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UNIVERSITYOFCAMBRIDGEINTERNATIONALEXAMINATIONS

GeneralCertificateofEducationAdvancedLevel

MATHEMATICS 9709/03

Paper3PureMathematics3(P3) October/November2008

1hour45minutes

AdditionalMaterials: AnswerBooklet/Paper

GraphPaper

ListofFormulae(MF9)

READTHESEINSTRUCTIONSFIRST

IfyouhavebeengivenanAnswerBooklet,followtheinstructionsonthefrontcoveroftheBooklet.WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin.

Writeindarkblueorblackpen.

Youmayuseasoftpencilforanydiagramsorgraphs.

Donotusestaples,paperclips,highlighters,glueorcorrectionfluid.

Answerallthequestions.

Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalceinthecaseofanglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion.

Theuseofanelectroniccalculatori pected,whereappropriate.Youareremindedoftheneedforclearpresentationinyouranswers.

At oftheexamination,fastenallyourworksecurelytogether.

Thenumberofmarksisgiveninbrackets[]at ofeachquestionorpartquestion.Thetotalnumberofmarksforthispaperis75.

Questionscarryingsmallernumbersofmarksareprintedearlierinthepaper,andquestionscarryinglargernumbersofmarkslaterinthepaper.

This consistsof4printedpages.

?UCLES2008 [Turnover

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Solvetheequation

ln(x+2)=2+lnx,

givingyouranswercorrectto3decimalces. [3]

(+)(? )

Expand1 x√1 2xinascendingpowersofx,uptoandincludingtheterminx2,simplifyingthe

coefficients. [4]

ex 1 1

Thecurvey=cosx,for?2<x<2,hasonestationarypoint.Findthex-coordinateofthispoint.

[5]

Theparametricequationsofacurveare

x=a(2?sin2), y=a(1?cos2).

=

Showthatdy cot. [5]

dx

Thepolynomial4x3?4x2+3x+a,whereaisaconstant,isdenotedbyp(x).Itisgiventhatp(x)isdivisibleby2x2?3x+3.

Findthevalueofa. [3]

Whenahasthisvalue,solvetheinequalityp(x)<0,justifyingyouranswer. [3]

+ (+ ) > < <

(i)Express5sinx 12cosxintheformRsinx ,whereR 0and0? 90?,givingthevalueofcorrectto2decimalces. [3]

(ii)Hencesolvetheequation

5sin2+12cos2=11,

givingallsolutionsintheinterval0?<<180?. [5]

Two neshaveequations2x?y?3=7andx+2y+2=0.

Findtheacuteanglebetweenthenes. [4]

Findavectorequationfortheirlineofintersection. [6]

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Anundergroundstorageisbeingfilledwithliquidasshowninthediagram.Initiallytheisempty.Attimethoursafterfillingbegins,thevolumeofliquidisVm3andthedepthofliquidishm.ItisgiventhatV=4h3.

= =

Theliquidispouredinatarateof20m3perhour,butowingtoleakage,liquidislostatarateproportionaltoh2.Whenh1,dh4.95.

dt

Showthathsatisfiesthedifferentialequation

dh=5

–1. [4]

dt h2 20

20h2

2000

Verifythat100?h2≡?20+(10?h)(10+h). [1]

Hencesolvethedifferentialequationinpart(i),obtaininganexpressionfortintermsofh. [5]

a

Theconstantaissuchthat

0

1x

xe2

dx=6.

Showthatasatisfiestheequation

2

x=2+e?1x. [5]

Bysketchingasuitablepairofgraphs,showthatthisequationhasonlyoneroot. [2]

Verifybycalculationthatthisrootliesbetween2and2.5. [2]

Useani tiveformulabasedontheequationinpart(i)tocalculatethevalueofacorrectto2decimal ces.Givetheresultofeachitionto4decimalces. [3]

?UCLES2008 9709/03/O/N/08 [Turnover

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Thecomplexnumberwisgivenbyw=?1+i√3.

Findthemodulusandargumentofw. [2]

ThecomplexnumberhasmodulusRandargument,where?1<<1.Statethemodulus

3 3

andargumentofwandthemodulusandargumentofw. [4]

w

Henceex inwhy,inanArganddiagram,thepointsrepresenting,wandarethevertices

ofanequilaltriangle. [2]

+

+

InanArganddiagram,theverticesofanequilaltrianglelieonacirclewithcentreattheorigin.Oneoftheverticesrepresentsthecomplexnumber42i.Findthecomplexnumbersrepresentedbytheothertwovertices.Giveyouranswersintheformxiy,wherexandyarerealandexact. [4]

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