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A LEVELSpecification

MATHEMATICS B (MEI)H640For first assessment in 2018

Version 2.0 (October 2018)

ocr.org.uk/alevelmathsmei

A LEVEL Mathematics B (MEI)

http://ocr.org.uk/alevelmathsmei

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We will inform centres about changes to specifications. We will also publish changes on our website. The latest version of our specifications will always be those on our website (ocr.org.uk) and these may differ from printed versions.

Cover Image: A Level students and teachers from Exeter College

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1© OCR 2018 A Level in Mathematics B (MEI)

Contents

1 OCR’sALevelinMathematicsB(MEI) 21a. WhychooseanOCRALevelinMathematicsB(MEI)? 31b. Whatarethekeyfeaturesofthisspecification? 41c. Aims and learning outcomes 51d. HowdoIfindoutmoreinformation? 6

2 Thespecificationoverview 72a. ContentofALevelinMathematicsB(H640) 72b. Pre-release material 82c. Use of technology 92d. Command words 102e. Overarching Themes 162f. ContentofALevelMathematicsB(MEI) 222g. Prior knowledge, learning and progression 66

3 AssessmentofALevelinMathematicsB(MEI) 673a. Forms of assessment 673b. Assessmentobjectives(AO) 683c. Assessment availability 693d. Retakingthequalification 693e. Assessment of extended response 693f. Synopticassessment 703g. Calculatingqualificationresults 70

4 Admin:whatyouneedtoknow 714a. Pre-assessment 714b. Specialconsideration 724c. Externalassessmentarrangements 724d. Resultsandcertificates 734e. Post-resultsservices 734f. Malpractice 73

5 Appendices 745a. Overlapwithotherqualifications 745b. Accessibility 745c. Mathematicalnotation 745d. Mathematicalformulaeandidentities 79

Summary of updates 86

2© OCR 2018

A Level in Mathematics B (MEI)

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Learnersmustcompletecomponents01,02and03tobeawardedOCRALevelinMathematicsB(MEI).Contents is in these three areas:1 Puremathematics2 Mechanics3 Statistics.

ContentOverview AssessmentOverview

Component01assessescontentfromareas 1 and 2

PureMathematicsandMechanics

(01)

100marks

2 hour writtenpaper

36.4%of totalA Level

Component02assessescontentfromareas 1 and 3

PureMathematicsandStatistics

(02)

100marks

2 hour writtenpaper


Component03assessescontentfromarea1(areas2and3areassumed

knowledge)

PureMathematicsandComprehension

(03)

75 marks

2 hour writtenpaper


Percentagesinthetableaboveareroundedto1decimalplace,exactcomponentproportionsare:

, , .36 36 27114

114

113

AllAlevelqualificationsofferedbyOCRareaccreditedbyOfqual,theRegulatorforqualificationsofferedinEngland.TheaccreditationnumberforOCR’sALevelinMathematicsB(MEI)isQN:603/1002/9.

Learnerswillbegivenformulaeineachassessmentonpages2and3ofthequestionpaper.Seesection5dforalist of these formulae.

1 OCR’sALevelinMathematicsB(MEI)


1

1a. WhychooseanOCRALevelinMathematicsB(MEI)?

OCRALevelMathematicsB(MEI)hasbeendevelopedinpartnershipwithMathematicsinEducationandIndustry(MEI).

OCRALevelinMathematicsB(MEI)providesaframework within which a large number of young peoplecontinuethesubjectbeyondGCSE(9–1). Itsupportstheirmathematicalneedsacrossabroadrange of subjects at this level and provides a basis for subsequentquantitativeworkinaverywiderangeofhighereducationcoursesandinemployment.Italsosupports the study of AS and A Level Further Mathematics.

OCRALevelinMathematicsB(MEI)buildsfromGCSE(9–1)levelmathematicsandintroducescalculusanditsapplications.Itemphasiseshowmathematicalideasareinterconnectedandhowmathematicscanbeappliedtomodelsituationsusingalgebraandotherrepresentations,tohelpmakesenseofdata,tounderstand the physical world and to solve problems in a variety of contexts, including social sciences and business.Itpreparesstudentsforfurtherstudyandemployment in a wide range of disciplines involving theuseofmathematics.

ASLevelMathematicsB(MEI),whichcanbeco-taughtwiththeALevelasaseparatequalification,consolidatesanddevelopsGCSElevelmathematicsandsupportstransitiontohighereducationoremployment in any of the many disciplines that make useofquantitativeanalysis,includingthoseinvolvingcalculus.

ThisqualificationispartofawiderangeofOCRmathematicsqualifications,whichallowsprogressionfromEntryLevelCertificateandFunctionalSkills,throughGCSE(9–1)andAdditionalMathstoCoreMaths, AS and A level.

Allofourmathematicsspecificationshave beendevelopedbyOCRandMEIwithsubject and teaching experts. We have worked in close consultationwithteachersfromlearnedsocieties,industryandrepresentativesfromHigher Education(HE).

We recognise that teachers want to be able to choose qualificationsthatsuittheirlearnerssoweoffertwosuitesofqualificationsinmathematicsandfurthermathematics.

MathematicsAbuildsonourexistingpopularcourse.We’vebasedtheredevelopmentofourcurrentsuitearound an understanding of what works well in centres and have updated areas of content and assessmentwherestakeholdershaveidentifiedthatimprovements could be made.

MathematicsB(MEI)isbasedontheexistingsuite ofqualificationsassessedbyOCR.MEIisalongestablished, independent curriculum development body.MEIprovidesadviceandCPDrelatingtoall the curriculum and teaching aspects of the course. Italsoprovidesteachingresources,whichforthisspecificationcanbefoundonthewebsite (www.mei.org.uk).

http://www.mei.org.uk

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1b. Whatarethekeyfeaturesofthisspecification?

Exemplarcontent.

Clear command words and guidance on calculator use.

SeparateQuestionPapersandAnswerBookletssothatstudentscanalwaysseethewholeofaquestionatonetimeandtoallowfordiagramsandtablesforthemtoworkon.

Easytofollowmarkschemeswithcompletesolutionsandclearguidance.

Stretchandchallengequestionsdesignedtoallowthemostablelearnerstheopportunitytodemonstratethe full extent of their knowledge and skills, and to support the awarding of A* grade at A Level.

Appliedcontent(statisticsandmechanics)assessedonseparatepaperssothatthecontentdomainsassessedonanygivenpaperdon’tcoverbothatonce.

MathematicsAH240 MathematicsB(MEI)H640

Single pre-release data set designed to last the life ofthequalification.

Components02and03areintwosections:sectionAonthePureMathematicscontent;sectionBoneitherStatisticsorMechanics.

Threedatasetsavailableatalltimes,sothatyoucanuse all three for teaching, but for each cohort of students just one will be the context for some of the questionsintheexam.Eachdatasetwillbeclearlylabelled as to when it is used.

Components01and02areintwosections: sectionAconsistsofshorterquestionswithminimalreadingandinterpretation; sectionBincludeslongerquestionsandproblemsolving.


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1c. Aimsandlearningoutcomes

OCRALevelinMathematicsB(MEI)encourageslearners to:

• understandmathematicsandmathematicalprocessesinawaythatpromotesconfidence,fosters enjoyment and provides a strongfoundationforprogresstofurtherstudy

• extendtheirrangeofmathematicalskillsandtechniques

• understandcoherenceandprogressioninmathematicsandhowdifferentareasofmathematicsareconnected

• applymathematicsinotherfieldsofstudyandbeawareoftherelevanceofmathematicstotheworldofworkandtosituationsinsocietyingeneral

• usetheirmathematicalknowledgetomakelogical and reasoned decisions in solvingproblemsbothwithinpuremathematicsandina variety of contexts, and communicate themathematicalrationaleforthesedecisionsclearly

• reasonlogicallyandrecogniseincorrectreasoning

• generalisemathematically

• constructmathematicalproofs

• usetheirmathematicalskillsandtechniquestosolve challenging problems which require themtodecideonthesolutionstrategy

• recognisewhenmathematicscanbeusedtoanalyse and solve a problem in context

• representsituationsmathematicallyandunderstandtherelationshipbetweenproblemsincontextandmathematicalmodelsthatmaybe applied to solve them

• drawdiagramsandsketchgraphstohelpexploremathematicalsituationsandinterpretsolutions

• makedeductionsandinferencesanddrawconclusionsbyusingmathematicalreasoning

• interpretsolutionsandcommunicatetheirinterpretationeffectivelyinthecontextoftheproblem

• readandcomprehendmathematicalarguments,includingjustificationsofmethodsand formulae, and communicate theirunderstanding

• readandcomprehendarticlesconcerningapplicationsofmathematicsandcommunicatetheir understanding

• usetechnologysuchascalculatorsandcomputerseffectivelyandrecognisewhensuchuse may be inappropriate

• takeincreasingresponsibilityfortheirownlearningandtheevaluationoftheirownmathematicaldevelopment.

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1IfyouarealreadyusingOCRspecificationsyoucancontact us at: www.ocr.org.uk.

IfyouarenotalreadyaregisteredOCRcentrethenyoucanfindoutmoreinformationonthebenefitsofbecoming one at: www.ocr.org.uk.

GetintouchwithoneofOCR’sSubjectAdvisors:

Email:[email protected]

Twitter:@OCR Maths

CustomerContactCentre:01223553998

Teacher resources, blogs and support: available from: www.ocr.org.uk

SignupforourmonthlyMathsnewsletter,Total Maths

AccessCPD,training,eventsandsupportthroughOCR’sCPDHub

Access our online past papers service that enables you to build your own test papers from past OCR examquestionsthroughOCR’sExamBuilder

Access our free results analysis service to help you review the performance of individual learners or whole schools through ActiveResults

1d. HowdoIfindoutmoreinformation?

http://www.ocr.org.uk



https://twitter.com/OCR_Maths


http://www.ocr.org.uk/qualifications/by-subject/mathematics/total-maths/

https://www.cpdhub.ocr.org.uk/desktopdefault.aspx

http://www.ocr.org.uk/qualifications/past-papers/exambuilder/

http://www.ocr.org.uk/administration/support-and-tools/active-results/

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ThisAlevelqualificationbuildsontheskills,knowledge and understanding set out in the whole GCSE(9–1)subjectcontentformathematicsforfirstteachingfrom2015.

ALevelMathematicsB(MEI)isalinearqualification,withnooptions.Thecontentislistedbelow,underthree areas:

1. Puremathematicsincludesproof,algebra,graphs, sequences, trigonometry, logarithms,calculus and vectors

2. Mechanicsincludeskinematics,motionundergravity,workingwithforcesincludingfriction,Newton’slawsandsimplemoments

3. Statisticsincludesworkingwithdatafromasampletomakeinferencesaboutapopulation,probabilitycalculations,usingbinomialandNormaldistributionsasmodelsandstatisticalhypothesistesting.

Therewillbethreeexaminationpapers,attheendofthe course, to assess all the content:

• PureMathematicsandMechanics(01),a2hourpaperassessingpuremathematicsandmechanics

• PureMathematicsandStatistics(02),a2hourpaperassessingpuremathematicsandstatistics

• PureMathematicsandComprehension(03),a2hourpaperassessingpuremathematics.

Although the content is listed under three separate areas, links should also be made between pure mathematicsandeachofmechanicsandstatistics;some of the links that can be made are indicated in the notes in the detailed content.

The overarching themes should be applied, along withassociatedmathematicalthinkingandunderstanding, across the whole of the detailed contentinpuremathematics,statisticsandmechanics.

2 Thespecificationoverview

2a. ContentofALevelinMathematicsB(H640)

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Pre-release material will be made available in advanceoftheexaminations.Itwillberelevanttosome(butnotall)ofthequestionsincomponent02.The pre-release material will be a large data set (LDS)thatcanbeusedasteachingmaterialthroughoutthecourse.Itiscomparabletoasettextforaliteraturecourse.Itwillbepublishedinadvanceofthecourse.Intheexaminationitwillbeassumedthat learners are familiar with the contexts covered by this data set and that they have used a spreadsheetorotherstatisticalsoftwarewhenworkingwiththedata.QuestionsbasedontheseassumptionswillbesetinPureMathematicsandStatistics(02).

Theintentionisthatthesequestionsshouldgiveamaterial advantage to learners who have studied, and are familiar with, the prescribed large data set. They mightincludequestionsrequiringlearnerstointerpret data in ways which would be too demanding in an unfamiliar context.

LearnerswillNOThaveaprintoutofthepre-releasedatasetavailabletothemintheexaminationbutselecteddataorsummarystatisticsfromthedata setmaybeprovided,withintheexamination paper.

DifferentLDSwillbeissuedfordifferentyears;threelargedatasetswillbeavailableatanytime.Onlyoneof these data sets will be used in a given series of examinations.Eachdatasetwillbeclearlylabelledwiththeyearoftheexaminationseriesinwhichitwillbeused.Theexpectationisthatteacherswilluseallthree data sets but they do have the choice of concentratingmoreontheexaminationdataset(orofusingjustthatoneiftheywish).Tosupportprogression and co-teachability, students taking AS LevelMathematicsB(MEI)willusethesamedatasetiftheytakeALevelMathematicsB(MEI)inthefollowing year.

Theintentionofthelargedatasetisthatitandassociated contexts are explored in the classroom using technology, and that learners become familiar with the context and main features of the data.

Tosupporttheteachingandlearningofstatisticswiththe large data set, we suggest that the following activitiesarecarriedoutthroughoutthecourse:

1. Exploratorydataanalysis:LearnersshouldexploretheLDSwithbothquantitativeandvisual techniques to develop insight intounderlyingpatternsandstructures,suggesthypotheses to test and to provide a reason forfurtherdatacollection.Thiswillincludetheuse of the following techniques.• Creatingdiagrams:Learnersshoulduse

spreadsheetsorstatisticalgraphingtoolsto create diagrams from data.

• Calculations:Learnersshoulduseappropriate technology to performstatisticalcalculations.

• Investigatingcorrelation:Learnersshoulduse appropriate technology to explorecorrelationbetweenvariablesintheLDS.

2. Modelling:LearnersshouldusetheLDStoprovideestimatesofprobabilitiesformodellingandtoexplorepossiblerelationshipsbetweenvariables.

3. Repeated sampling: Learners should use theLDSasamodelforthepopulationtoperformrepeatedsamplingexperimentstoinvestigatevariabilityandtheeffectofsamplesize.Theyshouldcomparetheresultsfromdifferentsamples with each other and with the resultsfromthewholeLDS.

4. Hypothesistesting:LearnersshouldusetheLDSasthepopulationagainstwhichtotesthypotheses based on their own sampling.

2b. Pre-releasematerial

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Itisassumedthatlearnerswillhaveaccesstoappropriate technology when studying this course suchasmathematicalandstatisticalgraphingtoolsand spreadsheets. When embedded in the mathematicsclassroom,theuseoftechnologycanfacilitatethevisualisationofabstractconceptsanddeepenlearners’overallunderstanding.Learnersarenotexpectedtobefamiliarwithanyparticularsoftware,buttheyareexpectedtobeabletousetheircalculatorforanyfunctionitcanperform,whenappropriate.Examinationquestionsmayincludeprintoutsfromsoftwarewhichlearnerswillneedtocomplete or interpret. They should be familiar with language used to describe spreadsheets such as row, column and cell.

Tosupporttheteachingandlearningofmathematicsusing technology, we suggest that the following activitiesarecarriedoutthroughthecourse:

1. Graphingtools:Learnersshouldusegraphingsoftwaretoinvestigatetherelationshipsbetween graphical and algebraicrepresentations,e.g.understandingtheeffectof changing the parameter k in the graphs of

y x k1= + or y x kx2= - ;e.g.investigating

tangents to curves.

2. Spreadsheets: Learners should usespreadsheets to generate tables of values forfunctions,toinvestigatefunctionsnumericallyand as an example of applying algebraicnotation.Learnersshouldalsousespreadsheetsoftwaretoinvestigatenumericalmethodsforsolvingequationsandformodellinginstatisticsand mechanics.

3. Statistics:Learnersshouldusespreadsheetsorstatisticalsoftwaretoexploredatasetsandstatisticalmodelsincludinggeneratingtablesand diagrams, and performing standardstatisticalcalculations.

4. Mechanics:Learnersshouldusegraphingand/orspreadsheetsoftwareformodelling,includingkinematicsandprojectiles.

5. ComputerAlgebraSystem(CAS):LearnerscoulduseCASsoftwaretoinvestigatealgebraicrelationships,includingderivativesandintegrals,andasaninvestigativeproblemsolvingtool.Thisisbestdoneinconjunctionwithothersoftwaresuchasgraphingtoolsandspreadsheets.

2c. Useoftechnology

Useofcalculators

Calculators must comply with the published Instructions for conducting examinations, which can be found at http://www.jcq.org.uk/

Itisexpectedthatcalculatorsavailableintheexaminationswillincludethefollowingfeatures:

• AniterativefunctionsuchasanANSkey.• Theabilitytocomputesummarystatisticsand

accessprobabilitiesfromthebinomialandNormaldistributions

When using calculators, candidates should bear in mind the following:

1. Candidates are advised to write down explicitlyany expressions, including integrals, that theyuse the calculator to evaluate.

2. Candidates are advised to write down thevalues of any parameters and variables thatthey input into the calculator. Candidates arenot expected to write down data transferredfromquestionpapertocalculator.

3. Correctmathematicalnotation(ratherthan“calculatornotation”)shouldbeused;incorrectnotationmayresultinlossofmarks.

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2d. Commandwords

Itisexpectedthatlearnerswillsimplifyalgebraicandnumericalexpressionswhengivingtheirfinalanswers,eveniftheexaminationquestiondoesnotexplicitlyaskthemtodoso.

Example1:

80 23

shouldbewrittenas40 3.

Example2:

21 1 2 2x 2

1#+ -^ h shouldbewrittenaseither 1 2x 2

1+ -^ h or

1 21

x+.

Example3:

ln 2 ln 3 ln1+ - shouldbewrittenasln 6.

Example4:

Theequationofastraightlineshouldbegivenintheformy mx c= + or ax by c+ = unless otherwise stated.

The meanings of someinstructionsandwordsusedinthisspecificationaredetailedbelow.

Othercommandwords,forexample“explain”or“calculate”,willhavetheirordinaryEnglishmeaning.

Exact

Anexactanswerisonewherenumbersarenotgiveninroundedform.Theanswerwilloftencontainanirrationalnumbersuchas 3, e or π and these numbers should be given in that form when an exact answer is required.

Theuseoftheword‘exact’alsotellslearnersthatrigorous(exact)workingisexpectedintheanswertothequestion.

Example1:

Findtheexactsolutionof1n x = 2.

The correct answer is e2andnot7.389056.

Example2:

Findtheexactsolutionof3 2x =

The correct answer is 32 or 0.6,x x= = o not x = 0.67 or similar.

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Prove

Learnersaregivenastatementandmustprovideaformalmathematicalargumentwhichdemonstratesitsvalidity.

Aformalproofrequiresahighlevelofmathematicaldetail,withcandidatesclearlydefiningvariables,correctalgebraicmanipulationandaconciseconclusion.

ExampleQuestion

Provethatthesumofthesquaresofanythreeconsecutivepositiveintegerscannot be divided by 3.

ExampleResponse

Letthethreeconsecutivepositiveintegersben – 1, n and n + 1

(n – 1)2 + n2 + (n + 1)2 = 3n2 + 2

This always leaves a remainder of 2 and so cannot be divided by 3.

Showthat

Learnersaregivenaresultandhavetoshowthatitistrue.Becausetheyaregiventheresult,theexplanationhastobesufficientlydetailedtocovereverystepoftheirworking.

ExampleQuestion

Show that the curve y = x ln xhasastationarypoint 1, 1e e-J

L

KKKKN

P

OOOO.

ExampleResponse

1. . 1 1ln lnxy

x x x xdd

= + = +

0xy

dd

= forstationarypoint

When 1 1 1 0lnx x

ye d

de&= = + = sostationary

When 1, 1 1 1lnx y ye e e e&= = =- so

1, 1e e-J

L

KKKKN

P

OOOOisastationarypointonthe

curve.

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Verify

Aclearsubstitutionofthegivenvaluetojustifythestatementisrequired.

ExampleQuestion

Verify that the curve y = x ln x hasastationarypointat 1 .x e=

ExampleResponse

1lnxy

xdd

= +

At 1, 1 1 1 1 0lnx xy

e dd

e= = + =- + = thereforeitisastationarypoint.

Find,Solve,Calculate

Thesecommandwordsindicatethat,whileworkingmaybenecessarytoanswerthequestion,nojustificationisrequired.Asolutioncouldbeobtainedfromtheefficientuseofacalculator,eithergraphicallyorusinganumerical method.

ExampleQuestion

Findthecoordinatesofthestationarypointofthecurvey = x ln x.

ExampleResponse

(0.368, –0.368)

Determine

Thiscommandwordindicatesthatjustificationshouldbegivenforanyresultsfound,includingworkingwhereappropriate.

ExampleQuestion

Determinethecoordinatesofthestationarypointofthecurvey = x ln x.

ExampleResponse

1. . 1 1ln lnxy

x x x xdd

= + = +

ln l 0.368...0 xx &+ = =

When 0.368..., 0.368... 0.368...1 0.368...lnx y #= = =-

So (0.368, –0.368)

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Give,State,Writedown

Thesecommandwordsindicatethatneitherworkingnorjustificationisrequired.

ExampleQuestion

Statethecoordinatesofthestationarypointofthecurvey=(x − 3)2 + 4.

ExampleResponse

(3, 4)

Inthisquestionyoumustshowdetailedreasoning.

Whenaquestionincludesthisinstructionlearnersmustgiveasolutionwhichleadstoaconclusionshowingadetailedandcompleteanalyticalmethod.Theirsolutionshouldcontainsufficientdetailtoallowthelineoftheirargumenttobefollowed.Thisisnotarestrictiononalearner’suseofacalculatorwhentacklingthequestion,e.g.forcheckingananswerorevaluatingafunctionatagivenpoint,butitisarestrictiononwhatwillbeaccepted as evidence of a complete method.

Intheseexamplesvariationsinthestructureoftheanswersarepossible,forexampleusingadifferentbaseforthelogarithmsinexample1,anddifferentintermediatestepsmaybegiven.

Example1:

Uselogarithmstosolvetheequation 3 4x2 1 100=+ ,givingyouranswercorrectto3significantfigures.

The answer is x = 62.6, but the learner must include the steps log 3 log 42 1 100x =+ , ( ) log logx2 1 3 4100+ =

andanintermediateevaluationstep,forexample . ...x2 1 126 18+ = . Usingthesolvefunctiononacalculatortoskiponeofthesestepswouldnotresultinacompleteanalyticalmethod.

Example2:

Evaluate .x x x4 1d3

0

1

2+ -y

The answer is 127

, but the learner must include at least x x x41 4

34 3

0

1+ -7 A andthesubstitution 14

134

+ - . Justwritingdowntheanswerusingthedefiniteintegralfunctiononacalculatorwouldthereforenotbeawarded any marks.

Example3:

Solvetheequation forsin cosx x x3 2 0 180c c# #= .

The answer is x = 9.59°, 90° or 170° (to3sf),butthelearnermust include … sin cos cosx x x6 0- = , ( ) ,cos sin cos sinx x x x6 1 0 0 or 6

1- = = = .

Agraphicalmethodwhichinvestigatedtheintersectionsofthecurves siny x3 2= and cosy x= would be acceptabletofindthesolutionat 90°ifcarefullyverified,buttheothertwosolutionsmustbefoundanalytically,notnumerically.

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Hence

Whenaquestionusestheword‘hence’,itisanindicationthatthenextstepshouldbebasedonwhathasgonebefore.Theintentionisthatlearnersshouldstartfromtheindicatedstatement.

You are given that ( ) 2 7 6x x x xf 3 2= - - + . Show that ( 1)x - is a factor of ( )xf .

Hencefindthethreefactorsof ( )xf .

Henceorotherwise

Thisisusedwhentherearemultiplewaysofansweringagivenquestion.Learnersstartingfromtheindicatedstatementmaywellgainsomeinformationaboutthesolutionfromdoingso,andmayalreadybesomewaytowardstheanswer.Thecommandphraseisusedtodirectlearnerstowardsusingaparticularpieceofinformationtostartfromortoaparticularmethod.Italsoindicatestolearnersthatvalidalternatemethodsexistwhichwillbegivenfullcredit,butthattheymaybemoretime-consumingorcomplex.

Example:

Show that ( ) 1 2x x xcos sin sin2+ = + for all x.

Henceorotherwise,findthederivativeof( )cos sinx x 2+ .

Youmayusetheresult

When this phrase is used it indicates a given result that learners would not normally be expected to know, but whichmaybeusefulinansweringthequestion.

Thephraseshouldbetakenaspermissive;useofthegivenresultisnotrequired.

Plot

Learners should mark points accurately on the graph in their printed answer booklet. They will either have been given the points or have had to calculate them. They may also need to join them with a curve or a straight line, ordrawalineofbestfitthroughthem.

Example:

Plotthisadditionalpointonthescatterdiagram.

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Sketch

Learners should draw a diagram, not necessarily to scale, showing the main features of a curve. These are likely to include at least some of the following.

• Turningpoints• Asymptotes• Intersectionwiththey-axis• Intersectionwiththex-axis• Behaviourforlargex(+ or –)

Any other important features should also be shown.

Example:

Sketchthecurvewithequation ( 1)1y x=-

Draw

Learners should draw to an accuracy appropriate to the problem. They are being asked to make a sensible judgement about this.

Example1:

Drawadiagramshowingtheforcesactingontheparticle.

Example2:

Drawalineofbestfitforthedata.

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TheseOverarchingThemesshouldbeapplied,alongwithassociatedmathematicalthinkingandunderstanding,acrossthewholeofthedetailedcontentinthisspecification.Thesestatementsareintendedtodirecttheteachingandlearningof ALevelMathematics,andtheywillbereflectedinassessmenttasks.

2e. OverarchingThemes

OT1Mathematicalargument,languageandproof

Knowledge/Skill

OT1.1 Constructandpresentmathematicalargumentsthroughappropriateuseofdiagrams;sketchinggraphs;logicaldeduction;precisestatementsinvolvingcorrectuseofsymbolsandconnectinglanguage,including:constant,coefficient,expression,equation,function,identity,index,term,variable

OT1.2 Understandandusemathematicallanguageandsyntaxassetoutinthecontent

OT1.3 Understand and use language and symbols associated with set theory, as set out in the contentApplytosolutionsofinequalitiesandprobability

OT1.4 Understandandusethedefinitionofafunction;domainandrangeoffunctions

OT1.5 Comprehendandcritiquemathematicalarguments,proofsandjustificationsofmethodsandformulae,includingthoserelatingtoapplicationsofmathematics

OT2Mathematicalproblemsolving

Knowledge/Skill

OT2.1 Recognisetheunderlyingmathematicalstructureinasituationandsimplifyandabstractappropriately to enable problems to be solved

OT2.2 Construct extended arguments to solve problems presented in an unstructured form, including problems in context

OT2.3 Interpretandcommunicatesolutionsinthecontextoftheoriginalproblem

OT2.4 Understandthatmanymathematicalproblemscannotbesolvedanalytically,butnumericalmethodspermitsolutiontoarequiredlevelofaccuracy

OT2.5 Evaluate,includingbymakingreasonedestimates,theaccuracyorlimitationsofsolutions,including those obtained using numerical methods

OT2.6 Understandtheconceptofamathematicalproblemsolvingcycle,includingspecifyingtheproblem,collectinginformation,processingandrepresentinginformationandinterpretingresults,whichmayidentifytheneedtorepeatthecycle

OT2.7 Understand,interpretandextractinformationfromdiagramsandconstructmathematicaldiagrams to solve problems, including in mechanics

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OT3Mathematicalmodelling

Knowledge/Skill

OT3.1 Translateasituationincontextintoamathematicalmodel,makingsimplifyingassumptions

OT3.2 Useamathematicalmodelwithsuitableinputstoengagewithandexploresituations(foragivenmodeloramodelconstructedorselectedbythestudent)

OT3.3 Interprettheoutputsofamathematicalmodelinthecontextoftheoriginalsituation(foragivenmodeloramodelconstructedorselectedbythestudent)

OT3.4 Understandthatamathematicalmodelcanberefinedbyconsideringitsoutputsandsimplifyingassumptions;evaluatewhetherthemodelisappropriate

OT3.5 Understandandusemodellingassumptions

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MathematicalProblemSolvingCycle

Mathematicalproblemsolvingisacorepartofmathematics.Theproblemsolvingcyclegivesageneralstrategyfordealingwithproblemswhichcanbesolvedusingmathematicalmethods;itcanbeusedforproblemswithinmathematicalcontextsandforproblemsinreal-worldcontexts.

Process Description

Problemspecificationandanalysis

Theproblemtobeaddressedneedstobeformulatedinawaywhichallowsmathematicalmethodstobeused.Itthenneedstobeanalysedsothataplancanbemadeastohowtogoaboutit.Theplanwillalmostalwaysinvolvethecollectionofinformationinsomeform.Theinformationmayalreadybeavailable(e.g.online)oritmaybenecessarytocarryoutsomeformofexperimentalorinvestigationalworktogatherit.

Insomecasestheplanwillinvolveconsideringsimplecaseswithaviewtogeneralisingfromthem.Inothers,physicalexperimentsmaybeneeded.Instatistics,decisionsneedtobemade at this early stage about what data will be relevant and how they will be collected.

The analysis may involve considering whether there is an appropriate standard model to use(e.g.theNormaldistributionortheparticlemodel)orwhethertheproblemissimilarto one which has been solved before.

Atthecompletionofthe problemsolvingcycle,thereneedstobeconsiderationofwhethertheoriginalproblemhasbeensolvedinasatisfactorywayorwhetheritisnecessarytorepeattheproblemsolvingcycleinordertogainabettersolution. Forexample,thesolutionmightnotbeaccurateenoughoronlyapplyinsomecases.

Informationcollection

Thisstageinvolvesgettingthenecessaryinputsforthemathematicalprocessingthatwilltake place at the next stage. This may involve deciding which are the important variables, findingkeymeasurementsorcollectingdata.

Processingandrepresentation

Thisstageinvolvesusingsuitablemathematicaltechniques,suchascalculations,graphsordiagrams,inordertomakesenseoftheinformationcollectedinthepreviousstage.Thisstageendswithaprovisionalsolutiontotheproblem.

Interpretation Thisstageoftheprocessinvolvesreportingthesolutiontotheprobleminawaywhichrelatestotheoriginalsituation.CommunicationshouldbeinclearplainEnglishwhichcan be understood by someone who has an interest in the original problem but is not anexpertinmathematics.Thisshouldleadintoreflectiononthesolutiontoconsiderwhetheritissatisfactoryoriffurtherworkisneeded.

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Theexaminationswillassumethatlearnershaveusedthe full modelling cycle during the course.

Mathematicscanbeappliedtoawidevarietyofproblemsarisingfromrealsituationsbutreallifeiscomplicated, and can be unpredictable, so some assumptionsneed to be made to simplify the situationandallowmathematicstobeused.Onceanswers have been obtained, we need to comparewithexperienceto make sure that the answers are useful. For example, the government might want to

know how many primary school children there will be in the future so that they can make sure that there areenoughteachersandschoolplaces.Tofindareasonableestimate,theymightassumethat the birthrateoverthenextfiveyearswillbesimilartothatforthelastfiveyearsandthosechildrenwillgoto school in the area they were born in. They would evaluatetheseassumptionsbycheckingwhethertheyfitinwithnewdataand review the estimatetoseewhetheritisstillreasonable.

TheModellingCycle

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Learning outcomes are designed to help users by clarifying the requirements, but the following points need to be noted:

• Contentthatiscoveredbyalearningoutcomewith a reference code may be tested in anexaminationquestionwithoutfurtherguidancebeing given.

• Learningoutcomesmarkedwithanasterisk*are assumed knowledge and will not form thefocusofanyexaminationquestions.Thesestatements are included for clarity andcompleteness.

• Manyexaminationquestionswillrequirelearners to use two or more learning outcomesatthesametimewithoutfurtherguidancebeing given. Learners are expected to be abletomakelinksbetweendifferentareasofmathematics.

• Learners are expected to be able to use theirknowledge and understanding to reasonmathematicallyandsolveproblemsbothwithinmathematicsandincontext.Contentthatiscovered by any learning outcome may berequired in problem solving, modelling andreasoning tasks even if that is not explicitlystated in the learning outcome.

• Learningoutcomeshaveanimpliedprefix:‘Alearnershould…’.

• Eachreferencecodeforalearningoutcomeisunique. For example, in the code Mc1, M referstoMathematics,crefersto‘calculus’(seebelow)and1meansthatitisthefirstsuchlearning outcome in the list.

• Thelettersusedinassigningreferencecodestolearning outcomes are common to allqualificationsinspecB(H630,H640,H635andH645).OnlythoseusedinAlevelMathematicsB(MEI)areshownbelow.

Learningoutcomes

2


a algebra Ab Bc calculus C Curves, Curve sketchingd D Datapresentation&interpretatione equations E Exponentialsandlogarithmsf functions F Forcesg geometry, graphs Gh H Hypothesistestingi Ij Jk kinematics Kl Lm Mn Newton’slaws No Op mathematicalprocesses(modelling,

proof,etc)P

q Qr R Random variabless sequences and series St trigonometry Tu probability(uncertainty) Uv vectors Vw Wx Xy projectiles Yz Z

Notes,notationandexclusions

Thenotes,notationandexclusionscolumnsinthespecificationareintendedtoassistteachersandlearners.

• Thenotescolumnprovidesexamplesandfurther detail for some learning outcomes. Allexemplarscontainedinthespecificationareforillustrationonlyanddonotconstituteanexhaustivelist

• Thenotationcolumnshowsnotationandterminology that learners are expected toknow, understand and be able to use

• Theexclusionscolumnlistscontentwhichwillnot be tested, for the avoidance of doubt wheninterpretinglearningoutcomes.

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2f. ContentofALevelMathematicsB(MEI)

Specification Ref. Learningoutcomes Notes Notation Exclusions

PUREMATHEMATICS:PROOF(1)

Proof Mp1 Understand and be able to use the structure of mathematicalproof.

Use methods of proof, including proof by deductionandproofbyexhaustion.

Proceedingfromgivenassumptionsthrough a series of logical steps to a conclusion.

p2 Beabletodisproveaconjecturebytheuseofacounter example.

PUREMATHEMATICS:PROOF(2)

Proof p3 Understand and be able to use proof by contradiction.

Includingproofoftheirrationalityof2andtheinfinityofprimes,and

applicationtounfamiliarproofs.

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PUREMATHEMATICS:ALGEBRA(1)

Algebraic language

Ma1 Knowandbeabletousevocabularyandnotationappropriate to the subject at this level.

Vocabularyincludesconstant,coefficient,expression,equation,function,identity,index, term, variable, unknown.

( )xf

Solutionofequations

* Beabletosolvelinearequationsinoneunknown. Includingthosecontainingbrackets,fractionsandtheunknownonbothsidesoftheequation.

* Beabletochangethesubjectofaformula. Includingcaseswherethenewsubjectappears on both sides of the original formula, and cases involving squares, square roots and reciprocals.

Ma2 Beabletosolvequadraticequations. Byfactorising,completingthesquare,using the formula and graphically.Includesquadraticequationsinafunctionoftheunknown.

a3 Beabletofindthediscriminantofaquadraticfunctionandunderstanditssignificance.

Theconditionfordistinctrealrootsofax bx c 02 + + = is: Discriminant> 0. Theconditionforrepeatedrootsis:Discriminant= 0. Theconditionfornorealrootsis:Discriminant< 0.

For ax bx c 02 + + = the discriminant is b ac42 - .

Complex roots.

a4 Beabletosolvelinearsimultaneousequationsintwo unknowns.

Byeliminationandbysubstitution.

a5 Beabletosolvesimultaneousequationsintwounknownswithoneequationlinearandonequadratic.

Byeliminationandbysubstitution.

a6 Knowthesignificanceofpointsofintersectionoftwographswithrelationtothesolutionofequations.

Includingsimultaneousequations.

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Inequalities Ma7 Beabletosolvelinearinequalitiesinonevariable.Beabletorepresentandinterpretlinearinequalitiesgraphicallye.g. .y x 1> +

Includingthosecontainingbracketsandfractions.

a8 Beabletosolvequadraticinequalitiesinonevariable.Beabletorepresentandinterpretquadraticinequalitiesgraphicallye.g.y ax bx c> 2 + + .

Algebraic and graphical treatment of solutionofquadraticinequalities.Forregionsdefinedbyinequalitieslearners must state clearly which regions are included and whether the boundaries areincluded.Noparticularshadingconventionisexpected.

Complex roots

a9 Beabletoexpresssolutionsofinequalitiesthroughcorrectuseof‘and’and‘or’,orbyusingsetnotation.

Learners will be expected to express solutionstoquadraticinequalitiesinanappropriate version of one of the following ways.

• x 1# or x 4$• : :x x x ax1 4,# $" ", ,• x2 5< < • x 5< and x 2>

• : :x x x x5 2< >+" ", ,

:x x 4>" ,

SurdsIndices

a10 Beabletouseandmanipulatesurds.

a11 Beabletorationalisethedenominatorofasurd.e.g.

5 31

225 3

+=

-

a12 Understand and be able to use the laws of indices forallrationalexponents.

,x x xa b a b# = + , ( )x x x x xa b a b a n an' = =-

a13 Understandandbeabletousenegative,fractionalandzeroindices.

,x x1aa=- x 10 = ( )x 0! , x xa a1

=

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Proportion a14 Understandanduseproportionalrelationshipsand their graphs.

For one variable directly or inversely proportionaltoapowerorrootofanother.


Partialfractions

a15 Beabletoexpressalgebraicfractionsaspartialfractions.

Fractionswithconstantorlinearnumerators and denominators up to threelinearterms.Includessquaredlinear terms in denominator.

Fractionswithaquadraticorcubic which cannot be factorised in the denominator.

Rationalexpressions

a16 Beabletosimplifyrationalexpressions. Includingfactorising,cancellingandsimple algebraic division. Any correct method of algebraic division may be used.

Divisionbynon-linear expressions.

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PUREMATHEMATICS:FUNCTIONS(1)

Polynomials Mf1 Beabletoadd,subtract,multiplyanddividepolynomials.

Expandingbracketsandcollectingliketerms. Divisionbynon-linearexpressions.

f2 Understand the factor theorem and be able to use it to factorise a polynomial or to determineitszeros.

( ) ( )a x a0f += - is a factor of ( )xf .Includingwhensolvingapolynomialequation.

Equationsofdegree> 4.

PUREMATHEMATICS:FUNCTIONS(2)

The language offunctions

f3 Understandthedefinitionofafunction,andbe able to use the associated language.

Afunctionisamappingfromthedomaintothe range such that for each x in the domain, there is a unique y in the range with ( ) .x yf =

The range is the set of all possible values of ( )xf .

Many-to-one, one-to-one, domain, range.

: x yf "

f4 Understandandusecompositefunctions. Includesfindingthecorrectdomainofgf given the domains of f and g.

( )xgf

f5 Understand and be able to use inverse functionsandtheirgraphs.Knowtheconditionsnecessaryfortheinverseofafunctiontoexistandhowtofindit.

Includesusingreflectionintheliney x= andfindingdomainandrangeofaninversefunction. e.g. ln x(x 0> )istheinverseofex.

( )xf 1-

The modulus function

f6 Understand and be able to use the modulus function.

Graphsofthemodulusoflinearfunctionsinvolving a single modulus sign.

f7 Beabletosolvesimpleinequalitiescontaining a modulus sign.

Includingtheuseofinequalitiesoftheformx a b#- to express upper and lower

bounds, a b! , for the value of x.

Inequalitiesinvolvingmore than one modulus sign or modulus of non-linear functions.

Modelling f8 Beabletousefunctionsinmodelling. Includingconsiderationoflimitationsandrefinementsofthemodels.

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PUREMATHEMATICS:GRAPHS(1)

Graphs MC1 Understandandusegraphsoffunctions.

Sketching curves

C2 Understandhowtofindintersectionpointsof a curve with coordinate axes.

Includingrelatingthistothesolutionofanequation.

C3 Understand and be able to use the method ofcompletingthesquaretofindthelineofsymmetry and turning point of the graph of a quadraticfunctionandtosketchaquadraticcurve(parabola).

The curve ( )y a x p q2= + + has• aminimum at ( , )p q- for a 0> or

a maximum at ( , )p q- for a < 0• aline of symmetry x p=- .

C4 Beabletosketchandinterpretthegraphsofsimplefunctionsincludingpolynomials.

Includingcasesofrepeatedrootsforpolynomials.

C5 Beabletousestationarypoints when curve sketching.

Includingdistinguishingbetweenmaximumand minimum turning points.

C6 Beabletosketchandinterpretthegraphsofy x

a= and y x

a2= .

Includingtheirverticalandhorizontalasymptotes and recognising them as graphs ofproportionalrelationships.

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Transformations MC7 Beabletosketchcurvesoftheforms( ), ( ) , ( )y a x y x a y x af f f= = + = + and

( )y axf= , given the curve of ( )y xf= and describetheassociatedtransformations.Beabletoformtheequationofagraphfollowingasingletransformation.

Includingworkingwithsketchesofgraphswherefunctionsarenotdefinedalgebraically.

Map(s)onto.Translation,stretch,reflection.


Transformations C8 Understandtheeffectofcombinedtransformationsonagraphandbeable toformtheequationofthenewgraph andtosketchit.Beabletorecognisethetransformationsthathavebeenappliedtoagraphfromthegraphoritsequation.

Vectornotationmaybe used for a translation. abJ

L

KKKKN

P

OOOO, a bi j+

Sketching curves C9 Beabletousestationarypointsofinflectionwhen curve sketching.

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PUREMATHEMATICS:COORDINATEGEOMETRY(1)

The coordinate geometry of straight lines

* Understandandusetheequation .y mx c= +

Mg1 Knowandbeabletousetherelationshipbetween the gradients of parallel lines and perpendicular lines.

For parallel lines m m1 2= . For perpendicular lines m m 11 2 =- .

g2 Beabletocalculatethedistancebetweentwo points.

g3 Beabletofindthecoordinatesofthemidpoint of a line segment joining two points.

g4 Beabletoformtheequationofastraightline.

Including ( )y y m x x1 1- = - and ax by c 0+ + =

g5 Beabletodrawalinegivenitsequation. Byusinggradientandinterceptorinterceptswithaxesaswellasbyplottingpoints.

g6 Beabletofindthepointofintersectionoftwo lines.

Bysolutionofsimultaneousequations.

g7 Beabletousestraightlinemodels. Inavarietyofcontexts;includesconsideringtheassumptionsthatleadtoastraightlinemodel.

Equationsofstraightlines

ManylearnerstakingALevelMathematicswillbefamiliarwiththeequationofastraightlineintheformy mx c= + . Their understanding at A Level should extend to

differentformsoftheequationofastraightlineincluding ( )y y m x x1 1- = - , ax by c 0+ + = and y yy y

x xx x

2 1

1

2 1

1

-

-= -

-.

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The coordinate geometry of curves

Mg8 Beabletofindthepoint(s)ofintersectionofalineand a curve or of two curves.

g9 Beabletofindthepoint(s)ofintersectionofalineand a circle.

g10 Understandandusetheequationofacircleintheform ( ) ( )x a y b r2 2 2- + - = .

Includescompletingthesquaretofindthecentreand radius.

g11 Knowandbeabletousethefollowingproperties:• theangleinasemicircleisarightangle;• theperpendicularfromthecentreofa

circletoachordbisectsthechord;• theradiusofacircleatagivenpointonits

circumference is perpendicular to thetangent to the circle at that point.

These results may be used in the context of coordinate geometry.


Parametric equations

g12 Understand the meaning of the terms parameter andparametricequations.

g13 Beabletoconvertbetweencartesianandparametricformsofequations.

Whenconvertingfromcartesiantoparametricform, guidance will be given as to the choice of parameter.

g14 Understandandusetheequationofacirclewrittenin parametric form.

g15 Beabletofindthegradientofacurvedefinedintermsofaparameterbydifferentiation.

xy

tx

ty

dd

dd

dd

= J

L

KKKKKK

J

L

KKKKKKKN

P

OOOOOO

N

P

OOOOOOO

Second and higher derivatives.

g16 Beabletouseparametricequationsinmodelling. Contextsincludekinematicsandprojectilesinmechanics.Includingmodellingwithaparameterwith a restricted domain.

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PUREMATHEMATICS:SEQUENCESANDSERIES(1)

Binomialexpansions

Ms1 Understand and use the binomial expansion of ( )a bx n+ where nisapositiveinteger.

s2 Knowthenotationsn! and Cn r and that Cn r is the numberofwaysofselectingrdistinctobjectsfrom n.

The meaning of the term factorial.napositiveinteger.Linktobinomialprobabilities.

! ( ) !!

! . . ...

!

,

r n rn

n nC C

Cnr

1 2 31

0 1

Cn r

n n n

nr

0

=-

=

= =

=J

L

KKKKKK

N

P

OOOOOO

Cn r will only be used in the context of binomial expansions and binomial probabilities.


Binomialexpansions

s3 Use the binomial expansion of ( )x1 n+ where n is anyrationalnumber.

For x 1< when nisnotapositiveinteger. Generalterm.

s4

Beabletowrite( )a bx n+ in the form an abx1

n

+

J

L

KKKKKK

N

P

OOOOOOand hence expand ( )a bx n+ .

abx 1< when nisnotapositiveinteger.

Proof of convergence.

s5 Beabletousebinomialexpansionswithnrationaltofindpolynomialswhichapproximate( )a bx n+ .

Includesfindingapproximationstorationalpowersofnumbers.

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Sequences Ms6 Know what a sequence of numbers is and the meaning of finiteandinfinitewithreferencetosequences.

s7 Beabletogenerateasequenceusingaformulaforthekthterm,orarecurrencerelationoftheform ( )a afk k1 =+ .

e.g. a k2 3k = + ;a a 3k k1 = ++ with a 51 = .

kth term: ak

s8 Knowthataseriesisthesumofconsecutivetermsofasequence.

Startingfromthefirstterm.

s9 Understand and use sigma notation....r n1 2

r

n

1= + + +

=

|

s10 Beabletorecogniseincreasing,decreasingandperiodicsequences.

s11 Knowthedifferencebetweenconvergentanddivergentsequences.

Includingwhenusingasequenceasamodel or when using numerical methods.

Limit to denote the value to which a sequence converges.

Formal tests for convergence.

Arithmeticseries

s12 Understandandusearithmeticsequencesandseries. Thetermarithmeticprogression(AP)mayalsobeusedforanarithmeticsequence.

First term, a Last term, l Common difference,d.

s13 Beabletousethestandardformulaeassociatedwitharithmeticsequencesandseries.

The nth term, the sum to n terms.Includingthesumofthefirstn natural numbers.

Sn

Geometricseries

s14 Understand and use geometric sequences and series. Thetermgeometricprogression(GP)mayalso be used for a geometric sequence.

First term, a Commonratio,r.

s15 Beabletousethestandardformulaeassociatedwithgeometric sequences and series.

The nth term, the sum to n terms. Sn

s16 Knowtheconditionforageometricseriestobeconvergentandbeabletofinditssumtoinfinity.

,S ra r1 11=-3

Modelling s17 Beabletousesequencesandseriesinmodelling.

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PUREMATHEMATICS:TRIGONOMETRY(1)

Basictrigonometry

* Know how to solve right-angled triangles using trigonometry.

Trig.functions Mt1 Beabletousethedefinitionsofsin i, cos i and tan i for any angle.

Byreferencetotheunitcircle,

, ,sin cos tany x xy

i i i= = = .

t2 Know and use the graphs of sin i, cos i and tan i for all values of i,theirsymmetriesandperiodicities.

Stretches,translationsandreflectionsofthese graphs.Combinationsofthesetransformations.

Period.

* Know and be able to use the exact values of sin i and cos i for i =0°,30°,45°,60°and90°andtheexactvalues of tan i for i =0°,30°,45°and60°.

Area of triangle;sineand cosine rules

t3 Know and be able to use the fact that the area of a triangle is given by ½ sinab C.

t4 Know and be able to use the sine and cosine rules. Use of bearings may be required.

Identities t5Understand and be able to use tan

cossin

iii

= .e.g. solve sin cos3i i= for0 360c c# #i .

t6 Understandandbeabletousetheidentitysin cos 12 2i i+ = .

e.g. solve sin cos2i i= for0 360c c# #i .

Equations t7 Beabletosolvesimpletrigonometricequationsingivenintervals and know the principal values from the inverse trigonometricfunctions.

e.g. .sin 0 5i = , in [0o, 360o],30 150+ c ci=

Includesequationsinvolvingmultiplesof the unknown angle e.g.

.sin cos2 3 2i i=

Includesquadraticequations.

arcsin x sin–1xarccos x cos–1xarctan x tan–1x

Generalsolutions.

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Trig.functions Mt8 Know and be able to use exact values of , ,sin cos tani i i

for , , , , 0 6 4 3ir r r

r= andmultiplesthereofand

,sin cosi i for 2ir

= andmultiplesthereof.

t9 Understandandusethedefinitionsofthefunctionsarcsin,arccosandarctan,theirrelationshiptosin,cosand tan, their graphs and their ranges and domains.

Radians t10 Understandandusethedefinitionofaradianandbeable to convert between radians and degrees.

t11 Knowandbeabletofindthearclengthandareaofasector of a circle, when the angle is given in radians.

The results s ri= and A r21 2i= where

i is measured in radians.

t12 Understand and use the standard small angle approximationsofsine,cosineandtangent. , ,sin cos tan1 2

2. . .i i i i i

i-

where i is in radians.

Secant, cosecant and cotangent

t13 Understandandusethedefinitionsofthesec,cosecandcotfunctions.

Includingknowledgeoftheanglesforwhichtheyareundefined.

t14 Understandrelationshipsbetweenthegraphsofthesin,cos,tan,cosec,secandcotfunctions.

Includingdomainsandranges.

t15 Understandandusetherelationshipstan sec12 2i i+ = and cot 1 cosec2 2i i+ = .

Radians

A radian is the angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.

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Compound angle formulae

Mt16 Understandandusetheidentitiesfor ( )sin !i z , ( )cos !i z , ( )tan !i z .

Includesunderstandinggeometricproofs.Thestartingpointfortheproofwill be given.

Proofs using de Moivre’stheorem will not be accepted.

t17 Knowanduseidentitiesforsin 2i, cos 2i, tan 2i. Includesunderstandingderivations

from ( )sin i z+ , ( )cos i z+ ,

( ) .tan i z+

cos cos sincos coscos sin

22 2 12 1 2

2 2

2

2

i i i

i i

i i

= -

= -

= -

t18 Understand and use expressions for cos sina b!i i in the equivalent forms ( )sinR !i a and ( )cosR !i a .

Includessketchingthegraphofthefunction,findingitsmaximumandminimumvaluesandsolvingequations.

Equations t19 Usetrigonometricidentities,relationshipsanddefinitionsinsolvingequations.

Proofs and problems

t20 Constructproofsinvolvingtrigonometricfunctionsandidentities.

t21 Usetrigonometricfunctionstosolveproblemsincontext,includingproblemsinvolvingvectors,kinematicsandforces.

The argument of the trigonometric functionsisnotrestrictedtoangles.

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PUREMATHEMATICS:EXPONENTIALSANDLOGARITHMS(1)

Exponentialsand Logarithms

ME1 Knowandusethefunctiony ax= and its graph. For a 0> .

E2 Beabletoconvertfromanindextoalogarithmicformand vice versa.

logx a y xya+= = for a 0> and .x 0>

E3 Understand a logarithm as the inverse of the appropriate exponentialfunctionandbeabletosketchthegraphsofexponentialandlogarithmicfunctions.

logy x a xay+= = for a 0> and .x 0>

Includesfindingandinterpretingasymptotes.

E4 Understand the laws of logarithms and be able to apply them, including to taking logarithms of both sides of an equation.

( )

( )

log log log

log log log

log log

xy x y

yx x y

x k x

a a a

a a a

ak

a

= +

= -

=

a k

Including,forexamplek 1=- and k 2

1=-

Change of base of logarithms.

E5 Know and use the values of log aa and log 1a . log a 1a = , log 1 0a =

E6 Beabletosolveanequationoftheforma bx = . Includessolvingrelatedinequalities.

E7 Knowhowtoreducetheequationsy axn= and y abx= to linear form and, using experimental data, to use a graph toestimatevaluesoftheparameters.

Bytakinglogarithmsofbothsidesandcomparingwiththeequationy mx c= + .Learners may be given graphs and asked to select an appropriate model.

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PUREMATHEMATICS:EXPONENTIALSANDLOGARITHMS(1)

Exponentialsand natural logarithms

ME8 Knowandbeabletousethefunctiony ex= and its graph.

E9 Know that the gradient of ekx is kekx and hence understandwhytheexponentialmodelissuitableinmanyapplications.

E10 Knowandbeabletousethefunction lny x= and its graph.Knowtherelationshipbetweenln x and ex.

ln xistheinversefunctionofex. log lnx xe =

Exponentialgrowth and decay

E11 Beabletosolveproblemsinvolvingexponentialgrowthanddecay;beabletoconsiderlimitationsandrefinementsofexponentialgrowthanddecaymodels.

Understandanduseexponentialgrowthanddecay:useinmodelling(examplesmayincludetheuseofeincontinuouscompoundinterest,radioactivedecay,drugconcentrationdecay,exponentialgrowthasamodelforpopulationgrowth);considerationoflimitationsandrefinementsofexponentialmodels. Finding long term values.

Graphswithgradientproportionaltooneofthecoordinates

xy

xdd? resultsinaquadraticgraph. x

yyd

d? resultsinanexponentialgraph.

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PUREMATHEMATICS:CALCULUS(1)

Basicdifferentiation

Mc1 Know and use that the gradient of a curve at a point is given by the gradient of the tangent at the point.

c2 Know and use that the gradient of the tangent at a point A on a curve is given by the limit of the gradient of chord AP as P approaches A along the curve.

The modulus function.

c3 Understandandusethederivativeoff(x) as the gradient of the tangent to the graph of y = f(x) at a general point (x, y).Knowthatthegradientfunction

xy

dd

gives the gradient of the curve and measures the

rate of change of y with respect to x.

Beabletodeducetheunitsofrate of change for graphs modellingrealsituations.Thetermderivativeofafunction.

xy

xy

dd

Limx 0 d

d=

"d

( )( ) ( )

x hx h x

f Limf f

h 0=

+ -

"l d n

c4 Beabletosketchthegradientfunctionforagivencurve.

Differentiationoffunctions

c5 Beabletodifferentiatey kxn= where k is a constant and nisrational,includingrelatedsumsanddifferences.

Differentiationfromfirstprinciplesforsmallpositiveinteger powers.

Applicationsofdifferentiationtofunctionsand graphs

c6 Understandandusethesecondderivativeastherateof change of gradient. ( )x x

yf d

d2

2

=ll

c7 Beabletousedifferentiationtofindstationarypointson a curve: maxima and minima.

Distinguishbetweenmaximumand minimum turning points.

c8 Understandthetermsincreasingfunctionanddecreasingfunctionandbeabletofindwherethefunctionisincreasingordecreasing.

Inrelationtothesignof xy

dd

.

c9 Beabletofindtheequationofthetangentandnormal at a point on a curve.

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Differentiationoffunctions

Mc10 Beabletodifferentiateekx, akx and ln x. Includingrelatedsums,differencesandconstantmultiples.

c11 Beabletodifferentiatethetrigonometricalfunctions:sin kx;cos kx;tan kx for x in radians.

Includingtheirconstantmultiples,sumsanddifferences.Differentiationfromfirstprinciples for sin x and cos x.

Product, quotientandchain rules

c12 Beabletodifferentiatetheproductoftwofunctions.

The product rule: y uv= ,

xy

u xv v x

udd

dd

dd

= +

Or [ ( ) ( )] ( ) ( ) ( ) ( )x x x x x xf g f g f g= +l l l

c13 Beabletodifferentiatethequotientoftwofunctions. ,y v

uxy

v

v xu u x

v

dd d

ddd

2= =-

Or ( )( )

[ ( )]( ) ( ) ( ) ( )

xx

xx x x x

gf

gg f f g

2=-l l l< F

c14 Beabletodifferentiatecompositefunctionsusingthe chain rule. ( ), ( ),y u u xf g= = x

yuy

xu

dd

dd

dd#= or

[ ( )] [ ( )] ( )x x xf g f g g=l l l" ,c15 Beabletofindratesofchangeusingthechain

rule, including connected rates of change and differentiationofinversefunctions.

1

xy

yxd

d

dd

= J

L

KKKKK

N

P

OOOOO

Implicitdifferentiation

c16 Beabletodifferentiateafunctionorrelationdefinedimplicitly.

e.g. ( )x y x22+ = . Second and higher derivatives.

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Applicationsofdifferentiationtofunctionsand graphs

c17 Understandthatasectionofcurvewhichhasincreasinggradient(andsopositivesecondderivative)isconcaveupwards.Understandthatasectionofcurvewhichhasdecreasinggradient(andsonegativesecondderivative)isconcavedownwards.

concave upwards (convexdownwards)

concave downwards (convexupwards)

The wording “concaveupwards”or “concave downwards”willbe used in examinationquestions.

c18 Understandthatapointofinflectiononacurveiswhere the curve changes from concave upwards to concavedownwards(orviceversa)andhencethatthesecondderivativeatapointofinflectioniszero.Beabletousedifferentiationtofindstationaryandnon-stationarypointsofinflection.

Learners are expected to be able to findandclassifypointsofinflectionasstationaryornon-stationary.Distinguishbetweenmaxima,minimaandstationarypointsofinflection.

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Integrationasreverse of differentiation

Mc19 Knowthatintegrationisthereverseofdifferentiation. Fundamental Theorem of Calculus.

c20 Beabletointegratefunctionsoftheformkxn where k is a constant and n 1!- .

Includingrelatedsumsanddifferences.

c21 Beabletofindaconstantofintegrationgivenrelevantinformation.

e.g. Find yasafunctionofx given that

xy

x 2dd

2= + and y 7= when x 1= .

Integrationtofindareaunder a curve

c22 Knowwhatismeantbyindefiniteanddefiniteintegrals.Beabletoevaluatedefiniteintegrals.

e.g. (3 5 1)x x xd2

1

3

+ -y .

c23 Beabletouseintegrationtofindtheareabetweenagraph and the x-axis.

Includesareasofregionspartlyaboveand partly below the x-axis.Generalunderstandingthattheareaunder a graph can be found as the limit of a sum of areas of rectangles.

Formal understanding of thecontinuityconditionsrequired for the Fundamental Theorem of Calculus.

TheFundamentalTheoremofCalculus

Onewaytodefinetheintegralofafunctionisasfollows.

Theareaunderthegraphofthefunctionisapproximatelythesumoftheareasofnarrowrectangles(asshown).Thelimitofthissumastherectanglesbecomenarrower(andtherearemoreofthem)istheintegral.Thefundamentaltheoremofcalculussaysthatthisisthesameasdoingthereverseofdifferentiation.

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Integrationasinverse of differentiation

Mc24Beabletointegrateekx, x

1, sin kx, cos kx and related

sums,differencesandconstantmultiples.,lnx x x c x1 0d != +y

x in radians for trigonometrical integrals.

Integralsinvolvinginverse trigonometrical functions.

Integrationtofindareaunder a curve

c25 Understandintegrationasthelimitofasum.Know that

a( ) ( )lim x x x xf f d

0x a

b b

d ="d| y

c26 Beabletouseintegrationtofindtheareabetweentwo curves.

Learnersshouldalsobeabletofindthe area between a curve and the y-axis,includingintegratingwithrespect to y.

Integrationbysubstitution

c27 Beabletouseintegrationbysubstitutionincaseswhere the process is the reverse of the chain rule (includingfindingasuitablesubstitution).

e.g. ( )x1 2 8+ , ( )x x1 2 8+ , xex2, x2 31+

Learners can recognise the integral,they need not show all the working forthesubstitution.

c28 Beabletouseintegrationbysubstitutioninothercases.

Learnerswillbeexpectedtofindasuitablesubstitutioninsimplecasese.g. ( )x

x1 3+

.

Integralsrequiringmore than one substitutionbefore they can be integrated.

Integrationbyparts

c29 Beabletousethemethodofintegrationbypartsinsimple cases.

Includescaseswheretheprocessisthe reverse of the product rule.e.g. xex.Morethanoneapplicationofthe method may be required.Includesbeingabletoapplyintegrationbypartstoln x.

Reductionformulae.

Partialfractions

c30 Beabletointegrateusingpartialfractionsthatarelinear in the denominator.

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Differentialequations

c31 Beabletoformulatefirstorderdifferentialequationsusinginformationaboutratesofchange.

Contextsmayincludekinematics,populationgrowthandmodellingtherelationshipbetweenpriceanddemand.

c32 Beabletofindgeneralorparticularsolutionsoffirstorderdifferentialequationsanalyticallybyseparatingvariables.

Equationsmayneedtobefactorisedusing a common factor before variables can be separated.

c33 Beabletointerpretthesolutionofadifferentialequationinthecontextofsolvingaproblem,includingidentifyinglimitationsofthesolution.

Includeslinkstokinematics.

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PUREMATHEMATICS:NUMERICALMETHODS(2)

Solutionofequations

Me1 Beabletolocatetherootsof ( )x 0f = by considering changes of sign of ( )xf in an interval of x in which ( )xf issufficientlywell-behaved.

Finding an interval in which a root lies.ThisisoftenusedasapreliminarysteptofindastartingvalueforthemethodsinMe3andMe4.

e2 Beawareofcircumstancesunderwhichchangeofsign methods may fail.

e.g. when the curve of ( )y xf=

touches the x-axis.e.g. when the curve of ( )y xf= has averticalasymptote.e.g. there may be several roots in theinterval.

e3 Beabletocarryoutafixedpointiterationafterrearranginganequationintotheform ( )x xg= and be able to draw associated staircase and cobweb diagrams.

e.g. write x x 4 03 - - = asx x 43= + andusetheiterationx x 4n n1

3= ++ with an appropriatestartingvalue.IncludesuseofANSkeyoncalculator.

iteration,iterate

e4 Be able to use theNewton-Raphsonmethod to find arootofanequationandrepresenttheprocessonagraph.

e5 Understandthatnotalliterationsconvergetoaparticularrootofanequation.

KnowhowNewton-Raphsonandfixedpointiterationcanfailandbeable to show this graphically.

Integration Mc34 Beabletofindanapproximatevalueofadefiniteintegralusingthetrapeziumrule,anddecidewhetheritisanover-oranunder-estimate.

Inanintervalwherethecurveiseither concave upwards or concave downwards.

Numberofstrips.

c35 Usethesumofaseriesofrectanglestofindanupperand/orlowerboundontheareaunderacurve.

Problem solving

Me6 Use numerical methods to solve problems in context.

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PUREMATHEMATICS:VECTORS(1)

Generalvectors

Mv1 Understand the language of vectors in two dimensions.

Scalar, vector, modulus, magnitude, direction,positionvector,unitvector,cartesian components, equal vectors, parallel vectors, collinear.

Vectors printed in bold. Unit vectors i, j, rt The magnitude of the vector a iswritten|a|ora.

aaa 1

2=J

L

KKKKK

N

P

OOOOO

v2 Beabletoaddandsubtractvectorsusingadiagramoralgebraically,multiplyavectorbyascalar,andexpressavectorasacombinationofothers.

Geometricalinterpretation.Includesgeneral vectors not expressed in component form.

v3 Beabletocalculatethemagnitudeanddirectionofavector and convert between component form and magnitude-directionform.

Magnitude-direction

Positionvectors

v4 Understandandusepositionvectors. Includinginterpretingcomponentsofapositionvectorasthecartesiancoordinates of the point.AB –b a=

OB or b.

xyr =J

L

KKKK

N

P

OOOO

v5 Beabletocalculatethedistancebetweentwopointsrepresentedbypositionvectors.

Using vectors v6 Beabletousevectorstosolveproblemsinpuremathematicsandincontext,includingproblemsinvolving forces.

Includesinterpretingthesumofvectorsrepresentingforcesastheresultant force.

PUREMATHEMATICS:VECTORS(2)

Generalvectors

Mv7 Understand the language of vectors in three dimensions.

ExtendtheworkofMv2toMv6toinclude vectors in three dimensions.

Unit vectors i, j,k, rt a

a

a

a

1

2

3

=

J

L

KKKKKKKKKK

N

P

OOOOOOOOOO

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STATISTICS:SAMPLING(1)

Populationand sample

Mp21 Understandandusethetermspopulationandsample.

p22 Beabletousesamplestomakeinformalinferencesaboutapopulation,recognisingthatdifferentsamplesmightleadtodifferentconclusions.

e.g. using sample mean or variance asanestimateofpopulationmeanorvariance.

Sampling techniques

p23 Understand and be able to use the concept of random sampling.

Simplerandomsampling.Everysampleoftherequiredsizehasthesameprobability of being selected.

p24 Understand and be able to use a variety of sampling techniques.

Opportunitysampling,systematicsampling,stratifiedsampling,quotasampling, cluster sampling, self-selected samples.Any other techniques will be explained inthequestion.

p25 Beabletoselectorevaluatesamplingtechniquesinthecontextofsolvingastatisticalproblem.

Includesrecognisingpossiblesourcesof bias and being aware of the practicalitiesofimplementation.

Populationandsample

Populationinstatisticsmeansalltheindividualsweareinterestedinforaparticularinvestigatione.g.allcodinanareaofthesea.Apopulationcanbeinfinitee.g.allpossibletossesofaparticularcoin.Aprobabilitydistributioncanbeusedtomodelsomecharacteristicofthepopulationwhichisofintereste.gaNormaldistributioncould be used to model lengths of cod.

A sampleisasetofitemschosenfromapopulation.Whensamplingfromaninfinitepopulationitdoesnotmatterwhetherthesamplingiswithorwithoutreplacement.Whentakingasampleofindividuals,e.g.forasamplesurvey,itisusualtosamplewithoutreplacementtoavoidgettingdatafromthesameindividualmore than once.

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STATISTICS:DATAPRESENTATIONANDINTERPRETATION(1)

Datapresentationfor single variable

MD1 Beabletorecogniseandworkwithcategorical,discrete,continuousandrankeddata.Beabletointerpret standard diagrams for grouped and ungrouped single-variable data.

Includesknowingthisvocabularyanddecidingwhatdatapresentationmethods are appropriate: bar chart, dotplot,histogram,verticallinechart,pie chart, stem-and-leaf diagram, box-and-whiskerdiagram(boxplot),frequency chart.

Learners may be asked to add to diagramsinexaminationsinordertointerpret data.

A frequency chart resembles a histogram with equal width bars butitsverticalaxisis frequency. A dot plot is similar to a bar chart but with stacks of dots in lines to represent frequency.

Comparativepiecharts with area proportionaltofrequency.

D2 Understand that the area of each bar in a histogram is proportionaltofrequency.Beabletocalculateproportionsfromahistogramandunderstandthemintermsofestimatedprobabilities.

Includesuseofareascaleandcalculationoffrequencyfromfrequency density.

D3 Beabletointerpretacumulativefrequencydiagram.

D4 Beabletodescribefrequencydistributions. Symmetrical, unimodal, bimodal, skewed(positivelyandnegatively).

Measures of skewness.

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Datapresentation

MD5 Understandthatdiagramsrepresentingunbiased samples become more representativeoftheoreticalprobabilitydistributionswithincreasingsamplesize.

e.g.Abarchartrepresentingtheproportionof heads and tails when a fair coin is tossedtendstohavetheproportionofheadsincreasinglycloseto50%asthesamplesizeincreases.

D6 Beabletointerpretascatterdiagramforbivariate data, interpret a regression line or otherbestfitmodel,includinginterpolationandextrapolation,understandingthatextrapolationmightnotbejustified.

Includingthetermsassociation,correlation,regression line.

Leaners should be able to interpret other best fitmodelsproducedbysoftware(e.g.acurve).

Learners may be asked to add to diagrams in examinationsinordertointerpretdata.

Calculationofequationofregression line from data or summarystatistics.

D7 Beabletorecognisewhenascatterdiagramappearstoshowdistinctsectionsinthepopulation.Beabletorecogniseandcommentonoutliersinascatterdiagram.

An outlier is an item which is inconsistent with the rest of the data.

Outliersinscatterdiagramsshouldbejudgedby eye.

D8 Beabletorecogniseanddescribecorrelationinascatterdiagramandunderstandthatcorrelationdoesnotimplycausation.

Positivecorrelation,negativecorrelation,nocorrelation,weak/strongcorrelation.

D9 Beabletoselectorcritiquedatapresentationtechniquesinthecontextofastatisticalproblem.

Includinggraphsfortimeseries.

Bivariatedata,associationandcorrelation

Bivariatedataconsistsoftwovariablesforeachmemberofthepopulationorsample.Anassociationbetweenthetwovariablesissomekindofrelationshipbetween them.Correlationmeasureslinearrelationships.AtALevel,learnersareexpectedtojudgerelationshipsfromscatterdiagramsbyeyeandmaybeaskedtointerpretgivencorrelationcoefficients–seeMAH10.

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Summary measures

MD10 Know the standard measures of central tendency and be able to calculate and interpret them and to decide when it is most appropriate to use one of them.

Median,mode,(arithmetic)mean,midrange.Themainfocusofquestionswillbeoninterpretationratherthancalculation.

Includesunderstandingwhenitisappropriate to use a weighted mean e.g. whenusingpopulationsasweights.

Mean x=

D11 Know simple measures of spread and be able to use and interpret them appropriately.

Range,percentiles,quartiles,interquartilerange.

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Summary measures

MD12 Know how to calculate and interpret variance andstandarddeviationforrawdata,frequencydistributions,groupedfrequencydistributions.

Beabletousethestatisticalfunctionsofacalculatortofindmeanandstandarddeviation.

sample variance: s nS

1xx2 =

- (†)

where ( )S x x 2

1xx i

i

n

= -=

|

samplestandarddeviation:

s variance= (‡)

s2

s

Correctionsforclass interval in thesecalculations.

D13 Understand the term outlier and be able to identifyoutliers.Knowthatthetermoutliercanbe applied to an item of data which is:• atleast2standarddeviationsfromthe

mean;OR• atleast1.5 × IQR beyond the nearer

quartile.

An outlier is an item which is inconsistent with the rest of the data.

D14 Beabletocleandataincludingdealingwithmissing data, errors and outliers.

Notationforsamplevarianceandsamplestandarddeviation

Thenotationss2 and sforsamplevarianceandsamplestandarddeviation,respectively,arewrittenintobothBritishStandards(BS3534-1,2006)andInternationalStandards(ISO3534).Thedefinitionsarethosegivenaboveinequations(†)and(‡).Thecalculationsarecarriedoutusingdivisor( )n 1- .

Inthisspecification,theusagewillbeconsistentwiththesedefinitions.Thusthemeaningsof‘samplevariance’,denotedbys2, and‘samplestandarddeviation’,denoted by s,aredefinedtobecalculatedwithdivisor( )n 1- .

Inearlyworkinstatisticsitiscommonpracticetointroducetheseconceptswithdivisorn rather than ( )n 1- .Howeverthereisnorecognisednotationtodenotethequantitiessoderived.

Studentsshouldbeawareofthevariationsinnotationusedbymanufacturersoncalculatorsandknowwhatthesymbolsontheirparticularmodelsrepresent.

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STATISTICS:PROBABILITY(1)

Probability of events in a finitesamplespace

* Beabletocalculatetheprobabilityofanevent. Usingmodellingassumptionssuchasequally likely outcomes.

( )AP

* Understand the concept of a complementary event and know that the probability of an event maybefoundbymeansoffindingthatofitscomplementary event.

A´ is the event “not-A”.

Probability of two or more events

* Beabletocalculatetheexpectedfrequencyofanevent given its probability.

Expectedfrequency= ( )n AP

* Beabletouseappropriatediagramstoassistinthecalculationofprobabilities.

E.g.treediagrams,samplespacediagrams, Venn diagrams.

Mu1 Understand and use mutually exclusive events and independent events.

u2 Knowtoaddprobabilitiesformutuallyexclusiveevents.

E.g.tofind ( ) .A BP or

u3 Knowtomultiplyprobabilitiesforindependentevents.

E.g.tofind ( ) .A BP and Includingtheuseofcomplementaryevents,e.g.findingtheprobabilityofatleastone6infivethrowsofadice.

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STATISTICS:PROBABILITY(2)

Probability of two or more events

u4 Understand and use mutually exclusive events andindependenteventsandassociatednotationanddefinitions.

For mutually exclusive events A B 0P + =^ h for any pair of events.

u5 BeabletouseVenndiagramstoassistinthecalculationsofprobabilities.Knowhowtocalculateprobabilitiesfortwoeventswhicharenot mutually exclusive.

Venn diagrams for up to three events.Learnersshouldunderstandtherelation:

( ) ( ) ( ) ( )A B A B A BP P P P, += + - .

Probability of a generalorinfinitenumber of events.Formal proofs.

Conditionalprobability

u6 Beabletocalculateconditionalprobabilitiesbyformula, from tree diagrams, two-way tables, Venn diagrams or sample space diagrams.

( | ) ( )( )

A B BA B

P PP +

=P( | )A B Finding reverse

conditionalprobability i.e. calculatingP( | )B A given P( | )A B andadditionalinformation.

u7 Know that P( | )B A = P(B) + B and A are independent.

Inthiscase( ) ( ) . ( )A B A BP P P+ = .

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STATISTICS:PROBABILITYDISTRIBUTIONS(1)

Situationsleading to a binomial distribution

MR1 Recognisesituationswhichgiverisetoabinomialdistribution.

R2 Beabletoidentifytheprobabilityofsuccess,p, for the binomialdistribution.

Thebinomialdistributionasamodel for observed data.

B(n, p), q p1= - ~ means ‘has the distribution’.

Calculationsrelatingtobinomial distribution

R3 Beabletocalculateprobabilitiesusingthebinomialdistribution.

Includinguseofcalculatorfunctions.

Mean and expected frequencies for binomial distribution

R4 Understand and use mean = np. Derivationofmean= np

R5 Beabletocalculateexpectedfrequenciesassociatedwiththebinomialdistribution.

Discreteprobability distributions

R6 Beabletouseprobabilityfunctions,givenalgebraicallyor in tables. Know the term discrete random variable.

Restrictedtosimplefinitedistributions.

X for the random variable.x or r for a value of the random variable.

R7 Beabletocalculatethenumericalprobabilitiesforasimpledistribution.Understandthetermdiscreteuniformdistribution.

Restrictedtosimplefinitedistributions.

( )X xP =

( )X xP #

CalculationofE(X) or Var(X).

Situationswhichgiverisetoabinomialdistribution

• Anexperimentortrialisconductedafixednumberoftimes.• Thereareexactly2outcomes,whichcanbethoughtofas“success”or“failure”.• Theprobabilityof“success”isthesameeachtime.• Theprobabilityof“success”onanytrialisindependentofwhathashappenedinprevioustrials.• Therandomvariableofinterestis“thenumberofsuccesses”.

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STATISTICS:PROBABILITYDISTRIBUTIONS(2)

Normaldistribution

MR8 BeabletousetheNormaldistributionasamodel. IncludesrecognisingwhenaNormaldistributionmaynotbeappropriate.

UnderstandhowandwhyacontinuitycorrectionisusedwhenusingaNormaldistributionasamodelforadistributionofdiscrete data.

RecognisefromtheshapeofthedistributionwhenabinomialdistributioncanbeapproximatedbyaNormaldistribution.

( , )X N 2+ n v Knowing conditionsforNormalapproximationto binomial.

R9 KnowtheshapeoftheNormalcurveandunderstandthat histograms from increasingly large samples from a NormaldistributiontendtotheNormalcurve.

IncludesunderstandingthattheareaundertheNormalcurverepresentsprobability.

R10 KnowthatlineartransformationofaNormalvariablegivesanotherNormalvariableandknowhowthemeanandstandarddeviationareaffected.BeabletostandardiseaNormalvariable.

, y a bx y a bx s b s2 2 2i i y x&= + = + = StandardNormal

( , )Z 0 1N+

ZXvn

=-

Proof

R11 KnowthatthelineofsymmetryoftheNormalcurveislocatedatthemeanandthepointsofinflectionarelocatedonestandarddeviationawayfromthemean.

R12 BeabletocalculateanduseprobabilitiesfromaNormaldistribution.

Includinguseofcalculatorfunctions.

Modelling with probability

R13 Beabletomodelwithprobabilityandprobabilitydistributions,includingrecognisingwhenthebinomialorNormalmodelmaynotbeappropriate.

Includingcritiquingassumptionsmadeandthelikelyeffectofmorerealisticassumptions.

MeanandvarianceofaNormaldistribution

TheNormaldistributionisaprobabilitymodel;itsmeanandvariancearecalculatedusingtechniquesbeyondthescopeofALevelMathematics.Atthislevel,studentsshouldunderstandthemeanandvarianceofaNormaldistributionasthelimitingvaluesfromcalculatingthemeanandvarianceofincreasinglylargesamplesfromaNormaldistribution.

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STATISTICS:STATISTICALHYPOTHESISTESTING(1)

Hypothesis testing

MH1 Understandtheprocessofhypothesistestingandthe associated language.

Nullhypothesis,alternativehypothesis. Significancelevel,teststatistic,1-tailtest, 2-tail test. Criticalvalue,criticalregion(rejectionregion),acceptanceregion,p-value.

H2 Understand when to apply 1-tail and 2-tail tests.

H3 Understand that a sample is being used to make aninferenceaboutthepopulationandappreciatethatthesignificancelevelistheprobabilityofincorrectlyrejectingthenullhypothesis.

For a binomial hypothesis test, the probabilityoftheteststatisticbeingintherejectionregionwillalwaysbelessthan or equal to the intended significancelevelofthetest,andwillusuallybelessthanthesignificancelevel of the test. Learners will not be testedonthisdistinction.Ifaskedtogive the probability of incorrectly rejectingthenullhypothesisforaparticularbinomialtest,eithertheintendedsignificancelevelortheprobabilityoftheteststatisticbeingintherejectionregionwillbeacceptable.

Nullandalternativehypotheses

Thenullhypothesisforahypothesistestisthedefaultpositionwhichwillonlyberejectedinfavourofthealternativehypothesisiftheevidenceisstrongenough.Assumingthenullhypothesisistrue,asadefaultposition,allowsthecalculationofvaluesoftheteststatisticwhichwouldbeunlikely(havelowprobability)ifthenullhypothesisweretrue;thisisthecriticalregion(rejectionregion).

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Hypothesis testingforabinomial probability p

H4 Beabletoidentifynullandalternativehypotheses(H0 and H1)whensettingupahypothesistestbasedon a binomial probability model.

H0 of form p =aparticularvalue,withpaprobabilityforthewholepopulation.

H0, H1

H5 Beabletoconductahypothesistestatagivenlevelofsignificance.Beabletodrawacorrectconclusionfrom the results of a hypothesis test based on a binomial probability model and interpret the results in context.

Normalapproximation.

H6 Beabletoidentifythecriticalandacceptanceregions.


Hypothesis testingforamean using Normaldistribution

MH7 Knowthatrandomsamplesofsizen from ( , )X N 2+ n v havethesamplemeanNormally

distributed with mean n and variance n2v.

Sample mean, X Particularvalueofsample mean, x Populationmean,n

Central Limit Theorem

H8 BeabletocarryoutahypothesistestforasinglemeanusingtheNormaldistributionandbeabletointerpret the results in context.

Insituationswhereeither (a) thepopulationvarianceisknownor(b) thepopulationvarianceisunknownbutthesamplesizeislargeLearners may be asked to use a p-valueoracriticalregion.

H0 of form n =aparticularvalue,where nisthepopulationmean.

Significancelevelwillbegiven.

H9 Beabletoidentifythecriticalandacceptanceregions.

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Informalhypothesis testingforcorrelation/association

MH10 Understandcorrelationasameasureofhowclosedata points lie to a straight line.

Understandthatarankcorrelationcoefficientmeasuresthecorrelationbetweenthedataranksrather than actual data values.

Learners are not required to know the namesofparticularcorrelationcoefficients.

r Calculationofcorrelationcoefficient

H11 Beabletouseagivencorrelationcoefficientforasampletomakeaninferenceaboutcorrelationorassociationinthepopulationforgivenp-value or criticalvalue.

Associationreferstoamoregeneralrelationshipbetweenthevariables.

The(oftenimplicit)nullhypothesisisoftheformeitherthatthereisnocorrelationornoassociationinthepopulation.Questionswilluseanappropriatecorrelationcoefficientandindicatewhethercorrelationorassociationisbeingtestedfor.

Questionsmayrequire understanding of notationfromsoftware;sufficientguidance will be given in the question.

Knowledge of bivariate Normaldistribution

Calculatingcorrelation

Learnersareexpectedtousetechnologytoworkwithrealdata,includingthepre-releasedata.Calculators,spreadsheetsandothersoftwarewillcalculatecorrelationcoefficients.Learnersmaybeaskedtointerpretsuchcorrelationcoefficientsintheexamination.Thefollowingpointsshouldbenoted:• Acorrelationcoefficientmeasuresthestrengthofalinearrelationship.Acorrelationbetweentheranksofthedatavaluesmaybeusedforamoregeneral

relationship.• Correlationcoefficientswillonlybeusedfordatawherebothvariablesarerandom(not,forexample,fortimeseriesdatawhereonevariableoccursatsetintervals).• Outliersordistinctsectionsofdatainthescatterdiagramcanaffectthevalueofthecorrelationcoefficient.

Conclusionfromahypothesistest

Learnersareexpectedtomakenon-assertiveconclusionsincontext.E.g.“Thereisnotenoughevidencetoconcludethattheproportionof...hasincreased.”E.g.“Thereisenoughevidencetoindicatethattheprobabilityof.....haschanged.”E.g.“Thereisinsufficientevidencetoindicatethatthetruemeanof.....islowerthan......”E.g.“Thereissufficientevidencetosuggestthatthereispositivecorrelationbetween.....and.....”E.g.“Thereisnotsufficientevidencetosuggestthatthereisassociationbetween...and....”

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MECHANICS:MODELSANDQUANTITIES(1)

Standard models in mechanics

Mp31 Know the language used to describe simplifying assumptionsinmechanics.

Includingthewords:light;smooth;uniform;particle;inextensible;thin;rigid;longterm.

p32 Understandandusetheparticlemodel.

Units and quantities

p33 UnderstandandusefundamentalquantitiesandunitsintheS.I.system:length,time,mass.

Metre(m),second(s),kilogram(kg).

p34 Understandandusederivedquantitiesandunits:velocity,acceleration,force,weight.

Metrepersecond(ms–1),metrepersecondpersecond(ms–2),newton(N).

MECHANICS:MODELSANDQUANTITIES(2)

Units and quantities

p35 Understandandusederivedquantitiesandunits:moment.

Newtonmetre(Nm).

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MECHANICS:KINEMATICSIN1DIMENSION(1)

Motionin1dimension

Mk1 Understandandusethelanguageofkinematics. Position,displacement,distancetravelled;speed,velocity;acceleration,magnitudeofacceleration;relativevelocity(in1-dimension).

Average speed = distance travelled ÷ elapsedtime

Average velocity = overall displacement ÷elapsedtime

k2 Knowthedifferencebetweenposition,displacement,distance and distance travelled.

k3 Knowthedifferencebetweenvelocityandspeed,andbetweenaccelerationandmagnitudeofacceleration.

Kinematicsgraphs

k4 Beabletodrawandinterpretkinematicsgraphsformotioninastraightline,knowingthesignificance(whereappropriate)oftheirgradientsandtheareasunderneath them.

Position-time,displacement-time,distance-time,velocity-time, speed-time,acceleration-time.

Calculus in kinematics

k5 Beabletodifferentiatepositionandvelocitywithrespecttotimeandknowwhatmeasuresresult.

,v tr a t

vtr

dd

dd

dd

2

2= = =

k6 Beabletointegrateaccelerationandvelocitywithrespecttotimeandknowwhatmeasuresresult.

,r v t v a td d= =y y

Constant accelerationformulae

k7 Beabletorecognisewhentheuseofconstantaccelerationformulaeisappropriate.

Learners should be able to derive the formulae.

s ut at21 2= +

s vt at21 2= -

v u at= +

( )s u v t21

= +

v u as22 2- =

Problem solving

k8 Beabletosolvekinematicsproblemsusingconstantaccelerationformulaeandcalculusformotioninastraight line.

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MECHANICS:KINEMATICSIN2DIMENSIONS(2)

Motionin2dimensions

Mk9 Understandthelanguageofkinematicsappropriatetomotionin2dimensions.Knowthedifferencebetween, displacement, distance from and distance travelled;velocityandspeed,andbetweenaccelerationandmagnitudeofacceleration.

Positionvector,relativeposition.

Average speed = distance travelled ÷ elapsedtime

Average velocity = overall displacement÷elapsedtime

Relativevelocity

k10 Beabletoextendthescopeoftechniquesfrommotionin1dimensiontothatin2dimensionsbyusing vectors.

The use of calculus and the use of constantaccelerationformulae.

,t tdd

dda v v v r r

= = = =o o

,t td dr v v a= = yyt ts u a2

1 2= +

t ts v a21 2= -

tv u a= +

( ) ts u v21

= +

Vector form ofv u as22 2- =

k11 Beabletofindthecartesianequationofthepathofaparticlewhenthecomponentsofitspositionvectoraregivenintermsoftime.

k12 Beabletousevectorstosolveproblemsinkinematics.

Includesrelativepositionofoneparticlefromanother.

Includesknowingthatthevelocityvectorgivesthedirectionofmotionandtheaccelerationvectorgivesthedirectionofresultantforce.

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MECHANICS:PROJECTILES(2)

Motionundergravity in 2 dimensions

My1 Beabletomodelmotionundergravityinaverticalplaneusingvectors.Beabletoformulatetheequationsofmotionofaprojectileusingvectors.

Standardmodellingassumptionsforprojectilemotionareasfollows.• Noairresistance.• Theprojectileisaparticle.• Horizontaldistancetravelledis

small enough to assume thatgravity is always in the samedirection.

• Verticaldistancetravelledissmallenough to assume that gravity isconstant.

Calculationsinvolving air resistance.

y2 Knowhowtofindthepositionandvelocityatanytimeofaprojectileandfindrangeandmaximumheight.

y3 Beabletofindtheinitialvelocityofaprojectilegivensufficientinformation.

y4 Beabletoeliminatetimefromthecomponentequationsthatgivethehorizontalandverticaldisplacementintermsoftimetoobtaintheequationofthetrajectory.

y5 Beabletosolvesimpleproblemsinvolvingprojectiles.

Maximum range on inclined planeBoundingparabola.

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MECHANICS:FORCES(1)

Identifyingandrepresentingforces

MF1 Understandthelanguagerelatingtoforces. Weight, tension, thrust or compression, normalreaction(ornormalcontactforce),frictionalforce,resistance,drivingforce.

Understand that the value of the normal reactiondependsontheotherforcesacting.

Understandthattheremaybefrictionalforcewhenthesurfaceisnotsmooth(i.e.isrough).

F2 Knowthattheaccelerationduetogravityisnotauniversalconstantbutdependsonlocationintheuniverse.Knowthatonearth,theaccelerationduetogravityisoftenmodelledtobeaconstant,g m s–2.

g ≈10, g ≈ 9.8Unlessotherwisespecified,inexaminationsthe value of gshouldbetakentobe9.8.

Accelerationdue to gravity, g m s–2.

Inversesquarelaw for gravitation.

F3 Beabletoidentifytheforcesactingonasystemandrepresent them in a force diagram. Understand the differencebetweenexternalandinternalforcesandbeabletoidentifytheforcesactingonpartofthesystem.

Vector treatment of forces

F4 Beabletofindtheresultantofseveralconcurrentforces when the forces are parallel or in two perpendiculardirectionsorinsimplecasesofforcesgivenas2-Dvectorsincomponentform.

F5 Understand the concept of equilibrium and know that a particleisinequilibriumifandonlyifthevectorsumoftheforcesactingonitiszerointhecaseswheretheforcesareparallelorintwoperpendiculardirectionsorinsimplecasesofforcesgivenas2-Dvectorsincomponentform.

Accelerationduetogravity

Theaccelerationduetogravity(g m s–2)variesonearthbetween9.76and9.83.Itdependsonlatitudeandheightabovesealevel.Thestandardaccelerationduetogravityisinternationallyagreedtobe9.80665;thisvalueisstoredinsomecalculators.

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MECHANICS:FORCES(2)

Vector treatment of forces

MF6 Beabletoresolveaforceintocomponentsandbeabletoselectsuitabledirectionsforresolution.Beabletofindtheresultantofseveralconcurrentforcesby resolving and adding components.

e.g.Horizontallyandvertically,orparallel and perpendicular to aninclined plane.

F7 Knowthataparticleisinequilibriumifandonlyiftheresultantoftheforcesactingonitiszero.Knowthatabody is in equilibrium under a set of concurrent forces ifandonlyiftheirresultantiszero.

F8 Knowthatvectorsrepresentingasetofforcesinequilibriumsumtozero.Knowthataclosedfiguremaybedrawntorepresenttheadditionoftheforceson an object in equilibrium.

F9 Beabletoformulateandsolveequationsforaparticlein equilibrium: by resolving forces in suitable directions;bydrawingandusingapolygonofforces.

For example, a triangle of forces. Non-coplanarforces.

Frictionalforceand normal contact force

F10 Understand that the overall contact force between surfacesmaybeexpressedintermsofafrictionalforce and a normal contact force and be able to draw an appropriate force diagram.

Understand that the normal contact force cannot be negative.

Understand the following modelling assumptions.• Smoothisusedtomeanthat

frictionmaybeignored.• Roughindicatesthatfrictionmust

be taken into account.

Normalreaction.

F11 Understandthatthefrictionalforcemaybemodelledby F R# n andthatfrictionactsinthedirectiontoopposesliding.ModelfrictionusingF Rn= when sliding occurs.

Coefficientoffriction= n Limitingfriction,staticequilibrium

The term angle of friction.

F12 BeabletoapplyNewton’sLawstoproblemsinvolvingfriction.

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MECHANICS:NEWTON’SLAWSOFMOTION(1)

Newton’slawsforaparticle

Mn1 KnowandunderstandthemeaningofNewton’sthreelaws.

Includesapplyingthelawstoproblems.

n2 Understandthetermequationofmotion.

n3 Beabletoformulatetheequationofmotionforaparticlemovinginastraightlinewhentheforcesactingareparallelorintwoperpendiculardirectionsorinsimplecasesofforcesgivenas2-Dvectorsincomponent form.

Includingmotionundergravity. F ma= where F is the resultant force.

mF a= where F is the resultant force.

Variable mass.

Connected particles

n4 Beabletomodelasystemasasetofconnectedparticles.

e.g. simple smooth pulley systems, trains.Internalandexternalforcesforthesystem.

n5 Beabletoformulatetheequationsofmotionfortheindividualparticleswithinthesystem.

n6 Know that a system in which none of its components haveanyrelativemotionmaybemodelledasasingleparticlewiththemassofthesystem.

e.g. Train.

MECHANICS:NEWTON’SLAWSOFMOTION(2)

Newton’slawsforaparticle

n7 Beabletoformulatetheequationofmotionforaparticlemovinginastraightlineorinaplane.

Includingmotionundergravity. F ma= where F is the resultant force.

mF a= where F is the resultant force.

Variable mass.

Newton’slawsofmotion

I Anobjectcontinuesinastateofrestoruniformmotioninastraightlineunlessitisactedonbyaresultantforce.II AresultantforceF actingonanobjectoffixedmassmgivestheobjectanaccelerationa given by F = ma.III Whenoneobjectexertsaforceonanother,thereisalwaysareactionwhichisequalinmagnitudeandoppositeindirectiontotheactingforce.

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MECHANICS:RIGIDBODIES(2)

Thissectionisanintroductiontomomentsinstaticcontexts.Theonlysituationsconsideredarebodiesthatmaybemodelledas(possiblynon-uniform)rodsandrectangular laminas. The only forces considered are coplanar, and act perpendicular to the rod or to an edge of the lamina. The learning outcome should be read inthelightofthisrestriction.

Inmoreadvancedwork,momentsaredescribedasactingaboutanaxis,andlearnersshouldbeawareofthis.Giventherestrictionsonthesituationsconsidered,however,momentsmaybedescribedasactingaboutapoint,withanimpliedaxisperpendiculartotheplaneinwhichtheforcesareacting.Thisisconsistentwiththeapproachusedtodescriberotationsin2-D.

Rigid bodies in equilibrium

MF13 Beabletocalculatethemomentofaforceaboutapoint or axis.

Units of moment areNm.

Vector treatment.

F14 Understand that a rigid body is in equilibrium when theresultantforceiszeroandthesumofthemomentsaboutanyonepointiszero.

F15 Understand that a system of forces can have a turning effectonarigidbody.

Moment

F16 Knowthat,forthepurposeofcalculatingitsmoment,theweightofabodycanbetakenasactingthroughapoint.

The point is the centre of mass of the body.

Questionswillberestrictedtocaseswhere the centre of mass is given or can be found using symmetry or can befoundfromconsiderationofmoments.

Uniform Finding the centre of mass of a composite body.

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2g. Priorknowledge,learningandprogression

• ItisassumedthatlearnersarefamiliarwiththecontentofGCSE(9–1)Mathematicsforfirstteachingfrom2015.

• ALevelMathematicsB(MEI)providestheframework within which a large number ofyoungpeoplecontinuethesubjectbeyondGCSElevel.Itsupportstheirmathematicalneeds across a broad range of other subjects atthis level and provides a basis for subsequentquantitativeworkinaverywiderangeofhighereducationcoursesandinemployment.ItalsosupportsthestudyofASandALevelFurtherMathematicsB(MEI).

• ALevelMathematicsB(MEI)buildsfromGCSELevelMathematicsandintroducescalculusanditsapplications.Itemphasiseshowmathematicalideasareinterconnectedandhowmathematicscanbeappliedtohelpmakesense of data, to understand the physical worldand to solve problems in a variety of contexts,includingsocialsciencesandbusiness.Itprepares students for further study and

employment in a wide range of disciplines involvingtheuseofmathematics.

• SomelearnersmaywishtofollowamathematicscourseonlyuptoASlevel,inorder to broaden their curriculum, and todevelop their interest and understanding ofdifferentareasofthesubject.Othersmayfollowaco-teachableroute,completingaoneyearAScourseandthencontinuingtocomplete the second year of the two yearA level course, developing a deeper knowledgeandunderstandingofmathematicsanditsapplications.

• LearnerswhowishtospecialiseinmathematicsorSTEMsubjectssuchasphysics or engineering can further extendtheir knowledge and understanding ofmathematicsanditsapplicationsbytakingFurtherMathematicsB(MEI)ASorALevel.

ThereareanumberofMathematicsspecificationsatOCR. Find out more at www.ocr.org.uk


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3a. Formsofassessment

OCRALevelinMathematicsB(MEI)consistsofthreecomponents that are externally assessed. The three externallyassessedcomponents(01–03)containsomesynopticassessment,someextendedresponsequestionsandsomestretchandchallengequestions.

Stretchandchallengequestionsaredesignedtoallowthe most able learners the opportunity to demonstrate the full extent of their knowledge and skills.

Stretchandchallengequestionswillsupporttheawarding of A* grade at A level, addressing the need forgreaterdifferentiationbetweenthemostablelearners.

PureMathematicsandMechanics(Component01)

Thiscomponentisworth36.4%ofthetotalAlevel.Allquestionsarecompulsoryandthereare100marksintotal.Theexaminationpaperhastwosections:AandB.

SectionAconsistsofshorterquestionswithminimalreadingandinterpretation;theaimofthisistoensure that all students feel as though they can do someofthequestionsonthepaper.SectionBincludeslongerquestionsandmoreproblemsolving.SectionBhasagradientofdifficulty.41to47ofthemarksareformechanics;mechanicsquestionscanbeineithersectionAorB.

SectionAwillhave20to25markswiththeremainderofthemarksallocatedtosectionB.

PureMathematicsandStatistics(Component02)

Thiscomponentisworth36.4%ofthetotalAlevel.Allquestionsarecompulsoryandthereare100marks

intotal.Theexaminationpaperhastwosections:AandB.

SectionAconsistsofshorterquestionswithminimalreadingandinterpretation;theaimofthisistoensure that all students feel as though they can do someofthequestionsonthepaper.SectionBincludeslongerquestionsandmoreproblemsolving.SectionBhasagradientofdifficulty.50to60ofthemarksareforstatisticswithsomeofthesebeingforquestionsbasedonthepre-releasedataset;statisticsquestionscanbeineithersectionAorB.

SectionAwillhave20to25markswiththeremainderofthemarksallocatedtosectionB.

PureMathematicsandComprehension(Component03)

Thiscomponentisworth27.3%ofthetotalAlevel.Allquestionsarecompulsoryandthereare75marksintotal.Theexaminationpaperhastwosections:AandB.

SectionAwillhave60marksonthepuremathematicscontent.

SectionBwillhave15marksworthofquestionsonapreviously unseen comprehension passage based on thepuremathematicscontentofthespecification,ratherthanonmechanicsorstatisticscontent,toensurethatmechanicsandstatisticsarenotover-assessed in some years. The passage may include examplesofapplicationsofthepurecontent(otherthanthosewhicharespecifiedinthemechanicsandstatisticssections).Themechanicsandstatisticscontentofthespecificationisassumedknowledgeforcomponent03butthisassumedknowledgewillnotbethefocusofanyofthequestions.

3 AssessmentofALevelinMathematicsB(MEI)

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3b. Assessmentobjectives(AO)

Thereare3AssessmentObjectivesinOCRASLevelMathematicsB(MEI).Thesearedetailedinthetablebelow.

AssessmentobjectivesWeightings

A level

AO1

UseandapplystandardtechniquesLearners should be able to:• selectandcorrectlycarryoutroutineprocedures;and• accuratelyrecallfacts,terminologyanddefinitions.

50%(±2%)

AO2

Reason,interpretandcommunicatemathematicallyLearners should be able to:• constructrigorousmathematicalarguments(includingproofs);• makedeductionsandinferences;• assessthevalidityofmathematicalarguments;• explaintheirreasoning;and• usemathematicallanguageandnotationcorrectly.

Where questions/tasks targeting this assessment objective will also credit Learners for the ability to ‘use and apply standard techniques’ (AO1) and/or to ‘solve problems within mathematics and other contexts’ (AO3) an appropriate proportion of the marks for the question/task will be attributed to the corresponding assessment objective(s).

25%(±2%)

AO3

SolveproblemswithinmathematicsandinothercontextsLearners should be able to:• translateproblemsinmathematicalandnon-mathematicalcontextsinto

mathematicalprocesses;• interpretsolutionstoproblemsintheiroriginalcontext,and,whereappropriate,

evaluatetheiraccuracyandlimitations;• translatesituationsincontextintomathematicalmodels;• usemathematicalmodels;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsof

modelsand,whereappropriate,explainhowtorefinethem.

Where questions/tasks targeting this assessment objective will also credit Learners for the ability to ‘use and apply standard techniques’ (AO1) and/or to ‘reason, interpret and communicate mathematically’ (AO2) an appropriate proportion of the marks for the question/task will be attributed to the corresponding assessment objective(s).

25%(±2%)

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AOweightingsinALevelinMathematicsB(MEI)

Therelationshipbetweentheassessmentobjectivesandthecomponentsareshowninthefollowingtable:

ComponentAOmarkspercomponent

AO1 AO2 AO3

PureMathematicsandMechanics(H640/01) 47–53marks 18–24marks 26–32marks

PureMathematicsandStatistics(H640/02) 47–53marks 22–28marks 22–28marks

PureMathematicsandComprehension(H640/03) 35–39marks 21–25marks 13–17marks

%ofoverallAlevelinMathematicsB(MEI)(H640) 48–52% 23–27% 23–27%

Across all three papers combined in any given series, AO totals will fall within the stated percentages for the qualification.Morevariationisallowedpercomponent,however,toallowforflexibilityinthedesignofitems.

3c. Assessmentavailability

TherewillbeoneexaminationseriesavailableeachyearinMay/Junetoall learners.

All examined components must be taken in the same examinationseriesattheendofthecourse.

ThisspecificationwillbecertificatedfromtheJune2018examinationseriesonwards.

3d. Retakingthequalification

Learnerscanretakethequalificationasmanytimesasthey wish. They must retake all components of the qualification.

3e. Assessmentofextendedresponse

Theassessmentmaterialsforthisqualificationprovide learners with the opportunity to demonstrate their ability to construct and develop a sustained and coherent line of reasoning and marks for extended

responses are integrated into the marking criteria. Taskswhichofferthisopportunitywillbefoundacross all three components.

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3f. Synopticassessment

• Synopticassessmentisthelearner’sunderstandingoftheconnectionsbetweendifferentelementsofthesubject.Itinvolvesthe explicit drawing together of knowledge,skillsandunderstandingwithindifferentpartsof the A level course.

• TheemphasisofsynopticassessmentistoencouragetheunderstandingofMathematicsas a discipline.

• Synopticassessmentallowslearnerstodemonstrate the understanding they haveacquired from the course as a whole andtheir ability to integrate and apply thatunderstanding. This level of understanding isneeded for successful use of the knowledgeand skills from this course in future life, workand study.

• Learnersarerequiredtoknowandunderstandthecontentofallthepuremathematicsandto be able to apply the overarching themes,alongwithassociatedmathematicalthinkingand understanding, in all the assessmentcomponentsofALevelMathematicsB(MEI).

• Inalltheexaminationpapers,learnerswillberequired to integrate and apply theirunderstanding in order to address problemswhich require both breadth and depth ofunderstandinginordertoreachasatisfactorysolution.

• Learnerswillbeexpectedtoreflectonandinterpretsolutions,drawingontheirunderstandingofdifferentaspectsofthecourse.

3g. Calculatingqualificationresults

Alearner’soverallqualificationgradeforALevelinMathematicsB(MEI)willbecalculatedbyaddingtogether their marks from the three components taken to give their total mark. This mark will then be

comparedtothequalificationlevelgradeboundariesfor the relevant exam series to determine the learner’soverallqualificationgrade.

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Theinformationinthissectionisdesignedtogiveanoverview of the processes involved in administering thisqualificationsothatyoucanspeaktoyourexamsofficer.AllofthefollowingprocessesrequireyoutosubmitsomethingtoOCRbyaspecificdeadline.

Moreinformationabouttheprocessesanddeadlinesinvolved at each stage of the assessment cycle can be foundintheAdministrationareaoftheOCRwebsite.OCR’sAdmin overview is available on the OCR website at http://www.ocr.org.uk/administration.

4 Admin:whatyouneedtoknow

4a. Pre-assessment

Estimatedentries

Estimatedentriesareyourbestprojectionofthenumber of learners who will be entered for a qualificationinaparticularseries.Estimatedentries

shouldbesubmittedtoOCRbythespecifieddeadline. They are free and do not commit your centre in any way.

Finalentries

Final entries provide OCR with detailed data for each learner,showingeachassessmenttobetaken.Itisessentialthatyouusethecorrectentrycode,considering the relevant entry rules.

FinalentriesmustbesubmittedtoOCRbythepublished deadlines or late entry fees will apply.

AlllearnerstakinganALevelinMathematicsB(MEI)mustbeenteredforH640.

Entrycode

Title Component code

Componenttitle Assessmenttype

H640MathematicsB

(MEI)

01 PureMathematicsandMechanics

ExternalAssessment

02 PureMathematicsandStatistics

ExternalAssessment

03 PureMathematicsandComprehension

ExternalAssessment

http://www.ocr.org.uk/administration

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4b. Specialconsideration

Specialconsiderationisapostassessmentadjustmenttomarksorgradestoreflecttemporaryinjury,illnessorotherindispositionatthetimetheassessmentwastaken.

DetailedinformationabouteligibilityforspecialconsiderationcanbefoundintheJCQpublicationA guide to the special consideration process.

4c. Externalassessmentarrangements

RegulationsgoverningexaminationarrangementsarecontainedintheJCQInstructions for conducting examinations.

Learnersarepermittedtouseascientificorgraphicalcalculator for all components. Calculators are subject to the rules in the document Instructions for Conducting ExaminationspublishedannuallybyJCQ(www.jcq.co.uk).

Headofcentreannualdeclaration

The Head of centre is required to provide a declarationtotheJCQaspartoftheannualNCNupdate,conductedintheautumnterm,toconfirmthatthecentreismeetingalloftherequirementsdetailedinthespecification.Anyfailurebyacentre

toprovidetheHeadofCentreAnnualDeclaration will result in your centre status being suspended and could lead to the withdrawal of our approval for you to operate as a centre.

Privatecandidates

Private candidates may enter for OCR assessments.

A private candidate is someone who pursues a course of study independently but takes an examinationorassessmentatanapprovedexaminationcentre.Aprivatecandidatemay beapart-timestudent,someonetakingadistancelearning course, or someone being tutored privately. They must be based in the UK.

Private candidates need to contact OCR approved centres to establish whether they are prepared to host them as a private candidate. The centre may charge for this facility and OCR recommends that the arrangement is made early in the course.

Further guidance for private candidates may be found on the OCR website: http://www.ocr.org.uk.

www.jcq.org.uk

http://www.ocr.org.uk/Images/257982-guidance-for-private-candidates-2015-16.pdf

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4d. Resultsandcertificates

GradeScale

Alevelqualificationsaregradedonthescale:A*,A,B,C,D,E,whereA*isthehighest.LearnerswhofailtoreachtheminimumstandardforEwillbe

Unclassified(U).OnlysubjectsinwhichgradesA*toEareattainedwillberecordedoncertificates.

Results

Results are released to centres and learners for informationandtoallowanyqueriestoberesolvedbeforecertificatesareissued.

Centres will have access to the following results informationforeachlearner:

• thegradeforthequalification

• therawmarkforeachcomponent

• thetotalweightedmarkforthequalification.

Thefollowingsupportinginformationwillbeavailable:

• rawmarkgradeboundariesforeachcomponent

• weightedmarkgradeboundariesforthequalification.

Untilcertificatesareissued,resultsaredeemedtobeprovisional and may be subject to amendment.

Alearner’sfinalresultswillberecordedonanOCRcertificate.Thequalificationtitlewillbeshown onthecertificateas‘OCRLevel3AdvancedGCEinMathematicsB(MEI)’.

4e. Post-resultsservices

A number of post-results services are available:

• Reviewofmarking–Ifyouarenothappywiththeoutcomeofalearner’sresults,centresmayrequest a review of marking. Full details of thepost-results services are provided on the OCRwebsite.

• Missingandincompleteresults–Thisserviceshould be used if an individual subject resultfor a learner is missing, or the learner has beenomittedentirelyfromtheresultssupplied.

• Accesstoscripts–Centrescanrequestaccessto marked scripts.

4f. Malpractice

Anybreachoftheregulationsfortheconductofexaminationsandnon-examassessmentworkmayconstitutemalpractice(whichincludesmaladministration)andmustbereportedtoOCRassoon as it is detected.

DetailedinformationonmalpracticecanbefoundintheJCQpublicationSuspected Malpractice in Examinations and Assessments: Policies and Procedures.

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5a. Overlapwithotherqualifications

ThecontentofthisspecificationoverlapswithASMathematicsB(MEI)andwithotherspecificationsinALevelMathematicsandASMathematics.

5 Appendices

5b. Accessibility

Reasonable adjustments and access arrangements allowlearnerswithspecialeducationalneeds,disabilitiesortemporaryinjuriestoaccesstheassessment and show what they know and can do, without changing the demands of the assessment. Applicationsfortheseshouldbemadebeforetheexaminationseries.Detailedinformationabouteligibility for access arrangements can be found in the

JCQAccess Arrangements and Reasonable Adjustments.

TheAlevelqualificationandsubjectcriteriahavebeenreviewedinordertoidentifyanyfeaturewhichcould disadvantage learners who share a protected CharacteristicasdefinedbytheEqualityAct2010.Allreasonable steps have been taken to minimise any such disadvantage.

5c. Mathematicalnotation

ThetablesbelowsetoutthenotationthatmaybeusedbyASandAlevelmathematicsspecifications.Studentswillbeexpectedtounderstandthisnotationwithoutneedforfurtherexplanation.

1 SetNotation

1.1 ! is an element of

1.2 " is not an element of

1.3 3 is a subset of

1.4 1 is a proper subset of

1.5 , ,x x1 2 f" , the set with elements , ,x x1 2 f

1.6 : ...x" , the set of all x such that f

1.7 ( )An the number of elements in set A

1.8 Q the empty set

1.9 f the universal set

1.10 Al the complement of the set A

1.11 N the set of natural numbers, , , ,1 2 3 f" ,1.12 Z the set of integers, , , , ,0 1 2 3! ! f" ,1.13 Z+ thesetofpositiveintegers, , , ,1 2 3 f" ,

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1.14 Z0+ thesetofnon-negativeintegers,{0, 1, 2, 3, …}

1.15 R the set of real numbers

1.16 Q thesetofrationalnumbers, : , qpp qZ Z! ! +' 1

1.17 , union

1.18 + intersection

1.19 ( , )x y the ordered pair x, y

1.20 [ , ]a b the closed interval :x a x bR! # #" ,1.21 [ , )a b the interval :x a x b<R! #" ,1.22 ( , ]a b the interval :x a x b<R! #" ,1.23 ( , )a b the open interval :x a x b< <R!" ,

2 MiscellaneousSymbols

2.1 = is equal to

2.2 ! is not equal to

2.3 / isidenticaltooriscongruentto

2.4 . is approximately equal to

2.5 3 infinity

2.6 ? isproportionalto

2.7 Ñ therefore

2.8 Ö because

2.9 < is less than

2.10 G, # is less than or equal to, is not greater than

2.11 > is greater than

2.12 H, $ is greater than or equal to, is not less than

2.13 p q& p implies q(ifp then q)

2.14 p q% p is implied by q(ifq then p)

2.15 p q+ p implies and is implied by q(p is equivalent to q)

2.16 a firsttermforanarithmeticorgeometricsequence

2.17 l lasttermforanarithmeticsequence

2.18 d commondifferenceforanarithmeticsequence

2.19 r commonratioforageometricsequence

2.20 Sn sum to n terms of a sequence

2.21 S3 sumtoinfinityofasequence

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3 Operations

3.1 a b+ a plus b

3.2 a b- a minus b

3.3 , , .a b ab a b# amultipliedbyb

3.4 a ÷ b, ba a divided by b

3.5ai

i

n

1=

| a a an1 2 f+ + +

3.6ai

i

n

1=

% a a an1 2# # #f

3.7 a thenon-negativesquarerootofa

3.8 a the modulus of a

3.9 !n n factorial: ! ( ) ... , ; !n n n n1 2 1 0 1N# # # # != - =

3.10,

nr Cr

nJ

L

KKKKN

P

OOOO , Cn rthebinomialcoefficient ! ( ) !

!r n rn-

for n, r ! Z0+, r G n

or !( ) ( )

rn n n r1 1f- - +

for n ! Q, r ! Z0+

4 Functions

4.1 ( )xf thevalueofthefunctionf at x

4.2 : x yf 7 thefunctionf maps the element x to the element y

4.3 f 1- theinversefunctionofthefunctionf

4.4 gf thecompositefunctionoff and gwhichisdefinedby( ) ( ( ))x xgf g f=

4.5 ( )lim x fx a" the limit of ( )xf as x tends to a

4.6 , x xdD an increment of x

4.7

xy

dd thederivativeofy with respect to x

4.8

xy

dd

n

n the nthderivativeofy with respect to x

4.9 ( ), ( ), , ( )x x xf f f( )nfl m thefirst,second,...,nthderivativesof ( )xf with respect to x

4.10 , , x x fo p thefirst,second,...derivativesofx with respect to t

4.11 y xdy theindefiniteintegralofy with respect to x

4.12 y xda

by thedefiniteintegralofy with respect to x between the limits x a= and x b=

5


5 ExponentialandLogarithmicFunctions

5.1 e base of natural logarithms

5.2 , exp xex exponentialfunctionofx

5.3 log xa logarithm to the base a of x

5.4 , ln logx xe natural logarithm of x

6 TrigonometricFunctions

6.1 , , ,

, ,

sin cos tan

cosec sec cot4

thetrigonometricfunctions

6.2 , , , ,

sin cos tanarcsin arccos arctan

1 1 1- - -

2 theinversetrigonometricfunctions

6.3 ° degrees

6.4 rad radians

9 Vectors

9.1 a, a, a~

the vector a, a, a~;thesealternativesapplythroughout

section9

9.2 AB thevectorrepresentedinmagnitudeanddirectionbythe directed line segment AB

9.3 at aunitvectorinthedirectionofa

9.4 , , i j k unitvectorsinthedirectionsofthecartesiancoordinateaxes

9.5 , a a the magnitude of a

9.6 ,AB AB the magnitude of AB

9.7 ,ab a bi j+

J

L

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N

P

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columnvectorandcorrespondingunitvectornotation

9.8 r positionvector

9.9 s displacement vector

9.10 v velocity vector

9.11 a accelerationvector

11 ProbabilityandStatistics

11.1 , , , .A B C etc events

11.2 A B, union of the events A and B

11.3 A B+ intersectionoftheeventsA and B

5

78© OCR 2018


11.4 ( )AP probability of the event A

11.5 Al complement of the event A

11.6 ( | )A BP probability of the event AconditionalontheeventB

11.7 , , , .X Y R etc random variables

11.8 , , , .x y r etc values of the random variables , , X Y R etc.

11.9 , , x x1 2 f valuesofobservations

11.10 , , f f1 2 f frequencieswithwhichtheobservations , , x x1 2 f occur

11.11 p(x), P(X = x) probabilityfunctionofthediscreterandomvariableX

11.12 , , p p1 2 f probabilitiesofthevalues , , x x1 2 f of the discrete random variable X

11.13 ( )XE expectationoftherandomvariableX

11.14 ( )XVar variance of the random variable X

11.15 ~ hasthedistribution

11.16 ( , )n pB binomialdistributionwithparameters n and p, where n is the number of trials and p is the probability of success in a trial

11.17 q q p1= - forbinomialdistribution

11.18 ( , )N 2n v Normaldistributionwithmeann and variance 2v

11.19 ( , )Z 0 1N+ standardNormaldistribution

11.20 z probabilitydensityfunctionofthestandardisedNormalvariablewithdistribution ( , )0 1N

11.21 U correspondingcumulativedistributionfunction

11.22 n populationmean

11.23 2v populationvariance

11.24 v populationstandarddeviation

11.25 x sample mean

11.26 s2 sample variance

11.27 s samplestandarddeviation

11.28 H0 null hypothesis

11.29 H1 alternativehypothesis

11.30 r productmomentcorrelationcoefficientforasample

11.31 t productmomentcorrelationcoefficientforapopulation

5


12 Mechanics

12.1 kg kilograms

12.2 m metres

12.3 km kilometres

12.4 m/s, m s–1 metrespersecond(velocity)

12.5 m/s2, m s–2 metrespersecondpersecond(acceleration)

12.6 F Force or resultant force

12.7 N newton

12.8 N m newtonmetre(momentofaforce)

12.9 t time

12.10 s displacement

12.11 u initialvelocity

12.12 v velocityorfinalvelocity

12.13 a acceleration

12.14 g accelerationduetogravity

12.15 µ coefficientoffriction

5d. Mathematicalformulaeandidentities

LearnersmustbeabletousethefollowingformulaeandidentitiesforALevelmathematics,withouttheseformulaeandidentitiesbeingprovided,eitherintheseformsorinequivalentforms.Theseformulaeandidentitiesmayonlybeprovidedwheretheyarethestartingpointforaprooforasaresulttobeproved.

PureMathematics

QuadraticEquations

ax bx c ab b ac

0 24

has roots22!

+ + =- -

LawsofIndices

a a ax y x y= +

a a ax y x y' = -

( )a ax y xy/

5

80© OCR 2018


LawsofLogarithms

logx a n xna+= = for a 0> and x 0>

( )

( )

log log log

log log log

log log

x y xy

x y yx

k x x

a a a

a a a

a ak

/

/

/

+

-

J

L

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N

P

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CoordinateGeometry

A straight line graph, gradient m passing through ( , )x y1 1 hasequation

( )y y m x x1 1- = -

Straight lines with gradients m1 and m2 are perpendicular when m m 11 2 =-

Sequences

Generaltermofanarithmeticprogression:

( )u a n d1n = + -

Generaltermofageometricprogression:

u arnn 1= -

Trigonometry

InthetriangleABC

Sine rule: sin sin sinAa

Bb

Cc

= =

Cosine rule: cosa b c bc A22 2 2= + -

Area sinab C21

=

cos sinA A 12 2 /+

sec tanA A12 2/ +

cosec cotA A12 2/ +

sin sin cosA A A2 2/

cos cos sinA A A2 2 2/ -

tan tantanA A

A2 12

2/-

Mensuration

Circumference and Area of circle, radius r and diameter d:

C r d A r2 2r r r= = =

Pythagoras’Theorem:Inanyright-angledtrianglewherea, b and c are the lengths of the sides and c is the hypotenuse:

c a b2 2 2= +

5


Areaofatrapezium= ( )a b h21

+ , where a and b are the lengths of the parallel sides and h is their perpendicularseparation.

Volume of a prism =areaofcrosssection× length

For a circle of radius r, where an angle at the centre of i radians subtends an arc of length s and encloses an associated sector of area A:

s r A r21 2i i= =

CalculusandDifferentialEquations

Differentiation

Function Derivative

xn nxn – 1

sin kx cosk kx cos kx sink kx-

ekx kekx

ln x x1

( ) ( )x xf g+ ( ) ( )x xf g+l l

( ) ( )x xf g ( ) ( ) ( ) ( )x x x xf g f g+l l

( ( ))xf g ( ( )) ( )x xf g gl l

Integration

Function Integral

xn , n x c n11 1≠n 1+

+ -+

cos kx sink kx c1+

sin kx cosk kx c1- +

ekx k c1 ekx +

x1 , ln x c x 0!+

( ) ( )x xf g+l l ( ) ( )x x cf g+ +

( ( )) ( )x xf g gl l ( ( ))x cf g +

Area under a curve ( )y x y 0da

b$= y

Vectors

x y z x y zi j k 2 2 2+ + = + +

5

82© OCR 2018


Mechanics

ForcesandEquilibrium

Weight = mass g#

Friction: F Rµ#

Newton’ssecondlawintheform:F ma =

Kinematics

Formotioninastraightlinewithvariableacceleration:

v tr a t

vtr

dd d

ddd

2

2= = =

r v t v a t d d= = yy

Statistics

The mean of a set of data: x nx

f

fx= =|

||

ThestandardNormalvariable: ZX

vn

=-

where ( , )X N 2+ n v

5


Learnerswillbegiventhefollowingformulaesheetineachquestionpaper:

FormulaeALevelMathematicsB(MEI)(H640)

Arithmeticseries

( ) { ( ) }S n a l n a n d2 1n 21

21

= + = + -

Geometricseries

( )S r

a r11

n

n

=-

-

S ra r1 1for 1=-3

Binomialseries

( ) ( )a b a a b a b a b b nC C C Nn n n n n n nr

n r r n1

12

2 2 –f f !+ = + + + + + +- - ,

where ! ( ) !!n

r r n rnC Cn

r n r= = =-

J

L

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N

P

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( ) !( )

!( ) ( )

, x nxn n

x rn n n r

x x n1 1 21 1 1

1 Rn r2 ff

f 1 !+ = + +-

+ +- - +

+ ^ h

5

84© OCR 2018


Differentiation

( )xf ( )xf l

tan kx seck kx2

sec x sec tanx x

cot x cosec x2-

cosec x cosec cotx x-

QuotientRuley vu

= ,xy

v

v xu u x

v

dd d

ddd

2=-

Differentiationfromfirstprinciples

( )( ) ( )

limx hf x h f x

fh 0

=+ -

"l

Integration

( )( )

ln ( )xxx x cf

fd f= +

ly

( ) ( ( )) ( ( ))x x x n x c11f f d fn n 1=+

++ly

Integrationbyparts u xv x uv v x

u xdd d d

d d= - yy

Smallangleapproximations

, ,sin cos tan1 21 2. . .i i i i i i- where i is measured in radians

Trigonometricidentities

( )sin sin cos cos sinA B A B A B! !=

( )cos cos cos sin sinA B A B A B! "=

( ) ( ( ) )tan tan tantan tanA B A BA B A B k1 2

1!

"!

! ! r= +

Numericalmethods

Trapeziumrule: {( ) ( ... )},y x h y y y y y21 2d

a

b

n n0 1 2 1. + + + + + -y where h nb a

=-

TheNewton-Raphsoniterationforsolving ( )x 0f = : ( )( )

x xxx

ff

n nn

n1 = -+ l

Probability

( ) ( ) ( ) ( )A B A B A BP P P P, += + -

( ) ( ) ( | ) ( ) ( | ) ( | ) ( )( )

A B A B A B A B A B BA B

P P P P P P PP

or++

= = =

5


Samplevariance

11 ( )s n S S x x x n

xx nxwhere 2 2 2

2

2 2xx xx i i

ii=

-= - = - = -

_ i| |||

Standarddeviation,s variance=

Thebinomialdistribution

If ( , )X n pB+ then ( )P X r p qCn rr n r–= = where q p1= - Mean of X is np

HypothesistestingforthemeanofaNormaldistribution

If ( , )X N 2+ n v then ,X nN2

+ nv

J

L

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N

P

OOOOOO and

/( , )

n

X0 1N+

v

n-

PercentagepointsoftheNormaldistribution

p 10 5 2 1

z 1.645 1.960 2.326 2.576

KinematicsMotioninastraightline Motionintwodimensions

v u at= + tv u a= +

s ut at21 2= + t ts u a2

1 2= +

( )s u v t21

= + ( ) ts u v21

= +

v u as22 2= +

s vt at21 2= - t ts v a2

1 2= -

86© OCR 2018


Summaryofupdates

Date Version Section Titleofsection Change

June2018 1.1 Front cover Disclaimer AdditionofDisclaimer

October2018 2.0 Multiple Revisedsections1and2withnewsubsectionsfocussingonkeyfeaturesandcommandwords.Correctionofminortypographicalerrors.Nochangeshavebeenmade to any assessment requirements.

88© OCR 2018


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