剑桥大学三一学院数学系面试题目.pdf

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1、TRINITYCOLLEGEADMISSIONSQUIZ(MATHEMATICS2)DECEMBER1997.Therearetenquestionsbelowwhichareonvariousareasofmathematics.Theyareofvaryinglevelsofdifficulty:someshouldbeeasyandothersratherhard.Youarenotexpectedtoanswerallofthem,ornecessarilytocompletequestions.Youshouldjustattempt

2、thosethatappealtoyou,andtheywillbeusedasabasisfordiscussionintheinterviewthatfollows.Youshouldbringthequestionpaperwithyoutotheinterviewafterwards.1.Inatennistournamentthereare2nparticipants.Inthefirstroundofthetournament,eachplayerplaysexactlyonce,sotherearengames.Showthat

3、thepairingsforthefirstroundcanbearrangedinexactly(2n−1)!/2n−1(n−1)!ways.2.LetL1andL2betwolinesintheplane,withequationsy=m1x+c1andy=m2x+c2respectively.Supposethattheyintersectatanacuteangleθ.Showthatm1−m2tan(θ)=.1+m1m2Rπ3.Calculate(xsinx)2dx.04.Ofthenumbers1,2,3,...,6000,how

4、manyarenotmultiplesof2,3or5?5.Thereisapileof129coinsonatable,allunbiasedexceptforonewhichhasheadsonbothsides.Bobchoosesacoinatrandomandtossesiteighttimes.Thecoincomesupheadseverytime.Whatistheprobabilitythatitwillcomeupheadstheninthtimeaswell?6.Apackingcaseisheldonthesideo

5、fahillandgivenakickdownthehill.Thehillmakesanangleofθtothehorizontal,andthecoefficientoffrictionbetweenthepackingcaseandthegroundisµ.Whatrelationshipbetweenµandθguaranteesthatthepackingcaseeventuallycomestorest?Letgravitationalaccelerationbeg.Iftherelationshipaboveissatisfied

6、,whatmusttheinitialspeedofthepackingcasebetoensurethatthedistanceitgoesbeforestoppingisd?1
n7.Letstandforthenumberofsubsetsofsizertakenfromasetofsizen.(Thisisthenumberofrwaysofchoosingrobjectsfromniftheorderofchoicedoesnotmatter.YoumaybemorefamiliarwiththenotationnC,inwhi

7、chcasefeelfreetouseit.)Everysubsetoftheset{1,2,...,n}eithercontainstherelement1oritdoesn’t.Byconsideringthesetwopossibilities,showthatn−1n−1n+=.r−1rrByusingasimilarmethod,orotherwise,provethatn−2n−2n−2n+2+=.r−2r−1rr8.Oneendofarodofuniformdensityisattachedtoth

8、eceilinginsuchawaythattherodcanswingaboutfreelywithnoresistance.Theotherendoftherodisheld