121908-advanced-problems-in-core-mathematics

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1、AdvancedProblemsinCoreMathematicsStephenSiklosLatestedit:September2010AdvancedProblemsinCoreMathematics:editionSeptember20102ABOUTTHISBOOKLETThisbookletisintendedtohelpyoutoprepareforSTEPexaminations.Itshouldalsobeusefulaspreparationforanyundergraduatemathematicscourse,ev

2、enifyoudonotplantotakeSTEP.ThequestionsareallbasedonrecentSTEPquestions.Ichosethequestionseitherbecausetheyare‘nice’–inthesensethatyoushouldgetalotofpleasurefromtacklingthem(Idid)–orbecauseIfeltIhadsomethinginterestingtosayaboutthem.Inthisbooklet,Ihaverestrictedmyself(rel

3、uctantly)tothesyllabusforPapersIandII,whichistheA-levelcore(i.e.C1toC4)withafewadditions.ThismaterialshouldbefamiliartoyouifyouaretakingtheInternationalBaccalaureate,ScottishAdvancedHighersorothersimilarcourses.Thefirsttwoquestions(thesampleworkedquestions)areina‘streamofc

4、onsciousness’format.Theyareintendedtogiveyouanideahowatrainedmathematicianwouldthinkwhentacklingthem.Thisapproachismuchtoolong-windedtosustain,butitshouldhelpyoutoseewhatsortofquestionsyoushouldbeaskingyourselfasyouworkthroughthelaterquestions.Ihavegiveneachofthesubsequen

5、tquestionsadifficultyratingrangingfrom(∗)to(∗∗∗).Aquestionlabelled(∗)mightbefoundonSTEPI;aquestionlabelled(∗∗)mightbefoundonSTEPII;aquestionlabelled(∗∗∗)mightbefoundonSTEPIII.Butdifficultyinmathematicsisintheeyeofthebeholder:youmightfindaquestiondifficultsimplybecauseyouoverlook

6、edsomekeystep,whichonanotherdayyouwouldnothavehesitatedover.Youshouldnotthereforebediscouragedifyouarestuckonaone-starquestion;thoughyoushouldprobablybeencouragedifyougetthroughthree-starquestionswithoutmishap.Eachquestionisfollowedbyacommentandafullsolution.Thecommentwil

7、lcontainhintsand/ordirectyourattentiontokeypoints;oritmightputthequestioninitstruemathematicalcontext.Thesolutionshouldgivesufficientworkingforyoutoreaditthroughandpickupthegistofthemethod.Itisofcoursejustonewayofproducingtherequiredresult:theremaybemanyotherequallygoodorbe

8、tterways.Calculatorsarenotrequiredforanyoftheproblemsinthisbooklet;andcalculatorsarenotpermitted