein616m²areaonly?(a)10m (b)12m (c)14m (d)15m28.Calculatethetotalsurfaceareaofacuboidwhosedimensionsare12m,10mand5m.(a)400m² (b)460m² (c)360m² (d)480m²29.Findthetotalsurfaceareaofaconewithheightas21cmandradiusofitsbasebeing28cm.(a)5042cm² (b)5544cm² (c)5142cm² (d)5000cm²30.Findtheinradiusofthetrianglewithsides5,12&13cm?(a)12 (b)11.5(c)2 (d)12.531.Atraintravels4milesnorthfromtheplatform,then4mileswest,then2milesagainnorthandthen4mileswest.Howfaristhetrainfromtheplatform?(a)14miles (b)10miles (c)12miles (d)12.5miles32.Findthecircumradiusofthetrianglewithsides5,12&13cm?(a)12 (b)11.5(c)3 (d)6.533.Aladderwhichis40mtshighisleaningagainstawallwhichis32mtshigh.Howfaristhewallfromthebaseoftheladder.(a)26 2mts (b)25mts (c)24mts (d)25 2mts34.Findthenumberofspokesinthewheelofacycle,giventheanglebetweentwoconsecutivespokesas20°.(a)18 (b)20 (c)36 (d)935.Anice-creamconehastheheightof7cmanddiameterof6cm.Calculatethevolumeoftheice-creamthatwillbefilledinthiscone.(a)164cm³ (b)66cm³ (c)124cm³ (d)98cm³

159

36.Anangleisequaltoone-fourthofitssupplement.Theangleis:(a)42° (b)37° (c)57° (d)36°37.Awirewasintheshapeofrectangle,withlengthas14cmandbreadthas11cm.Thewireisthenmoldedintoacircle.Findthecircumferenceofthecircle.(a)44cm (b)54cm (c)50cm (d)40cm38.Twosquareshaveeachsideas20cmand21cmrespectively.Findthesideofthirdsquarewhoseareaisequaltothesumoftheareasofothertwosquares.(a)28cm (b)29cm (c)30cm (d)32cm39.Findtheareaofatrianglehavingsides7m,8m,and9m.(a)12 5m² (b)30m² (c)12 3m² (d)8 5m²40.Findthecircumradiusofthetrianglewithsides3,4&5cm?(a)2.5(b)2 (c)3 (d)12.541.InatrapeziumABCD,AB+CD=24.ColumnX ColumnYLengthofAB LengthofCD(a)ThequantityincolumnXisgreater(b)ThequantityincolumnYisgreater.(c)Boththequantitiesareequal.(d)Therelationshipcannotbedetermined.42.Aplotoflandisintheshapeofatrapeziumwhosedimensionsaregiveninthefigurebelow:

Hencetheperimeterofthefieldis(a)50m(b)64m(c)72m(d)Noneoftheabove43.Findtheareaofthesectorcoveredbythehourhandafterithasmovedthrough3hoursandthelengthofthehourhandis7cm.(a)77sq.cm(b)38.5sq.cm (c)35sq.cm(d)70sq.cm44.Whatisthemeasureofthecircumradiusofatrianglewhosesidesare9,40and41?(a)6cm (b)4cm (c)24.5cm(d)20.5cmQ45.Ifthesumoftheinterioranglesofaregularpolygonmeasuresupto1440degrees,howmanysidesdoesthepolygonhave?

160

(a)10 (b)8 (c)12 (d)9Q46.Whatisthemeasureofinradiusofthetrianglewhosesidesare24,7and25?(a)12.5 (b)3 (c)6(d)noneoftheseQ47.Whatisthecircumradiusofatrianglewhosesidesare7,24and25respectively?(a)18 (b)12.5 (c)12(d)14Q48.

ABCDisaparallelogram.BD=2.TheanglesoftriangleBCDareallequal.Whatistheperimeteroftheparallelogram?(a)9 (b)10 (c)11(d)noneofthese49.

PQRSisaparallelogramandST=TR.WhatistheratiooftheareaoftriangleQSTtotheareaoftheparallelogram?(a)1:2(b)2:3 (c)5:6(d)noneofthese50.

Twoequalcirclesarecutoutofarectangleofcardofdimensions16by8.Thecircleshavethemaximumdiameterpossible.Whatistheapproximateareaofthepaperremainingafterthecircleshavebeencutout?(a)21 (b)23(c)25(d)noneofthese

161

51.

ASBisaquartercircle.PQRSisarectanglewithsidesPQ=8andPS=6.WhatisthelengthofthearcAQB?(a)5pi (b)6pi (c)7pi(d)noneofthese52.

RadiusofcirclecenterOis3timestheradiusofcirclecenterC.∠C=∠O.IftheshadedareaofcircleCis2thenwhatistheareaoftheshadedpartofcircleO?(a)6 (b)12 (c)18 (d)noneofthese53.

IntriangleABC,AD=DB,DEisparalleltoBC,andtheareaoftriangleABCis40.WhatistheareaoftriangleADE?(a)10 (b)15 (c)20 (d)30 54.

162

RectangleABCDhasaperimeterof26.ThehalfcirclewithdiameterADhasanareaof8π.Whatistheperimeterofthepartofthefigurethatisnotshaded?(a)26+4π (b)18+8π (c)18+4π(d)noneofthese55.Findthenumberoftrianglesinanoctagon.(a)326(b)120 (c)56 (d)cannotbedetermined56:Findtheareaofthesectorcoveredbythehourhandafterithasmovedthrough3hoursandthelengthofthehourhandis7cm.(a)77 (b)38.5 (c)35(d)70 57.IfmedianADofanequilateraltriangleABCis9cmandGiscentroid.FindAG?(a)3 (b)6 (c)12 (d)1158.Theratiooftheside&heightofanequilateraltriangleis(a)1:1(b)2: 3(c) 3:2(d)2:159.InarightangletrianglePQR,rightangleatQ,PS=SQ=SRandangleSPQ=540.FindangleRSQ?(a)720(b)1080(c)360(d)540

60.Theareaofatrianglewithbasexunitsisequaltoareaofasquarewithsidexunits.Thenthealtitudeofthetriangleis:(a)x(b)2x(c)3x(d)4x61.Theratiobetweenlength&perimeterofarectangleplotis1:3.Whatistheratiobetweenlength&breadthoftheplot?(a)2:1(b)3:2(c)datainadequate(d)1:262.ThediagonalofasquareAis(x+y).ThediagonalofasquareBwithtwicetheareaofAis:(a) 2(x+y)(b)2(x+y)(c)2x+4y(d)4x+2y63.Theareaofarightangledtriangleis20sq.cm.andoneofthesidescontainingrightangleis4cm.thealtitudeonthehypotenuseis(a)20/ 29(b)8cm(c)10(d) 40/41

163

64.Thewholesurfaceofacubeis216cm2.Thevolumeincm3ofcubeis:(a)108(b)54(c)432(d)21665.TheradiusofbaseofarightcircularconeisR&isheightis2H,thenitsvolumeis:(a)<

C𝜋𝑅<𝐻(b)𝜋𝑅<𝐻 (c)2𝜋𝑅<𝐻(d)$

C𝜋𝑅<𝐻

66.Ifthecurvedsurfaceofacylinderbedoubletheareaoftheendsthentheratioofitsheightandradiusis:(a)2:3 (b)1:1(c)2:1(d)1:267.Theradiusofthebaseofacylinderis2cm&itsheight7cm,thenitscurvedsurfaceincm2is:(a)44 (b)22 (c)88 (d)5668.Eachedgeofacubeisincreasedby50%.Thepercentofincreaseinthesurfaceareaofthecubeis:(a)50(b)100 (c)120 (d)12569.Thesidesofatriangleare3cm,4cm&5cm.Itsareaincm2is(a)12(b)15(c)20(d)670.Theradiusofacircleisdiminishedby10%,theareaisdiminishedby:(a)10%(b)21%(c)19%(d)20%71.IfthecostofwhitewashingthefourwallsofarectangularroomisRs.25,thenthecostofwhitewashingaroomtwicethelength,breadthandtheheightwillbeRs.:(a)50(b)100(c)150(d)20072.Thedifferencebetweenthelength&breadthofarectangleis23m.Iftheperimeteroftherectangleis206m,finditsareainsq.cm.(a)2420(b)2480(c)2520(d)152073.Thelength&breadthofarectangleisinratio3:2.Ifcostoffencingit@Rs.12.5permeterisRs.2000.Byhowmuchitslengthexceeditsbreadthinmeters?(a)16(b)32(c)80(d)16074.Thevolumeofacubeis216cm3,itssideis:(a)16(b)6(c)26(d)3275.Whenrecast,theradiusofanironrodismadeone-fourth.Ifitsvolumeremainsconstant,thenthenewlengthwillbecome(a)¼timesoforiginal(b)1/16timesoftheoriginal(c)16timesoforiginal(d)4timesoforiginal

164

76.Arightcircularcone&arightcircularcylinderhaveequalbase&equalheight.Iftheradiusofthebase&theheightareintheratio5:12,thenratiooftotalsurfaceareaofthecylindertothatoftheconeis(a)3:1(b)13:9(c)17:9(d)34:977.Eachedgeofacubeisincreasedby20%.Thepercentofincreaseinthesurfaceareaofthecubeis:(a)43(b)45 (c)41 (d)4478.Thesidesofatriangleare9cm,12cm&15cm.Itsareaincm2is(a)12(b)15(c)50(d)5479.Theradiusofacircleisdiminishedby20%,theareaisdiminishedby:(a)40%(b)44%(c)36%(d)20%80.IfthecostofwhitewashingthefourwallsofarectangularroomisRs.50,thenthecostofwhitewashingaroomtwicethelength,breadthandtheheightwillbeRs.:(a)50(b)100(c)150(d)20081.Asolidmetallicconeismelted&recastintoasolidcylinderofthesamebaseasthatofthecone.Iftheheightofcylinderis7cm,theheightoftheconewas(a)20cm (b)21cm (c)28cm (d)24cm82.Themeasures(incm)ofsidesofarightangledtrianglearegivenbyconsecutiveintegers.Itsarea(incm2)isgivenby(a)8 (b)9 (c)5 (d)683.Ifatrianglewithsamebase8cmhasthesameareaasacirclewithradius8cm,thecorrespondingaltitude(incm)ofthetriangleis(a)12π(b)20π(c)16π(d)32π84.Theradiusofthebase&heightofarightcircularconeareinratio5:12.Ifthevolumeoftheconeis314cm3,theslantheightincmis(take𝜋 = 3.14):(a)12 (b)13 (c)15 (d)1785.Thearea(inm2)ofthesquarewhichhasthesameperimeterasarectanglewhoselengthsis48m&is3timesitsbreadthis:(a)1000(b)1024 (c)1600 (d)102586.Theareaofanequilateraltriangleis400 3sqm.Itsperimeteris(a)120(b)150 (c)90 (d)13587.Diameterofawheelis3m.Thewheelrevolves28timesinaminute.Tocover5.280kmdistance,thewheelwilltake

165

(a)10min (b)20min (c)30min (d)40min88.Theperimeterofarhombusis40m&itsheight5m.Itsareainsq.mis:(a)60 (b)50(c)45 (d)5589.Theareaofthebiggestcircleinsq.cm.,whichcanbedrawninsideasquareofside21cmis(a)344.5 (b)364.5 (c)346.5 (d)366.590.Theareaofrhombusis150cm2.Thelengthofoneofitsdiagonalis10cm.Thelengthoftheotherdiagonalincmis(a)25 (b)30 (c)35 (d)3691.Acircularwireoftheradius42cmisbentintheformofarectanglewhosesidesarein6:5.Thesmallestsideoftherectangleis(a)60 (b)30 (c)25 (d)3692.Ifradiusofthebaseofaconebedoubled&heightleftunchangedthentheratioofthevolumeofthenewconetothatofoneoriginalconewillbe:(a)1:4 (b)2:1 (c)1:2 (d)4:193.Theareaoftheincircleofanequilateraltriangleofside42cmis(a)231 (b)462 (c)22 3 (d)92494.Thediagonalsoftherhombusare32cm&24cmrespectively.Theperimeteroftherhombusincmis(a)80 (b)72 (c)68 (d)6495.Arectangularwatertankis2.1mlong&1.5mbroad.If630litresofwaterarepouredintotank,howmuchwillthewaterlevelrise?(a)0.2m (b)2m (c)0.63m (d)1.5m96.Howmanysidesdoesaregularpolygonhavewhoseinteriorandexterioranglesareintheratio2:1?(a)3(b)5(c)6(d)1297.ABCisatrianglewithbaseAB.DisapointonABsuchthatAB=5andDB=3.Whatistheratiooftheareaof∆ADCTotheareaof∆ABC?a.3/2b.2/3c.3/5d.4/2598.Intwotriangles,theratiooftheareais4:3andratiooftheirheightsis3:4.Findtheratiooftheirbases?a.16:9b.9:16c.9:12d.16:1299.Thecircumradiusofanequilateraltriangleis8cm.theinradiusofthetriangleis?a.3.25cmb.3.50cmc.4cmd.4.25cm

166

100.Fourequalcircleseachofradius“A”unitstouchoneanother.Theareaenclosedbetweenthem(π=22/7)insquareunits,is?a.3A2b.6A2/7c.41A2/7d.A2/7101.ThelengthsoftheperpendicularsdrawnfromanypointintheinteriorofanequilateraltriangletotherespectivesidesareA,BandC.thelengthofeachsideofthetriangleis?a. <

C(A+B+C)b.$

C(A+B+C)c. $

C(A+B+C)d. >

C(A+B+C)

102.Theareaofaregularhexagonofside2 3cmis?a.18 3b.12 3c.36 3d.27 3103.Thediagonalsofarhombusare24m&10m,itsslantheightis(a)60/13 (b)120/13(c)45(d)55104.Theperimeterofarhombusis80m&itsheight5m.Itsareainsq.mis:(a)60 (b)100(c)45(d)55105.Thechordoflength16cmisatadistanceof15cmfromthecentreofthecirclethenthelengthofthechordofthesamecirclewhichisatthedistanceof8cmfromthecentreisEqualto?a.10cmb.20cmc.30cmd.40cm106.Theratiooftheareasoftwoisoscelestriangleshavingthesameverticalangle(anglebetweenequalsides)is1:4,theratiooftheirheightsis?a.1:4b.2:5c.1:2d.3:4107.Threecirclesofdiameter10cmeach,areboundtogetherbyarubberband,incmifitisstretchedasshown,is?a.30b.30+10πc.10πd.60+20π108.Thelengthoftheeachsideofanequilateraltriangleis14 3.Theareaofthein-circle,incm2isa.450b.308c.154d.77109.Eachinteriorangleofaregularpolygonis18ᵒmorethaneighttimesanexteriorangle.Thenumberofsidesofthepolygonis?a.10b.15c.20d.25110.Ifthesumofthreedimensionsandthetotalsurfaceareaofarectangularboxare12cmand94cm2respectively,thanthemaximumlengthofastickthatcanbeplacedinsidetheboxis?a.5 2cmb.5cmc.6cmd.2 5cmg111.Thediagonalofacubeis15 3cm.findthesideofcube?

167

a.120b.14c.15d.12112.Thetotalsurfaceareaofasolidrightcircularcylinderistwicethatofasolidsphere.Iftheyhavethesameradii,theratioofthevolumeofthecylindertothatofthesphereisgivenby?a.9:4b.2:1c.3:1d.4:9113.Theradiusoftheincircleofatriangleis2cm.iftheareaofthetriangleis6cm2,thenitsperimeteris?a.2cmb.3cmc.6cmd.9cm114.Thediagonalofacubeis15 3cm.findtheratioofitstotalsurfaceareaandvolume?a.2:5b.5:2c.3:5d.5:3115.Waterisflowingattherateof3km/hrthroughacircularpipeof20cminternaldiameterintoacircularcisternofdiameter10manddepth2m.inhowmuchtimewillthecisternbefilled?a.1hrb.1hr,40minc.1hr,20mind.2hr,40min116.Ifthesideofasquareisincreasedby50%,itsareaisincreasedby?a.125%b.100%c.75%d.50%117.Ifawireisbentintotheshapeofasquare.Thentheareaofthesquaresoformedis81cm2.Whenthewireisbentintoasemicircularshape.Thenthearea,(incm2)ofthesemicirclewillbe?a.22b.44c.77d.154118.Iftheradiusofacircleisincreasedby50%,itsareaisincreasedby?a.125%b.100%c.75%d.50%119.Abicyclewheelmakes5000revolutionsinmoving11km.thentheradiusofthewheelis(incm)?a.70b.35c.17.5d.140120.Ariver3mdeepand40mwideisflowingattherateof2km.hr.Howmuchwater(inliters)willfallintotheseainaminute?a.4,00,000b.40,00,000c.40,000d.4,000121.Theperimeterofatriangleis40cmanditsareais60cm2.Ifthelargestsidemeasures17cm,thenthelength(incm)ofthesmallestsideofthetriangleis?a.4b.6c.8d.15122.Thevolume(inm3)oftherainwaterthatcanbecollectedfrom1.5hectaresofgroundinarainfallof5cmis?a.75b.750c.7500d.75000

168

123.Theareaoftheeconsecutivefacesofacuboidare12cm2,20cm2and15cm2,thenthevolumeofthecuboidis?a.3600b.100c.80d.60124.Waterisflowingattherateof5km/hrthroughapipeofdiameter14cmintoaRectangulartankwhichis50mlong,44mwide.Thetaken,inhours,fortheriseinthelevelofwaterinthetanktobe7cmis?a.2b.3/2c.3d.5/2125.Thesidesofatriangleareintheratio2:3:4,theperimeterofthetriangleis18cm.Thearea(incm2)ofthetriangleis?a.9b.36c. 42d.3 15126.Acopperwireisbentintheformofanequilateraltriangleandhasarea121 3cm2.Ifthesamewireisbentintotheformofacircle,theareais(incm2)enclosedbythewireis?a.364.5b.693.5c.346.5d.639.5127.Meetingpointofallperpendicularbisectorsiscalledasa.centroidb.orthocenterc.circumcenterd.incenter128.Waterflowsintoatankwhichis200mlongand150mwide,throughapipeofcross-section0.3m×0.2mat20km/hr.thenthetime(inhr)forthewaterlevelinthetanktoreach8mis?a.50b.120c.150d.200129.InanequilateraltriangleABCofside10cm,thesideBCistrisectedatD.Thelength(incm)ofADis?a.3 7b.7 3c.10 7/3d.7 10/3130.Thefloorofaroomisofsize4m×3𝑚anditsheightis3m.Thewallsandceilingoftheroomrequirepainting.Theareatobepaintedis?a.66m2b.54m2c.43m2d.33m2131.Whatistheratioofcircumradius&inradiusofanequilateraltriangle?a.2:1b.1:2c.1:1d.2:3132.Thelength(incm)ofachordofacircleofradius13cmatadistanceof12cmformitscentreis?a.5b.8c.10d.12133.Theradiusofbaseandslantheightofaconeareintheratio4:7.ifitscurvedsurfaceareais792cm2,thentheradius(incm)ofitsbaseis?a.8b.12c.14d.16134.Theperimeterofarhombusis146cmandoneofitsdiagonalis55cm.Theotherdiagonalis?

169

a.92cmb.73cmc.48cmd.72cm135.Theratiooftheareasoftwoisoscelestriangleshavingequalverticalanglesis1:4.Theratiooftheirheightswillbe?a.1:2b.3:4c.2:3d.6:7136.Ifeachoftheradiusofbaseandheightofarightcircularconeisincreasedby10%,thenthepercentageofincreaseinthevolumeoftheconewillbe?a.20b.33.1c.44.2d.100137.Meetingpointofallthealtitudesinatriangleiscalledasa.centroidb.orthocenterc.circumcenterd.incenter138.Meetingpointofallanglebisectorsiscalledasa.centroidb.orthocenterc.circumcenterd.incenter139.Anequilateraltriangleofside6cmhasitscornerscutofftoformaregularhexagon.Area(incm2)ofthisregularhexagonwillbe?a.3 3b.3 6c.6 3d.5 3/2140.Thelength(inmeter)ofthelongestrodthatcanbeputinaroomofdimensions10m×10m×5mis?a.15 3b.15c.10 2d.5 3141.Thelateralsurfaceareaofacylinderis1056cm2anditsheightis16cm.finditsvolume?a.4545cm3b.4455cm3c.5445cm3d.5544cm3142.Thelargestsphereiscurvedoutofacubeofside7cm.Thevolumeofthesphere(incm3)willbe?a.718.66b.543.72c.481.34d.179.67143.TheradiusofcircleAistwicethatofcircleBandtheradiusofthecircleBistwicethatofcircleC.Theirareawillbeintheratio?a.16:4:1b.4:2:1c.1:2:4d.1:4:16144.Througheachvertexofatriangle,alineparalleltotheoppositesideisdrawn.Theratiooftheperimeterofthenewtriangle,thusformed,withthatoftheoriginaltriangleis?a.3:2b.4:1c.2:1d.2:3145.TheradiiofthebasesoftwocylinderAandBareintheratio3:2andtheirheightsintheration:1.IfthevolumeofcylinderAis3timesthatofcylinderB,thevalueofnis?a.4/3b.2/3c.¾d.3/2

170

146.Thevolumeofarightcircularcylinderandthatofasphereareequalandtheirradiiarealsoequal.IftheheightofthecylinderbeHandthediameterofthesphereD,thenwhichofthefollowingrelationiscorrect?a.H=Db.2H=Dc.2H=3Dd.3H=2D147.Thecircumferenceofacircleis100cm.Themeasureofasideofthesquareinscribedinthiscircleis?a.25 2πb.50 2/πc.50 2πd.25 2/π148.Theradiioftwocircleare5cmand3cm,thedistancebetweentheircentersis24cm.Thenthelengthofthetransversecommontangentis?a.16cmb.15 2cmc.16 2cmd.15cm149.Eachoftheradiusofthebaseandtheheightofarightcircularcylinderisincreasedby10%.Thevolumeofthecylinderisincreasedby?a.3.31%b.14.5%c.33.1%d.19.5%150.Theheightofacylinderandthatofaconeareintheratio2:3andtheradiioftheirbasesintheratio3:4.Theratiooftheirvolumeswillbe?a.1:9b.2:9c.9:8d.3:8151.Ifthelengthandtheperimeterofarectangleareintheratio5:16.Thenitslengthandbreadthwillbeintheratio?a.5:11b.5:8c.5:4d.5:3152.Iftheperimeterofasemicircularfieldis144m,thenthediameterofthefieldis?a.55mb.30mc.28md.56m153.Theperimeterofarightangledtriangleis30cm.ifitshypotenuseis13cm,andthenfindstheothertwosides(incm)?a.6,11b.5,12c.7,8d.6,9154.Twocircletouchexternally.Thesumoftheirareasis130πcm2andthedistancebetweentheircentersis14cm.findtheradiiofthecircles?a.11cm,15cmb.11cm,4cmc.11cm,6cmd.11cm,3cm155.Theminutehandofaclockis10cmlong.Findtheareaofthefaceoftheclockdescribedbytheminutehandbetween9AMand9:35AM?a.180.5cm2b.183.3cm2c.182.3cm2d.187.3cm2

156.Asolidmetallicsphereofradius3decimetersismeltedtoformacircularsheetof1millimeterthickness.Thediameterofthesheetsoformedis?a.26mb.24mc.12md.6m157.Theheightandtheradiusofthebaseofarightcircularconeare12cmand6cmrespectively.Theradiusofthecircularcross-sectionoftheconecutbyaplaneparalleltoits

171

baseatadistanceof3cmfromthebaseis?a.4cmb.5.5cmc.4.5cmd.3.5cm158.Waterflowsthroughacylindricalpipe,whoseradiusis7cm,at5m/sec.Thetime,ittakestofillanemptywatertankwithheight1.54mandareaofthebase(3×5)m2is?a.6minb.5minc.10mind.9min159.Ifthedifferencebetweenareasofthecircumcircleandthein-circleofanequilateraltriangleis44cm2,thentheareaofthetriangleis?a.28cm2b.7 3cm2c.14 3cm2d.21cm2160.Awire,whenbentintheformofasquare,enclosesaregionhavingarea121cm2.Ifthesamewireisbentintotheformofacircle,thentheareaofthecircleis?a.144cm2b.180cm2c.154cm2d.176cm2161.Iftheareaofacircleinscribedinasquareis9𝜋cm2,thentheareaofthesquareis?a.24cm2b.30cm2c.36cm2d.81cm2162.ABCisanequilateraltriangleofside2cm.withA,B,Cascentersandradius1cmthreearcsaredrawn.Theareaoftheregionwithinthetriangleboundedbythethreearcsis?a.(3 3-Ú

<)cm2b.( 3-CÚ

<)cm2c.( 3-Ú

<)cm2d.(Ú

<- 3)

163.Whatistheeachinteriorangleofadecagon?a.360 b.1080 c.1120 d.1440164.ABCisanisoscelestriangleinwhichAB=AC.IfDandEarethemid-pointsofABandACrespectively.ThepointB,C,D,Eare?a.collinearb.non-collinearc.concyclicd.noneofthese165.Whatisthecircumradiusofanequilateraltriangleofside6cm?A.2 2b.3 2c.2 3d.4 2166.Iftwocirclearesuchthatthecentreofoneliesonthecircumferenceoftheotherthentheratioofthecommonchordofthetwocirclestotheradiusofanyoneofthecircleis?a.2:1b. 3:1c. 5:1d.4:1167.Ifoneangleofacyclictrapeziumistripleoftheother,thenthegreateronemeasures?a.90ᵒb.105ᵒc.120ᵒd.135ᵒ 168.InacyclicquadrilateralABCD,if⦞B-⦞D=60ᵒthenthemeasureofthesmallerofthetwois?a.60ᵒb.40ᵒc.38ᵒd.30ᵒ169.Thenumberofthecommontangentsthatcanbedrawntotwogivencirclesisatthemost?

172

a.1b.2c.3d.4170.ACBisatangenttoacircleatC.CDandCEarechordssuchthatthat⦞ACE>⦞ACD.if⦞ACD=⦞BCE=50ᵒthen?a.CD=CEb.EDisnotparalleltoABc.EDpassesthroughthecentreofthecircled.∆CDEisarightangledtriangle171.Inacircleofradius17cm,twoparallelchordsaredrawnonoppositesidesofadiameter.Thedistancebetweenthechordsis23cm.ifthelengthofonechordis16cm,thenthelengthoftheotheris?a.23cmb.30cmc.15cmd.noneofthese172.Ifanglebetweentwosidesof3cm&4cmofatriangleis300..whatistheareaofthetriangle?a.6cmb.12cmc.15cmd.noneofthese173.ABCisarightangledtriangleAB=3cm,BC=5cmandAC=4cm,thentheinradiusofthecircle?a.1cmb.1.25cmc.1.5cmd.noneofthese174.Acirclehastwoparallelchordsoflengths6cmand8cm.ifthechordsare1cmapartandthecentreisonthesamesideofthechords,thenadiameterofthecircleisoflength?a.5cmb.6cmc.8cmd.10cm175.Apointwhichisequidistantfromallvertexofatriangleiscalledasa.centroidb.orthocenterc.circumcenterd.incenter176.InacircleOisthecentreandABisachord⦞AOB=50ᵒthenfind⦞OAB=?a.50ᵒb.60ᵒc.55ᵒd.65ᵒ177.INacircleOisthecentre,ADisthediameterandAB,BC,CDarethechord.∠A=50ᵒthen∠O=?a.130ᵒb.50ᵒc.100ᵒd.80ᵒ178.INacirclewithcentreOandradius5cm,ABisachordoflength8cm.ifOMisperpendicularonAB,whatisthelengthofOM?a.4cmb.5cmc.3cmd.noneofthese179.Anequilateral∆ABCisinscribedinacirclewithcentreO.then⦞BOOCisequalto?a.120ᵒb.75ᵒc.180ᵒd.60ᵒ180.Inwhichofthefollowingarethelengthsofdiagonalsequal?a.Rhombusb.Rectanglec.Parallelogramd.Trapezium

173

181.Inacircle,PQisthediameterofacirclewithcentreatO.OSisperpendiculartoPR.ThenOSisequalto?a.¼QRb.1/3QRc.½QRd.QR182.InacircleOMandONaretheperpendiculardrawnonthechordsPQandRsifOM=ON=6cm.then?a.PQ≥RSb.PQ<RSc.PQ≤RSd.PQ=RS183.DiameterABandCDofacircleintersectatO.ifangleBOD=50ᵒthenangleAOD?a.50ᵒb.180ᵒc.130ᵒd.310ᵒ184.INacirclewithcentreO,AOCisadiameterofthecircle,BDisachordandOBandCDarejoined.If⦞AOB=130ᵒthenangleBDC=?a.30ᵒb.25ᵒc.50ᵒd.60ᵒ185.ABandCDareequalchordsofacirclewhosecentreisO.WhenproducedthesechordsmeetatEthen?a.EB=EDb.EA=ECc.EA=EDd.both(a)and(b)186.∆ABCisinscribedinacircle⦞P,⦞Qand⦞RareanglesinscribedinthearcscutoffbysideBC,ACandABrespectively.Then⦞P+⦞Q+⦞Rareequalto?a.180ᵒb.360ᵒc.240ᵒd.noneofthese187.Meetingpointofallmediansiscalledasa.centroidb.orthocenterc.circumcenterd.incenter188.OisthecircumcentreofthetriangleABCwithcircumradius13cm.LetBC=24cmandODisperpendiculartoBC.ThenthelengthofODis.a.7cmb.3cmc.4cmd.5cm189.ABisadiameterofacirclewithcentreO.CDisachordequaltotheradiusofthecircle.ACandBDareproducedtomeetatP.thenthemeasureofangleAPBis?a.120ᵒb.30ᵒc.60ᵒd.90ᵒ190.Pisthepointoutsideacircleandis13cmawayfromitscentre.AsecantdrawnfromthepointPintersectsthecircleatpointsAandBinsuchawaythatPA=9cmandAB=7cm.theradiusofthecircleis?a.5.5cmb.5cmc.4cmd.4.5cm191.ABCDisacyclicquadrilateral.sidesABandDC,whenproducedmeetatthepointPandsidesADandBC,whenproducedmeetatthepointQ.ifangleADC=85ᵒandangleBPC=40ᵒthenangleCQDisequalto?a.30ᵒb.40ᵒc.55ᵒd.85ᵒ192.Twocircleofradii8cmand2cmrespectivelytoucheachotherexternallyatthepointA.

174

PQisthedirectcommontangentofthosetwocirclesofcentersXandYrespectively.ThenlengthofPQisequaltoa.2cmb.3cmc,4cmd.8cm193.A,B,Carethreepointsonacircle.ThetangentatAmeetBCproducedatT,angleBTA=40ᵒ,anglecat=44ᵒ.TheanglesubtendedbyBCatthecentreofthecircleis?a.84ᵒb.92ᵒc.96ᵒd.104ᵒ194.PQisdirectcommontangentoftwocirclesofradiiR1andR2touchingeachotherexternallyatA.thenthevalueofPQ2is?a.R1×R2b.2R1R2c.3R1R2d.4R1R2195.Twocirclewithradii5cmand8cmtoucheachotherexternallyatapointA.ifastraightlinethroughthepointAcutsthecirclesatpointPandQrespectively,thenAP:AQis?a.8:5b.5:8c.3:4d.4:5196.ABandCDaretwoparallelchordsdrownontwooppositesidesoftheirparalleldiametersuchthatAB=6cm,CD=8cm.iftheradiusofthecircleis5cm,thedistancebetweenthechords,incmis?a.2b.7c.5d.3197.Theradiusofacircleis6cm.anexternalpointisatadistanceof10cmfromthecentre.Thenthelengthofthetangentdrawntothecirclefromtheexternalpointuptothepointofcontactis?a.8cmb.10cmc.6cmd.12cm198.Twocircleofradii4cmand9cmrespectivelytoucheachotherexternallyatapointandacommontangenttouchesthematthepointsPandQrespectively.ThentheareaofasquarewithonesidePQ,isa.81cm2b.121cm2c.196cm2d.144cm2199.TwotangentaredrawnfromapointptoacircleatAandB.Oisthecentreofthecircle.ifangleAOP=60ᵒ,thenangleAPBis?a.120ᵒb.90ᵒc.60ᵒd.30ᵒ200.Ifthelengthofachordofacircle,whichmakesanangle45ᵒwiththetangentdrawnatoneendpointofthechord,is6cm,thentheradiusofthecircleis?a.6 2cmb.5cmc.3 2cmd.6cm201.Theradiusofacircleis13cmandXYisachordwhichisatadistanceof12cmfromthecentre.Thelengthofthechordis?a.15cmb.12cmc.10cmd.20cm202.SRisadirectcommontangenttothecirclesofradii8cmand3cmrespectively,theircentersbeing13cmapart.Ifthepointsofcontact,thanthelengthofSRis?a.12cmb.11cmc.17cmd.10cm

175

203.Theradiusoftwoconcentriccirclesare9cmand15cm.ifthechordofthegreatercirclebeatangenttothesmallercircle,thenthelengthofthatchordis?a.24cmb.12cmc.30cmd.18cm204.OandCarerespectivelytheorthocenterandthecircumcentreofanacute-angledtrianglePQR.ThepointPandOarejoinedandproducedtomeetthesideQRatS.ifanglePQS=60ᵒandangleQCR=130ᵒ,thenangleRPS=?a.30ᵒb.35ᵒc.100ᵒd.60ᵒ205.Theratioofcircumradiiandtheinradiiofanequilateraltriangleis?a.2:1b.4:1c.8:1d.1:2206.Theratiooftheareasofthecircumcircleandthein-circleofanequilateraltriangleis?a.2:1b.4:1c.8:1d.3:2207.AB=8cmandCD=6cmaretwoparallelchordsonthesamesideofthecentreofacircle.Thedistancebetweenthemis1cm.theradiusofthecircleis?a.5cmb.4cmc.3cmd.2cm208.Meetingpointofallprependicularsiscalledasa.centroidb.orthocenterc.circumcenterd.incenter209.TWOequalcirclesofradius4cmintersecteachothersuchthateachpassesthroughthecentreoftheother.Thelengthofthecommonchordis?a.2 3cmb.4 3cmc.2 2cmd.8cm210.Thelengthofeachsideofanequilateraltriangleis14 3𝑐𝑚.Theareaofthein-circle,incm2,isa.450b.308c.154d.77211.ThecircumcentreofatriangleABCisO.ifangleBAC=85ᵒandangle=75ᵒ,thenthevalueofangleOACis?a.40ᵒb.60ᵒc.70ᵒd.90ᵒ212.IfIistheincentreof∆ABCandangleA=50ᵒ,thenthevalueofangleBICis?a.25ᵒb.115ᵒc.105ᵒd.80ᵒ213.IfSisthecircumcentreof∆ABCandangleA=50ᵒ,thenthevalueofangleBCSis?a.20ᵒb.40ᵒc.60ᵒd.80ᵒ214.IfIisthein-centreof∆ABCandangleBIC=135ᵒthen∆ABCis?a.acuteangledb.equilateralc.rightangledd.obtuseangled215.IfOisthecircumcentreof∆ABCandangleOBC=35ᵒ,thentheangleBACisequalto?

176

a.55ᵒb.110ᵒc.70ᵒd.35ᵒ216.Circumcentreof∆ABCisO.ifangleBAC=85ᵒ,angleBCA=80ᵒ,thenangleOACis?a.80ᵒb.30ᵒc.60ᵒd.75ᵒ217.OisthecentreofacircleandarcABCsubtendsanangleof130ᵒatO.ABisextendedtoP.thenanglePBCis?a.75ᵒb.70ᵒc.65ᵒd.80ᵒ218.ChordsABandCDofacircleintersectexternallyatP.IfAB=6cm,CD=18cmandPD=40cm,thenthelengthofPAis?a.5cmb.6cmc.4cmd.5.5cm219.TwocirclestoucheachexternallyatpointAandPQisadirectcommontangentwhichtouchesthecirclesatPandQrespectively.ThenanglePAQ=?a.45ᵒb.90ᵒc.80ᵒd.100ᵒ220.ThelengthoftwochordsABandACofacircleare8cmand6cmandangleBAC=90ᵒ,thentheradiusofcircleis?a.25cmb.20cmc.4cmd.5cm221.InternalbisectorsofanglesBandCofatriangleABCmeetatO.IfangleBAC=80ᵒ,thenthevalueofangleBOCis?a.120ᵒb.140ᵒc.110ᵒd.130ᵒ222.Theangleofatriangleare(x+5)ᵒ,(2x-3)ᵒand(3x+4)ᵒ.Thevalueofxis?a.30b.31c.29d.28223.InatriangleABC,incentreisOandangle110ᵒ,thenthemeasureofangleBACis?a.20ᵒb.40ᵒc.55ᵒd.110ᵒ224.DisanypointonsideACof∆ABC.ifP,Q,x,Y,arethemid-pointsofAB,BC,AD,andDCrespectively,thentheratioofPXandQYis?a.1:2b.1:1c.2:1d.2:3225.LetObetheincentreofatriangleABCandDbeapointonthesideBCof∆ABC,suchthatODisperpendicularonBC.IfangleBOD=15ᵒ,thenangleABC=?a.75ᵒb.45ᵒc.150ᵒd.90ᵒ226.In∆ABC,PQisparalleltoBC.IfAP:PB=1:2andAQ=3cm,ACisequalto?a.6cmb.9cmc.12cmd.8cm227.Theratiobetweenthenumberofsidesoftwopolygonis2:1andtheratiobetweentheirinteriorangleis4:3.Thenumberofsidesofthesepolygonsisrespectively?a.8,4b.10,5c.12,6d.14,7

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228.Oistheincentreof∆ABCandangleA=30ᵒ,thenangleBOCis?a.100ᵒb.105ᵒc.110ᵒd.90ᵒ229.In∆ABC,angleBAC=90ᵒandAB=½BC.ThenthemeasureofangleABCis?a.60ᵒb.30ᵒc.45ᵒd.15ᵒ230.AstraightlineparalleltothebaseBCofthetriangleABCintersectsABandAcatthepointDandErespectively.Iftheareaofthe∆ABEbe36cm2.Thentheareaofthe∆ACDis?a.18cm2b.36cm2c.18cm2d.36cm2231.In∆ABC,ADistheinternalbisectorofangleA.metingthesideBCatD.ifBD=5cm,BC=7.5cm.ThenAB:ACis?a.2:1b.1:2c.4:5d.3:5232.Gisthecentroidof∆ABC.IfAG=BC,thenangleBGCis?a.90ᵒb.30ᵒc.60ᵒd.120ᵒ233.In∆ABC,angleB=60ᵒ,anglec=40ᵒ.ifADbisectsangleBACandAEisperpendicularonBC,thenangleEADis?a.10ᵒb.20ᵒc.40ᵒd.80ᵒ234.Consider∆ABDsuchthatangleADB=20ᵒandCisapointonBDsuchthatAB=ACandCD=CA.ThenthemeasureofangleABCis?a.40ᵒb.45ᵒc.60ᵒd.30ᵒ235.The3mediansAD,BEandCFof∆ABCintersectatpointG.iftheareaof∆ABCis60cm2.thentheareaofthequadrilateralBDGFis?a.10cm2b.15cm2c.20cm2d.30cm2236.IfGisthecentroidandAD,BE,CFarethreemediansof∆ABCwitharea72cm2,,thentheareaof∆BDGis?a.12cm2b.16cm2c.24cm2d.8cm2237.In∆ABC,ADisthemedianandAD=½BC.IfangleBAD=30ᵒ,thenmeasureofangleACBis?a.90ᵒb.45ᵒc.30ᵒd.60ᵒ238.IfGbethecentroidof∆ABCandtheareaof∆GBDis6cm2.WhereDisthemid-pointofsideBC,thentheareaof∆ABCis?a.18cm2b.12cm2c.24cm2d.36cm2

239.Iftheratioofareasoftwosimilartrianglesis4:9,thentheratiooftheircorrespondingsidesis?a.2:3b.4:3c.4:5d.4:3

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240.LetBEandCFbethe2mediansofa∆ABCandGbetheirintersection.AlsoletEFcutAGatO.ThenAO:OGis?a.1:1b1:2c.2:1d.3:1241.Iftheratioofareasoftwosimilartrianglesis9:16,thentheratiooftheircorrespond-dingsidesis?a.3:5b.3:4c.4:5d.4:3242.In∆ABC,PandQarethemiddlepointsofthesidesABandACrespectively.RisapointonthesegmentPQsuchthatPR:RQ=1:2.ifPR=2cm,thenBC=?a.4cmb.2cmc.12cmd.6cm243.ABCisatriangle.theinternalbisectoroftheanglesA,angleBandangleCintersectthecircumcircleatX,YandZrespectively.IfangleA=50ᵒ,angleCZY=30ᵒ,thenangleBYZ,willbe?a.45ᵒb.55ᵒc.35ᵒd.30ᵒ244.ABCisaright-angledtriangle.ADisperpendiculartothehypotenuseBC.IfAC=2AB,thenthevalueofBDis?a.BC/2b.BC/3c.BC/4d.BC/5245.IFGisthecentroidandADbeAmedianwithlength12cmof∆ABC.ThenthevalueofAGis?a.4cmb.8cmc.10cmd.6cm246.Theperimetersof2similartriangles∆ABCand∆PQRare36cmand24cmrespectively.IfPQ=10cm,thenABis?a.25cmb.10cmc.15cmd.20cm247.DandEarethemid-pointsofABandACof∆ABC,BCisproducedtoanypointPDE,DPandEParejoined.Then?a.∆PED=¼∆ABCb.∆PED=∆BECc.∆ADE=∆BECd.∆BDE=∆BEC248.IfGisthecentroidof∆ABCandAG=BC,thenangleBGCis?a.75ᵒb.45ᵒc.90ᵒd.60ᵒ249.∆ABCand∆DEFaresimilarandtheirareasarerespectively64cm2and121cm2.IfEF=15.4cmBCis?a.12.3cmb.11.2cmc.12.1cmd.11.0cm250.Whatistheratioofin-radiustothecircumradiusofarightangledtriangle?a.1:2b.1: 2c.2:5d.can’tbedetermined251.InABC,Gisthecentroid,AB=15cm,BC=18cmandAC=25cm,findGD,whereDisthemid-pointofBC?

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a. 86/3cmb.2 86/3c.8 15/3cmd.noneofthese252.In∆ABC,AB2+AC2=2500cm2andmedianAD=25cm,findBC?a.25cmb.40cmc.50cmd.48cm253.Areaof∆ABC=30cm2.DandEarethemid-pointsofBCandABrespectively.Findareaof∆ADE?a.10cm2b.7.5cm2c.15cm2d.noneofthese254.ADisthemedianofatriangleABCandOisthecentroidsuchthatAO=10cm.thelengthofODincmis?a.4b.5c.6d.8255.Inanequilateraltriangleofside2a,calculatethelengthofitsaltitude?a.2a 3b.a 3c.a 3/2d.noneofthese256.TheinternalbisectorsofangleBandangleCof∆ABCmeetatO.ifangleA=80ᵒthenangleBOCis?a.50ᵒb.160ᵒc.100ᵒd.130ᵒ257.Onesideotherthanthehypotenuseofrightangleisoscelestriangleis6cm.Thelengthoftheperpendicularonthehypotenusefromtheoppositevertexis?a.6cmb.6 2cmc.4cmd.3 2cm258.Thetriangleisformedbyjoiningthemid-pointsofthesidesAB,BCandCAof∆ABCandtheareaof∆PQRis6cm2,thentheareaof∆ABCis?a.36cm2b.12cm2c.18cm2d.24cm2259.Whatistheratioofsideandheightofanequilateraltriangle?a.2:1b.1:1c.2: 3d. 3:2260.Thedifferencebetweenaltitudeandbaseofarightangledtriangleis17cmanditshypotenuseis25cm.whatisthesumofthebaseandaltitudeofthetriangle?a.24cmb.31cmc.34cmd.can’tbedetermined261.In∆ABC,AB=5cm,AD=7cm.IfADistheanglebisectorofangleA.thenBD:CDis?a.25:49b.49:25c.6:1d.5:7262.ABCDisarhombuswhosesideAB=4cmandangleABC=120ᵒ,thenthelengthofdiagonalBDisequalto?a.1cmb.2cmc.3cmd.4cm263.Ifanexteriorangleofacyclicquadrilateralbe50ᵒ,thentheinterioroppositeangleis?a.130ᵒb.40ᵒc.50ᵒd.90ᵒ

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264.Measureofeachinteriorangleofaregularpolygoncanneverbe?a.150ᵒb.105ᵒc.108ᵒd.144ᵒ265.ABCDisacyclictrapeziumwhosesidesADandBCareparalleltoeachother.ifangleABC=72ᵒ,thenthemeasureoftheangleBCDis?a.162ᵒb.18ᵒc.108ᵒd.72ᵒ266.ThelengthofthediagonalBDoftheparallelogramABCDis18cm.ifPandQarethecentroidofthe∆ABCand∆ADCrespectivelythenthelengthofthelinesegmentPQis?a.4cmb.6cmc.9cmd.12cm267.Eachinteriorangleofaregularpolygonis18ᵒmorethaneighttimesanexteriorangle.thenumberofsideofthepolygonis?a.10b.15c.20d.25268.ThesideABofAparallelogramABCDisproducedtoEinsuchwaythatBE=AB.DEintersectsBCatQ.ThepointQdividesBCintheratio?a.1:2b.1:1c.2:3d.2:1269.Ifthemeasureofeachinteriorangleofaregularpolygonbe144ᵒ,thenumberofsidesofthepolygonis?a.10b.20c.24d.36270.Eachinteriorangleofaregularpolygonis144ᵒ.Thenumberofsidesofthepolygonis?a.8b.9c.10d.11271.IfthelengthofthesidePQoftherhombusPQRSis6cmandanglePQR=120ᵒ,thenthelengthofQSincm,is?a.4b.6c.3d.5272.Ifeachinteriorangleisdoubleofeachexteriorangleofaregularpolygonwithnsides,thenthevalueofnis?a.8b.10c.5d.6273.Ifaregularpolygonhaseachofitsanglesequalto3/5timesoftworightangles,thenthenumberofsidesis?a.3b.5c.6d.8274.Ifeachinteriorangleofaregularpolygonis150ᵒ,whatisthenumberofsidesofapolygon?a.4b.8c.12d.16275.InaquadrilateralABCD,IfAOandBOarethebisectorsofangleAandangleBrespectively,angleC=70ᵒandangleD=30ᵒ,thenangleAOB=?a.40ᵒb.50ᵒc.80ᵒd.100ᵒ

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276.Howmanysidesdoesaregularpolygonhavewhoseinteriorandexterioranglesareintheratio2:1?a.3b.5c.6d.12277.Twoparallelogramsstandonequalbasesandbetweenthesameparallels.Theratiooftheirareasis?a.1:2b.2:1c.1:3d.1:1278.Anycyclicparallelogramisa?a.rhombusb.trapeziumc.quadrilaterald.rectangle279.Theexteriorangleofaregularpolygonisonethirdofitsinteriorangle,thenumberofsidesofthepolygonis?a.2b.4c.6d.8280.Apolygonhas35diagonals,thenumberofsidesofthepolygonis?a.4b.6c.8d.10281.InaparallelogramABCD,thebisectorofangleAandangleBmeetatO.thenangleAOBisequalto?a.85ᵒb.90ᵒc.110ᵒd.noneofthese282.Howmanydiagonalsarethereinanoctagon?a.8b.16c.18d.20283.Aregularpolygonisinscribedinacircle.ifasidesubtendsanangleof72ᵒatthecentre,thenthenumberofsidesofthepolygonis?a.5b.7c.6d.8284.AregularhexagonisinscribedinacirclewithcentreO.thentheanglesubtendedbyeachsideofthesquareatthecentreOis?a.80ᵒb.90ᵒc.60ᵒd.45ᵒ285.Ifanangleofaparallelogramis2/3ofitsadjacentangle,thesmallestangleoftheparallelogramis?a.108ᵒb.54ᵒc.72ᵒd.81ᵒ286.IfABCDisaparallelograminwhichPandQarethecentroidsof∆ABDand∆BCD,thenPQequals?a.AQb.APc.BPd.DQ287.ABCDisaparallelogramandBDisadiagonalangleBAD=65ᵒandangleDBC=45ᵒthenangleBDCis?a.65b.70c.20d.noneofthese288.Theratioofaninteriorangletotheexteriorangletotheexteriorangleofaregular

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polygonis4:1.Thenumberofsidesofpolygonis?a.10b.11c.12d.14289.Thedifferencebetweentheinteriorangleandexterioranglesofaregularpolygonis60ᵒ.Thenumberofsidesofpolygonis?a.4b.5c.6d.8290.Theratioofaninteriorangletotheexteriorangletotheexteriorangleofaregularpolygonis5:1.Thenumberofsidesofpolygonis?a.10b.11c.12d.14291.Theexteriorangleofaregularpolygonisonethirdofitsinteriorangle,thenumberofsidesofpolygonis?a.2b.4c.6d.8292.Theareaofthelargesttrianglethatcanbeinscribedinasemicircleofradiusxinsquareunitis?a.4x2b.x2c.2x2d.3x2293.Theexteriorangleofaregularpolygonisonefourthofitsinteriorangle,thenumberofsidesofpolygonis?a.7b.5c.10d.8294.Findouttheratiooftheareaoftheinscribedandcircumscribedcircleofthesquare?a. 2:1b.1: 2c. 2:1d.1:2295.Areaofthetrapeziumformedbyx-axis;y-axisandthelines3x+4y=12and6x+8y=60is:a.37.5sq.unitb.31.5sq.unitc.48sq.unitd.36.5sq.unit296.Thelengthofthesideofasquareis14cm.Findouttheratiooftheradiioftheinscribedandcircumscribedcircleofthesquare?a. 2:1b.1: 2c. 2:1d.2:1297.Theperimeterofarhombusis146cmandoneofitsdiagonalsis55cm.theotherdiagonalis?a.92cmb.73cmc.48cmd.72cm298.Ifacirclewithradiusof10cmhastwoparallelchords16cmand12cmandtheyareonthesamesideofthecentreofthecircle,thenthedistancebetweenthetwoparallelchordsis?a.2cmb.3cmc.5cmd.8cm299.Ifthelengthofachordofacircleatadistanceof12cmfromthecentreis10cm,thenthediameterofthecircleis?a.13cmb.15cmc.26cmd.30cm

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300.Areaofthein-circleofanequilateraltrianglewithside6cmis?a.π/2cm2b. 3πcm2c.6πcm2d.3πcm2301.Ifthecircumradiusofanequilateraltrianglebe10cm,thenthemeasureofitsin-radiusis?a.5cmb.10cmc.20cmd.15cm302.Atthecentresoftwocircles,twoarcsofequallengthsubtendanglesof60ᵒand75ᵒrespectively.Theratiooftheradiiofthetwocirclesis?a.5:2b.5:4c.3:2d.2:1303.Ametalwirewhenbentintheformofasquareenclosesanarea484cm2,ifthesamewireisbentintheformofacircle,thenitsareais?a.308cm2b.506cm2c.600cm2d.616cm2304.Sidesofaparallelogramareintheratio5:4.itsareais1000sq.units.Altitudeonthegreatersideis20units.Altitudeonthesmallersideis?a.30unitsb.25unitsc.10unitsd.15units305.Theperimeterofarhombusis40cmandthemeasureofanangleis60ᵒ,thentheareaofitis?a.100 3cm2b.50 3cm2c.160 3cm2d.100cm2306.Theadjacentsideofaparallelogramare36cmand27cminlength.Ifthedistancebetweentheshortersidesis12cm.thenthedistancebetweenthelongersidesis?a.10cmb.12cmc.16cmd.9cm307.Thelengthofaroomfloorexceedsitsbreadthby20m.Theareaofthefloorremainsunalteredwhenthelengthisdecreasedby10mbutthebreadthisincreasedby5m.theareaofthefloor(inm2)is?a.280b.325c.300d.420308.Arightangledisoscelestriangleisinscribedinasemi-circleofradius7cm.theareaenclosedbythesemi-circlebutexteriortothetriangleis?a.14cm2b.28cm2c.44cm2d.68cm2309.Theradiioftwocirclesare5cmand3cm,thedistancebetweentheircentresis24cm.Thenthelengthofthetransversecommontangentis?a.16cmb.15 2cmc.16 2cmd.15cm310.Eachoftheheightandradiusofthebaseofarightcircularconeisincreasedby100%.Thevolumeoftheconewillbeincreasedby?a.700%b.500%c.300%d.100%311.Theperimeterofarhombusis150cmandoneofitsdiagonalsis10cm.theotherdiagonal

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is?a.30cmb.60cmc.75cmd.70cm312.Theratioofthevolumeofacubetothatofasphere,whichwillexactlyfitinsidethecubeis?a.2:πb.π:6c.6:πd.8:π313.Acubeofside1meterisreduced3timesintheratio1:2.theareaofonefaceofthereducedcubetothatoftheoriginalcubeisintheratio?a.1:4b.1:8c.1:16d.1:64314.Iftheradiusofasphereisincreasedby2m,itssurface-areaisincreasedby704m2.Whatwastheradiusoftheoriginalsphere?a.16mb.15mc.14md.12m315.IfasphereofradiusRisdividedintofouridenticalparts,thenthetotalsurfaceareaofthefourpartsis?a.4πR2b.2πR2c.8πR2d.3πR2316.Threesphericalballsofradius1cm,2cm,and3cmaremeltedtoformasinglesphericalball.Intheprocess,thelossofmaterialis25%.Theradiusofthenewballis?a.6cmb.5cmc.3cmd.2cm317.Thebaseofarightprismisanequilateraltriangleofarea173cm2andthevolumeoftheprismis10380cm3.Theareaofthelateralsurfaceoftheprismis(use 3=1.73)?a.1200cm2b.2400cm2c.3600cm2d.4380cm2318.Thereisapyramidonabasewhichisaregularhexagonofside2acm.ifeveryslantedgeofthispyramidisoflength5a/2cm,thenthevolumeofthispyramidis?a.3a3cm3b.3 2a3cm3c.3 3a3cm3d.6a3cm3319.Thebaseofarightprismisanequilateraltriangleofside8cmandheightoftheprismis10cm.Thenthevolumeoftheprismis?a.320 3cm3b.160 3cm3c.150 3cm3d.300 3cm3320.Theheightofaconeis30cm.Asmallconeiscutoffatthetopbyaplaneparalleltothebase.Ifitsvolumebe1/27ofthevolumeofthegivencone,atwhatheightabovethebase,thesectionhasbeenmade?a.10cmb.12cmc.16cmd.20cm

Answers

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1.D 2.C 3.A 4.B 5.C 6.D 7.A 8.A 9.D 10.C11.D 12.C 13.C 14.B 15.C 16.A 17.A 18.C 19.D 20.B21.A 22.B 23.D 24.B 25.D 26.D 27.C 28.B 29.B 30.C31.B 32.D 33.C 34.A 35.B 36.D 37.C 38.B 39.A 40.A41.D 42.C 43.B 44.D 45.A 46.B 47.B 48.D 49.D 50.D51.A 52.C 53.A 54.D 55.C 56.B 57.B 58.B 59.B 60.B61.A 62.A 63.A 64.D 65.A 66.C 67.C 68.D 69.D 70.C71.B 72.C 73.A 74.B 75.C 76.C 77.D 78.D 79.C 80.D81.B 82.D 83.C 84.B 85.B 86.A 87.B 88.B 89.C 90.B91.A 92.D 93.B 94.A 95.A 96.C 97.D 98.A 99.C 100.B101.A 102.A 103.B 104.B 105.C 106.C 107.B 108.C 109.C 110.A111.C 112.A 113.C 114.A 115.B 116.B 117.C 118.A 119.B 120.B121.C 122.B 123.D 124.A 125.D 126.C 127.C 128.D 129.C 130.B131.A 132.C 133.B 134.C 135.A 136.B 137.B 138.D 139.C 140.B141.D 142.D 143.A 144.C 145.A 146.D 147.D 148.C 149.C 150.D151.D 152.D 153.B 154.D 155.B 156.C 157.C 158.D 159.B 160.C161.C 162.C 163.D 164.C 165.C 166.B 167.D 168.A 169.D 170.B171.B 172.A 173.A 174.A 175.C 176.D 177.A 178.C 179.A 180.B181.C 182.D 183.C 184.B 185.B 186.B 187.A 188.D 189.C 190.A191.A 192.D 193.D 194.D 195.A 196.B 197.A 198.D 199.D 200.C201.C 202.A 203.A 204.B 205.A 206.A 207.A 208.B 209.B 210.C211.A 212.B 213.B 214.C 215.A 216.D 217.C 218.C 219.B 220.D221.D 222.C 223.B 224.B 225.C 226.B 227.B 228.B 229.A 230.D231.A 232.A 233.A 234.A 235.C 236.A 237.D 238.D 239.A 240.D241.B 242.C 243.C 244.D 245.B 246.C 247.A 248.C 249.B 250.D251.D 252.C 253.B 254.B 255.B 256.D 257.D 258.D 259.C 260.B261.D 262.D 263.C 264.B 265.C 266.B 267.C 268.B 269.A 270.A271.B 272.D 273.B 274.C 275.B 276.C 277.D 278.D 279.D 280.D281.B 282.D 283.A 284.C 285.C 286.B 287.B 288.A 289.C 290.C291.D 292.B 293.C 294.D 295.B 296.B 297.C 298.A 299.C 300.D301.A 302.B 303.D 304.B 305.B 306.D 307.C 308.B 309.C 310.A311.A 312.B 313.C 314.D 315.A 316.C 317.C 318.C 319.B 320.D

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TrigonometryIfoneangleofatriangleis90degreesandoneoftheotheranglesisknown,thethirdistherebyfixed,becausethethreeanglesofanytriangleaddupto180degrees.Thetwoacuteanglesthereforeaddupto90degrees:theyarecomplementaryangles.Theshapeofatriangleiscompletelydetermined,exceptforsimilarity,bytheangles.Oncetheanglesareknown,theratiosofthesidesaredetermined,regardlessoftheoverallsizeofthetriangle.Ifthelengthofoneofthesidesisknown,theothertwoaredetermined.TheseratiosaregivenbythefollowingfunctionsoftheknownangleA,wherea,bandcrefertothelengthsofthesidesintheaccompanyingfigure:

Sinefunction(sin),definedastheratioofthesideoppositetheangletothehypotenuse.

SinA=opposite/hypotenuse=a/cCosinefunction(cos),definedastheratiooftheadjacentlegtothehypotenuse.CosA=adjacent/hypotenuse=b/cTangentfunction(tan),definedastheratiooftheoppositelegtotheadjacentleg.

TanA=opposite/adjacent=a/b=sinA/cosA

Thehypotenuseisthesideoppositetothe90degreeangleinarighttriangle;itisthelongestsideofthetriangle,andoneofthetwosidesadjacenttoangleA.TheadjacentlegistheothersidethatisadjacenttoangleA.TheoppositesideisthesidethatisoppositetoangleA.Thetermsperpendicularandbasearesometimesusedfortheoppositeandadjacentsidesrespectively.Thereciprocalsofthesefunctionsarenamedthecosecant(cosecorcsc),secant(sec),andcotangent(cot),respectively:

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CosecA=1/sinA=c/aSecA=1/cosA=c/bCotA=1/tanA=cosA/sinA=b/aComplementaryanglesinTrigonometryComplementaryanglesintrigonometry:Twoanglesaresaidtobecomplementary,iftheirsumis900.Itfollowsfromtheabovedefinitionthatθand(90-θ)arecomplementaryanglesintrigonometryforanacuteangleθInΔABC,∠B=900∴∠A+∠C=900∠C=900-∠Asin(900-A)=cosAtan(900-A)=cotAsec(900-A)=cosecAcos(900-A)=sinAcot(900-A)=tanAcosec(900-A)=secATrigonometricEquationsAnequationinvolvingtrigonometricratiosofanangleθ(say)issaidtobeatrigonometricequations,ifitissatisfiedforallvaluesofθforwhichthegiventrigonometricratiosaredefined.SomeTrigonometricequations(Identities)areasfollows:1)Sin²θ+cos2θ=12)sin2θ=1-cos2θ3)cos2θ=1-sin2θ4)1+tan2θ=sec2θ5)tan2θ=sec2θ-16)sec2θ-tan2θ=17)1+cot2θ=cosec2θ8)cot2θ=cosec2θ-19)cosec2θ-cot2θ=1TheseTrigonometricequationsaretrueforanyangleθforwhichthetrigonometricratiosaremeaningful.

188

QuadrantsBelowisasimplediagramtohelpyoudeterminethesign(positiveornegative)ofthetrigratiointheirrespectivequadrants.Wecallthisdiagramthe‘ASTC'diagram

Forexampleweneedtofindvalueofsine1300

θIsinthe2ndquadrantandthebasicangleofθis1300.ThebasicangleismeasuredastheacuteanglewhichOPmakeswiththex-axis.Sincesineispositiveinthe2ndquadrantasseeninthe‘ASTC'diagram,sin 130° = sin 50°.Thusthetrigratiowouldbesine130°

189

Trigonometricvalues

RadiansIntrigonometryweconsiderπ=180°

Theradianmeasure,θ,oftheangleAOBisdefinedby:

S=lengthofarcr=radiusofcircleToconvertbetweendegreesandradians:

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1.Multiplyadegreemeasureby 0180rad π andsimplifytoconverttoradians.

2.Multiplyaradianmeasurebyrad

1800

πandsimplifytoconverttodegrees.

Chartofsomepopularradians

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Someimportantpoints:-

1.Thelengthsofthesidesofa450-450-900triangleareintheratioof1:1:√2.

2.Thelengthsofthesidesofa30°-60°-90°triangleareintheratioof1:√3:2

3.𝝅radian=𝟏𝟖𝟎°

4.If𝒙 + 𝟏𝒙=2then𝒙=1&If𝒙 + 𝟏

𝒙=−2then𝒙=−1

5.Pythagorastriplet:(3,4,5),(5,12,13),(8,15,17),(7,24,25),(9,40,41),(20,21,29)

6.Twoanglesaresaidtobecomplementary,iftheirsumis900.

7.Twoanglesaresaidtobesupplementary,iftheirsumis1800.

8.IftanA.tanB=1thenA&BarecomplementaryanglesitmeansA+B=900

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Example1:-cosec(90o−A)=a.cosecA b.sinA c.cosA d.tanA e.secASolution:cosec(90o−A)=1/sin(90o−A)=1/cosA=secA.Example2:-sec(90o−A)=a.secA b.sinA c.cosA d.cosecA e.tanASolution:sec(90o−A)=1/cos(90o−A)=1/sinA=cosecA.Example3:-cot(90o−A)=a.tanA b.cosA c.sinA d.secA e.cosecASolution:cot(90o−A)=1/tan(90o−A)=1/cotA=tanA.Example4:-sin(90o−A)/cos(90o−A)=a.tanA b.cot(90o−A) c.cotA d.sinA e.noneSolution:Weknowsin(90o−A)=cosA.Similarlycos(90o−A)=sinA.Substitutingthesevaluesintheexpression,wegetsin(90o−A)/cos(90o−A)=cosA/sinA=cotA.Example5:-cos(90o−A)/sin(90o−A)=a.tanA b.cotA c.tan(90o−A) d.sec(90o−A) e.noneSolution:Substitutingthevaluescos(90o−A)=sinAandsin(90o−A)=cosAintheexpression,wegetcos(90o−A)/sin(90o−A)=sinA/cosA=tanA.Example6:-[cosec2A−1]scos(90o−A)/sin(90o−A)=a.tanA b.sinA c.cotA d.cosASolution:Thevalueofcosec2A−1=cot2A.Alsothevalueofcos(90o−A)/sin(90o−A)=sinA/cosA=tanA.Substitutingthesevalues,theanswerisfoundtobecot2AtanA=cotA.Example7:-cotA[cos(90o−A)/sin(90o−A)]=a.tanA b.cot2A c.tan2A d.cotA e.1Solution:Thevalueofcos(90o−A)/sin(90o−A)=sinA/cosA=tanA.ThereforethegivenexpressionreducestocotAtanAwhichequals1.Example8:-Thevalueoftan45o−cos45osin45oisa.1/2 b.1 c.0 d.3/4 e.1/4Solution:Fromthetableofvaluesoftrigonometricfunctions,tan45o−cos45osin45o=1−(1/2)=1/2.Example9:-Thevalueofsin230o+cos230oisa.0 b.1 c.1/2d.3/2 e.1/4Solution:Fromthetableofvaluesoftrigonometricfunctions,sin230o+cos230o=¼+¾=1.Butitshouldberememberedthatsin2A+cos2A=1,forallvaluesofA.

193

Example10:-Thevalueoftan45o+cos0+sin90oisa.2 b.1 c.1/2d.0 e.3Solution:Fromthetableofvaluesoftrigonometricfunctions,tan45o+cos0+sin90o=1+1+1=3.Example11:-Thevalueoftan60ocos30o−sin60otan30oisa.0 b.1/2 c.3/2 d.1 e.2Solution:Fromthetableofvaluesoftrigonometricfunctions,tan60ocos30o−sin60otan30o=(3/2)−(1/2)=1.Example12:-Thevalueof[1+sin60o+sin230o+sin260o+sin445o][cos30o−sin60o]isa.1 b.13/12 c.12/13 d.1/2 e.0Solution:Thevalueofthesecondterm[cos30o−sin60o]is0.Hencethevalueoftheentireexpressionis0,irrespectiveofthevalueofthefirstterm.Example13:-(sinA+cosA)2−2sinAcosA=a.2 b.0 c.1 d.tanA e.sin2A−cos2ASolution:(sinA+cosA)2−2sinAcosA=sin2A+cos2A=1.Example14:-sin2A−sec2A+cos2A+tan2A=a.1 b.0 c.cotA d.cosecA e.cosec2ASolution:Herethetermsneedtobegroupedproperly.Thegivenexpressioncanbewrittenas(sin2A+cos2A)−(sec2A−tan2A)=1−1=0.Example15:-1/(1+cot2A)+1/(1+tan2A)=a.0 b.sin2A c.1 d.cos2A e.sin2A/cos2ASolution:Theexpression1/(1+cot2A)+1/(1+tan2A)=1/cosec2A+1/sec2A=sin2A+cos2A=1.Example16:-cotAtanA=a.sinA b.cosA c.sinAcosA d.1 e.1/(sinAcosA)Solution:cotA=1/tanA.HencecotAtanA=1.AlternativelycotA=cosA/sinAandtanA=sinA/cosA.SocotAtanA=(cosA/sinA)(sinA/cosA)=1.Example17:-

Fromthefigure,thevalueofcosecA+cotAisa.(a+b)/c b.(b+c)/a c.a/(b+c) d.b/(a+c)e.(a+c)/bSolution:-WeknowcosecA=b/aandcotA=c/a.HencecosecA+cotA=(b+c)/a.

194

Example18:-Whichofthefollowingrelationshipsistrue:a.sinAcotA=1b.sinA+cosecA=1c.cosAsecA=1d.secA-cosA=1e.secAcotA=1Solution:cosAsecA=1Bydefinition,secA=1/cosA.SocosAsecA=1istrue.Example19:-

Fromthefigure,thevalueofsin2A+cos2Aisa.a/b+c/b b.b/a+c/b c.1 d.(a/b+c/b)2 e.(b/a+c/b)2Solution:Thisquestionisabittricky.WeknowsinA=a/bandcosA=c/b.Sosin2A+cos2A=(a2+c2)/b2.ByPythagorasTheorem,a2+c2=b2foraright-angledtriangle.Hencesin2A+cos2A=1,whichisafamousidentity.Example20:-Fromthefigure,thevalueofcotC+cosecCis

a.(a+c)/b b.(a+b)/c c.(c+b)/a d.a/c+c/b e.c/a+b/cSolution:cotCisBase/OppositeSideandcosecCisHypotenuse/OppositeSide.Fromthesedefinitions,thevaluesofcotCandcosecCaregivenbya/candb/crespectively.Hencetheansweris(a+b)/c.Example21:-cosecA/secA=a.tanA b.sinA c.cosA d.cotA e.sinA+cosASolution:Bydefinition,cosecA=1/sinAandsecA=1/cosA.SocosecA/secA=cosA/sinA=cotA.Example22:-Forthefiguregivenontheright,thevalueofcotAis

195

a.sinA/cosA b.tanC c.cosC/sinCd.a/c e.c/bSolution:ThevalueofcotAisc/a.SimilarlythevalueoftanCisc/a.HencecotA=tanC.Example23:-Theangleofelevationofthetopofatower30mhigh,fromtwopointsonthelevelgroundonitsoppositesidesare45degreesand60degrees.Whatisthedistancebetweenthetwopoints?a.30 b.51.96 c.47.32 d.81.96Solution:LetOTbetetower.Therefore,Heightoftower=OT=30mLetAandBbethetwopointsonthelevelgroundontheoppositesideoftowerOT.Then,angleofelevationfromA= TAO=45oandangleofelevationfromB= TBO=60oDistancebetweenAB=AO+OB=x+y(say)Now,inrighttriangleATO,AO=OT=30&inrighttriangleBTO OB=30/√3=30√3/3=10√3=10×1.732=17.32Hence,therequireddistance=x+y=30+17.32=47.32m

Example24:-IfSin(A+B)= 3/2andSin(A-B)=1/2thenwhatarethevaluesofAandB?(giventhatbothAandBareacuteanglesandA>B)

a.30,60 b.45,45 c.45,15 d.noneofabove

Solution:

Sin(A+B)=root3/2(thisisgiveninthequestionitself).

Ifyoulookatthetable,sin60= 3/2.ThatmeansA+B=60………eq1

Similarlywe’llgetA-B=30……….eq2

Sowe’vetwoequations:

A+B=60&A-B=30

Nowaddthesetwoequationseq1+eq2

(A+B)+(A-B)=60+30

2A=90

A=45

Putthisvaluebackineq1(oreq2).AndyougetB=15

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FinalanswerC:45,15

Example25:-Findthevalueofcos2×cos4×cos6×cos8×….×cos92

Solution:

Asyoucanseetheanglesareincreasingasperthemultiplicationtableof2.Sointhechain,you’llalsogetcos90(because2x45=90).Sowecanwritethechainas

cos2×cos4×cos6×cos8×….cos88×cos90×cos92

butweknowthatcos90=0,hencethewholemultiplicationwillbecomezero.

Example26:-Theangleofelevationofthetopofatowerfromapointontheground,whichis30mawayfromthefootofthetower,is30°.Findtheheightofthetower.

Solution:

tan30°=p/30orp=30/ 3=10 3=17.32

Example27:-Akiteisflyingataheightof60mabovetheground.Thestringattachedtothekiteistemporarilytiedtoapointontheground.Theinclinationofthestringwiththegroundis60°.Findthelengthofthestring,assumingthatthereisnoslackinthestring.

Solution:

Sin60°=p/h=60/ℎ

Or 3/2=60/horh=120/ 3=40 3

Example28:-A1.5mtallboyisstandingatsomedistancefroma30mtallbuilding.Theangleofelevationfromhiseyestothetopofthebuildingincreasesfrom30°to60°ashewalkstowardsthebuilding.Findthedistancehewalkedtowardsthebuilding.

Solution:AB=30m(heightofbuilding)

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DC=EG=1.5m(heightofboy)AngleD=30°AngleE=60°AF=30-1.5=28.5mIn∆AFD,tan30°=AF/FD=28.5/FDor1/ 3=28.5/FDFD=28.5 3In∆AFE,tan60°=AF/FE=28.5/FEor1/ 3=28.5/FEFD=28.5/ 3Requireddistance=ED=28.5 3-28.5/ 3=19 3Example29:-Fromapointontheground,theanglesofelevationofthebottomandthetopofatransmissiontowerfixedatthetopofa20mhighbuildingare45°and60°respectively.Findtheheightofthetower.

Solution:Heightofbuilding=DB=20mAngleDCB=45°In∆DBC,tan45°=1=DB/BCorDB=BC=20mIn∆ABC,tan60°= 3=AB/BCorAB=20 3NowAD=AB–DB=20 3 − 20=14.64mExample30:-Astatue,1.6mtall,standsonthetopofapedestal.Fromapointontheground,theangleofelevationofthetopofthestatueis60°andfromthesamepointtheangleofelevationofthetopofthepedestalis45°.Findtheheightofthepedestal.

Solution:Heightofstatute=AD=1.6mAngleACB=60°AngleDCB=45°In∆DBC,tan45°=1=DB/BC

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OrBC=DBIn∆ABC,tan60°= 3=AB/BCorBC=AB/ 3=(BC+16)/ 3orBC 3-BC=16orBC=16/0.732=2.18mExample31:-Theangleofelevationofthetopofabuildingfromthefootofthetoweris30°andtheangleofelevationofthetopofthetowerfromthefootofthebuildingis60°.Ifthetoweris50mhigh,findtheheightofthebuilding.

Solution:heightoftower=AB=50mAngleACB=60°AngleDBC=60°In∆ABC,tan60°= 3=50/BCorBC=50/ 3In∆DCB,tan30°=1/ 3=DC/BCorDC=50/3mExample32:-Twopolesofequalheightsarestandingoppositeeachotheroneithersideoftheroad,whichis80mwide.Fromapointbetweenthemontheroad,theanglesofelevationofthetopofthepolesare60°and30°,respectively.Findtheheightofthepolesandthedistancesofthepointfromthepoles.

Solution:BD=widthofroad=80m

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AngleACB=60°AngleECD=30°AB=ED=heightofpolesIn∆ABC,tan60°= 3=AB/BCorAB=BC 3……..(i)In∆EDC,tan30°=1/ 3=ED/CD=AB/(80-BC)OrAB=(80-BC)/ 3……(ii)OrBC 3=(80-BC)/ 3Or3BC=80–BCOr4BC=80OrBC=20soAB=20 3Example33:-ATVtowerstandsverticallyonabankofacanal.Fromapointontheotherbankdirectlyoppositethetower,theangleofelevationofthetopofthetoweris60°.Fromanotherpoint20mawayfromthispointonthelinejoiningthispointtothefootofthetower,theangleofelevationofthetopofthetoweris30°.Findtheheightofthetowerandthewidthofthecanal.

Solution:widthofcanal=BCCD=20mAngleACB=60°AngleADB=30°In∆ABC,AB=BC 3In∆ABD,AB=(BC+20)/ 3or3BC=BC+20or2BC=20orBC=10(widthofcanal)Example34:-Fromthetopofa7mhighbuilding,theangleofelevationofthetopofacabletoweris60°andtheangleofdepressionofitsfootis45°.Determinetheheightofthetower.

200

Solution: AB=heightofbuilding=7m AngleACB=45°AngleEAD=60°In∆ABC,AB=BCorAB=BC=AD=7m,In∆EAD,ED= 3AD=7 3Soheightoftower=DC+ED=7+7 3=7(1+ 3)=19.124mExample35:-Asobservedfromthetopofa75mhighlighthousefromthesea-level,theanglesofdepressionoftwoshipsare30°and45°.Ifoneshipisexactlybehindtheotheronthesamesideofthelighthouse,findthedistancebetweenthetwoships.

Solution:AB=75mAngleEAC=AngleACB=45°AngleEAD=AngleADB=30°In∆ABC,AB=AC=75m,In∆ABD,BD=AB 3=75 3SodistancebetweenshipsC&D=CD=75 3-75=75( 3 − 1)=54.9mExample36:-A1.2mtallgirlspotsaballoonmovingwiththewindinahorizontallineataheightof88.2mfromtheground.Theangleofelevationoftheballoonfromtheeyesofthegirlatanyinstantis60°.Aftersometime,theangleofelevationreducesto30°.Findthedistancetravelledbytheballoonduringtheinterval.

Solution:

201

AC=GH=heightofballoon=88.2mBC=EF=heightofGirl=1.2mAngleGFB=60°AngleAFB=30°In∆ABF,BF=87 3In∆GFI,FI=87/ 3Distancetraveledbyballoon=BI=58 3Example37:-Theanglesofelevationofthetopofatowerfromtwopointsatadistanceof4mand9mfromthebaseofthetowerandinthesamestraightlinewithitarecomplementary.Provethattheheightofthetoweris6m.

Solution:BD=4mangleAngleADB=𝜃AngleACB=90–𝜃In∆ABD,AB=tan𝜃×4In∆ABC,tan(90-𝜃)=AB/BCORAB=Cot𝜃×9or4tan𝜃=9Cot𝜃ottan2𝜃=9/4ortan𝜃=3/2=AB/4orAB=6m

Example38:-Amaniswalkingalongastraightroad.HenoticesthetopofatowersubtendinganangleA=60owiththegroundatthepointwhereheisstanding.Iftheheightofthetowerish=30m,thenwhatisthedistance(inmeters)ofthemanfromthetower?Solution:

LetBCrepresentthetowerwithheighth=30m,andArepresentthepointwherethemanisstanding.AB=ddenotesthedistanceofthemanfromtower.TheanglesubtendedbythetowerisA=60o.Fromtrigonometry,

tanA=tan60o=h/d=√3Sod=30/√3sm.Hencethedistanceofthemanfromthetoweris17.32m.

202

Example39:-Alittleboyisflyingakite.Thestringofthekitemakesanangleof30owiththeground.Iftheheightofthekiteish=18m,findthelength(inmeters)ofthestringthattheboyhasused.Solution:

IfthekiteisatCandtheboyisatA,thenAC=lrepresentsthelengthofthestringandBC=hrepresentstheheightofthekite.Fromthefigure,sinA=sin30o=h/l=1/2.Hencethelengthofthestringusedbythelittleboyisl=2h=2(18)=36m.

Example40:-Twotowersfaceeachotherseparatedbyadistanced=45m.Asseenfromthetopofthefirsttower,theangleofdepressionofthesecondtower'sbaseis60oandthatofthetopis30o.Whatistheheight(inmeters)ofthesecondtower?Solution:

ThefirsttowerABandthesecondtowerCDaredepictedinthefigureontheleft.FirstconsiderthetriangleBAC.AngleC=60o.tanBCA=tan60o=AB/AC.ThisgivesAB=dtan60o.SimilarlyforthetriangleBED,BE=dtan30o.NowheightofthesecondtowerCD=AB−BE=d(tan60o−tan30o)=45(√3−1/√3)=45×2/√3=51.96m.

Example41:-Ashipofheighth=12missightedfromalighthouse.Fromthetopofthelighthouse,theangleofdepressiontothetopofthemastandthebaseoftheshipequal30oand45orespectively.Howfaristheshipfromthelighthouse(inmeters)?Solution:LetABrepresentthelighthouseandCDrepresenttheship.Fromthefigure,tanBCA=tan45o=AB/AC.SimilarlyforthetriangleBED,tanBDE=tan30o=BE/ED.Now,AC=ED=d.Heightoftheship=CD=AB−BE=d(tan45o−tan30o)=12m.Thusdistanceoftheshipfromthelighthoused=12/(1−1/√3)=28.39m

203

Example42:-TwomenonoppositesidesofaTVtowerofheight28mnoticetheangleofelevationofthetopofthistowertobe45oand60orespectively.Findthedistance(inmeters)betweenthetwomen.Solution:

ThesituationisdepictedinthefigurewithCDrepresentingthetowerandABbeingthedistancebetweenthetwomen.FortriangleACD,tanA=tan60o=CD/AD.SimilarlyfortriangleBCD,tanB=tan45o=CD/DB.ThedistancebetweenthetwomenisAB=AD+DB=(CD/tan60o)+(CD/tan45o)=(28/√3)+(28/1)=44.17m.

Example43:-Twomenonthesamesideofatallbuildingnoticetheangleofelevationtothetopofthebuildingtobe30oand60orespectively.Iftheheightofthebuildingisknowntobeh=50m,findthedistance(inmeters)betweenthetwomen.Solution:

Inthefigure,AandBrepresentthetwomenandCDthetallbuilding.tanA=tan30o=DC/AC=h/AC;andtanB=tan60o=DC/BC=h/BC.NowthedistancebetweenthemenisAB=x=AC−BC=(h/tan30o)−(h/tan60o)=(50√3)−(50/√3)=57.73m.

Example44:-Apoleofheighth=40fthasashadowoflengthl=40.00ftataparticularinstantoftime.Findtheangleofelevation(indegrees)ofthesunatthispointoftime.Solution:

204

Inthefigure,BCrepresentsthepoleandABitsshadow.tanA=BC/AB=h/l=40/40.00=1.000Fromtrigonometrictables,wenotethattanA=1.000forA=45o.Hencetheangleofelevationofthesunatthispointoftimeis45o.

Example45:-Youarestationedataradarbaseandyouobserveanunidentifiedplaneatanaltitudeh=6000mflyingtowardsyourradarbaseatanangleofelevation=30o.Afterexactlyoneminute,yourradarsweeprevealsthattheplaneisnowatanangleofelevation=60omaintainingthesamealtitude.Whatisthespeed(inm/s)oftheplane?Solution:

Inthefigure,theradarbaseisatpointA.TheplaneisatpointDinthefirstsweepandatpointEinthesecondsweep.ThedistanceitcoversintheoneminuteintervalisDE.Fromthefigure,tanDAC=tan30o=DC/AC=h/AC.Similarly,tanEAB=tan60o=EB/AB=h/AB.Distancecoveredbytheplaneinoneminute=DE=AC−AB=(h/tan30o)−(h/tan60o)=(6000√3)−(6000/√3)=6928.20m.ThevelocityoftheplaneisgivenbyV=distancecovered/timetaken=DE/60=115.47m/s.

Example46:-cos2Ú[+ D

<-sin2 Ú

[+ D

<isequalto

Solution:-Weknowthatcos2A–sin2B=cos(A+B)cos(A–B),therefore

cos2Ú[+ D

<–sin2

Ú[− D

<=cos

Ú[+ D

<+ Ú

[− D

<×cos

Ú[+ D

<− Ú

[+ D

<

=cosÚ>cosx= $

<cosx.

Example47:-Ifsecθ-tanθ=½,thenθliesinwhichquadrant?Solution:-secθ–tanθ=½secθ+tanθ=2( sec2θ–tan2θ=1)secθ=5/4andtanθ=¾.cosθ=4/5andsinθ=4/5 ¾=3/5

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Asbothsinθandcosθare+ve theangleθliesinthe1stquadrant.Example48:-Ifsin(A+B+C)=1,tan(A–B)=1/ 3,sec(A+C)=2,thenSolution:-Assin(A+B+C)=1A+B+C=90oAs,tan(A–B)=1/ 3&sec(A+C)=2A–B=30&A+C=60oFromtheabovestatementswecanconclude,A=60o,B=30o,C=0oORAssin(A+B+C)=1

A+B+C=900{Note:wecaneasilyruleoutoptions(1)&(3)asA+B+C>900}Nowcheckingoption(2)sin900=1,tan30=1/ 3,sec60=2,thussatisfy.

Example49:-Findthevalueoftan0×tan1×tan2×tan3×..….tan89?

Solution:-

Fromthetableweknowthattan0=0.Sonomatterwhatyoumultiplywithzero,finalanswerwillalwaysbezero.

Example50:-Findvalueoftan1xtan2xtan3x….x…tan88xtan89

Solution:-

youmakepairsofcomplimentarynumbers:1+89=90,2+88=90….Thereisonlyoneangleleftwhodoesn’tgetapair(45)

Soit’lllooklikethis=(tan1xta89)(tan2xtan88)x…xtan45

Ineachofthosepairs,youconvertonetanintoitscomplimentarycot.

=(tan1xtan(90-89))x(tan2xcot(90-88))…..x1;becausetan45=1

=(tan1xcot1)x(tan2xcot2)x…..x1=1becausetanandcotareinverseofeachother.

Example51:-.Findvalueoftan48xtan23xtan42xtan67

Solution:-

206

Whenyougetthistypemultiplicationchainquestion,you’vetofindoutthepairofcomplimentaryangles.Here48+42=90and23+67=90soI’llclubthemtogether

Inbothparts,we’llconvertanyonetanintocot,thentanxcot=1(becausethey’reinverseofeachother).

Finalanswer=1.

Example52:-Onthelevelground,theangleofelevationofthetopofthetoweris30o.Onmoving2metersnearer,theangleofelevationbecomes60o.Whatistheheightofthetower?Solution:-

�D=tan600

�DE<

=tan300

DE<D=3 x=1 h= 3

Example53:-TheangleofelevationofthetopofthetowerobservedfromeachofthethreepointsA,BandContheground,formingatriangleisthesameangleα,IfRisthecircum-radiusofthetriangleABC,thentheheightofthetowerisSolution:-Sincethetowermakesequalanglesattheverticesofthetriangle,thereforefootofthetowerisatthecircumcentre.FromΔOAP,wehavetanα=OP/A

207

orOP=OAtanAorOP=Rtan𝛼OP=OAtanAorOP=RtanαExample54:-Ifthelengthofachordofacircleisequaltothatoftheradiusofthatcircle,thentheanglesubtended(inradians)atthecentreofthecirclebythechordisSolution:-Inthiscase,thechordandthetworadiijoiningthecentretotheendsofthechordmakeanequilateraltriangle;sotheanglesubtendedis60o=π/3radians.Example55:-InatriangleABC,A=45°,thenthevalueof(1+cotB)(1+cotC)isequaltoSolution:-A=45oorB+C=180o–45=135otan(B+C)=–1ortanB+tanC=–1+tanBtanC

or $�|bº

+ $�|b¼

=-1+ $�|bº�|b¼

orcotB+cotC+cotBcotC=1or1+cotB+cotC+cotBcotC=2.or(1+cotB)(1+cotC)=2.orA=450A+B+C=1800B+C=180–45o=135oLetstake,B=900&C=450or(1+cotB)(1+cotC)=(1+cot900)(1+cot450)=(1+0)(1+1)=1×2=2Example56:-Ifcos(A-B)=3/5andtanAtanB=2,thena.cosAcosB=1/5b.sinAsinB=–2/5c.cos(A+B)=1/5d.sinAsinB=4/5Solution:-cos(A–B)=3/5andtanAtanB=2orcosAcosB+sinAsinB=3/5andsinAsinB=2cosAcosBor3cosAcosB=3/5 cosAcosB=1/5.Example57:-cos1°cos2°cos3°….cos179°isequalto

208

Solution:-Thegivenproductcontainsthefactorcos90o=0.Thusthevalueoftheproductbecomes0.Example58:-Ifsin2θ+3cosθ–2=0,thencos3θ+sec3θisequaltoSolution:-Givensin2θ+3cosθ–2=0cos2θ–3cosθ+1=0orcos2θ+1=3cosθorcosθ+ $

�|�Û=3

Cubingbothsides,wegetcos3θ+ $

�|�³Û+3(cosθ+ $

�|�Û)=27

orcos3θ+sec3θ=27–3×3=18

Example59:-.Findthevalueofcos18°/sin72°

Solution:-

Weknowthatcosandsinarecomplimentary.cosA=sin(90-A)

SoforCos18,Youcanwritecos18=sin(90-18)=sin72.Let’suseit

Cos18/sin72=sin72/sin72(becausecos18=sin72)=1(becausenumeratoranddenominatoraresamesothey’llcanceleach other.)

Example60:-Findvalueofsec70°xsin20°–cos20°xcosec70°

Solution:-

Ifyouconvertallfour(sin,cos,cosec,andsec)intotheircomplimentary(cos,sin,secandcosec)thenyou’llrunintoinfiniteloop.

InthepartA,ifIconvertsinintoitscomplimentarycos,thensecxcos=1becausesecandcosareinverseofeachother.SamewayinpartBifIconvertcosecintoitscomplimentarysec,thattoowillleadto1.Let’ssee

=1-1=0isthefinalanswer.

So,tanandcotarecomplimentary

1. tanA=cot(90-A)butnotvalidfor90degreesbecausetan90isnotdefined.2. cotA=tan(90-A)butthisisnotvalidfor0degreebecausecot0isnotdefined.

Thetanandcot’scomplimentaryrelationshipisalsoimportformultiplicationchainquestionsbecausetanandcotarealsoinverseofeachother.ThatistanAxcotA=tanAx1/tanA=1.

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Exercise

Q1.IfCosA=1−2sin230°,thenfindvalueofA?

a.30 b.45 c.60 d.90

Q2.Ifcos60°=cos2A−sin230°,thenfindvalueofA?

a.30 b.45 c.60 d.90

Q3.IfsinA=2sin30°×cos30°,thenfindvalueofA?

a.30 b.45 c.60 d.90

Q4.IfsinA=2tan30°/(1+tan230°),thenfindvalueofA?

a.30 b.45 c.60 d.90

Q5.IfcosA=(1−tan230°)/(1+tan230°),thenfindvalueofA?

a.30 b.45 c.60 d.90

Q6.IfcosA=4cos330°−3cos30°,thenfindvalueofA?

a.30 b.45 c.60 d.90

Q7.Findvalueof(5cos260°+4sec230°−tan245°)/(sin230°+cos230°)?

a.67 b.12 c.67/12 d.12/67

Q8.Findvalueof3cos230°+sec230°+2cos0°+3sin90° −tan260°?

a.67 b.12 c.67/12 d.12/67

Q9.IfsinA=cosA,whatisthevalueof2tan2A+sin2A+1

210

a.2 b.7 c.7/2 d.2/7

Q10.WhatisthevalueofsinAcosB+cosAsinB,ifA=30°andB=60°

a.0 b.½ c.2 d.1

Q11.WhatisthevalueofcosAcosB−sinAsinB,ifA=30°andB=60°

a.0 b.½ c.2 d.1

Q12.secA=cosec60°,whatisthevalueof2cos2A−1

a.0 b.½ c.2 d.1

Q13.(tan10°xtan15°xtan75°xtan80°)

a.0 b.1 c.3 d.2

Q14.Whatisthevalueof5cos90°−cot30°+(sin60°/cos245°)

a.1 b.3 c.0 d.5

Q15.Findthevalueof(cos30°+sin60°)/(sin30°+cos60°+1)

a.½ b. 2 c. 3/2 d.2/ 3

Q16.Findthevalueoftan260°/(sin245°+cos245°)

a.3 b.1/2 c.2/3 d.1/3

Q17.Ifcot(A+B)=1/ 3andCot(A−B)= 3.FindthevaluesofAandB

a.30,60 b.60,30 c.15,45 d.45,15

Q18.Findthevalueofcos10°×cos20°×cos30°×………×cos90°

a.1 b.½ c.0 d.Notdefined.

Q19.IfsecA−cosecA=0thenfindvalueofsecA+cosecA,giventhatAisanacuteangle.

a.2 b.2 2 c.0 d.Notdefined.

Q20.Giventhat4Aisanacuteangleandsec4A=cosec(A−20),whatisthevalueofA?

a.22 b.23 c.24 d.25

211

Q21.Findvalueofcosec32-sec58?

a.1 b.2 c.3 d.0

Q22.Findvalueofcot12°xcot48°xcot52°xcot60°xcot78°

a.1 b.0 c. 3 d.1/ 3

Q23.(tan10°xtan45°xtan40°xtan50°xtan80°)

a.0 b.1 c.3 d.2

Q24.Cos48°xcosec42°+sin48°xsec42°

a.0 b.1 c.2 d.3

Q25.(cos70°/sin20°)+(cos59°xcosec31°)

a.1 b.2 c.3 d.0

Q26.(sec20°/cosec70°)+[(cos55°xcosec35°)/(tan5°xtan25°xtan45°xtan65°xtan85°)]

a.1 b.2 c.3 d.0

Q27.(sin70°/cos20°)+(cosec20°/sec70°)-(2cos70°xcosec20°)

a.1 b.2 c.3 d.0

Q28.Ifsin3A=cos(A−26),thenwhatisthevalueofA?giventhat3Aisanacuteangle.

a.21 b.23 c.24 d.29

Q29.Ifsec2A=cosec(A−42),thenwhatisthevalueofA?giventhat2Aisanacuteangle.

a.22 b.33 c.44 d.55

Q30.IfsinA=cos30°,whatisthevalueof2tan2A−tan45,giventhatAisanacuteangle.a.3 b.4 c.5 d.7

Q31.Ifsin7x=cos11xthenthevalueoftan9x+cot9xisa.1b.2c.3d.4

Q32.Ifcot2A=tan(A+6),findthevalueofA.giventhat2AandA+6areacuteangles?a.18 b.28 c.30 d.noneofabove

Q33.tan10×tan20×tan880×tan890

212

a.0 b.1 c.3 d.1/ 3Q34.tan40×tan430×tan470×tan860a.0 b.1 c.3 d.1/ 3

Q35.Ifõö÷ θEøùõθõö÷ θWøùõθ

=2thenthevalueofsinθisa.2/ 3 b. 3/2 c.1/2 d.1

Q36.Ifcosec390=xthenthevalueof$

�|���;?$°+sin239°+tan251°−

$�z9;?$°���;C_°

is

a. 𝑥< − 1 b. 1 − 𝑥< c.𝑥< − 1 d.1 − 𝑥<Q37.IftanA=2/3findthevalueof(3sinA+4cosA)/(3sinA–4cosA)a.-3 b.-4 c.-5 d.NoneoftheseQ38.sin21+sin23+sin25+sin27+….+sin289=a.22.5 b.23 c.24 d.25Q39.sin4A−cos4A=a.1 b.0c.sin2A−cos2Ad.tan2A Q40.[(secA−tanA)(secA+tanA)]+[(cosecA−cotA)(cosecA+cotA)]a.1 b.0 c.2 d.1/2

Q41.Thevalueof$

$E�|b;¸+ $

$Eby9;¸

a.2b.1c.1/4d.1/2Q42.IfsinA=1/3thencosAcosecA+tanAsecA=

a.$` <EC[

b.$` <E?

[ c.7 d.

$` <ECX

Q43.IfsinA=3/5,thenfindthevalueof4tanA+3sinAisequaltoa.6cosA b.6secA c.6secA d.tanAQ44.IfcosA=0.96then $

�z9¸+ $

by9¸isequalto

a.0.98b.2c.4d.7Q45.Ifsec𝜃-cos𝜃=3/2(𝜃isapositiveacuteangle),thensec𝜃isequaltoa.2 b.–1/2 c.0 d.1/2 Q46.Thevalueof[1+sin60o+sin230o+sin260o+sin445o][cos30o−sin60o]isa.1 b.13/12 c.12/13 d.0

213

Q47.Iftan2𝜃tan4𝜃=1thenthevalueoftan3𝜃=a.1/√3 b.0 c.1 d.√3Q48.Iftan22$

<°=xthencos67$

<°

a. $$ED;

b. D$ED;

c.𝑥 d. 1 + 𝑥< Q49.Ifcos43°=

DD;E{;

,thenthevalueoftan47°

a.D

D;E{; b. {

D;E{; c.{

D d.D

{

Q50.úûø ÛEby9Ûúûø ÛWby9Û

=<E C<W C

thenthevalueof𝜃inthecircularmeasurewillbe

a.𝜋/6 b.𝜋/4 c.𝜋/12 d.𝜋/3Q51.Iftan𝜃+cot𝜃=2where0<𝜃<900;findthevalueoftan17𝜃+cot19𝜃.a.1 b.2 c.3 d.4Q52.sin22+sin24+sin26+sin28+………+sin288=a.22.5b.23c.24d.22Q53.tanA/sinA=a.cosecA b.sinA c.secA d.1/sinAQ54.(sinA/tanA)+cosA=a.2secA b.secA c.2cosA d.2cosecAQ55.tan10×tan20×…………………….tan880×tan890=a.0b.1c.2d.3Q56.Iftan(x+y)tan(x-y)=1thenthevalueoftanx=a.1/√3b.0c.1d.√3Q57.1/(tan10×tan20×…………………….tan1880×tan1890)=a.0b.1c.2d.3Q58.2cosec2230cot2670–sin2230-sin2670–cot2670=a.0b.1c.sec2230d.tan2230Q59.Thelengthoftheshadowofaverticaltoweronlevelgroundincreasesby10mwhenthealtitudeofthesunchangesfrom450to300.Thentheheightofthetowerisa.10 3b.5 3c.10( 3 + 1)d.5( 3+1)

214

Q60.If5tan𝜃=4then?�z9ÛWC�|�Û?�z9ÛE<�|�Û

=a.1/3b.2/3c.1/4d.1/6Q61.If2(cos2𝜃−sin2𝜃)=1(𝜃isapositiveacuteangle),thencot𝜃isequaltoa. 3b.− 3c.1/ 3d.1Q62.IftanC=11,thenfindvalueofsin2C+cos2Cisequaltoa.2 b.0 c.1 d.1/2Q63.InarightangletriangleABC,rightangledatB,theratioofABtoACis1: 2then2tanA/(1-tan2A)isequalsto:a.2 b.1 c.3 d.undefinedQ64.IfSinA=(a2-b2)/(a2+b2);thentanA=a.a2+b2/2ab b.2ab/(a2-b2) c.a2-b2/2ab d.tanAQ65.cos(90o−A)/sin(90o−A)=a.tanA b.cotA c.tan(90o−A) d.sec(90o−A)Q66.[cosec2A−1]cos(90o−A)/sin(90o−A)=a.tanAb.sinAc.cotA d.cosAQ67.cotA[cos(90o−A)/sin(90o−A)]=a.tanA b.cot2A c.tan2Ad.1Q68.Thevalueoftan45o−cos45osin45oisa.1/2 b.1c.0 d.3/4Q69.sin2A−sec2A+cos2A+tan2A=a.1 b.0 c.cotA d.cosecAQ70.cotAtanA=a.sinA b.cosA c.sinAcosA d.1Q71.Findthevalueofsin21+sin22+sin288+sin289a.3 b.2 c.3 d.noneoftheseQ72.Atreebreaksduetostormandthebrokenpartbendssothatthetopofthetreetouchesthegroundmakinganangle30°withit.Thedistancebetweenthefootofthetreetothepointwherethetoptouchesthegroundis8m.Findtheheightofthetree.a.8 b.16 c.8 3 d.NoneoftheseQ73.Upperpartofatreebrokenoverbythewindmakesanangleof45°withtheground,andthehorizontaldistancefromthefootofthetreetothepointwherethetopofthetreetouchesthegroundis12m.Findtheheightofthetreebeforeitwasbroken.

215

a.12m b.12+12/√3 c.12(1+√3) d.NoneoftheseQ74.Fromthetopofa7mhighbuilding,theangleofelevationofthetopofacabletoweris60oandtheangleofdepressionofthefootofthetoweris30o.Findtheheightofthetower.a.15 b.28 c.32 d.NoneoftheseQ75.Amanstandingonthedeckofaship,whichis16mabovethewaterlevel,observetheangleofelevationofthetopofcliffas60°andtheangleofdepressionofthebaseofthecliffas30°.Calculatethedistanceoftheclifffromtheshipandtheheightofthecliff.a.16√3m,h=64mb.20√3m,h=32mc.15√3m,h=64md.noneoftheseQ76.Anaeroplanewhenflyingataheightof5000mfromthegroundpassesverticallyaboveanotheraeroplaneataninstantwhentheanglesoftheelevationofthetwoplanesfromthesamepointonthegroundare60°and45°respectively.Findtheverticaldistancebetweentheaeroplanesattheinstant.a.2116.5 b.2115 c.2113.5 d.noneoftheseQ77.Ifdistancebetweentwopillarsoflength16&9misxmeters.Iftwoangleofelevationoftheirrespectivetopfromapointongroundofotherarecomplementarytoeachother,thenvalueofxisa.7 b.16 c.12 d.9Q78.Evaluate:sin225o+sin265o.a.2 b.0 c.1 d.-1 Q79.Theangleofelevationofthetopofatower30mhigh,fromtwopointsonthelevelgroundonitsoppositesidesare45degreesand60degrees.Whatisthedistancebetweenthetwopoints?a.30 b.51.96 c.47.32 d.81.96Q80.Theangleofelevationofthetopofatowerfromapointontheground,whichis30mawayfromthefootofthetower,is30°.Findtheheightofthetower.a.10 b.10 3 c.11.32 d.41.96Q81.Fromapointontheground,theanglesofelevationofthebottomandthetopofatransmissiontowerfixedatthetopofa20mhighbuildingare45°and60°respectively.Findtheheightofthetower.a.20( 3 − 1) b.20 3 c.20/ 3 d.41.96Q82.Aladder15mlongjustreachesthetopofaverticalwall.Iftheladdermakesanangleof60°withthewall,findtheheightofthewall.a.7.5√3 b.7.5 c.5√3 d.Noneofthese

216

Q83.Apole12mhighcastsashadow4√3mlongontheground.Findtheangleofelevationa.30° b.60° c.90° d.NoneoftheseQ84.Theangleofelevationofthetopofatowerfromapointonthegroundis30°ifonwalking30mtowardsthetower,theangleofelevationbecomes60°.Findtheheightofthetower.a.15√3 b.7√3 c.5√3 d.NoneoftheseQ85.Theangleofelevationoftheaeroplanefromapointonthegroundis60°.After15secondsflight,theangleofelevationchangesto30°.Iftheaeroplaneisflyingataheightof1500√3m.Findthespeedoftheplanea.200m/s b.450m/s c.250m/s d.NoneoftheseQ86.Anobserver1.5mtallis20.5mawayfromatower22mhigh.Determinetheangleofelevationofthetopofthetowerfromtheeyeoftheobserver.a.75° b.60° c.45° d.NoneoftheseQ87.If3sin2α+7cos2α=4,thenthevalueoftanα-is(where0<α<90°)a. 3b. 6c. 2d. 5Q88.Iftanθ–cotθ=0,0°<θ<90°,thevalueof(sinθ–cosθ)isa.1b.2c.–2d.0Q89.If0<θ<$

<,thensinθ+cosθisalways

a.greaterthan1b.lessthan1c.equalto1d.greaterthan2Q90Ifsec2θ+tan2θ=7,thenthevalueofθ,when0°<θ<90°,isa.60°b.30°c.0° d.90°Q91.If�z9ÚE�|�Ú

�z9ÚW�|�Ú=3,thenthevlaueofsin4θ–cos4θis

a.$?b.<

?c.C

? d.>

?

Q92If2cosθ–sinθ= $

<,(0°<θ<90°)thevalueof2sinθ+cosθis

a. $<b. 2c. C

< d. <

C

Q93.If0°<θ<90°andsinθ+cosθ=$X

$Cthenthevlaueofsinθ–cosθis

a. C$Cb.$%

$Cc. ?

$C d. X

$C

Q94.If�z9ÛE�|�Û

�z9ÛW�|bÛ=7,thenthevalueoftanθisequalto:

a.<Cb.>

C c.$

C d. X

$C

217

Q95.Ifby9ÛE�|bÛ

�|bÛW�|bÛ=2,(0<θ<90°),thenthevalueofsinθis

a. <Cb. C

< c.$

< d.1

Q96.Ifsec(4θ–50°)=cosec(50°–θ),thenthevalueofθ,when0°<θ <90°,isa.33$

Cb.18°c.3$

Cd.30°

Q97.Ifcosθ+secθ= 3,thenthevalueofcos3θ+sec3θisa.–1b. 3 c.0 d.1 Q98.Themaximumvalueof24sinθ+7cosθisa.24b.25c.7 d.17 Q99.Iftan2θ.tan4θ=1,thenthevalueoftan3θisa.$

<b.2c.0 d.1

Q100.In"ABC,∠AisarightangleandADisperpendiculartoBC.IfAD=4cm,Bc=12cm,thenthevalueof(costBcotC)isa.4b.C

<c.6 d.3

Q101.Ifα+β=90°andα:β=2:1,thenthevalueofsinα:sinβis:a. 1:1 b. 2:1c. 3:1d.2:1 Q102.Ifsecθ+tanθ=2,thensecθisequaltoa.?

<b.?

> c. X

>d.X

<

Q103.Thevalueofθ[0°<θ<90°]forwhich �|�Û

$W�z9Û+ �|�Û

$E�z9Û=4is

a.45°b.60°c.30°d.noneofthese Q104.If2ycosθ=xsinθand2xsecθ–ycosecθ=3,thentherelationbetweenxandyisa.2x2+y2=2b.x2+4y2=4c. x2+4y2=1 d.4x2+y2=4 Q105.Inaright-angledtriangleABC,AB=2.5cm,cosB=0.5,∠ACB=90°.ThelengthofsideAC,incm,isa.?

>3 b. ?

$`3c.5 3d.?

<3

Q106.Iftanθ–cotθ=aandcosθ–sinθ=b,thenthevalueof(a2+4)(b2–1)2is:a.1b.2c.3d.4

218

Q107.Ifx=cosecθ=sinθandy=secθ–cosθ,thenthevalueofx2y2(x2+y2+3)is:a.0b.1c.2d.3 Q108.Inaright-angledtriangleABC,∠BistherightangledandAC=2 5cm.IfAB–BC=2cm,thenthevalueof(cos2A–cos2C)is:a.C

?b.`

?c. C

$%d.<

?

Q109.If0<θ <

Ú<,2ycosθ=xSinθand2xsecθ–ycosecθ=3,thenthevalueofx2+4y2is:

a.1b.2c.3d.4 Q110.Ifcosθ.cosec23°=1,thevalueofθisa.23°b.37°c.63°d.67° Q111.Ifsin(3x–20°)=cos(3y+20°),thenthevalueof(x+y)isa.20°b.30°c.40°d.45° Q112.Ifcosθ=>

?,thenthevalueof�|���Û

$E�|bÛis

a.X?b.<

Xc.?

Xd.>

X

Q113.Ifsin2θ=$

<,thenthevalueofcos(90–θ)is

a.1b.$<c. C

<d. $

<

Q114.If�z9Û

�|�Û+ �|�Û

�z9Û=2with0<θ<90°,thenwhatisθequalto?

a.30°b.45°c.60°d.75° Q115.If5tanθ=4,thenthevalueof?�z9ÛWC�|�Û

�z9ÛEC�|�Ûis:

a.$Xb.<

Xc.?

Xd.<

?

Q116. Ifx,yarepositiveacuteangles,x+y<90°andsin(2x–20)=cos(2y+20°),thenthe

valueofsec(x+y)is

a. 2𝑏. $<c. C

<d. $

<

Q117.If(a2+b2)sinθ+2abcos=a2+b2,thenthevalueoftanθisa.$

<(a2–b2)b. $

<y�(a2–b2)c.$

<(a2+b2)d. $

<y�(a2+b2)

Q118.IfA+B=90°,thenthevalueofsec2A+sec2B–sec2A.sec2Bis:a.0b.1c.2d.3

219

Q119.Ifcosθ+secθ= 3thenthevalueofcos3θ+sec3θis:a.0b.1c.–1 d. 3 Q120.Ifsin(x+y)=cos[3(x+y)]thenthevalueoftan[2(x+y)]isa. 3b.1c.0d. $

C

Q121.IfX+1/X=2thenthevalueofcos2θ+sec2θis:a.0b.1 c.2 d. 3 Q122.Theleastvalueof4cosec2a+9sin2ais:a.14b.10c.11d.12 Q123.Ifsecθ–cosecθ=0,thenthevalueof(secθ+cosecθ)is:

a. C<b. <

Cc.0d.2 2

Q124.Ifcosx=sinyandcot(x–40)=tan(50–y)thenthevalueofxandyare:a.1b.1c.1d.2 Q125.Ifxsin3θ+ycos3θ=sinθcosθ,andxSinθ–ycosθ=0thenvalueof(x2+y2)a.1b.sinθ–cosθc.sinθ+cosθd.0 Q126.Ifsinθ+cosθ= 2cos(90–θ),thencotθisa. 2+1b.0c. 2d. 2–1 Q127.Ifsecθ+cosec(90– θ)=4,(0<θ<90°)thenthevalueoftanθis:a. $

Cb.1c. 3d. $

<

Q128.If 2.cos(5x+5°)=cot45°,thenthevalueofxindegreeis:a.10b.8c.11d.0 Q129.Ifsinθ–cosθ=0,findthevlaueofsin Ú

<− 𝜃 +cos Ú

<+ 𝜃

a.0b.1c. 2d.2 2

Q130.Ifsinθ=y;W$y;E$

,thenthevalueofsecθ+cosθwillbe:

a. 2ab.ac. y<d. y

y;E$

Q131.Ifsin(x–2y)=cos(4y–x),thenthevalueofcot2yis:a.0b.1c. $

Cd.undefined

220

Q132.Iftan Ú

<− Û

<= 3,valueofcosθis:

a.0b. $<c.$

<d.1

Answers1.C 2.A 3.C 4.C 5.C 6.D 7.C 8.C 9.C 10.D11.A 12.B 13.B 14.C 15.A 16.A 17.D 18.C 19.B 20.A21.D 22.D 23.B 24.C 25.A 26.B 27.D 28.D 29.C 30.C31.B 32.B 33.B 34.B 35.B 36.C 37.A 38.A 39.C 40.C41.B 42.A 43.A 44.D 45.A 46.D 47.C 48.B 49.C 50.D51.B 52.D 53.C 54.C 55.B 56.C 57.A 58.C 59.D 60.D61.C 62.C 63.D 64.B 65.A 66.C 67.D 68.A 69.B 70.D71.B 72.C 73.D 74.B 75.A 76.C 77.C 78.C 79.C 80.B81.A 82.B 83.B 84.A 85.A 86.C 87.A 88.D 89.A 90.A91.C 92.C 93.D 94.B 95.B 96.D 97.C 98.B 99.D 100.D101.C 102.B 103.B 104.B 105.A 106.D 107.B 108.A 109.D 110.D111.B 112.C 113.B 114.B 115.A 116.A 117.B 118.A 119.A 120.B121.C 122.D 123.D 124.B 125.A 126.D 127.C 128.B 129.A 130.B131.D 132.C

221