2022澳大利亚高考数学试题答案

2022HIGHERSCHOOLCERTIFICATEEXAMINATIONMathematicsExtension2GeneralInstructionsWriteusingblackorbluepenBlackpenispreferredBoard-approvedcalculatorsmaybeusedAtableofstandardintegralsisprovidedatthebackofthispaperInQuestions1116,showrelevantmathematicalreasoningand/orcalculationsTotalmarks-lOOSectionlPages2-610marksAttemptQuestions1-10Allowabout15minutesforthissectionSectionIIPages7-1790marksAttemptQuestions11-16Allowabout2hoursand45minutesforthissectionSectionI10marksAttemptQuestions1-10Allowabout15minutesforthissectionUsethemultiple-choiceanswersheetforQuestions1-10.Whichexpressionisequaltotanxdx?sec2x+c-ln(cosx)+c2tanx(C)+c(D)ln(secx+tannx)+c2Whichpairofequationsgivesthedirectricesof4x2-25y2=100?(A)x=(B)x=±129x=±29(D)x=±2925Whichdiagrambestrepresentsz2?(B)(C)(D)4Thepolynomialequation4x3+x23x+5=0hasrootsa，BandY・Whichpolynomialequationhasrootsa+1，B+1andY+1?(A)4x311x2+7x+5=0(B)4x3+x23x+6=0(C)4x3+13x2+11x+7=0(D)4x32x22x+8=05WhichregionontheArganddiagramisdefinedbynn<z1<?436dx?Whichexpressionisequaltox26x+5x3(A)sin1+C2x3(B)cos1+C2TOC\o"1-5"\h\zlnx3+lnx3++Cx3)2+42x34+C)7Theangularspeedofadiscofradius5cmis10revolutionsperminute.Whatisthespeedofamarkonthecircumferenceofthedisc?(A)50cmmin-1(B)cmmin-12(C)IOOncmmin-1(D)cmmin-14n8Thebaseofasolidistheregionboundedbythecirclex2+y2=16.VerticalWhichintegralrepresentsthevolumeofthesolid?4x2dx44nx2dx4416x2dx44n16x2dx44444()()9Whichdiagrambestrepresentsthegraphy=sinx?SectionII90marksAttemptQuestionsll-16Allowabout2hoursand45minutesforthissectionAnswereachquestioninaSEPARATEwritingbooklet.Extrawritingbookletsareavailable.InQuestions11-16,yourresponsesshouldincluderelevantmathematicalreasoningand/orcalculations.Question11(15marks)UseaSEPARATEwritingbooklet.w=1+i3.(a)z=2i3(i)z+.Expresswinmodulus-argumentform.(iii)Writew24initssimplestform.122FindnumbersA,BandCsuchthat2=Ax3+Bx+Cx+22x3x+22x2+8x+11(c)Factorisez2+4iz+5.232Evaluatex1xdx.03SketchtheregionontheArganddiagramdefinedbyz2+2s8.3Question12(15marks)UseaSEPARATEwritingbooklet.Usingthesubstitutiont二tanxn22,orotherwise,evaluatedx.04+5cosxTheequationlogxeyloge(1000y)=50loge3implicitlydefinesyasafunctionofx.Showthatysatisfiesthedifferentialequationdydx=y50y1100011x11x.(c)Thediagramshowstheregionboundedbythegraphy=ex,thex-axisandthelinesx=1andx=3.Theregionisrotatedaboutthelinex=4toformasolid.Findthevolumeofthesolid.Question12continuesonpage9424Question12(continued)(d)ThepointsPcp,candQcq,c,wherepHq,lieontherectangularhyperbolawithequationxy=c2.ThetangenttothehyperbolaatPintersectsthex-axisatAandthey-axisatB.Similarly,thetangenttothehyperbolaatQintersectsthex-axisatCandthey-axisatD.ShowthattheequationofthetangentatPisx+p2y=2cp.(ii)ShowthatA,BandOareonacirclewithcentreP.(iii)ProvethatBCisparalleltoPQ.EndofQuestion12221Question13(15marks)UseaSEPARATEwritingbooklet.nLetIn=1x2)2dx,wherennOisaninteger.O((i)ShowthatIn=nn+1In2foreveryintegernn2.(ii)EvaluateI5.Thediagramshowsthegraphofafunction(x).Sketchthefollowingcurvesonseparatehalfdiagrams.(i)y2=(x)(ii)y=321321=0=0Question13continuesonpage113223Question13(continued)ThepointsA,B,CandDlieonacircleofradiusr,formingacyclicquadrilateral.ThesideABisadiameterofthecircle.ThepointEischosenonthediagonalACsothatDE丄AC.Leta二乙DACandB二乙ACD.ShowthatAC=2rsin(a+B).ByconsideringABD,orotherwise,showthatAE=2rcosasinB.(iii)Hence,showthatsin(a+B)二sinacosB+sinRcosa.EndofQuestion13221Question14(15marks)UseaSEPARATEwritingbooklet.(a)Thediagramshowsthegraphy=lnx.=lnxlnt>2t1t+1,fort>1.(b)Letzi2=1+iand,forn>2,letzn=zn11+zn1.Usemathematicalinductiontoprovethatzn二nforallintegersnn2.Question14continuesonpage1333Question14(continued)(i)Givenapositiveintegern,showthatsec2ne=k^nnktan2k0.(ii)Hence,bywritingsec80assec60sec20,findsec80d0.(d)AtrianglehasverticesA,BandC.ThepointDliesontheintervalABsuchthatAD=3andDB=5.ThepointEliesontheintervalACsuchthatAE=4,DE=3andEC=2.B5D3A34NOTTOSCALE2C(i)ProvethatABCandAEDaresimilar.(ii)ProvethatBCEDisacyclicquadrilateral.ShowthatCD=21.FindtheexactvalueoftheradiusofthecirclepassingthroughthepointsB,C,EandD.EndofQuestion1421122Question15(15marks)UseaSEPARATEwritingbooklet.respectively,where^<0.TheareaofthetriangleformedbyO,wandzisA.Showthat=4iA.(b)ThepolynomialP(x)=ax4+bx3+cx2+ehasremainder-3whendividedbyx-l.Thepolynomialhasadoublerootatx=-1.(i)Showthat4a+2c=92.(ii)Hence,orotherwise,findtheslopeofthetangenttothegraphy=P(x)whenx=1.Question15continuesonpage1512Question15(continued)Aballofmassmisprojectedverticallyintotheairfromthegroundwithinitialvelocityu.AfterreachingthemaximumheightHitfallsbacktotheground.Whileintheair,theballexperiencesaresistiveforcekv2,wherevisthevelocityoftheballandkisaconstant.Theequationofmotionwhentheballfallscanbewrittenasmv=mgkv2(DoNOTprovethis.)(i)ShowthattheterminalvelocityvTShowthatwhentheballgoesup,themaximumheightHis2H=v2Tu2gln1+v2TWhentheballfallsfromheightHithitsthegroundwithvelocityw.Showthatw2u2+1v2.TEndofQuestion15132Question16(15marks)UseaSEPARATEwritingbooklet.(a)(i)FindtheminimumvalueofP(x)=2x315x2+24x+16,forxn0.(ii)Hence,orotherwise,showthatforxnO,(x+1)x2+(x+4)2n25x2.(iii)Hence,orotherwise,showthatformn0andnn0,(m+n)2+(m+n+4)2>100mnm+n+1.(b)AsmallbeadPofmassmcanfreelymovealongastring.TheendsofthestringS.Thebeadundergoesuniformcircularmotionwithradiusrandconstantangularvelocitywinahorizontalplane.Theforcesactingonthebeadarethegravitationalforceandthetensionforcesalongthestring.ThetensionforcesalongPSandPS'havethesamemagnitudeT.Thelengthofthestringis2aandSS'=2ae,where0vev1.ThehorizontalplanethroughPmeetsSS'atQ.ThemidpointofSS'isOandB二乙S'PQ.TheparameterBischosensothatOQ=acos0.Question16continuesonpage17212Question16(contin