内容文稿教案

*

*

UNIVERSITYOFCAMBRIDGEINTERNATIONALEXAMINATIONS

GeneralCertificateofEducationAdvancedSubsidiaryLevel

MATHEMATICS 9709/23

Paper2PureMathematics2(P2) May/June2010

1hour15minutes

AdditionalMaterials: AnswerBooklet/Paper

GraphPaper

ListofFormulae(MF9)

READTHESEINSTRUCTIONSFIRST

IfyouhavebeengivenanAnswerBooklet,followtheinstructionsonthefrontcoveroftheBooklet.WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin.

Writeindarkblueorblackpen.

Youmayuseasoftpencilforanydiagramsorgraphs.

Donotusestaples,paperclips,highlighters,glueorcorrectionfluid.

Answerallthequestions.

Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalceinthecaseofanglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion.

Theuseofanelectroniccalculatoripected,whereappropriate.Youareremindedoftheneedforclearpresentationinyouranswers.

Atoftheexamination,fastenallyourworksecurelytogether.

Thenumberofmarksisgiveninbrackets[]atofeachquestionorpartquestion.Thetotalnumberofmarksforthispaperis50.

Questionscarryingsmallernumbersofmarksareprintedearlierinthepaper,andquestionscarryinglargernumbersofmarkslaterinthepaper.

Thisconsistsof3printedpagesand1blankpage.

©UCLES2010 [Turnover

PAGE

2

©UCLES2010

9709/23/M/J/10

Giventhat13x=(2.8)y,uselogarithmstoshowthaty=kxandfindthevalueofkcorrectto3significantfigures. [3]

y

R

O

1

2

3

x

Thediagramshowspartofthecurvey=xe−x.TheshadedregionRisboundedbythecurveandbythelinesx=2,x=3andy=0.

UsethetrapeziumrulewithtwointervalstoestimatetheareaofR,givingyouranswercorrect

to2decimalces. [3]

State,withareason,whetherthetrapeziumrulegivesanunder-estimateoranover-estimateofthetruevalueoftheareaofR. [1]

Solvetheinequality|2x−1|<|x+4|. [4]

ã

1

(a)Showthat 4

0

cos2xdx=1. [2]

2

(b)Byusinganappropriatetrigonometricalidentity,findtheexactvalueof

ã

[]

1

33tan2xdx. 4

1

6

Theequationofacurveisy=x3e−x.

Showthatthecurvehasastationarypointwherex=3. [3]

Findtheequationofthetangenttothecurveatthepointwherex=1. [4]

(i)Bysketchingasuitablepairofgraphs,showthattheequation

lnx=2−x2

hasonlyoneroot. [2]

Verifybycalculationthatthisrootliesbetweenx=1.3andx=1.4. [2]

Showthat,ifasequenceofvaluesgivenbytheitiveformula

xn+1=√(2−lnxn)

converges,thenitconvergestotherootoftheequationinpart(i). [1]

to

Usethei tiveformulaxn+1=√(2−lnxn) determinetherootcorrectto2decimal ces.

Givetheresultofeachi tionto4decimal ces. [3]

Thepolynomial2x3+ax2+bx+6,whereaandbareconstants,isdenotedbyp(x).Itisgiventhatwhenp(x)isdividedby(x−3)theremainderis30,andthatwhenp(x)isdividedby(x+1)theremainderis18.

Findthevaluesofaandb. [5]

Whenaandbhavethesevalues,verifythat(x−2)isafactorofp(x)andhencefactorisep(x)compley. [4]

(i)Provetheidentity

sin(x−30◦)+cos(x−60◦)≡(√3)sinx. [3]

(ii)Hencesolvetheequation

2

sin(x−30◦)+cos(x−60◦)=1secx,

for0◦<x<360◦. [6]

PAGE

4

BLANKPAGE

Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonableefforthasbeenmadebythepublisher(UCLES)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbepleasedtomakeamendsattheearliestpossibleopportunity.

UniversityofCambridgeInternationalExaminationsispartoftheCambrid