牛津大学本科招生笔试试题(数学).docx

牛津大学本科招生笔试试题(数学)

第一篇：牛津大学本科招生笔试试题(数学)MathematicsInterviewQuestions1.Differentiatexx2.Integratecos2(x)andcos3(x).3.Whatisthesquarerootofi?4.IfIhadaDifferentiatexx5.cubeandsixcoloursandpaintedeachsideadifferentcolour,howmany(different)wayscouldIpaintthecube?WhataboutifIhadncoloursinsteadof6?6.Provethatroot2isirrational.7.Integratelnx.8.Sketchthecurve(y2-2)2+(x2-2)2=2.Whatdoesitlooklike?9.3girlsand4boyswerestandinginacircle.Whatistheprobabilitythattwogirlsaretogetherbutoneisnotwiththem?10.Provethat1+1/2+1/3+...+1/100018.prove4nR(x))2,whereR(x)isxroundedupordownintheusualway.thensketchg(x)=f(1/x)34.(a+b)/2isaninteger,istherelationshiptransitive?(a+b)/3?35.Differentiate1/1+(1/1+(1/1+1/(1+x))))36.Sketchgraphof1/x,1/x2,1/(1+x2)37.Integrate1/(1+x2)38.Integrateexx2betweenlimitsof1and0.Drawthatgraph.39.Integratex-2betweenlimitsof1and-1.Drawthegraph.Whyisit-2andnotinfinity,asitappearstobeonthegraph?40.Writedown3digits,andthenwritethenumberagainnexttoitself,eg:145145.Whyisitpisibleby13?41.Youaregivenatrianglewithafixedperimeterbutyouwanttomaximisethearea.Whatshapewillitbe?Proveit.42.Nextyouaregivenaquadrilateralwithfixedperimeterandyouwanttomaximisethearea.Whatshapewillitbe?Proveit.43.Integrate(1)/(x+x3),(1)/(1+x3),(1)/(1+xn)44.Howmany0'sarein100!45.Provethattheangleatthecentreofacircleistwicethatatthecircumference.46.Howmanywaysarethereinwhichyoucancolourthreeequalportionsofadisc?47.Integrate1/(9+x2)48.Drawy=ex,thendrawy=kx,thendrawagraphofthenumbersofsolutionsofxagainstxforex=kx,andthenfindthevalueofkwherethereisonly1solution.49.Rubik'scubeandhelditbytwodiagonallyoppositeverticesandrotatedittillitreachedthesameposition,byhowmanydegreesdidittakeaturn?50.Solveab=baforallrealaandb.51.Thereisagamewith2players(A&B)whotaketurnstorolladieandhavetorollasixtowin.WhatistheprobabilityofpersonAwinning?52.Sketchy=x3andy=x5onthesameaxis.53.Whatthe2sidesofarectangle(aandb)wouldbetomaximisetheareaifa+b=2C(whereCisaconstant).54.Can1000003bewrittenasthesumof2squarenumbers?55.Showthatwhenyousq