讲义课件成果

UNIVERSITYOFCAMBRIDGEINTERNATIONALEXAMINATIONS

GCEAdvancedSubsidiaryLevelandGCEAdvancedLevel

MARKSCHEMEfortheOctober/November2011questionpaperfortheguidanceofteachers

9709MATHEMATICS

9709/11

Paper1,

umrawmark75

Thismarkschemeispublishedasanaidtoteachersandcandidates,toindicatetherequirementsoftheexamination.ItshowsthebasisonwhichExaminerswereinstructedtoawardmarks.ItdoesnotindicatethedetailsofthediscussionsthattookceatanExaminers’meetingbeforemarkingbegan,whichwouldhaveconsideredtheacceptabilityofalternativeanswers.

Markschemesmustbereadinconjunctionwiththequestionpapersandthereportontheexamination.

Cambridgewillnotenterintodiscussionsorcorrespondenceinconnectionwiththesemarkschemes.

CambridgeispublishingthemarkschemesfortheOctober/November2011questionpapersformostIGCSE,GCEAdvancedLevelandAdvancedSubsidiaryLevelsyllabusesandsomeOrdinaryLevelsyllabuses.

com

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MarkSchemeNotes

Marksareofthefollowingthreetypes:

MMethodmark,awardedforavalidmethodappliedtotheproblem.Methodmarksarenotlostfornumericalerrors,algebraicslipsorerrorsinunits.However,itisnotusuallysufficientforacandidatejusttoindicateanintentionofusingsomemethodorjusttoquoteaformula;theformulaorideamustbeappliedtothespecificprobleminhand,e.g.bysubstitutingtherelevanttiesintotheformula.CorrectapplicationofaformulawithouttheformulabeingquotedobviouslyearnstheMmarkandinsomecasesanMmarkcanbeimpliedfromacorrectanswer.

Accuracymark,awardedforacorrectanswerorintermediatestepcorrectlyobtained.Accuracymarkscannotbegivenunlesstheassociatedmethodmarkisearned(orimplied).

Markforacorrectresultorstatementindependentofmethodmarks.

Whenapartofaquestionhastwoormore“method”steps,theMmarksaregenerallyindependentunlesstheschemespecificallysaysotherwise;andsimilarlywhenthereareseveralBmarksallocated.ThenotationDMorDB(ordep*)isusedtoindicatethataparticularMorBmarkisdependentonanearlierMorB(asterisked)markinthescheme.Whentwoormorestepsareruntogetherbythecandidate,theearliermarksareimpliedandfullcreditisgiven.

Thesymbol√impliesthattheAorBmarkindicatedisallowedforworkcorrectlyfollowingonfrompreviouslyincorrectresults.Otherwise,AorBmarksaregivenforcorrectworkonly.AandBmarksarenotgivenforfortuitously“correct”answersorresultsobtainedfromincorrectworking.

Note: B2orA2meansthatthecandidatecanearn2or0.

B2/1/0meansthatthecandidatecanearnanythingfrom0to2.

Themarksindicatedintheschememaynotbesubdivided.Ifthereisgenuinedoubtwhetheracandidatehasearnedamark,allowthecandidatethebenefitofthedoubt.Unlessotherwiseindicated,marksoncegainedcannotsubsequentlybelost,e.g.wrongworkingfollowingacorrectformofanswerisignored.

Wrongormissingunitsinananswershouldnotleadtothelossofamarkunlesstheschemespecificallyindicatesotherwise.

Foranumericalanswer,allowtheAorBmarkifavalueisobtainedwhichiscorrectto3s.f.,orwhichwouldbecorrectto3s.f.ifrounded(1d.p.inthecaseofanangle).Asstatedabove,anAorBmarkisnotgivenifacorrectnumericalanswerarisesfortuitouslyfromincorrectworking.ForMechanicsquestions,allowAorBmarksforcorrectanswerswhicharisefromtakinggequalto9.8or9.81insteadof10.

Thefollowingabbreviationsmaybeusedinamarkschemeorusedonthescripts:AEF AnyEquivalentForm(ofanswerisequallyacceptable)

AG AnswerGivenonthequestionpaper(soextracheckingisneededtoensurethatthedetailedworkingleadingtotheresultisvalid)

BODBenefitofDoubt(allowedwhenthevalidityofasolutionmaynotbeabsoluyclear)

CAOCorrectAnswerOnly(emphasisingthatno“followthrough”fromapreviouserrorisallowed)

CWO CorrectWorkingOnly–oftenwrittenbya‘fortuitous’answerISW IgnoreSubsequentWorking

MR Misread

PA PrematureApproximation(resultinginbasicallycorrectworkthatisinsufficientlyaccurate)

SOS SeeOtherSolution(thecandidatemakesabetttemptatthesamequestion)

SR SpecialRuling(detailingthemarktobegivenforaspecificwrongsolution,oracasewheresomestandardmarkingpracticeistobevariedinthelightofaparticularcircumstance)

Penalties

MR–1ApenaltyofMR–1isdeductedfromAorBmarkswhenthedataofaquestionorpartquestionaregenuinelymisreadandtheobjectanddifficultyofthequestionremainunaltered.InthiscaseallAandBmarksthen e“followthrough√”marks.MRisnotappliedwhenthecandidatemisreadshisownfigures–thisisregardedasanerrorinaccuracy.AnMR–2penaltymaybeappliedinparticularcasesifagreedatthecoordinationmeeting.

PA–1ThisisdeductedfromAorBmarksinthecaseofprematureapproximation.ThePA–1penaltyisusuallydiscussedatthemeeting.

1

2

4 1

6C4×[2(x)]×(x2)

240

B2

B1for2/3termscorrect

B1

[3]

Identifiedasanswer.Allow240x0

2

δy 9x2 12x+4

δx

M1A1

(3x 2)2≥0

A1

[3]

3

(i)Correctcosinecurveforatleast1oscillation

Exactly2completeoscillationsin[0,2π]

1

Liney 2correct

B1

Range

θaxis

1→1.Ignorelabelson

B1

B1

[3]

(ii)4

B1√

[1]

Fttheirgraph.Accept30°,150°,

210°,330°

(iii)20

B1√

[1]

Or5×theirpart(ii)

4

(i)3

B1

[1]

(ii)f(x) x2 6x(+c)

Subst(3,4)

c 5→f(x) x2 6x+5

M1A1

Dependentoncpresent

M1

cao

A1

[4]

5

(i)ArcAB rθ

OC rsinθorBC rcosθ

r(1+θ+cosθ+sinθ)

correctlyderived

M1M1

A1

[3]

θ

oeegBC rsin etc

tanθ

OC&BCreversedlosesM1A1

1 2 π

(ii)SectorOAB 2×10×5(31.42)

∆OCB 1

210cosπ10sinπ

5 5

(23.78)

Totalarea 55.2

M1

oe∆intermsofπand10

M1

AllowOC&BCreversed(iemax4/6)

A1

[3]

6

(a)a+5d 23

5(2a+9d) 200

Attemptsolution,expectd 6a

29

7

B1

Solutionof2linearequations

B1

M1

A1

[4]

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1 4

(b)1 r() 1

1 r

4

r 4oeS 5

5

M1

A1A1

[3]

UseofS∞formulatwice

7

1

(i) y 6(48 8x)oe

B1

[1]

(ii)A 4xy+2xyor3xy+3xy 6xy

A x(48 8x) 48x 8x2

M1

A1

[2]

AG

δA

(iii)δx 4816x

A 72cao

δ2A

16(<0)⇒ um

δx2

B1M1A1

B1

[4]

Attempttosolvederivative 0

Expectx 3

wwwAcceptothercompletemethods

8

(i)(4i+7j pk).)8i j pk(25+p2

M1A1

[2]

x1x2+y1y2+z1z2

(Not25+( p)2)

(ii)25+p2 0⇒norealsolutions

B1√

[1]

Ftprovidedequationhasnorealsolutions

OA.OB

(iii)cos60 OAOBused

OA 65+p2orOB 65+p2

25+p2 1 hisscalar(i) 1

or

65+p2 2 65+p2 2

p ±3.87or± 15

M1

OA.OBmustbescalar

M1

Not 65 p2

unlessfollows 65+( p)2

A1√

Scalarproduct 25+p2canscorehereifnotscoredinpart(i)

A1

[4]

9

(i) x2+3x+4 2x+6⇒x2+x 2(0)

(x 1)(x+2( 0)→1,)8, 2),2 (

AB 32+62 6.71or45or35

1,5

2

M1

3-termsimplification

DM1A1

DM1forattemptedsolutionforx

B1

cao(45fromwrongpointsscoresB0)

B1√

[5]

Fttheircoordinates

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(ii)x2+(3 k)x+2k 6(0)

(3 k)2 4(2k 6) 0

(3 k)(11 k) 0

k 3or11

M1DM1

DM1

A1

[4]

Simplifiedto3-termquadratic

2

Applyb 4ac 0asfunctionofk

only

AttemptfactorisationoruseformulaBothcorrect

NBAlternativemethodsfor(ii)

possible

10

(i)B (0,1)C (4,3)

B1,B1

[2]

IfB0B0thenSCB1forbothy 1&

x 4

δy 1×2(1+2)1

(ii)δx 2 x

2

Grad.ofnormal 3

y 3 3(x 4)ory 3x+15oe

M1A1B1

B1