IGCSE数学试卷-past-paper

Thisdocumentconsistsof16printedpages.DC(NH/JG)50034/5©UCLES2012[Turnover*0835058084*ADDITIONALMATHEMATICS0606/11Paper1October/November20122hoursCandidatesanswerontheQuestionPaper.AdditionalMaterials:Electroniccalculator.READTHESEINSTRUCTIONSFIRSTWriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin.Writeindarkblueorblackpen.Youmayuseapencilforanydiagramsorgraphs.Donotusestaples,paperclips,highlighters,glueorcorrectionfluid.Answerallthequestions.Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalplaceinthecaseofanglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion.Theuseofanelectroniccalculatorisexpected,whereappropriate.Youareremindedoftheneedforclearpresentationinyouranswers.Attheendoftheexamination,fastenallyourworksecurelytogether.Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion.Thetotalnumberofmarksforthispaperis80.ForExaminer’sUse123456789101112TotalUNIVERSITYOFCAMBRIDGEINTERNATIONALEXAMINATIONSInternationalGeneralCertificateofSecondaryEducation2©UCLES20120606/11/O/N/12ForExaminer’sUseMathematicalFormulae1.ALGEBRAQuadraticEquationFortheequationax2+bx+c=0,xbbaca=−−242BinomialTheorem(a+b)n=an+(n1)an–1b+(n2)an–2b2+…+(nr)an–rbr+…+bn,wherenisapositiveintegerand(nr)=n!(n–r)!r!2.TRIGONOMETRYIdentitiessin2A+cos2A=1sec2A=1+tan2Acosec2A=1+cot2AFormulaefor∆ABCasinA=bsinB=csinCa2=b2+c2–2bccosA∆=12bcsinA3[Turnover©UCLES20120606/11/O/N/12ForExaminer’sUse1(i)Sketchthegraphofy=|3+5x|,showingthecoordinatesofthepointswhereyourgraphmeetsthecoordinateaxes.[2](ii)Solvetheequation|3+5x|=2.[2]2Findthevaluesofkforwhichtheliney=k–6xisatangenttothecurvey=x(2x+k).[4]4©UCLES20120606/11/O/N/12ForExaminer’sUse3Giventhatp=logq32,express,intermsofp,(i)logq4,[2](ii)logq16q.[2]4Usingthesubstitutionu=5x,orotherwise,solve52x+1=7(5x)–2.[5]5[Turnover©UCLES20120606/11/O/N/12ForExaminer’sUse5Giventhaty=x2cos4x,find(i)dydx,[3](ii)theapproximatechangeinywhenxincreasesfromπ4toπ4+p,wherepissmall.[2]6©UCLES20120606/11/O/N/12ForExaminer’sUse6(i)Findthefirst3terms,indescendingpowersofx,intheexpansionofx+2x26.[3](ii)Hencefindthetermindependentofxintheexpansionof2–4x3x+2x26.[2]7[Turnover©UCLES20120606/11/O/N/12ForExaminer’sUse7Donotuseacalculatorinanypartofthisquestion.(a)(i)Showthat35–22isasquarerootof53–1210.[1](ii)Statetheothersquarerootof53–1210.[1](b)Express63+7243+52intheforma+b6,whereaandbareintegerstobefound.[4]8©UCLES20120606/11/O/N/12ForExaminer’sUse8ThepointsA(–3,6),B(5,2)andClieonastraightlinesuchthatBisthemid-pointofAC.(i)FindthecoordinatesofC.[2]ThepointDliesonthey-axisandthelineCDisperpendiculartoAC.(ii)FindtheareaofthetriangleACD.[5]9[Turnover©UCLES20120606/11/O/N/12ForExaminer’sUse9Afunctiongissuchthatg(x)=12x–1for1x3.(i)Findtherangeofg.[1](ii)Findg–1(x).[2](iii)Writedownthedomainofg–1(x).[1](iv)Solveg2(x)=3.[3]10©UCLES20120606/11/O/N/12ForExaminer’sUse10Thetableshowsvaluesofthevariablesxandy.x10°30°45°60°80°y11.21619.522.424.7(i)Usingthegraphpaperbelow,plotasuitablestraightlinegraphtoshowthat,for10°x80°,y=Asinx+B,whereAandBarepositiveconstants.[4]11[Turnover©UCLES20120606/11/O/N/12ForExaminer’sUse(ii)UseyourgraphtofindthevalueofAandofB.[3](iii)Estimatethevalueofywhenx=50.[2](iv)Estimatethevalueofxwheny=12.[2]12©UCLES20120606/11/O/N/12ForExaminer’sUse11(a)Solvecosec2x–π3=2for0xπradians.[4](b)(i)Giventhat5(cosy+siny)(2cosy–siny)=7,showthat12tan2y–5tany–3=0.[4]13[Turnover©UCLES20120606/11/O/N/12ForExaminer’sUse(ii)Hencesolve5(cosy+siny)(2cosy–siny)=7for0°x180°.[3]14©UCLES20120606/11/O/N/12ForExaminer’sUse12Answeronlyoneofthefollowingtwoalternatives.EITHERyy=(12–6x)(1+x)2xAOCBThediagramshowspartofthegraphofy=(12–6x)(1+x)2,whichmeetsthex-axisatthepointsAandB.ThepointCisthemaximumpointofthecurve.(i)FindthecoordinatesofeachofA,BandC.[6](ii)Findtheareaoftheshadedregion.[5]ORyxAOBThediagramshowspartofacurvesuchthatdydx=3x2–6x–9.PointsAandBarestationarypointsofthecurveandlinesfromAandBaredrawnperpendiculartothex-axis.Giventhatthecurvepassesthroughthepoint(0,30),find(i)theequationofthecurve,[4](ii)thex-coordinateofAandofB,[3](iii)theareaoftheshadedregion.[4]15[Turnover©UCLES20120606/11/O/N/12ForExaminer’sUseStartyouranswertoQuestion12here.Indicatewhichquestionyouareanswering.EITHEROR.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................