Mathematics1HLPaper1spa.docx

Mathematics1HLPaper1spa.docx

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Mathematics1HLPaper1spa

IBDIPLOMAPROGRAMMEPROGRAMMEDUDIPLÔMEDUBIPROGRAMADELDIPLOMADELBI

MATHEMATICSHIGHERLEVELPAPER1

Wednesday3November2004（afternoon）

2hours

N04/5/MATHL/HP1/ENG/TZ0/XX

88047401

Schoolcode

Candidatecode

INSTRUCTIONSTOCANDIDATES

•Writeyourschoolcodeandcandidatecodeintheboxesabove.

•Donotopenthisexaminationpaperuntilinstructedtodoso.

•Answerallthequestionsinthespacesprovided.

•Unlessotherwisestatedinthequestion,allnumericalanswersmustbegivenexactlyortothreesignificantfigures.

•Indicatethemakeandmodelofyourcalculatorintheappropriateboxonyourcoversheet.

Maximummarkswillbegivenforcorrectanswers.Whereanansweriswrong,somemarksmaybegivenforcorrectmethod,providedthisisshownbywrittenworking.Workingmaybecontinuedbelowthebox,ifnecessary.Solutionsfoundfromagraphicdisplaycalculatorshouldbesupportedbysuitableworking,e.g.ifgraphsareusedtofindasolution,youshouldsketchtheseaspartofyouranswer.

1.Consider

f（x）=x3-2x2-5x+k.Findthevalueofkif（x+2）isafactorof

f（x）.

Working:

Answer:

⎛1-12⎞

2.Giventhatthematrix

A=⎜2p3⎟

issingular,findthevalueofp.

⎜⎟

⎜1-25⎟

⎝⎠

Working:

Answer:

3.Thesumofthefirstntermsofaseriesisgivenby

n

S=2n2-n,wheren∈®+.

（a）Findthefirstthreetermsoftheseries.

（b）Findanexpressionforthenthtermoftheseries,givingyouranswerintermsofn.

Working:

Answers:

（a）

（b）

4.Giventhat（a+i）（2-bi）=7-i,findthevalueofaandofb,wherea,b∈®.

Working:

Answer:

5.

If

y=ln（2x-1）,find

d2y

.

dx2

Working:

Answer:

6.Afairsix-sideddie,withsidesnumbered1,1,2,3,4,5isthrown.Findthemeanandvarianceofthescore.

Working:

Answer:

7.（a）FindthelargestsetSofvaluesofxsuchthatthefunctionvalues.

（b）FindtherangeofthefunctionfdefinedonthedomainS.

f（x）=1

takesreal

Working:

Answers:

（a）

（b）

8.（a）Findtheexpansionof（2+x）5,givingyouranswerinascendingpowersofx.

（b）Bylettingx=0.01orotherwise,findtheexactvalueof2.015.

Working:

Answers:

（a）

（b）

9.

B

Findtheareaoftheshadedregion.

Working:

Answer:

10.Considertheequatione-x=cos2x,for0≤x≤2π.

（a）Howmanysolutionsaretheretothisequation

（b）Findthesolutionclosestto2π,givingyouranswertofourdecimalplaces.

Working:

Answers:

（a）

（b）

11.ConsiderthefourpointsA（1,4,-1）,B（2,5,-2）,C（5,6,3）andD（8,8,4）.Findthepointof

intersectionofthelines（AB）and（CD）.

Working:

Answer:

12.AcontinuousrandomvariableXhasprobabilitydensityfunctiongivenby

f（x）=k（2x-x2）,for0≤x≤2

f（x）=0,elsewhere.

（a）Findthevalueofk.

（b）FindP（0.25≤x≤0.5）.

Working:

Answers:

（a）

（b）

13.Giventhat

z∈%,solvetheequation

z3-8i=0,givingyouranswersintheform

z=r（cosθ+isinθ）.

Working:

Answer:

14.Findthetotalareaofthetworegionsenclosedbythecurvey=x3-3x2-9x+27andtheline

y=x+3.

Working:

Answer:

15.Findtherangeofvaluesofmsuchthatforallx

m（x+1）≤x2.

Working:

Answer:

16.

Findtheequationofthenormaltothecurve

x3+y3-9xy=0

atthepoint（2,4）.

17.Usingthesubstitution2x=sinθ,orotherwise,find⎰（1-4x2）dx.

18.Aclosedcylindricalcanhasavolumeof500cm3.Theheightofthecanishcmandtheradiusofthebaseisrcm.

（a）FindanexpressionforthetotalsurfaceareaAofthecan,intermsofr.

（b）GiventhatthereisaminimumvalueofAforr>0,findthisvalueofr.

Working:

Answers:

（a）

（b）

19.（a）FindthecartesianequationoftheplanethatcontainstheoriginOandthetwopoints

A（1,1,1）andB（2,-1,3）.

（b）FindthedistancefromthepointC（10,5,1）totheplaneOAB.

Working:

Answers:

（a）

（b）

20.Thefollowingdiagramshowsthelines

x-2y-4=0,x+y=5

andthepoint

P（1,1）.Aline

isdrawnfromPtointersectwithmidpointof[QR].

x-2y-4=0

y

10

atQ,andwith

x+y=5

atR,sothatPisthe

8

6

2（1,

–10x

–2

–4

–6

–8

FindtheexactcoordinatesofQandofR.

Working:

Answer: