Syllabus Cambridge International AS & A Level Further Mathematics 9231

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SyllabusCambridge International AS & A LevelFurther Mathematics 9231For examination in June and November 2020, 2021 and 2022.Version 1

Why choose Cambridge?Cambridge Assessment International Education prepares school students for life, helping them develop an informedcuriosity and a lasting passion for learning. We are part of the University of Cambridge.Our international qualifications are recognised by the world’s best universities and employers, giving students awide range of options in their education and career. As a not-for-profit organisation, we devote our resources todelivering high-quality educational programmes that can unlock learners’ potential.Our programmes and qualifications set the global standard for international education. They are created by subjectexperts, rooted in academic rigour and reflect the latest educational research. They provide a strong platform forstudents to progress from one stage to the next, and are well supported by teaching and learning resources.We review all our syllabuses regularly, so they reflect the latest research evidence and professional teachingpractice – and take account of the different national contexts in which they are taught.We consult with teachers to help us design each syllabus around the needs of their learners. Consulting withleading universities has helped us make sure our syllabuses encourage students to master the key concepts in thesubject and develop the skills necessary for success in higher education.Our mission is to provide educational benefit through provision of international programmes and qualifications forschool education and to be the world leader in this field. Together with schools, we develop Cambridge learnerswho are confident, responsible, reflective, innovative and engaged – equipped for success in the modern world.Every year, nearly a million Cambridge students from 10 000 schools in 160 countries prepare for their future withan international education from Cambridge International.‘We think the Cambridge curriculum is superb preparation for university.’Christoph Guttentag, Dean of Undergraduate Admissions, Duke University, USAQuality managementOur systems for managing the provision of international qualifications and education programmes for studentsaged 5 to 19 are certified as meeting the internationally recognised standard for quality management, ISO9001:2008. Learn more at www.cambridgeinternational.org/ISO9001Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name ofthe University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.UCLES retains the copyright on all its publications. Registered centres are permitted to copy material from this booklet for their owninternal use. However, we cannot give permission to centres to photocopy any material that is acknowledged to a third party even forinternal use within a centre.

Contents1 Why choose this syllabus? .22 Syllabus overview . 6Aims6Content overview7Structure8Assessment overview9Assessment objectives113 Subject content .12Prior knowledge121Further Pure Mathematics 1 (for Paper 1)132Further Pure Mathematics 2 (for Paper 2)173Further Mechanics (for Paper 3)214Further Probability & Statistics (for Paper 4)244 Details of the assessment . 26Planning for assessment of AS & A Level Further Mathematics (9231)26Examination information26Command words285 List of formulae and statistical tables (MF19). 296 What else you need to know .42Before you start42Making entries43After the exam44How students, teachers and higher education can use the grades45Grade descriptions45Changes to this syllabus for 2020, 2021 and 202246Changes to this syllabusFor information about changes to this syllabus for 2020, 2021 and 2022, go to page 46.

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022.1 Why choose this syllabus?Key benefitsThe best motivation for a student is a real passion for the subjectthey’re learning. By offering students a variety of CambridgeInternational AS & A Levels, you can give them the greatest chanceof finding the path of education they most want to follow. Withover 50 subjects to choose from, students can select the onesthey love and that they’re best at, which helps motivate themthroughout their studies.Following a Cambridge International AS & A Level programmehelps students develop abilities which universities value highly,including: a deep understanding of their subjects higher order thinking skills – analysis, critical thinking,problem solving presenting ordered and coherent arguments independent learning and research.Cambridge International AS & A Level Further Mathematics develops a set of transferable skills. Theseinclude the skill of working with mathematical information, as well as the ability to think logically andindependently, consider accuracy, model situations mathematically, analyse results and reflect on findings.Learners can apply these skills across a wide range of subjects and the skills equip them well for progression tohigher education or directly into employment. Learners will find that the additional time spent studying thissubject will support their understanding of A Level Mathematics.Our approach in Cambridge International AS & A Level Further Mathematics encourages learners to be:confident, using and sharing information and ideas, and using mathematical techniques to solve problems. Theseskills build confidence and support work in other subject areas as well as in mathematics.responsible, through learning and applying skills which prepare them for future academic studies, helping them tobecome numerate members of society.reflective, through making connections between different branches of mathematics and considering the outcomesof mathematical problems and modelling.innovative, through solving both familiar and unfamiliar problems in different ways, selecting from a range ofmathematical and problem-solving techniques.engaged, by the beauty and structure of mathematics, its patterns and its many applications to real life situations.‘Cambridge students develop a deep understanding of subjects and independent thinking skills.’Tony Hines, Principal, Rockledge High School, USA2www.cambridgeinternational.org/alevelBack to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Why choose this syllabus?Key conceptsKey concepts are essential ideas that help students develop a deep understanding of their subject and make linksbetween different aspects. Key concepts may open up new ways of thinking about, understanding or interpretingthe important things to be learned.Good teaching and learning will incorporate and reinforce a subject’s key concepts to help students gain: a greater depth as well as breadth of subject knowledge confidence, especially in applying knowledge and skills in new situations the vocabulary to discuss their subject conceptually and show how different aspects link together a level of mastery of their subject to help them enter higher education.The key concepts identified below, carefully introduced and developed, will help to underpin the course you willteach. You may identify additional key concepts which will also enrich teaching and learning.The key concepts for Cambridge International AS & A Level Further Mathematics are: Problem solvingMathematics is fundamentally problem solving and representing systems and models in different ways. Theseinclude:–– Algebra: this is an essential tool which supports and expresses mathematical reasoning and provides ameans to generalise across a number of contexts.–– Geometrical techniques: algebraic representations also describe a spatial relationship, which gives us a newway to understand a situation.–– Calculus: this is a fundamental element which describes change in dynamic situations and underlines thelinks between functions and graphs.–– Mechanical models: these explain and predict how particles and objects move or remain stable under theinfluence of forces.–– Statistical methods: these are used to quantify and model aspects of the world around us. Probabilitytheory predicts how chance events might proceed, and whether assumptions about chance are justified byevidence. CommunicationMathematical proof and reasoning is expressed using algebra and notation so that others can follow each lineof reasoning and confirm its completeness and accuracy. Mathematical notation is universal. Each solution isstructured, but proof and problem solving also invite creative and original thinking. Mathematical modellingMathematical modelling can be applied to many different situations and problems, leading to predictions andsolutions. A variety of mathematical content areas and techniques may be required to create the model. Oncethe model has been created and applied, the results can be interpreted to give predictions and informationabout the real world.Back to contents pagewww.cambridgeinternational.org/alevel3

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Why choose this syllabus?Recognition and progressionEvery year thousands of students with Cambridge International AS & A Levels gain places at leading universitiesworldwide. Cambridge International AS & A Levels are accepted across 195 countries. They are valued by topuniversities around the world including those in the UK, US (including Ivy League universities), Europe, Australia,Canada and New Zealand.UK NARIC, the national agency in the UK for the recognition and comparison of international qualifications andskills, has carried out an independent benchmarking study of Cambridge International AS & A Level and found it tobe comparable to the standard of AS & A Level in the UK. This means students can be confident that their CambridgeInternational AS & A Level qualifications are accepted as equivalent, grade for grade, to UK AS & A Levels by leadinguniversities worldwide.Cambridge International AS Level Further Mathematics makes up the first half of the Cambridge InternationalA Level course in further mathematics and provides a foundation for the study of further mathematics atCambridge International A Level. Depending on local university entrance requirements, students may be able to useit to progress directly to university courses in mathematics or some other subjects. It is also suitable as part of acourse of general education.Cambridge International A Level Further Mathematics provides an excellent foundation for the study ofmathematics or related courses in higher education. Equally it is suitable as part of a course of general education.For more information about the relationship between the Cambridge International AS Level and CambridgeInternational A Level see the ‘Assessment overview’ section of the Syllabus overview.We recommend learners check the Cambridge recognitions database and the university websites to find the mostup-to-date entry requirements for courses they wish to study.Learn more at www.cambridgeinternational.org/recognition‘The depth of knowledge displayed by the best A Level students makes them prime targets forAmerica’s Ivy League universities’Yale University, USA4www.cambridgeinternational.org/alevelBack to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Why choose this syllabus?Supporting teachersWe provide a wide range of practical resources, detailed guidance, and innovative training and professionaldevelopment so that you can give your learners the best possible preparation for Cambridge International AS & ALevel.Teaching resourcesExam preparation resources The School Support Hubwww.cambridgeinternational.org/support Question papers Syllabus Scheme of work Example candidate responses to understandwhat examiners are looking for at key grades Learner guide   Examiner reports to improve future teaching Mark schemes Discussion forum Resource list Endorsed textbooks and digital resourcesTraining Face-to-face workshops around the world Online self-study training Online tutor-led training Cambridge Professional DevelopmentQualificationsSupportfor CambridgeInternationalAS & A LevelCommunityYou can find useful information, as well asshare your ideas and experiences with otherteachers, on our social media channels andcommunity forums.Find out more mbridge International AS & A Levels prepare students well for university because they’velearnt to go into a subject in considerable depth. There’s that ability to really understand thedepth and richness and the detail of a subject. It’s a wonderful preparation for what they aregoing to face at university.’US Higher Education Advisory CouncilBack to contents pagewww.cambridgeinternational.org/alevel5

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022.2 Syllabus overviewAimsThe aims describe the purposes of a course based on this syllabus.The aims are to enable students to: further develop their mathematical knowledge and skills in a way which encourages confidence and providessatisfaction and enjoyment develop a greater understanding of mathematical principles and a further appreciation of mathematics as alogical and coherent subject acquire a greater range of mathematical skills, particularly those which will enable them to use applications ofmathematics in the context of everyday situations and of other subjects they may be studying further develop the ability to analyse problems logically recognise when and how a situation may be represented mathematically, identify and interpret relevant factorsand select an appropriate mathematical method to solve the problem use mathematics fluently as a means of communication with emphasis on the use of clear expression acquire the mathematical background necessary for further study in mathematics or related subjects.Support for Cambridge International AS & A Level Further MathematicsOur School Support Hub www.cambridgeinternational.org/support provides Cambridge schools with asecure site for downloading specimen and past question papers, mark schemes, grade thresholds and othercurriculum resources specific to this syllabus. The School Support Hub community offers teachers theopportunity to connect with each other and to ask questions related to the k to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Syllabus overviewContent overviewContent sectionAssessmentcomponentTopics included1  Further Pure Mathematics 1Paper 11.1Roots of polynomial equations1.2Rational functions and graphs1.3Summation of series1.4 Matrices1.5Polar coordinates1.6 Vectors2 Further Pure Mathematics 2Paper 21.7Proof by induction2.1Hyperbolic functions2.2 Matrices2.3 Differentiation2.4 Integration2.5 Complex numbers2.6 Differential equations3 Further MechanicsPaper 33.1Motion of a projectile3.2 Equilibrium of a rigid body3.3 Circular motion3.4 Hooke’s law3.5 Linear motion under a variable force3.6 Momentum4  Further Probability &StatisticsPaper 44.1 Continuous random variables4.2 Inference using normal and t-distributions4.3 2 -tests4.4 Non-parametric tests4.5 Probability generating functionsBack to contents pagewww.cambridgeinternational.org/alevel7

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Syllabus overviewStructureThere are four components that can be combined in specific ways (please see below):Paper 1: Further Pure Mathematics 1Paper 2: Further Pure Mathematics 2Paper 3: Further MechanicsPaper 4: Further Probability & StatisticsAll AS Level candidates take two written papers.All A Level candidates take four written papers.AS Level Further MathematicsThe Cambridge International AS Level Further Mathematics qualification offers two different options: Further Pure Mathematics 1 and Further Mechanics (Paper 1 and Paper 3)or Further Pure Mathematics 1 and Further Probability & Statistics (Paper 1 and Paper 4).A Level Further MathematicsCambridge International A Level Further Mathematics includes all four components: Paper 1: Further Pure Mathematics 1 Paper 2: Further Pure Mathematics 2 Paper 3: Further Mechanics Paper 4: Further Probability & Statistics.See page 10 for a table showing all possible assessment routes.Structure of AS Level and A Level Further MathematicsAS Level Further MathematicsPaper 1 and Paper 3Further PureMathematics 1 andFurther MechanicsPaper 1 and Paper 4Further PureMathematics 1 andFurther Probability &Statistics8www.cambridgeinternational.org/alevelA Level Further MathematicsPaper 1, 2, 3 and 4Further PureMathematics 1 and 2,Further Mechanics andFurther Probability &StatisticsBack to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Syllabus overviewAssessment overviewPaper 1Further Pure Mathematics 1Paper 32 hoursFurther Mechanics1 hour 30 minutes75 marks50 marks6 to 8 structured questions based on theFurther Pure Mathematics 1 subject content5 to 7 structured questions based on theFurther Mechanics subject contentAnswer all questionsAnswer all questionsWritten examinationWritten examinationExternally assessedExternally assessed60% of the AS Level40% of the AS Level30% of the A Level20% of the A LevelCompulsory for AS Level and A LevelOffered as part of AS Level or A LevelPaper 2Paper 4Further Pure Mathematics 22 hours75 marks7 to 9 structured questions based on theFurther Pure Mathematics 2 subject contentAnswer all questionsWritten examinationExternally assessed30% of the A Level onlyCompulsory for A LevelFurther Probability &Statistics1 hour 30 minutes50 marks5 to 7 structured questions based on theFurther Probability & Statistics subject contentAnswer all questionsWritten examinationExternally assessed40% of the AS Level20% of the A LevelOffered as part of AS Level or A LevelBack to contents pagewww.cambridgeinternational.org/alevel9

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Syllabus overviewThree routes for Cambridge International AS & A Level Further MathematicsCandidates may combine components as shown below.Route 1AS Level only(Candidates take the AScomponents in the sameseries)Paper 1Paper 2Further PureMathematics 1Further PureMathematics 2Either Or Route 2A Level (staged overtwo years)Paper 1Paper 2Further PureMathematics 1Further PureMathematics 2Either Year 1 AS LevelYear 2 Completethe A LevelYear 2 Completethe A LevelA Level(Candidates take theA Level components inthe same series) Paper 3 Paper 1Paper 2Paper 3Further PureMathematics 1Further PureMathematics 2 www.cambridgeinternational.org/alevelPaper 4Further Mechanics Further Probability& Statistics Year 2 full A Level10 Year 1 AS LevelPaper 4Further Mechanics Further Probability& Statistics OrRoute 3Not available forAS LevelPaper 3Paper 4Further Mechanics Further Probability& Statistics Back to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Syllabus overviewAssessment objectivesThe assessment objectives (AOs) are:AO1 Knowledge and understanding Show understanding of relevant mathematical concepts, terminology and notation Recall accurately and use appropriate mathematical manipulative techniquesAO2 Application and communication Recognise the appropriate mathematical procedure for a given situation Apply appropriate combinations of mathematical skills and techniques in solving problems Present relevant mathematical work, and communicate corresponding conclusions, in a clear and logical wayWeighting for assessment objectivesThe approximate weightings ( 5%) allocated to each of the assessment objectives (AOs) are summarised below.Assessment objectives as an approximate percentage of each componentAssessment objectiveWeighting in components %Paper 1Paper 2Paper 3Paper 4AO1 Knowledge and understanding45454545AO2 Application and communication55555555Assessment objectives as an approximate percentage of each qualificationAssessment objectiveWeighting in AS Level %Weighting in A Level %AO1 Knowledge and understanding4545AO2 Application and communication5555Back to contents pagewww.cambridgeinternational.org/alevel11

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022.3 Subject contentThe mathematical content for each component is detailed below. You can teach the topics in any order you findappropriate. However, please note the prior knowledge requirements below, and the information about calculatoruse found in 4 Details of the assessment.Notes and examples are included to clarify the syllabus content. Please note that these are examples only andexamination questions may differ from the examples given.Prior knowledgeIt is expected that learners will have studied the majority of the Cambridge International AS & A Level Mathematics(9709) subject content before studying Cambridge International AS & A Level Further Mathematics (9231).The prior knowledge required for each Further Mathematics component is shown in the following table.12Component in AS & A LevelFurther Mathematics (9231)Prior knowledge required fromAS & A Level Mathematics (9709)9231 Paper 1:Further Pure Mathematics 1 9709 Papers 1 and 39231 Paper 2:Further Pure Mathematics 2 9709 Papers 1 and 39231 Paper 3:Further Mechanics 9709 Papers 1, 3 and 49231 Paper 4:Further Probability & Statistics 9709 Papers 1, 3, 5 and 6www.cambridgeinternational.org/alevelBack to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Subject content1 Further Pure Mathematics 1 (for Paper 1)1.1Roots of polynomial equationsCandidates should be able to:Notes and examples recall and use the relations between the rootsand coefficients of polynomial equationse.g. to evaluate symmetric functions of the roots orto solve problems involving unknown coefficients inequations; restricted to equations of degree 2, 3 or 4only. use a substitution to obtain an equation whoseroots are related in a simple way to those of theoriginal equation.Substitutions will not be given for the easiest cases,e.g. where the new roots are reciprocals or squares ora simple linear function of the old roots.1.2 Rational functions and graphsCandidates should be able to:Notes and examples sketch graphs of simple rational functions,including the determination of obliqueasymptotes, in cases where the degree of thenumerator and the denominator are at most 2Including determination of the set of values taken bythe function, e.g. by the use of a discriminant. understand and use relationships between the1graphs of y f(x), y2 f(x), y , y f xh f xhand y f x h .Including use of such sketch graphs in the course ofsolving equations or inequalities.Detailed plotting of curves will not be required,but sketches will generally be expected to showsignificant features, such as turning points,asymptotes and intersections with the axes.1.3 Summation of seriesCandidates should be able to:Notes and examples use the standard results for / r , / r 2 , / r3 to findrelated sums use the method of differences to obtain the sumof a finite seriesUse of partial fractions to express a general term in asuitable form may be required. recognise, by direct consideration of a sum to nterms, when a series is convergent, and find thesum to infinity in such cases.Back to contents pagewww.cambridgeinternational.org/alevel13

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Subject content1Further Pure Mathematics 11.4 MatricesCandidates should be able to:Notes and examples carry out operations of matrix addition,subtraction and multiplication, and recognise theterms zero matrix and identity (or unit) matrixIncluding non-square matrices. Matrices will have atmost 3 rows and columns. recall the meaning of the terms ‘singular’ and‘non-singular’ as applied to square matricesand, for 2 # 2 and 3 # 3 matrices, evaluatedeterminants and find inverses of non-singularmatricesThe notations det M for the determinant of a matrixM, and I for the identity matrix, will be used. understand and use the result, for non-singularmatrices, (AB)–1 B–1A–1Extension to the product of more than two matricesmay be required. understand the use of 2 # 2 matrices torepresent certain geometric transformations inthe x-y plane, in particularUnderstanding of the terms ‘rotation’, ‘reflection’,‘enlargement’, ‘stretch’ and ‘shear’ for 2Dtransformations will be required.–– understand the relationship between thetransformations represented by A and A–1Other 2D transformations may be included, but noparticular knowledge of them is expected.–– recognise that the matrix product ABrepresents the transformation that resultsfrom the transformation represented by Bfollowed by the transformation representedby A–– recall how the area scale factor of atransformation is related to the determinantof the corresponding matrix–– find the matrix that represents agiven transformation or sequence oftransformations understand the meaning of ‘invariant’ asapplied to points and lines in the context oftransformations represented by matrices, andsolve simple problems involving invariant pointsand invariant lines.e.g. to locate the invariant points of the6 5o , or to findtransformation represented by e2 34 -1o,the invariant lines through the origin for e2 1or to show that any line with gradient 1 is invariant2 0o.for e1 114www.cambridgeinternational.org/alevelBack to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Subject content1Further Pure Mathematics 11.5 Polar coordinatesCandidates should be able to:Notes and examples understand the relations between Cartesianand polar coordinates, and convert equationsof curves from Cartesian to polar form and viceversaThe convention r H 0 will be used. sketch simple polar curves, for 0 G i 2ror r 1 i G r or a subset of either of theseintervalsDetailed plotting of curves will not be required,but sketches will generally be expected to showsignificant features, such as symmetry, coordinatesof intersections with the initial line, the form of thecurve at the pole and least/greatest values of r. recall the formula12y r2di for the area of asector, and use this formula in simple cases.1.6 VectorsCandidates should be able to:Notes and examples use the equation of a plane in any of the formsax by cz d or r.n p or r a λb μcand convert equations of planes from one formto another as necessary in solving problems recall that the vector product a # b of twovectors can be expressed either as a b sin int ,where nt is a unit vector, or in component form as a2 b3 a3 b2j i a3 b1 a1 b3j j a1 b2 a2 b1j k use equations of lines and planes, together withscalar and vector products where appropriate, tosolve problems concerning distances, angles andintersections, including–– determining whether a line lies in a plane, isparallel to a plane or intersects a plane, andfinding the point of intersection of a line anda plane when it exists–– finding the foot of the perpendicular from apoint to a plane–– finding the angle between a line and a plane,and the angle between two planes–– finding an equation for the line ofintersection of two planes–– calculating the shortest distance betweentwo skew lines–– finding an equation for the commonperpendicular to two skew lines.Back to contents pagewww.cambridgeinternational.org/alevel15

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Subject content1Further Pure Mathematics 11.7 Proof by inductionCandidates should be able to: use the method of mathematical induction toestablish a given resultNotes and examplesn12e.g. / r 4 n 2 n 1h ,4r 11un 2 1 3 n - 1i for the sequence given byun 1 3un 1 and u1 1,n4 13 # 2n 2 1 2nfp fp,6 13 # 2 n 1 6 3 2 n 13 2n 2 # 5 n 3 is divisible by 8. recognise situations where conjecture based ona limited trial followed by inductive proof is auseful strategy, and carry this out in simple cases.16www.cambridgeinternational.org/alevele.g. find the nth derivative of x ex,nfind / r # r! .r 1Back to contents page

Cambridge International AS & A Level Further Mathematics 9231 syllabus for 2020, 2021 and 2022. Subject content2 Further Pure Mathematics 2 (for Paper 2)Knowledge of Paper 1: Further Pure Mathematics 1 subject content from this syllabus is assumed for thiscomponent.2.1 Hyberbolic functionsCandidates should be able to:Notes and examples understand the definitions of the hyperbolicfunctions sinh x, cosh x, tanh x, sech x, cosech x,coth x in terms of the exponential function sketch the graphs of hyperbolic functions prove and use identities involving hyperbolicfunctionse.g. cosh2 x – sinh2 x 1,

Cambridge International AS Level Further Mathematics makes up the first half of the Cambridge International A Level course in further mathematics and provides a foundation for the study of further mathematics at Cambridge International A Level. Depending on local university entrance requirements, students may be able to use

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