Specification MATHEMATICS B (MEI)ocr.org.uk/Images/308740-specification-accredited-a-level-gce... · A LEVEL Specification MATHEMATICS B (MEI) H640 For first assessment in 2018 ocr.org.uk/alevelmathsmei

A LEVELSpecification

MATHEMATICS B (MEI)

ocr.org.uk/alevelmathsmei

A LEVEL Mathematics B (MEI)

H640For first assessment in 2018

Version 1.1 (June 2018)

http://ocr.org.uk/alevelmathsmei

Registered office: 1 Hills Road Cambridge CB1 2EU

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Disclaimer Specifications are updated over time. Whilst every effort is made to check all documents, there may be contradictions between published resources and the specification, therefore please use the information on the latest specification at all times. Where changes are made to specifications these will be indicated within the document, there will be a new version number indicated, and a summary of the changes. If you do notice a discrepancy between the specification and a resource please contact us at: [email protected]

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.

mailto:[email protected]

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1© OCR 2017 A Level in Mathematics B (MEI)

Contents

1 WhychooseanOCRALevelinMathematicsB(MEI)? 21a. WhychooseanOCRqualification? 21b. WhychooseanOCRALevelinMathematicsB(MEI) 31c. Whatarethekeyfeaturesofthisspecification? 51d. HowdoIfindoutmoreinformation? 5

2 Thespecificationoverview 62a. OCRALevelinMathematicsB(MEI)(H640) 62b. ContentofALevelinMathematicsB(H640) 72c. ContentofALevelMathematicsB(MEI) 182d. Prior knowledge, learning and progression 62

3 AssessmentofALevelinMathematicsB(MEI) 633a. Forms of assessment 633b. Assessmentobjectives(AO) 643c. Assessment availability 653d. Retakingthequalification 653e. Assessment of extended response 653f. Synopticassessment 663g. Calculatingqualificationresults 66

4 Admin:whatyouneedtoknow 674a. Pre-assessment 674b. Specialconsideration 684c. Externalassessmentarrangements 684d. Resultsandcertificates 694e. Post-resultsservices 694f. Malpractice 69

5 Appendices 705a. Overlapwithotherqualifications 705b. Accessibility 705c. Mathematicalnotation 705d. Mathematicalformulaeandidentities 75

Summary of updates 82

2© OCR 2017

A Level in Mathematics B (MEI)

11a. WhychooseanOCRqualification?

Choose OCR and you’ve got the reassurance that you’re working with one of the UK’s leading exam boards.OurnewALevelinMathematicsB(MEI)coursehasbeendevelopedinconsultationwithteachers,employersandHigherEducationtoprovidelearnerswithaqualificationthat’srelevanttothemand meets their needs.

We’re part of the Cambridge Assessment Group, Europe’slargestassessmentagencyandadepartment of the University of Cambridge. Cambridge Assessment plays a leading role in developing and delivering assessments throughout theworld,operatinginover150countries.

Weworkwitharangeofeducationproviders,including schools, colleges, workplaces and other institutionsinboththepublicandprivatesectors.Over13,000centreschooseourALevels,GCSEsandvocationalqualificationsincludingCambridgeNationalsandCambridgeTechnicals.

OurSpecifications

Webelieveindevelopingspecificationsthathelpyoubring the subject to life and inspire your students to achieve more.

We’vecreatedteacher-friendlyspecificationsbasedon extensive research and engagement with the teachingcommunity.They’redesignedtobestraightforwardandaccessiblesothatyoucantailorthe delivery of the course to suit your needs. We aim to encourage learners to become responsible for their ownlearning,confidentindiscussingideas,innovativeandengaged.

We provide a range of support services designed tohelpyouateverystage,frompreparation throughtothedeliveryofourspecifications. Thisincludes:

• Awiderangeofhigh-qualitycreativeresourcesincluding:o Delivery Guideso TransitionGuideso TopicExplorationPackso LessonElementso …and much more.

• Access to Subject Advisors to support you throughthetransitionandthroughoutthelifetimeofthespecification.

• CPD/Trainingforteacherstointroducethequalificationsandprepareyouforfirstteaching.

• ActiveResults–ourfreeresultsanalysis service to help you review the performance of individual learners or whole schools.

• ExamBuilder–ournewfreeonlinepastpapersservice that enables you to build your own test papersfrompastOCRexamquestionscanbefoundonthewebsiteat: www.ocr.org.uk/exambuilder

AllALevelqualificationsofferedbyOCRareaccreditedbyOfqual,theRegulatorforqualificationsofferedinEngland.Theaccreditationnumber forOCRALevelinMathematicsB(MEI)is603/1002/9

1 WhychooseanOCRALevelinMathematicsB(MEI)?

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


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1b. 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.

ALevelMathematicsB(MEI),whichcanbe co-taughtwiththeASLevelasaseparatequalification,consolidatesanddevelopsGCSELevelmathematicsandsupportstransitiontohighereducationoremploymentinanyofthemanydisciplinesthatmakeuseofquantitativeanalysis,including those involving calculus.

ThisqualificationispartofawiderangeofOCRmathematicsqualifications,allowingprogression

fromEntryLevelCertificate,throughGCSEto Core Maths, AS and A Level.

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

MathematicsB(MEI)buildsonourexistingpopularcourse. We’ve based the redevelopment of our current suite around an understanding of what works well in centres and have updated areas of content andassessmentwherestakeholdershaveidentifiedthat improvements could be made. We’ve undertaken asignificantamountofconsultationthroughourmathematicsforums(whichincluderepresentativesfromlearnedsocieties,HE,teachingandindustry)and through focus groups with teachers.

MathematicsB(MEI)isbasedontheexistingsuite ofqualificationsassessedbyOCR.MEIisalongestablished, independent curriculum development body;indevelopingthisspecification,MEIhasconsultedwithteachersandrepresentativesfromHigherEducationtodecidehowbesttomeetthelong-termneedsoflearners.MEIprovidesadvice andCPDrelatingtoallthecurriculumandteachingaspectsofthecourse.Italsoprovidesteachingresources,whichforthisspecificationcanbefoundonthewebsite(www.mei.org.uk).

Allofourspecificationshavebeendevelopedwithsubject and teaching experts. We have worked in closeconsultationwithteachersandrepresentativesfromHigherEducation(HE).

http://www.mei.org.uk

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Aimsandlearningoutcomes

OCRALevelinMathematicsB(MEI)encourageslearnersto:

• understandmathematicsandmathematicalprocessesinawaythatpromotesconfidence,fosters enjoyment and provides a strong foundationforprogresstofurtherstudy

• extendtheirrangeofmathematicalskillsandtechniques

• understand coherence and progression in mathematicsandhowdifferentareasofmathematicsareconnected

• applymathematicsinotherfieldsofstudyandbeawareoftherelevanceofmathematicstotheworldofworkandtosituationsinsocietyingeneral

• usetheirmathematicalknowledgetomakelogical and reasoned decisions in solving problemsbothwithinpuremathematicsandina variety of contexts, and communicate the mathematicalrationaleforthesedecisionsclearly

• reason logically and recognise incorrect reasoning

• generalisemathematically

• constructmathematicalproofs

• usetheirmathematicalskillsandtechniquestosolve challenging problems which require them todecideonthesolutionstrategy

• recognisewhenmathematicscanbe used to analyse and solve a problem in context

• representsituationsmathematicallyandunderstandtherelationshipbetweenproblemsincontextandmathematicalmodelsthatmaybe applied to solve them

• draw diagrams and sketch graphs to help exploremathematicalsituationsandinterpretsolutions

• makedeductionsandinferencesand drawconclusionsbyusingmathematicalreasoning

• interpretsolutionsandcommunicatetheirinterpretationeffectivelyinthecontextoftheproblem

• readandcomprehendmathematicalarguments,includingjustificationsofmethodsand formulae, and communicate their understanding

• readandcomprehendarticlesconcerningapplicationsofmathematicsandcommunicatetheir understanding

• use technology such as calculators and computerseffectivelyandrecognisewhen such use may be inappropriate

• take increasing responsibility for their own learningandtheevaluationoftheirownmathematicaldevelopment.


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1c. Whatarethekeyfeaturesofthisspecification?

OCRALevelinMathematicsB(MEI)hasbeendesignedtohelplearnerstofulfiltheirpotentialinmathematicsandtosupportteachersinenablingthemtodothis.Thespecification:

• encourages learners to develop a deep understandingofmathematicsandanability to use it in a variety of contexts

• encourages learners to use appropriate technologytodeepentheirmathematicalunderstanding and extend the range of problems which they are able to solve

• usespre-releasedatainstatisticstoenablelearners to develop an understanding of working with real data to solve real problems

• is assessed in a way which is designed to enable all learners to show what they are able to do

• includesmathematicalcomprehensionin the assessment to prepare learners to use mathematicsinavarietyofcontextsinHE and future employment

• is clearly laid out with detailed guidance regarding what learners need to know, understand and be able to do

• isresourcedandsupportedbyMEIinline with the aims and learning outcomes of the qualification.

Thisspecificationisdesignedtobeco-teachablewithASLevelMathematicsB(MEI).Toassistteacherswhoareco-teachingtheALevelandtheAS,sectionsofthespecificationnumbered(1)couldbetaughtinthefirstyearofthecourse,sectionsnumbered(2)canbeleftuntilthesecondyear.

1d. HowdoIfindoutmoreinformation?

IfyouarealreadyusingOCRspecificationsyoucancontactusat:www.ocr.org.uk.

IfyouarenotalreadyaregisteredOCRcentrethenyoucanfindoutmoreinformationonthebenefitsofbecomingoneat:www.ocr.org.uk.

Ifyouarenotyetanapprovedcentreandwouldliketobecomeonegoto:www.ocr.org.uk.

Wanttofindoutmore?

GetintouchwithoneofOCR’sSubjectAdvisors:

Email:[email protected]

CustomerContactCentre:01223553998

Teachersupport:www.ocr.org.uk

AdviceandsupportisalsoavailablefromMEI;contactdetails can be found on www.mei.org.uk.

http://www.ocr.org.uk



mailto:[email protected]


http://www.mei.org.uk

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2a. OCRALevelinMathematicsB(MEI)(H640)

LearnersmustcompleteComponents01,02and03tobeawardedOCRALevelinMathematicsB(MEI).

Contentisinthreeareas:

1 Puremathematics2 Mechanics 3 Statistics.

ContentOverview AssessmentOverview

Component01assessescontentfromareas 1 and 2

PureMathematicsandMechanics

(01)

100marks

2 hours

36.4%of totalA Level

Component02assessescontentfromareas 1 and 3

PureMathematicsandStatistics

(02)

100marks

2 hours


Component03assessescontentfromarea 1 (areas2and3areassumed

knowledge)

PureMathematicsandComprehension

(03)

75 marks

2 hours


Percentagesinthetableaboveareroundedto1decimalplace,exactcomponentproportionsare:

, , .36 36 27114

114

113

2 Thespecificationoverview

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2b. ContentofALevelinMathematicsB(H640)

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

ALevelMathematicsB(MEI)isalinearqualification,withnooptions.Thecontentislistedbelow,underthreeareas:

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

2. Mechanicsincludeskinematics,motionundergravity,workingwithforcesincludingfriction,Newton’s laws and simple moments

3. Statisticsincludesworkingwithdatafromasampletomakeinferencesaboutapopulation,probabilitycalculations,usingbinomialandNormaldistributionsasmodelsandstatisticalhypothesistesting.

Therewillbethreeexaminationpapers,attheendofthecourse,toassessallthecontent:

• PureMathematicsandMechanics(01),a 2hourpaperassessingpuremathematicsandmechanics

• PureMathematicsandStatistics(02),a 2hourpaperassessingpuremathematicsandstatistics

• 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.

Theoverarchingthemesshouldbeapplied, alongwithassociatedmathematicalthinkingandunderstanding, across the whole of the detailed contentinpuremathematics,statisticsandmechanics.

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Pre-releasematerial

Pre-releasematerialwillbemadeavailableinadvanceoftheexaminations.Itwillberelevanttosome(butnotall)ofthequestionsincomponent 02.Thepre-releasematerialwillbealargedataset(LDS)thatcanbeusedasteachingmaterialthroughoutthecourse.Itiscomparabletoaset text for a literature course. Itwillbepublishedinadvanceofthecourse.Intheexaminationitwillbeassumed that 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. Theymightincludequestionsrequiringlearnerstointerpret data in ways which would be too demanding in an unfamiliar context.

LearnerswillNOThaveaprintoutofthepre-releasedatasetavailabletothemintheexaminationbutselecteddataorsummarystatisticsfromthe datasetmaybeprovided,withintheexaminationpaper.

Differentdatasetswillbeissuedfordifferentyears;threelargedatasetswillbeavailableatanytime. Only one of these data sets will be used in a given seriesofexaminations.Eachdatasetwillbeclearlylabelledwiththeyearoftheexaminationseriesinwhichitwillbeused.Theexpectationisthatteacherswill use all three data sets but they do have the choiceofconcentratingmoreontheexaminationdataset(orofusingjustthatoneiftheywish).Tosupportprogressionandco-teachability,studentstakingASLevelMathematicsB(MEI)willusethesamedatasetiftheytakeALevelMathematicsB(MEI)inthefollowingyear.

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 into underlyingpatternsandstructures,suggesthypotheses to test and to provide a reason for furtherdatacollection.Thiswillincludetheuseof the following techniques.• Creatingdiagrams:Learnersshoulduse

spreadsheetsorstatisticalgraphingtoolsto create diagrams from data.

• Calculations:Learnersshoulduseappropriate technology to perform statisticalcalculations.

• Investigatingcorrelation:Learnersshoulduse appropriate technology to explore correlationbetweenvariablesintheLDS.

2. Modelling:LearnersshouldusetheLDStoprovideestimatesofprobabilitiesformodellingandtoexplorepossiblerelationshipsbetweenvariables.

3. Repeatedsampling:LearnersshouldusetheLDSasamodelforthepopulationtoperformrepeatedsamplingexperimentstoinvestigatevariabilityandtheeffectofsamplesize.Theyshouldcomparetheresultsfromdifferentsamples with each other and with the results from the whole LDS.

4. Hypothesistesting:LearnersshouldusetheLDSasthepopulationagainstwhichtotesthypotheses based on their own sampling.

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Useoftechnology

Itisassumedthatlearnerswillhaveaccesstoappropriate technology when studying this course suchasmathematicalandstatisticalgraphingtoolsand spreadsheets. When embedded in the mathematicsclassroom,theuseoftechnologycanfacilitatethevisualisationofabstractconceptsanddeepen learners’ overall understanding. Learners are notexpectedtobefamiliarwithanyparticularsoftware,buttheyareexpectedtobeabletousetheircalculatorforanyfunctionitcanperform,whenappropriate.Examinationquestionsmayincludeprintoutsfromsoftwarewhichlearnerswillneedtocompleteorinterpret.Theyshouldbefamiliarwithlanguage used to describe spreadsheets such as row, column and cell.

Tosupporttheteachingandlearningofmathematicsusing technology, we suggest that the following activitiesarecarriedoutthroughthecourse:

1. Graphingtools:Learnersshoulduse graphingsoftwaretoinvestigatetherelationshipsbetweengraphicalandalgebraicrepresentations,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:Learnersshouldusespreadsheets to generate tables of values for functions,toinvestigatefunctionsnumericallyand as an example of applying algebraic notation.Learnersshouldalsousespreadsheetsoftwaretoinvestigatenumericalmethodsforsolvingequationsandformodellinginstatisticsand mechanics.

3. Statistics:Learnersshouldusespreadsheets orstatisticalsoftwaretoexploredatasetsandstatisticalmodelsincludinggeneratingtablesand diagrams, and performing standard statisticalcalculations.

4. Mechanics:Learnersshouldusegraphingand/orspreadsheetsoftwareformodelling,includingkinematicsandprojectiles.

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

Useofcalculators

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

Itisexpectedthatcalculatorsavailableintheexaminationswillincludethefollowingfeatures:

• Aniterativefunctionsuch as an ANS key.• Theabilitytocomputesummarystatisticsand

accessprobabilitiesfromthebinomialandNormaldistributions.

When using calculators, candidates should bear in mindthefollowing:

1. Candidates are advised to write down explicitly any expressions, including integrals, that they use the calculator to evaluate.

2. Candidates are advised to write down the values of any parameters and variables that they input into the calculator. Candidates are not expected to write down data transferred fromquestionpapertocalculator.

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

Formulae

Learners will be given formulae in each assessment onpage2ofthequestionpaper.Seesection5dforalist of these formulae.

Simplifyingexpressions

Itisexpectedthatlearnerswillsimplifyalgebraicandnumericalexpressionswhengivingtheirfinal

http://www.jcq.org.uk

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10© OCR 2017


answers,eveniftheexaminationquestiondoesnotexplicitly ask them to do so.

• 80 23

shouldbewrittenas40 3,

• ( )x21 1 2 2

21 #+ - shouldbewrittenas

either ( )x1 2 21

+ - or x1 2

1+

,

• ln ln ln2 3 1+ - shouldbewrittenasln 6,

• Theequationofastraightlineshouldbegiven in the form y mx c= + or ax by c+ = unless otherwise stated.

Themeaningsofsomeinstructionsusedinexaminationquestions

Ingeneral,learnersshouldshowsufficientdetailoftheir working and reasoning to indicate that a correct methodisbeingused.Thefollowingcommandwordsareusedtoindicatewhenmore,orless,specificdetail is required.

ExactAn exact answer is one where numbers are not given inroundedform.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.e.g.Findtheexactsolutionof ln x 2= .Thecorrectanswerise2andnot7.389056.e.g.Findtheexactsolutionof3 2.x =

Thecorrectansweris 32x = or 0.6x = o , not 0.67x =

or similar.

ShowthatLearners are given a result and have to get to thegivenresultfromthestartinginformation.Becausetheyaregiventheresult,theexplanation hastobesufficientlydetailedtocovereverystep of their working.

e.g.Showthattheequation ( )x x 3 22- = can be expressed as x x x6 9 2 03 2- + - = .

DetermineThiscommandwordindicatesthatjustificationshouldbe given for any results found, including working where appropriate.

Give,State,WritedownThesecommandwordsindicatethatneitherworkingnorjustificationisrequired.

Inthisquestionyoumustshowdetailedreasoning. Whenaquestionincludesthisinstructionlearnersmustgiveasolutionwhichleadstoaconclusionshowingadetailedandcompleteanalyticalmethod.Theirsolutionshouldcontainsufficientdetailtoallowthelineoftheirargumenttobefollowed.Thisisnotarestrictiononalearner’suseofacalculatorwhentacklingthequestion,e.g.forcheckingananswerorevaluatingafunctionatagivenpoint,butitisarestrictiononwhatwillbeacceptedasevidenceofacompletemethod.Intheseexamplesvariationsinthestructure of the answers are possible, for example usingadifferentbaseforthelogarithmsinexample1,anddifferentintermediatestepsmaybegiven.

Example1:Uselogarithmstosolvetheequation3 4x2 1 100=+ , givingyouranswercorrectto3significantfigures.Theanswerisx=62.6,butthelearnermust include the steps log log3 4x2 1 100=+ , ( ) log logx2 1 3 4100+ = andanintermediateevaluationstep,forexample

. ...x2 1 126 18+ = .Usingthesolvefunctiononacalculator to skip one of these steps would not result inacompleteanalyticalmethod.

Example2:

Evaluate .x x x4 1d3

0

1

2+ -y

Theansweris 712 , but the learner must include at

least x x x41

344 3

0

1

+ -; E andthesubstitution41

34 1+ - .

Justwritingdowntheanswerusingthedefiniteintegralfunctiononacalculatorwouldtherefore not be awarded any marks.

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Example3: Solvetheequation forsin cosx x x3 2 0 180c c# #= . Theanswerisx = 9.59°, 90° or 170°, but the learner must include … sin cos cosx x x6 0- = ,

( ) ,cos sin cos sinx x x x6 1 0 0 61or- = = = .

Agraphicalmethodwhichinvestigatedtheintersectionsofthecurves siny x3 2= and cosy x= wouldbeacceptabletofindthesolutionat90°ifcarefullyverified,buttheothertwosolutionsmust befoundanalytically,notnumerically.

HenceWhenaquestionusestheword‘hence’,itisanindicationthatthenextstepshouldbebasedonwhathasgonebefore.Theintentionisthatlearnersshouldstart from the indicated statement.e.g. You are given that f ( ) 2 7 6x x x x3 2= - - + . Show that ( 1)x - is a factor of f ( )x .Hencefindthethreefactors of f ( )x .

Henceorotherwiseisusedwhentherearemultiplewaysofansweringagivenquestion.Learnersstartingfrom the indicated statement may well gain some informationaboutthesolutionfromdoingso,andmayalreadybesomewaytowardstheanswer.Thecommand phrase is used to direct learners towards usingaparticularpieceofinformationtostartfromortoaparticularmethod.Italsoindicatestolearnersthatvalidalternativemethodsexistwhichwillbegivenfullcredit,butthattheymaybemoretime-consuming or complex.

e.g. Show that ( )cos sin sinx x x1 22+ = + for all x. Hence,orotherwise,findthederivativeof( )cos sinx x 2+ .

YoumayusetheresultWhen this phrase is used it indicates a given result that learners would not always be expected

to know, but which may be useful in answering the question.Thephraseshouldbetakenaspermissive;use of the given result is not required.

PlotLearners should mark points accurately on graph paperprovidedinthePrintedAnswerBooklet.Theywill either have been given the points or have had to calculatethem.Theymayalsoneedtojointhemwitha curve or a straight line.e.g.Plotthisadditionalpointonthescatterdiagram.

Sketch(agraph)Learners should draw a diagram, not necessarily to scale,showingthemainfeaturesofacurve.Theseare 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.

e.g.Sketchthecurvewithequation ( )y x 11

=-

.

DrawLearners should draw to an accuracy appropriate to theproblem.Theyarebeingaskedtomakeasensiblejudgement about the level of accuracy which is appropriate.e.g.Drawadiagramshowingtheforcesactingontheparticle.e.g.Drawalineofbestfitforthedata.

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

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OverarchingThemes

TheseOverarchingThemesshouldbeapplied,alongwithassociatedmathematicalthinkingandunderstanding,acrossthewholeofthedetailedcontentinthisspecification.ThesestatementsareintendedtodirecttheteachingandlearningofALevelMathematics,andtheywillbereflectedinassessmenttasks.

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

Problemspecificationand analysis

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.

Theanalysismayinvolveconsideringwhetherthereisanappropriatestandardmodeltouse(e.g.theNormaldistributionortheparticlemodel)orwhethertheproblemissimilarto one which has been solved before.

Atthecompletionoftheproblemsolvingcycle,thereneedstobeconsiderationofwhethertheoriginal problemhasbeen solved in a satisfactorywayorwhether it is necessary torepeattheproblemsolvingcycleinordertogainabettersolution.Forexample,thesolutionmight not be accurate enough or only apply in some cases.

Informationcollection

Thisstageinvolvesgettingthenecessaryinputsforthemathematicalprocessingthatwilltakeplaceatthenextstage.Thismayinvolvedecidingwhicharetheimportantvariables,findingkeymeasurementsorcollectingdata.

Processingandrepresentation

Thisstageinvolvesusingsuitablemathematicaltechniques,suchascalculations,graphsordiagrams,inordertomakesenseoftheinformationcollectedinthepreviousstage.Thisstageendswithaprovisionalsolutiontotheproblem.

Interpretation Thisstageoftheprocessinvolvesreportingthesolutiontotheprobleminawaywhichrelatestotheoriginalsituation.CommunicationshouldbeinclearplainEnglishwhichcanbe understood by someone who has an interest in the original problem but is not an expert inmathematics.Thisshouldleadintoreflectiononthesolutiontoconsiderwhetheritissatisfactoryoriffurtherworkisneeded.

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TheModellingCycle

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 thereareenoughteachersandschoolplaces.Tofindareasonableestimate,theymightassumethat the birthrateoverthenextfiveyearswillbesimilartothatforthelastfiveyearsandthosechildrenwillgotoschoolintheareatheywerebornin.Theywould evaluatetheseassumptionsbycheckingwhethertheyfitinwithnewdataand review the estimatetoseewhetheritisstillreasonable.

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Learningoutcomes

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

• Content that is covered by a learning outcome with a reference code may be tested in an examinationquestionwithoutfurtherguidancebeing given.

• Learning outcomes marked with an asterisk * are assumed knowledge and will not form the focusofanyexaminationquestions.Thesestatements are included for clarity and completeness.

• Manyexaminationquestionswillrequirelearners to use two or more learning outcomes atthesametimewithoutfurtherguidance being given. Learners are expected to be able tomakelinksbetweendifferentareasofmathematics.

• Learners are expected to be able to use their knowledge and understanding to reason mathematicallyandsolveproblemsbothwithinmathematicsandincontext.Contentthatiscovered by any learning outcome may be required in problem solving, modelling and reasoning tasks even if that is not explicitly stated in the learning outcome.

• Learning outcomeshaveanimpliedprefix: ‘Alearnershould…’.

• Eachreferencecodeforalearning outcome is unique. For example, in the code Mc1, M referstoMathematics,crefersto‘calculus’ (seebelow)and1meansthatitisthefirst such learning outcome in the list.

• Thelettersusedinassigningreferencecodes to learning outcomes are common to all qualificationsinspecB(H630,H640,H635andH645).OnlythoseusedinALevelMathematicsB(MEI)areshownbelow.

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’s laws 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

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Notes,notationandexclusions

Thenotes,notationandexclusionscolumnsinthespecificationareintendedtoassistteachersandlearners.

• Thenotescolumnprovidesexamples and further detail for some learning outcomes. Allexemplarscontainedinthespecificationareforillustrationonlyanddonotconstituteanexhaustivelist

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

• Theexclusionscolumnlistscontentwhichwillnot be tested, for the avoidance of doubt when interpretinglearning outcomes.

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2c. 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.

Division by non-linearexpressions.

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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)

Thelanguageoffunctions

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 =

Therangeisthesetofallpossiblevalues 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-

Themodulusfunction

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 modulusofnon-linearfunctions.

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).

Thecurve ( )y a x p q2= + + has• a minimum at ( , )p q- for a 0> or a maximum at ( , )p q- for a < 0• a line 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)

Thecoordinategeometry 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= + .TheirunderstandingatALevelshouldextendto

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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Thecoordinategeometry 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: • the angle in a semicircle is a right angle; • the perpendicular from the centre of a

circle to a chord bisects the chord; • the radius of a circle at a given point on its

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

Theseresultsmaybeusedinthecontextofcoordinate 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.

Themeaningofthetermfactorial.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. General term.

s4Beabletowrite( )a bx n+ in the form an a

bx1

n

+

J

L

KKKKKK

N

P

OOOOOO

and 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 = .

kthterm: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.

Thenth term, the sum to n terms.Includingthesumofthefirstn natural numbers.

Sn

Geometric series

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.

Thenth 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

* Knowhowtosolveright-angledtrianglesusingtrigonometry.

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; sine and 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= for 0 360c c# #i .

t6 Understandandbeabletousetheidentitysin cos 12 2i i+ = .

e.g. solve sin cos2i i= for 0 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=

Includesequationsinvolvingmultiples of the unknown angle e.g.

.sin cos2 3 2i i=

Includesquadraticequations.

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

General solutions.

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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.

Theresultss 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’s theorem 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.

Theargumentofthetrigonometricfunctionsisnotrestrictedtoangles.

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PUREMATHEMATICS:EXPONENTIALSANDLOGARITHMS(1)

Exponentialsand Logarithms

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

E2 Beabletoconvertfromanindextoalogarithmicform and 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.

Themodulusfunction.

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 Beabletousedifferentiationtofindstationarypointsonacurve:maximaandminima.

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.

Theproductrule: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.

xy

yx

1dd

dd

= J

L

KKKKKKK

N

P

OOOOOOO

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.

concaveupwards(convexdownwards)

concave downwards (convexupwards)

Thewording“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. FundamentalTheoremofCalculus.

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( )x x x3 5 1 d

12 + -y .

c23 Beabletouseintegrationtofindtheareabetweenagraph and the x-axis.

Includesareasofregionspartlyaboveand partly below the x-axis.General understanding that the area under a graph can be found as the limit of a sum of areas of rectangles.

Formal understanding of thecontinuityconditionsrequired for the Fundamental TheoremofCalculus.

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

b( ) ( )lim x x x xf f d

x a

b

a0d =

"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 a verticalasymptote.e.g. there may be several roots in the interval.

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

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

3= ++ with an appropriate startingvalue.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.

Number of strips.

c35 Usethesumofaseriesofrectanglestofindanupperand/or lower bound on the area under a curve.

Problem solving

Me6 Use numerical methods to solve problems in context.

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PUREMATHEMATICS:VECTORS(1)

General vectors

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 Themagnitudeofthevector a iswritten|a|ora.

aaa 1

2=

J

L

KKKKKK

N

P

OOOOOO

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

KKKKKK

N

P

OOOOOO

v5 Beabletocalculatethedistancebetweentwopointsrepresentedbypositionvectors.

Using vectors v6 Beabletousevectorstosolveproblemsinpuremathematicsandincontext,includingproblemsinvolving forces.

Includesinterpretingthesumofvectorsrepresentingforcesastheresultant force.

PUREMATHEMATICS:VECTORS(2)

General vectors

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 as anestimateofpopulationmeanorvariance.

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,clustersampling,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)

Data presentationfor single variable

MD1 Beabletorecogniseandworkwithcategorical,discrete,continuousandrankeddata.Beabletointerpret standard diagrams for grouped and ungroupedsingle-variabledata.

Includesknowingthisvocabularyanddecidingwhatdatapresentationmethodsareappropriate:barchart,dotplot,histogram,verticallinechart,piechart,stem-and-leafdiagram,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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Data presentation

MD5 Understandthatdiagramsrepresentingunbiased samples become more representativeoftheoreticalprobabilitydistributionswithincreasingsamplesize.

e.g.Abarchartrepresentingtheproportionof heads and tails when a fair coin is tossed tendstohavetheproportionofheadsincreasinglycloseto50%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.Themainfocusofquestions willbeoninterpretationratherthancalculation.

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.

samplevariance:s nS

1xx2 =

- (†)

where ( )S x xxx ii

n2

1= -

=

|

samplestandarddeviation:

s variance= (‡)

s2

s

Correctionsforclass interval in thesecalculations.

D13 Understand the term outlier and be able to identifyoutliers.Knowthatthetermoutliercanbeappliedtoanitemofdatawhichis:• atleast2standarddeviationsfromthe

mean;OR• at least 1.5 × 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,fromtreediagrams,two-waytables,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‘hasthedistribution’.

Calculationsrelatingtobinomial distribution

R3 Beabletocalculateprobabilitiesusingthebinomialdistribution.

Includinguseofcalculatorfunctions.

Mean and expected frequencies for binomial distribution

R4 Understand and use mean = np. Derivationofmean= np

R5 Beabletocalculateexpectedfrequenciesassociatedwiththebinomialdistribution.

Discrete probability 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)

Normal distribution

MR8 BeabletousetheNormaldistributionasamodel. IncludesrecognisingwhenaNormaldistributionmaynotbeappropriate.UnderstandhowandwhyacontinuitycorrectionisusedwhenusingaNormaldistributionasamodelforadistributionofdiscrete data.RecognisefromtheshapeofthedistributionwhenabinomialdistributioncanbeapproximatedbyaNormaldistribution.

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

R9 Know the shape of the Normal curve and understand that histograms from increasingly large samples from a NormaldistributiontendtotheNormalcurve.

Includesunderstandingthattheareaunderthe Normal curve represents probability.

R10 KnowthatlineartransformationofaNormalvariablegives another Normal variable and know how the meanandstandarddeviationareaffected.Beabletostandardise a Normal variable.

, y a bx y a bx s b si i y x2 2 2&= + = + = Standard Normal

( , )Z 0 1N+

ZXvn

=-

Proof

R11 Know that the line of symmetry of the Normal curve is locatedatthemeanandthepointsofinflectionarelocatedonestandarddeviationawayfromthemean.

R12 BeabletocalculateanduseprobabilitiesfromaNormaldistribution.

Includinguseofcalculatorfunctions.

Modelling with probability

R13 Beabletomodelwithprobabilityandprobabilitydistributions,includingrecognisingwhenthebinomialor Normal model may not be appropriate.

Includingcritiquingassumptionsmade andthelikelyeffectofmorerealisticassumptions.

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-tailtest. Criticalvalue,criticalregion(rejectionregion),acceptanceregion,p-value.

H2 Understandwhentoapply1-tailand2-tailtests.

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,withp aprobabilityforthewholepopulation.

H0, H1

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

Normal approximation.

H6 Beabletoidentifythecriticalandacceptanceregions.


Hypothesis testingforamean using Normal distribution

MH7 Knowthatrandomsamplesofsizen from ( , )X N 2+ n v have the sample mean Normally

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)thepopulationvarianceisknown or (b)thepopulationvarianceisunknownbutthesamplesizeislarge Learners 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-valueorcriticalvalue.

Associationreferstoamoregeneralrelationshipbetweenthevariables.

The(oftenimplicit)nullhypothesisisoftheformeitherthatthereisnocorrelationornoassociationinthepopulation.Questionswilluseanappropriatecorrelationcoefficientandindicatewhethercorrelationorassociationisbeingtestedfor.

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

Knowledge of bivariate Normal distribution

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; long term.

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 ÷ elapsedtimeAverage 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 t

r a tv

tr

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; velocity and speed, and between accelerationandmagnitudeofacceleration.

Positionvector,relativeposition.Average speed = distance travelled ÷ elapsedtimeAverage velocity = overall displacement÷elapsedtime

Relativevelocity

k10 Beabletoextendthescopeoftechniquesfrommotionin1dimensiontothatin2dimensionsbyusing vectors.

Theuseofcalculusandtheuseofconstantaccelerationformulae. ,t td

ddda 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.• No air resistance.• Theprojectileisaparticle.• Horizontaldistancetravelledis

small enough to assume that gravity is always in the same direction.

• Verticaldistancetravelledissmallenough to assume that gravity is constant.

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 g should be taken to be 9.8.

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 an inclined plane.

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

F8 Knowthatvectorsrepresentingasetofforcesinequilibriumsumtozero.Knowthataclosedfiguremaybedrawntorepresenttheadditionoftheforceson an object in equilibrium.

F9 Beabletoformulateandsolveequationsforaparticleinequilibrium:byresolvingforcesinsuitabledirections;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.• Smooth is used to mean that

frictionmaybeignored.• Roughindicatesthatfrictionmust

be taken into account.

Normalreaction.

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

Coefficientoffriction= n Limitingfriction,staticequilibrium

Thetermangleoffriction.

F12 BeabletoapplyNewton’sLawstoproblemsinvolvingfriction.

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MECHANICS:NEWTON’SLAWSOFMOTION(1)

Newton’s laws foraparticle

Mn1 Know and understand the meaning of Newton’s three laws.

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’s laws foraparticle

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)rodsandrectangularlaminas.Theonlyforcesconsideredarecoplanar,andactperpendiculartotherodortoanedgeofthelamina.Thelearningoutcomeshouldberead inthelightofthisrestriction.

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

Rigid bodies in equilibrium

MF13 Beabletocalculatethemomentofaforceaboutapoint or axis.

Units of moment are N m.

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.

Thepointisthecentreofmassofthe 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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2d. Priorknowledge,learningandprogression

• ItisassumedthatlearnersarefamiliarwiththecontentofGCSE(9–1)Mathematicsforfirstteachingfrom2015.

• ALevelMathematicsB(MEI)providestheframework within which a large number of youngpeoplecontinuethesubjectbeyondGCSElevel.Itsupportstheirmathematicalneeds across a broad range of other subjects at this level and provides a basis for subsequent quantitativeworkinaverywiderangeofhighereducationcoursesandinemployment.ItalsosupportsthestudyofASandALevelFurtherMathematicsB(MEI).

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

employment in a wide range of disciplines involvingtheuseofmathematics.

• Some learners may wish to follow a mathematicscourseonlyuptoASlevel,inorder to broaden their curriculum, and to develop their interest and understanding of differentareasofthesubject.Othersmayfollowaco-teachableroute,completinga oneyearAScourseandthencontinuingtocomplete the second year of the two year A level course, developing a deeper knowledge andunderstandingofmathematicsanditsapplications.

• Learners who wish to specialise in mathematicsorSTEMsubjectssuchas physics or engineering can further extend their knowledge and understanding of mathematicsanditsapplicationsbytakingFurtherMathematicsB(MEI)ASorALevel.

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


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3a. Formsofassessment

OCRALevelinMathematicsB(MEI)consistsofthreecomponentsthatareexternallyassessed.Thethreeexternallyassessedcomponents(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-assessedinsomeyears.Thepassagemayincludeexamplesofapplicationsofthepurecontent(otherthanthosewhicharespecifiedinthemechanicsandstatisticssections).Themechanicsandstatisticscontentofthespecificationisassumedknowledgeforcomponent03butthisassumedknowledgewillnotbethefocusofanyofthequestions.

3 AssessmentofALevelinMathematicsB(MEI)

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3b. Assessmentobjectives(AO)

Thereare3AssessmentObjectivesinOCRASLevelMathematicsB(MEI).Thesearedetailedinthetablebelow.

AssessmentobjectivesWeightings

A level

AO1

UseandapplystandardtechniquesLearnersshouldbeableto:• selectandcorrectlycarryoutroutineprocedures;and• accuratelyrecallfacts,terminologyanddefinitions.

50%(±2%)

AO2

Reason,interpretandcommunicatemathematicallyLearnersshouldbeableto:• constructrigorousmathematicalarguments(includingproofs);• makedeductionsandinferences;• assessthevalidityofmathematicalarguments;• explain their reasoning; 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

SolveproblemswithinmathematicsandinothercontextsLearnersshouldbeableto:• 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. Assessment availability

Therewillbeoneexaminationseriesavailableeachyear in May/June to all learners.

All examined components must be taken in the same examinationseriesattheendofthecourse.

ThisspecificationwillbecertificatedfromtheJune2018examinationseriesonwards.

3d. Retakingthequalification

Learnerscanretakethequalificationasmanytimesas theywish.Theymustretakeallcomponentsofthe 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 have acquired from the course as a whole and their ability to integrate and apply that understanding.Thislevelofunderstandingisneeded for successful use of the knowledge and skills from this course in future life, work and study.

• Learners are required to know and understand thecontentofallthepuremathematicsand to be able to apply the overarching themes, alongwithassociatedmathematicalthinkingand understanding, in all the assessment componentsofALevelMathematicsB(MEI).

• Inalltheexaminationpapers,learnerswillberequired to integrate and apply their understanding in order to address problems which require both breadth and depth of understandinginordertoreachasatisfactorysolution.

• Learnerswillbeexpectedtoreflectonandinterpretsolutions,drawingontheirunderstandingofdifferentaspectsofthecourse.

3g. Calculatingqualificationresults

Alearner’soverallqualificationgradeforALevelinMathematicsB(MEI)willbecalculatedbyaddingtogether their marks from the three components takentogivetheirtotalmark.Thismarkwillthenbe

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

4


Theinformationinthissectionisdesignedtogiveanoverview of the processes involved in administering thisqualificationsothatyoucanspeaktoyourexamsofficer.AllofthefollowingprocessesrequireyoutosubmitsomethingtoOCRbyaspecificdeadline.

Moreinformationabouttheprocessesanddeadlinesinvolved at each stage of the assessment cycle can be foundintheAdministrationareaoftheOCRwebsite.OCR’s Admin 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.Theyarefreeanddonotcommityourcentre 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.

Entry code

Title Component code

Componenttitle Assessment type

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

Regulationsgoverningexaminationarrangementsarecontained in the JCQ Instructions for conducting examinations.

Learnersarepermittedtouseascientificorgraphicalcalculator for all components. Calculators are subject to the rules in the document Instructions for Conducting Examinations published annually by JCQ (www.jcq.org.uk).

Headofcentreannualdeclaration

TheHeadofcentreisrequiredtoprovideadeclarationtotheJCQaspartoftheannualNCNupdate,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. TheymustbebasedintheUK.

Private candidates need to contact OCR approved centres to establish whether they are prepared to hostthemasaprivatecandidate.Thecentremaycharge for this facility and OCR recommends that the arrangement is made early in the course.

Further guidance for private candidates may be found ontheOCRwebsite:http://www.ocr.org.uk

www.jcq.org.uk


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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

• the raw mark for each component

• thetotalweightedmarkforthequalification.

Thefollowingsupportinginformationwillbeavailable:

• raw mark grade boundaries for each component

• weighted mark grade boundaries for the qualification.

Untilcertificatesareissued,resultsaredeemedtobeprovisional and may be subject to amendment.

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

4e. Post-resultsservices

Anumberofpost-resultsservicesareavailable:

• Reviewofmarking–Ifyouarenothappywiththe outcome of a learner’s results, centres may request a review of marking. Full details of the post-resultsservicesareprovidedontheOCRwebsite.

• Missingandincompleteresults–Thisserviceshould be used if an individual subject result for a learner is missing, or the learner has been omittedentirelyfromtheresultssupplied.

• 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

JCQ Access 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" ,

5


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

3.10,

nr Cn r

J

L

KKKKKK

N

P

OOOOOO, Cn r

thebinomialcoefficient ! ( ) !!

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

4.7xy

dd thederivativeofy with respect to x

4.8xy

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

KKKKKK

N

P

OOOOOO

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

74© OCR 2017


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

76© OCR 2017


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

KKKKKK

N

P

OOOOOO

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

Sinerule: sin sin sinAa

Bb

Cc

= =

Cosinerule: 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 x

1 ( ) ( )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

78© OCR 2017


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

Themeanofasetofdata: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 rnCn r = =-

J

L

KKKKKK

N

P

OOOOOO

( ) !( )

!( ) ( )

, x nxn n

x rn n n r

x x n1 1 21 1 1

1 Rn r2 ff

f 1 !+ = + +-

+ +- - +

+ ^ h

5

80© OCR 2017


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

KKKKKK

N

P

OOOOOO and

/( , )

n

X0 1N+

v

n-

PercentagepointsoftheNormaldistribution

p 10 5 2 1

z 1.645 1.960 2.326 2.575

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= -

82© OCR 2017


Summaryofupdates

Date Version Section Titleofsection ChangeJune2018 1.1 Front cover Disclaimer AdditionofDisclaimer


84© OCR 2017


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Specification MATHEMATICS B (MEI)ocr.org.uk/Images/308740-specification-accredited-a-level-gce... · A LEVEL Specification MATHEMATICS B (MEI) H640 For first assessment in 2018 ocr.org.uk/alevelmathsmei