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9231_w02_er

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SRINFOREWORDFURTHERMATHEMATICSGCEAdvancedLevelPaperPaperPaperPaperThisbookletcontainsreportswrittenbyExaminersontheworkofcandidatesincertainpapersItscontentsareprimarilyfortheinformationofthesubjectteachersconcernedwwwXtremePaperscomGCEAdvancedLevelPaperPaperGeneralcommentsTheoverallqualityofworksubmittedinresponsetothisPaperwasgoodandprovidedclearevidenceofawellpreparedcandidatureMostcandidatesmadeaseriousattemptatallthequestions,submittedresponsesinorderandsetouttheirworkinaclearwayTheonlyopportunityforrubricinfringementoccurredinQuestion,butveryfewcandidateswastedtimebysubmittingresponsestobothoptionsofthatquestionThereweresomemisreads,especiallyinQuestion,andquitealotofelementaryarithmeticandalgebraicerrorsSucherrorscanleadtosevereconsequencesforaquestionresponse,fornotonlydotheynecessarilyleadtoincorrectresults,butalso,theycanincertainsituations,drivethecandidateintounworkable,oratleasttimeconsuming,strategiesInorder,therefore,toavoidtheseunfortunatesituations,whichinextremecasescanseriouslyunderminetheoverallexaminationperformance,itisessentialthatworkischeckedateachstageofitsdevelopmentAfeatureofthisnewsyllabusisthatcandidatesmustanswerallquestionsiftheyaretoobtainfullcreditInthiscase,itisgoodtobeabletorecordthatwiththeexceptionofthevectorproductandlinearspacesalmostallcandidatesgaveevidenceofhavinganindepthknowledgeoftheentiresyllabusInparticular,knowledgeofcomplexnumbers,testedinQuestion,wasgoodAsinthecaseofpreviousALevelFurtherMathematicsPaperexaminations,thecalculustopics,perse,werewellunderstoodandfailuresinquestions,suchas,,,andwhichrelatedtothismaterialwereusuallyduetononcalculuserrorsCommentsonspecificquestionsQuestionThisshortintroductoryquestiondidnotgoaswellasexpectedItcontainedasomewhatunusualelementandthisseemstohavebaffledmostofthecandidatureAlmostallcandidatesobtainedacorrectresultforSn=��Nnnu,byapplicationofthedifferencemethodAfewtreatedthisseriesasthedifferenceoftwogeometricseriesandsoobtainedacorrectbutunsimplifiedresultforSnandthis,moreoftenthannot,ledtodifficultiesinthefinalpartofthequestionThemajoritystatedthatthegivenseriesisconvergentifx<,butmadenomentionofx=Likewise,moststated,orimplied,thatS∞=exforx<,butveryfewidentifiedS∞=whenx=Answers:SN=ex�e(N)xinfiniteseriesisconvergentforx<S∞=ex,forx<,=forx=QuestionIngeneralthisquestionwaswellansweredAlmostallcandidatesbeganbysubstitutingyforxinthegivenpolynomialequation,andsowentontoobtainthecorrectpolynomialequationinyAsoftenasnot,however,thetermsofthisequationwerenotorderedinthestandardwayFortherest,themajorityvalidlyobtained����=–Aand��������=A,andsogenerallywentontoobtainthecorrectvalueofANeverthelessthereweresomewho,apparently,wereunabletosolvetheequation�A=AcorrectlyHowever,thereweresomewhoattemptedtoevaluatethesecondofthesesumsintermsofAbyconsideringthecoefficientsofthexequationAlthough,inprinciple,thiscomplicatedstrategyisfeasible,hardlyanycandidateshadthenecessaryalgebraicexpertisetoarguesuccessfullyinthiswayAnswer:A=�QuestionThereweremanyessentiallycorrectresponsestothisquestion,butonlyaminorityofthesewerecompleteMostresponsesshowedthesimplificationof�=an�antoaformwhichclearlyindicatesdivisibilitybyFortherestofthequestion,themaindeficiencieswerethegeneralfailuretostateattheoutsetwhattheinductivehypothesisactuallyis,failuretoverifythat│�(veryoftenitwasshowninsteadthat│a),failuretoestablishthekeyresult,namely,that│�k�│ak,andalsofailuretocompletetheinductionargumentinasatisfactorywayOfcourse,thiswasimpossibleif│�hadnotbeenpreviouslyestablishedAnswer:an�an=()()nnQuestionThisturnedouttobeasuccessfullyansweredquestionThemajorityofcandidatesproducedacompleteandcorrectresponse(i)Noteveryoneintegratedxex�correctlyandmoreoverthereweresomeerrorsintheapplicationofthelimits(ii)MostresponsesshowedcorrectworkingtoestablishthedisplayedreductionformulaNevertheless,veryfewresponsesshowedtheoptimalstrategy,basedonaconsiderationofD(xnex�)InsteaditwascommonforcandidatestobeginbyconsideringtheintegrandofInas(xnl)(xex�)andthentoapplytheintegrationbypartsruletothissituationSome,byasimilarargument,obtainedInintermsofIn�andthenchangedntonsoastoprovetherequiredresultThisisasatisfactorywaytoproceed,thoughitshouldbeemphasisedthatifnisgeneralthensoisn(iii)Surprisingly,thereweremoreerrorsherethanineitheroftheprecedingpartsofthisquestionAllthatwasrequiredwastouseI=I�e,I=I�e(commonerroneousvariantsoftheseequationswhereI=I�e,I=I�e)soastoexpressIintermsofIlandthentoapplytheresultobtainedinpart(i)Answers:(i)I=e��(iii)I=�eQuestionTherewereveryfewcompleteandcorrectresponsestothisquestion(i)Onlyabouthalfofallcandidatescouldproduceasatisfactoryargumenttoshowthatlim��y=Moreover,fewunderstoodthatinordertoshowthattheliney=isanasymptoteofCitisalsonecessarytoprovethatrorx��as��,yetonlyasmallnumberofcandidatesproducedsuchaproof(ii)Again,manycandidateswerenotabletoproduceacorrectsketchofCHereitwasexpectedofcandidatesthattheliney=wouldappear,butthiswasfrequentlyomittedItwasalsocommonforthecurvetobedrawnstartingfromthepole(iii)AlmostallresponsesbeganwithacorrectintegralrepresentationoftheareaofthesectorOPQandwentontocarryouttheintegrationof��correctlyItwasinthecompletion,whichatthepurelymanipulativelevelinvolvedonlysubALevelmathematics,thatsomeresponsesfellapartsothatagaintherewasevidenceoflackofbasicmathematicalskills(iv)ThisconcludingsectionofthequestionwasverywellansweredMostcandidatesstarted(correctly)withsomethinglike�ddr=����s=�����d����andsowentontoobtaintherequiredresultAnswers:(i)y=rsin�=��sin�as��,r=���as��(iii)AreaofsectorOPQ=�QuestionMostresponsesshowedagoodunderstandingofimplicitdifferentiationand,generally,theworkingwasaccurateNevertheless,fewcandidatesproducedacompleteandcorrectresponsetothisquestion(i)Mostresponsesgotasfarasexhibitingthepreliminaryresultxyxy������xyddy�y������xydd=Fromthisthemajorityofcandidatesargued(essentially)thatxydd=�xy=(*)whichisimpossiblesincexandyarenecessarilyrealThisisanincorrectargumentsinceinfact(*)doeshavethe(unique)solution,x=y=Thustocompletetheargumentitisnecessarytosaythatas(,)isnotonthecurve(thisrequiresformalverification)thenxyddisnotzeroatanypointonitHowever,relativelyfewcandidatesarguedinthisway(ii)Thelevelofaccuracy,bothinthefurtherdifferentiationandinthearithmetic,wasgenerallyimpressiveThemostpersistenterrorwasthewritingofD(y������xydd)asy������xyddy��������ddxyandnotasydd������xyy��������ddxyAnswers:(ii)At(,�),xydd=,dd�xyQuestionThisquestionwasgenerallywellansweredandtheworkingwas,forthemostpart,accurateStrangely,itwasinparts(i)and(ii),ratherthaninthelatermaterial,thattheworkingsometimesranintounnecessarycomplicationsInpart(i),forexample,itwascommontosee��������������������sinicossinicossincossinicossinicossinicossinicossinicos�������������zwhereasinitsplaceallthatwasrequiredwas�����������������sinicossincosiz(bydeMoivre’stheorem)=cos��isin�andtherewerealsosimilarcomplicationsinsomeresponsestopart(ii)Fortherest,responsesdevelopedalongcorrectlinesinthewaydemandedbythequestion,andtheworkingwasgenerallyaccurateAnswers:a=,b=QuestionThisquestionshowedthatalmostallcandidateshadagoodunderstandingofhowtosolvealinearsecondorderdifferentialequationNevertheless,thereseemedtobeagenerallylimitedunderstandingoftheformofthesolutionobtained,sothatrelativelyfewcompletelycorrectresponsesactuallyappearedAlmostallcandidatesobtainedtheAQE,solveditandformedthecorrectcomplementaryfunctionWorkingfortheobtainingoftheparticularintegralwasusuallyaccuratesothatacorrectgeneralsolutionforthegivendifferentialequationwasafeatureofmostscriptsThemajorityofcandidatesunderstood,inprinciple,howtoapplythegiveninitialconditionsandhereagainmostresponsesledtoacorrectresultforyintermsoftAsmallminorityofcandidates,however,appliedthegiveninitialconditionstothecomplementaryfunctionandthenaddedtheparticularintegralinanattempttofindtherequiredsolutionMostcandidatescomprehendedthatthecomplementaryfunctiontendstozeroasttendstopositiveinfinityandsoconcluded(correctly)thaty�sintwhentislargeandpositiveHowever,fromthispointonwards,veryfewmadeanysignificantprogressEventhepreliminaryinequalitysint<�(*)wasobtainedbyonlyaminority,eventhoughitfollowsalmostimmediatelyfromy<xTheconcludingargumentwouldthenfollowimmediatelybyobservingthatfor�t��,(*)��<t<�andthat����=NeverthelesssuchreasoningappearedinfewscriptseventhoughitrequirednomorethanaknowledgeofverybasicALevelmathematicsAnswer:y=e�t�e�tsintQuestionTheresponsestothisquestionshowedthatthemajorityofcandidateshadasoundunderstandingoftheapplicationofscalarandvectorproductstoproblemsinvolvingdimensionalmetricvectorsTheyalsoshowedsomeunderstandingofthegeometryspecifictothisquestionMostargumentsbeganwiththeuseofvectorproductstodeterminevectors,nandn,perpendiculartotheplanes�and�andthenwentonevaluatenxnsoastoobtainthevectorequationoflThisstrategygeneratedveryfewerrorsIncontrast,aminorityofcandidatesequatedthecomponentsofthevectorequationsof�and�andsoexpressedthreeof�,�,�,�intermsofthefourthandagainthiswillleadtotherequiredresultHowever,thisstrategyinevitablygeneratedmoreerrorsthanthefirst,andthusshowedtheclearadvantagethatderivesfromahavingacompleteunderstandingofallthesyllabusmethodologyThisquestion,andotherslikeit,becomemuchmoreformidableifnousecanbemadeofthevectorproductInthemiddlepartofthequestion,almostallcandidatesobtainedavectornotparalleltol(ijkwasthemostpopular)intheplane�Also,theyusuallywentontoobtainbothavectorequationof�,oftengivingitintheformrn=p,andascalarequationof�,though,ofcourse,lackofknowledgeofthevectorproductmadethisexercisemoredifficultthanitneedhavebeenInthefinalpartofthisquestion,manycandidatesarguedthatasthethreegivenlinearequationsrepresent�,�,�,respectively,andasalsothelineliscommontothethreeplanes,thenthegivensystem,S,hasaninfinitenumberofsolutionsThisiscorrectasfarasitgoes,butitalsorequiresthatthefirsttwoequationsbeshowntorepresent�,�,andthisvitalpartoftheargumentwasomittedbysomecandidatesOfcourse,intheprecedingpartofthequestion,thethirdequationwouldhavebeenshowntorepresent�Thealternativestrategy,adoptedbymanyothercandidates,wastoreducetheaugmentedmatrix,A,totheechelonformandthissimpleoperationwasusuallycarriedoutaccuratelyNevertheless,itwascommonfortheconcludingargumenttobedeficientinsomewayThustosaythatastherankofAisequalto,thenShasaninfinitenumberofsolutionsisincorrect,forSmighthavenosolutionsAgain,andsimilar,thefactthattheechelonformcontainsarowofzerosdoesnotofitselfimplythatShasaninfinitenumberofsolutionsThatdependsonthepositionofthezerosintherowswhicharenonzerorowvectorsFinally,thestillweakerargumentthatdetB=(*),whereBisthematrixofcoefficientsappertainingtoS,isclearlyfalse,for(*)isnecessaryfortheconclusionbutnotsufficientAnswers:Avectorequationoflisr=ijks(ijk)Avectorequationof�isr=ijks(ijk)t(ijk)Ascalarequationof�isxyz=QuestionThiswastheleastsuccessfulquestionofthePaper,byquitealongwayAlthoughsomeresponsesshowedcorrectstrategiesforparts(i)and(ii),onlyaminorityofallcandidatesmadesignificantprogresswithpart(iii)(i)AnotuncommonerrorwasthemisreadingofatleastoneoftheelementsofthematrixHand,ofcourse,suchaninaccuracyunderminedmuchofthesubsequentworkingItmustbeemphasisedtherefore,thatinsituations,suchasthis,wherethereisalotofnumericaldata,itisessentialthatthecandidatecarriesoutathoroughcheckofthecopyingofthisinformationbeforebecominginvolvedinthesubsequentworkingMoreover,therewerealsoanumberofarithmeticerrorsinattempts(notalwayscomplete)toobtainavalidechelonformandinsomesuchcasestheendproductturnedouttobeamatrixofrankThisshows,yetagain,theneedforcontinuouschecking(ii)MostcandidatesunderstoodthatabasisforthespaceofTcouldbeobtainedfromthreerelevantlinearequationsInthiscontext,noteveryoneusedtherowechelonformtheyhadobtainedinpart(i)fromwhichtherequiredbasiscanreadilybeobtainedInstead,asubstantialminorityusedthegivenformofHandsoembarkedonamoreextendedstrategyNeverthelesstheoverallstandardofworkingaccuracywasverysatisfactorysothatmostcandidatesobtainedtherequiredbasis(iii)HeretherewasasharpdichotomyofthecandidatureintothemajoritywhohadnoideahowtobeginandtheminoritywhoknewavalidmethodandapplieditinanaccuratewaySuchamethodstartswithaconsiderationofthegeneralsolutionofthegivenvectorequation,namelyx=��������������������������������FromthisitfollowsimmediatelythatIxI=(��)(��)(��)andtherestisthenasimpleminimisationexercisewhichinthemajorityofeffectiveresponseswascarriedoutbywritingtheaboveasaquadraticpolynomialin�andthenbyapplyingthecompletionofthesquaretechniqueThustheobtainingofanotherparticularsolutionandthenhopingforthebest,astrategyemployedbysomecandidates,isnotavalidmethodAmongthosewhousedcalculusthereweresomewhoconsideredIxIratherthanIxIandwhothusbecameinvolvedinunnecessarycomplicationAnswers:(i)DimensionofrangespaceofTis(ii)AbasisforthespaceofTis���������������(iii)LeastpossiblevalueofIxIisQuestionEITHERAlthoughlesspopularthanthealternativeforQuestion,itnonethelessgeneratedmuchgoodworkThelevelofnumericalaccuracyinresponsestothelaterpartofthisquestionwasimpressive,especiallyincaseswherecandidateswereinvolvedinunnecessarilycomplicatedstrategiesForresponsestopart(i)somethinglikethefollowingwasexpectedGe=�e�,(GkI)e=GekIe=�eke=(�k)e,andnocommentwouldthenbenecessarySimilarlyinpart(ii)thefollowingdetailshouldappearinacompleteresponseGe=G(Ge)=G(�e)=�(Ge)=�(�e)=�e,andagainnocommentwouldberequiredHowever,fewcandidatesproducedcompleteresponsestobothpartsFortherestofthisquestion,mostresponsesshowedworkingleadingtoacorrectcharacteristicequationforthematrixAfromwhichtheeigenvaluesandasetofcorrespondingeigenvectorswereobtainedwithouterrorItwasalsogoodtoseethatcarewastakentoensurethateigenvaluesandeigenvectorswerepairedoffcorrectlyManycandidatesfailedtoperceivethatB=A–Iandhencecouldnotexploittheresultsofparts(i)and(ii)soastoobtaintheeigenvaluesandeigenvectorsofBinaverysimplewayInstead,theyattemptedtofindthesescalarsandvectorsforthematrixBandthenwentontouse(ii)toobtaintherequiredresultsforBFinallyitmustberemarkedthattherewasasmallsubsetofcandidateswhoevaluatedBandthenattemptedtofinditscharacteristicequationandsotoworkontothefinaldestinationGenerallytheseattemptsperishedinthelargeamountofeffortrequiredAnswers:TheeigenvaluesofAare,,Correspondingeigenvectorsare,,���������������������������������TheabovearealsoeigenvectorsofBThecorrespondingeigenvaluesofBare,,,respectivelyQuestionORIntheworkforthismorepopularoptionofQuestionthereweremanytechnicaldeficienciesVeryfewcompletelycorrectresponsesappeared(i)UsuallycandidatesusedanalgebraicdivisionprocessinordertodeterminePandQHowever,eventhiselementaryprocesswasnotalwaysappliedcorrectlyThesimplestrategyofputtingx=cin(x�c)(xP)Q≡(x�a)(xb),soastoobtainQandthenconsideringthecoefficientofxsoastoobtainP,wasattemptedbyveryfew(ii)Mostresponsesshowedcorrectequationsfortheasymptotesandthiswasoftenthecase

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