9709_s03_qp_1

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MATHEMATICS

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May/June20031hour45minutes

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21151Findthevalueofthecoefficientofintheexpansionof󰀁2x−󰀂.xx[3]2Findallthevaluesofxintheinterval0◦≤x≤180◦whichsatisfytheequationsin3x+2cos3x=0.[4]6withrespecttox.x23(a)Differentiate4x+[2](b)Find󰀎󰀁4x+

6󰀂dx.x2[3]4Inanarithmeticprogression,the1sttermis−10,the15thtermis11andthelasttermis41.Findthesumofallthetermsintheprogression.[5]Thefunctionfisdefinedbyf:x→ax+b,forx∈󰀁,whereaandbareconstants.Itisgiventhatf(2)=1andf(5)=7.(i)Findthevaluesofaandb.(ii)Solvetheequationff(x)=0.(i)Sketchthegraphofthecurvey=3sinx,for−π≤x≤π.5[2][3][2]6Thestraightliney=kx,wherekisaconstant,passesthroughthemaximumpointofthiscurvefor−π≤x≤π.(ii)Findthevalueofkintermsofπ.[2](iii)Statethecoordinatesoftheotherpoint,apartfromtheorigin,wherethelineandthecurveintersect.[1]7ThelineL1hasequation2x+y=8.ThelineL2passesthroughthepointA(7,4)andisperpendiculartoL1.(i)FindtheequationofL2.(ii)GiventhatthelinesL1andL2intersectatthepointB,findthelengthofAB.[4][4]8ThepointsA,B,CandDhavepositionvectors3i+2k,2i−2j+5k,2j+7kand−2i+10j+7krespectively.(i)UseascalarproducttoshowthatBAandBCareperpendicular.[4](ii)ShowthatBCandADareparallelandfindtheratioofthelengthofBCtothelengthofAD.[4]9709/01/M/J/03

39ThediagramshowsasemicircleABCwithcentreOandradius8cm.AngleAOB=θradians.(i)Inthecasewhereθ=1,calculatetheareaofthesectorBOC.[3](ii)FindthevalueofθforwhichtheperimeterofsectorAOBisonehalfoftheperimeterofsectorBOC.[3](iii)Inthecasewhereθ=1π,showthattheexactlengthoftheperimeteroftriangleABCis3√

(24+83)cm.[3]10√

Theequationofacurveisy=(5x+4).(i)Calculatethegradientofthecurveatthepointwherex=1.[3](ii)Apointwithcoordinates(x,y)movesalongthecurveinsuchawaythattherateofincreaseofxhastheconstantvalue0.03unitspersecond.Findtherateofincreaseofyattheinstantwhenx=1.[2](iii)Findtheareaenclosedbythecurve,thex-axis,they-axisandthelinex=1.[5]11Theequationofacurveisy=8x−x2.(i)Express8x−x2intheforma−(x+b)2,statingthenumericalvaluesofaandb.(ii)Hence,orotherwise,findthecoordinatesofthestationarypointofthecurve.(iii)Findthesetofvaluesofxforwhichy≥−20.[3][2][3]Thefunctiongisdefinedbyg:x→8x−x2,forx≥4.(iv)Statethedomainandrangeofg−1.(v)Findanexpression,intermsofx,forg−1(x).[2][3]9709/01/M/J/03

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