# SPECIFICATION. Mathematics General Certificate of Education

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1 Version 1.0: 0913 General Certificate of Education Mathematics Material accompanying this Specification Specimen and Past Papers and Mark Schemes Reports on the Examination Teachers Guide SPECIFICATION

3 Advanced Subsidiary and Advanced, Mathematics Contents Background Information 1 Advanced Subsidiary and Advanced Level Specifications 6 Specification at a Glance 7 3 Availability of Assessment Units and Entry Details 11 Scheme of Assessment 4 Introduction 15 5 Aims 16 6 Assessment Objectives 17 7 Scheme of Assessment Advanced Subsidiary in Mathematics 18 8 Scheme of Assessment Advanced GCE in Mathematics 0 9 Scheme of Assessment Advanced Subsidiary and Advanced GCE in Pure Mathematics 3 10 Scheme of Assessment Advanced Subsidiary and Advanced GCE in Further Mathematics 6 Subject Content 11 Summary of Subject Content 9 1 AS Module - Pure Core AS Module - Pure Core A Module - Pure Core

4 Mathematics - Advanced Subsidiary and Advanced, A Module - Pure Core AS Module - Further Pure A Module - Further Pure A Module - Further Pure A Module - Further Pure AS Module - Statistics A Module - Statistics 6 A Module - Statistics A Module - Statistics AS Module - Mechanics A Module - Mechanics 74 6 A Module - Mechanics A Module - Mechanics A Module - Mechanics AS Module - Decision A Module - Decision 86 Key Skills and Other Issues 31 Key Skills Teaching, Developing and Providing Opportunities for Generating Evidence 87 3 Spiritual, Moral, Ethical, Social, Cultural and Other Issues 93 4

5 Advanced Subsidiary and Advanced, Mathematics Centre-assessed Component 33 Nature of Centre-assessed Component Guidance on Setting Centre-assessed Component Assessment Criteria Supervision and Authentication Standardisation Administrative Procedures Moderation 10 Awarding and Reporting 40 Grading, Shelf-life and Re-sits 103 Appendices A Grade Descriptions 104 B Formulae for AS/A level Mathematics Specifications 106 C Mathematical Notation 108 D Record Forms 113 E Overlaps with other Qualifications 114 F Relationship to other AQA GCE Mathematics and Statistics Specifications 115 5

7 Advanced Subsidiary and Advanced, Mathematics Specification at a Glance.1 General All assessment units are weighted at 16.7% of an A Level (33.3% of an AS). Three units are required for an AS subject award, and six for an A Level subject award. Each unit has a corresponding teaching module. The subject content of the modules is specified in Section 11 and following sections of this specification. The unit Statistics1 is available with coursework. This unit has an equivalent unit without coursework. The same teaching module is assessed, whether the assessment unit with or without coursework is chosen. So, Module Statistics 1 (Section 0) can be assessed by either unit MS1A or unit MS1B. For the unit with coursework, the coursework contributes 5% towards the marks for the unit, and the written paper 75% of the marks. Pure Core, Further Pure, Mechanics and Decision Mathematics units do not have coursework. The papers for units without coursework are 1 hour 30 minutes in duration and are worth 75 marks. The paper for MS1A (with coursework) is 1 hour 15 minutes in duration and is worth 60 marks. For units in which calculators are allowed (ie all except MPC1) the rules ( regarding what is permitted for GCE Maths and GCE Statistics are the same as for any other GCE examination. Most models of scientific or graphical calculator are allowed. However, calculators that feature a 'Computer Algebra System' (CAS) are not allowed. It is usually clear from the manufacturer's specifications whether a model has this feature.. List of units for AS/A Level Mathematics The following units can be used towards subject awards in AS Mathematics and A Level Mathematics. Allowed combinations of these units are detailed in the sections.3 and.4. Pure Core 1 MPC1 AS Pure Core MPC AS Pure Core 3 MPC3 A Pure Core 4 MPC4 A Statistics 1A MS1A AS with coursework Statistics 1B MS1B AS without coursework Statistics B MSB A Mechanics 1B MM1B AS Mechanics B Decision 1 Decision MMB MD01 MD0 A AS A 7

8 Mathematics - Advanced Subsidiary and Advanced, AS Mathematics Comprises 3 AS units. Two units are compulsory. MPC1* + MPC + MS1A or MS1B or MM1B or MD01.4 A Level Mathematics Comprises 6 units, of which 3 or 4 are AS units. Four units are compulsory. MPC1* + MPC + MS1A or MS1B or MM1B or MD01 together with MPC3 + MPC4 + MS1A or MS1B or MM1B or MD01 or MSB or MMB or MD0 Notes.5 List of units for AS/A Level Pure Mathematics * calculator not allowed unit includes coursework assessment Many combinations of AS and A optional Applied units are permitted for A Level Mathematics. However, the two units chosen must assess different teaching modules. For example, units MS1B and MM1B assess different teaching modules and this is an allowed combination. However, units MS1A and MS1B both assess module Statistics 1, and therefore MS1A and MS1B is not an allowed combination. Also a second Applied unit (MSB, MMB and MD0) can only be chosen in combination with a first Applied unit in the same application. For example, MSB can be chosen with MS1A (or MS1B), but not with MM1B or MD01. The following units can be used towards subject awards in AS Pure Mathematics and A Level Pure Mathematics. Allowed combinations of these units are detailed in the sections.6 and.7. Pure Core 1 MPC1 AS Pure Core MPC AS Pure Core 3 MPC3 A Pure Core 4 MPC4 A Further Pure MFP1 AS Further Pure MFP A Further Pure MFP3 A Further Pure MFP4 A 8

9 Advanced Subsidiary and Advanced, Mathematics.6 AS Pure Mathematics Comprises 3 compulsory AS units. MPC1* + MPC + MFP1.7 A Level Pure Mathematics Comprises 3 AS units and 3 A units. Five are compulsory. MPC1* + MPC + MFP1 together with MPC3 + MPC4 + MFP or MFP3 or MFP4 Notes.8 AS and A Level Further Mathematics * calculator not allowed The units in AS/A Level Pure Mathematics are common with those for AS/A Level Mathematics and AS/A Level Further Mathematics. Therefore there are restrictions on combinations of subject awards that candidates are allowed to enter. Details are given in section 3.4. Many combinations of units are allowed for AS and A Level Further Mathematics. Four Further Pure units are available. (Pure Core Units cannot be used towards AS/A Level Further Mathematics.) Any of the Applied units listed for AS/A Level Mathematics may be used towards AS/A Level Further Mathematics and there are additional Statistics and Mechanics units available only for Further Mathematics. Some units which are allowed to count towards AS/A Level Further Mathematics are common with those for AS/A Level Mathematics and AS/A Level Pure Mathematics. Therefore there are restrictions on combinations of subject awards that candidates are allowed to enter. Details are given in section 3.4. The subject award AS Further Mathematics requires three units, one of which is chosen from MFP1, MFP, MFP3 and MFP4, and two more units chosen from the list below. All three units can be at AS standard: for example, MFP1, MM1B and MS1A could be chosen. All three units can be in Pure Mathematics: for example, MFP1, MFP and MFP4 could be chosen. The subject award A Level Further Mathematics requires six units, two of which are chosen from MFP1, MFP, MFP3 and MFP4, and four more units chosen from the list below. At least three of the six units for A Level Further Mathematics must be at A standard, and at least two must be in Pure Mathematics. 9

10 Mathematics - Advanced Subsidiary and Advanced, List of units for AS/A Level Further Mathematics Notes The following units can be used towards subject awards in AS Further Mathematics and A Level Further Mathematics. Further Pure 1 MFP1 AS Further Pure MFP A Further Pure 3 MFP3 A Further Pure 4 MFP4 A Statistics 1A Statistics 1B Statistics B Statistics 3 Statistics 4 Mechanics 1B Mechanics B Mechanics 3 Mechanics 4 Mechanics 5 Decision 1 Decision MS1A MS1B MSB MS03 MS04 MM1B MMB MM03 MM04 MM05 MD01 MD0 AS with coursework AS without coursework Only one unit from MS1A and MS1B can be counted towards a subject award in AS or A Level Further Mathematics. MFP, MFP3 and MFP4 are independent of each other, so they can be taken in any order. MS03 and MS04 are independent of each other, so they can be taken in any order. MM03, MM04 and MM05 are independent of each other, so they can be taken in any order. A A A AS A A A A AS A 10

11 Advanced Subsidiary and Advanced, Mathematics 3 Availability of Assessment Units and Entry Details 3.1 Availability of Assessment Units Examinations based on this specification are available as follows: Availability of Units Availability of Qualification AS A AS A Level June series All All All All 3. Sequencing of Units There are no restrictions on the order in which assessment units are taken. However, later teaching modules assume some or all of the knowledge, understanding and skills of earlier modules. For example, some material in MPC depends on material in MPC1 and some material in MPC4 depends on material in MPC3. Some of the additional units available for Further Mathematics are exceptions to this general rule (see Section.9). Details of the prerequisites for each module are given in the introductions to the individual modules. It is anticipated that teachers will use this and other information to decide on a teaching sequence. 11

12 Mathematics - Advanced Subsidiary and Advanced, Entry Codes Normal entry requirements apply, but the following information should be noted. The following unit entry codes should be used. Module assessed by Unit Standard of assessment With or without coursework Unit Entry Code Pure Core 1 AS without MPC1 Pure Core AS without MPC Pure Core 3 A without MPC3 Pure Core 4 A without MPC4 Further Pure 1 AS without MFP1 Further Pure A without MFP Further Pure 3 A without MFP3 Further Pure 4 A without MFP4 Statistics 1 AS with MS1A Statistics 1 AS without MS1B Statistics A without MSB Statistics 3 A without MS03 Statistics 4 A without MS04 Mechanics 1 AS without MM1B Mechanics A without MMB Mechanics 3 A without MM03 Mechanics 4 A without MM04 Mechanics 5 A without MM05 Decision 1 AS without MD01 Decision A without MD0 The Subject Code for entry to the Mathematics AS only award is The Subject Code for entry to the Pure Mathematics AS only award is The Subject Code for entry to the Further Mathematics AS only award is The Subject Code for entry to the Mathematics Advanced Level award is The Subject Code for entry to the Pure Mathematics Advanced Level award is The Subject Code for entry to the Further Mathematics Advanced Level award is

13 Advanced Subsidiary and Advanced, Mathematics 3.4 Rules for Combinations of Awards and Unit Entries Combinations of subject awards for this specification are subject to the following restrictions. Awards in the following pairs of subjects titles will not be allowed: AS Mathematics and AS Pure Mathematics; AS Mathematics and A Level Pure Mathematics; A Level Mathematics and AS Pure Mathematics; A Level Mathematics and A Level Pure Mathematics. AS Pure Mathematics and AS Further Mathematics; AS Pure Mathematics and A Level Further Mathematics; A Level Pure Mathematics and AS Further Mathematics; A Level Pure Mathematics and A Level Further Mathematics. Units that contribute to an award in A Level Mathematics may not also be used for an award in Further Mathematics. Candidates who are awarded certificates in both A Level Mathematics and A Level Further Mathematics must use unit results from 1 different teaching modules. Candidates who are awarded certificates in both A Level Mathematics and AS Further Mathematics must use unit results from 9 different teaching modules. Candidates who are awarded certificates in both AS Mathematics and AS Further Mathematics must use unit results from 6 different teaching modules. Concurrent entries for: MS1A and will not be accepted. MS1B 13

15 Advanced Subsidiary and Advanced, Mathematics Scheme of Assessment 4 Introduction AQA offers one specification in GCE Mathematics, and a separate specification in GCE Statistics. This specification is a development from both the AQA GCE Mathematics Specification A (6300) (which was derived from the School Mathematics Project (SMP) syllabus) and the AQA GCE Mathematics and Statistics Specification B (630). It includes optional assessed coursework in one Statistics unit, but coursework is not a compulsory feature. This specification is designed to encourage candidates to study mathematics post-16. It enables a variety of teaching and learning styles, and provides opportunities for students to develop and be assessed in five of the six Key Skills. This GCE Mathematics specification complies with: the Common Criteria; the Subject Criteria for Mathematics; the GCSE, GCE, Principal Learning and Project Code of Practice, April 013; the GCE Advanced Subsidiary and Advanced Level Qualification-Specific Criteria. The qualifications based on this specification are a recognised part of the National Qualifications Framework. As such, AS and A Level provide progression from Key Stage 4, through post-16 studies and form the basis of entry to higher education or employment. Prior Level of Attainment Mathematics is, inherently, a sequential subject. There is a progression of material through all levels at which the subject is studied. The Subject Criteria for Mathematics and therefore this specification build on the knowledge, understanding and skills established at GCSE Mathematics. There is no specific prior requirement, for example, in terms of tier of GCSE entry or grade achieved. Teachers are best able to judge what is appropriate for different candidates and what additional support, if any, is required. 15

16 Mathematics - Advanced Subsidiary and Advanced, Aims The aims set out below describe the educational purposes of following a course in Mathematics/Further Mathematics/Pure Mathematics and are consistent with the Subject Criteria. They apply to both AS and Advanced specifications. Most of these aims are reflected in the assessment objectives; others are not because they cannot be readily translated into measurable objectives. The specification aims to encourage candidates to: a. develop their understanding of mathematics and mathematical processes in a way that promotes confidence and fosters enjoyment; b. develop abilities to reason logically and to recognise incorrect reasoning, to generalise and to construct mathematical proofs; c. extend their range of mathematical skills and techniques and use them in more difficult unstructured problems; d. develop an understanding of coherence and progression in mathematics and of how different areas of mathematics can be connected; e. recognise how a situation may be represented mathematically and understand the relationship between real world problems and standard and other mathematical models and how these can be refined and improved; f. use mathematics as an effective means of communication; g. read and comprehend mathematical arguments and articles concerning applications of mathematics; h. acquire the skills needed to use technology such as calculators and computers effectively, to recognise when such use may be inappropriate and to be aware of limitations; i. develop an awareness of the relevance of mathematics to other fields of study, to the world of work and to society in general; j. take increasing responsibility for their own learning and the evaluation of their own mathematical development. 16

17 Advanced Subsidiary and Advanced, Mathematics 6 Assessment Objectives AO1 AO AO3 AO4 AO5 The assessment objectives are common to both AS and A Level. The schemes of assessment will assess candidates ability to: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of contexts; construct rigorous mathematical arguments and proofs through use of precise statements, logical deduction and inference and by the manipulation of mathematical expressions, including the construction of extended arguments for handling substantial problems presented in unstructured form; recall, select and use their knowledge of standard mathematical models to represent situations in the real world; recognise and understand given representations involving standard models; present and interpret results from such models in terms of the original situation, including discussion of the assumptions made and refinement of such models; comprehend translations of common realistic contexts into mathematics; use the results of calculations to make predictions, or comment on the context; and, where appropriate, read critically and comprehend longer mathematical arguments or examples of applications; use contemporary calculator technology and other permitted resources (such as formulae booklets or statistical tables) accurately and efficiently; understand when not to use such technology, and its limitations; give answers to appropriate accuracy. The use of clear, precise and appropriate mathematical language is expected as an inherent part of the assessment of AO. 17

18 Mathematics - Advanced Subsidiary and Advanced, Scheme of Assessment Mathematics Advanced Subsidiary (AS) The Scheme of Assessment has a modular structure. The Advanced Subsidiary (AS) award comprises two compulsory core units and one optional Applied unit. All assessment is at AS standard. For the written papers, each candidate will require a copy of the AQA Booklet of formulae and statistical tables issued for this specification. 7.1 Compulsory Assessment Units Unit MPC /3% of the total AS marks Written Paper Core 1 All questions are compulsory. Calculators are not permitted. Unit MPC 33 1 /3% of the total AS marks Written Paper 1hour 30 minutes 75 marks 1 hour 30 minutes 75 marks Core All questions are compulsory. A graphics calculator may be used. 7. Optional Assessment Units Unit MS1A 33 1 /3% of the total AS marks Written Paper + Coursework 1 hour 15 minutes 60 marks Statistics 1A The written paper comprises 5% of the AS marks. All questions are compulsory. A graphics calculator may be used. The coursework comprises 8 1 / 3 % of the AS marks. One task is required. Unit MS1B 331/3% of the total AS marks Written Paper 1 hour 30 minutes 75 marks Statistics 1B All questions are compulsory. A graphics calculator may be used. Unit MM1B 33 1 /3% of the total AS marks Written Paper 1 hour 30 minutes 75 marks Mechanics 1B All questions are compulsory. A graphics calculator may be used. Unit MD /3% of the total AS marks Written Paper 1 hour 30 minutes 75 marks Decision 1 All questions are compulsory. A graphics calculator may be used. 18

19 Advanced Subsidiary and Advanced, Mathematics 7.3 Weighting of Assessment Objectives for AS The approximate relationship between the relative percentage weighting of the Assessment Objectives (AOs) and the overall Scheme of Assessment is shown in the following table: Assessment Objectives Unit Weightings (range %) MPC1 MPC Applied unit Overall Weighting of AOs (range %) AO AO AO AO AO Overall Weighting of Units (%) 33 1 / / / (maximum) Candidates marks for each assessment unit are scaled to achieve the correct weightings. 7.4 Progression to Advanced GCE in Mathematics Unit results counted towards an AS award in Mathematics may also be counted towards an Advanced award in Mathematics. Candidates who have completed the units needed for the AS qualification and who have taken the additional units necessary are eligible for an Advanced award. 19

20 Mathematics - Advanced Subsidiary and Advanced, Scheme of Assessment Mathematics Advanced Level (AS + A) The Scheme of Assessment has a modular structure. The A Level award comprises four compulsory Core units, one optional Applied unit from the AS scheme of assessment, and one optional Applied unit either from the AS scheme of assessment or from the A scheme of assessment. See section.4 on page 8 for permitted combinations. For the written papers, each candidate will require a copy of the AQA Booklet of formulae and statistical tables issued for this specification. 8.1 AS Compulsory Assessment Units Unit MPC1 16 /3% of the total A level marks Written Paper Core 1 All questions are compulsory. Calculators are not permitted. 1hour 30 minutes 75 marks Unit MPC 16 /3% of the total A level marks 8. AS Optional Assessment Units Unit MS1A 16 /3% of the total A level marks Written Paper 1hour 30 minutes 75 marks Core All questions are compulsory. A graphics calculator may be used. Written Paper + Coursework 1 hour 15 minutes 60 marks Statistics 1A The written paper comprises 1½% of the A Level marks. All questions are compulsory. A graphics calculator may be used. The coursework comprises 4 1 / 6 % of the A Level marks. One task is required. Unit MS1B 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Statistics 1B All questions are compulsory. A graphics calculator may be used. Unit MM1B 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Mechanics 1B All questions are compulsory. A graphics calculator may be used. 0

21 Advanced Subsidiary and Advanced, Mathematics Unit MD01 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Decision 1 All questions are compulsory. A graphics calculator may be used. 8.3 A Compulsory Assessment Units Unit MPC3 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Core 3 All questions are compulsory. A graphics calculator may be used. Unit MPC4 16 /3% of the total A level marks 8.4 A Optional Assessment Units Unit MSB 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Core 4 All questions are compulsory. A graphics calculator may be used. Written Paper 1 hour 30 minutes 75 marks Statistics All questions are compulsory. A graphics calculator may be used. Unit MMB 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Mechanics All questions are compulsory. A graphics calculator may be used. Unit MD0 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Decision All questions are compulsory. A graphics calculator may be used. 8.5 Synoptic Assessment The GCE Advanced Subsidiary and Advanced Level Qualificationspecific Criteria state that A Level specifications must include synoptic assessment (representing at least 0% of the total A Level marks). Synoptic assessment in mathematics addresses candidates understanding of the connections between different elements of the subject. It involves the explicit drawing together of knowledge, understanding and skills learned in different parts of the A level course, focusing on the use and application of methods developed at earlier stages of the course to the solution of problems. Making and understanding connections in this way is intrinsic to learning mathematics. The requirement for 0% synoptic assessment is met by synoptic assessment in: Core, Core 3, Core 4. There is no restriction on when synoptic units may be taken. 1

22 Mathematics - Advanced Subsidiary and Advanced, Weighting of Assessment Objectives for A Level The approximate relationship between the relative percentage weighting of the Assessment Objectives (AOs) and the overall Scheme of Assessment is shown in the following table. A Level Assessment Units (AS + A) Overall Unit Weightings (range %) Weighting Assessment Objectives of AOs (range %) MPC1 MPC Applied unit MPC3 MPC4 Applied unit AO AO AO AO AO ½ ½ 1 1 1½ ½ 6 11 Overall Weighting of Units (%) 16 / 3 16 / 3 16 / 3 16 / 3 16 / 3 16 / (maximum) Candidates marks for each assessment unit are scaled to achieve the correct weightings.

23 Advanced Subsidiary and Advanced, Mathematics 9 Scheme of Assessment Pure Mathematics Advanced Subsidiary (AS) Advanced Level (AS and A) The Pure Mathematics Advanced Subsidiary (AS) award comprises three compulsory assessment units. The Pure Mathematics A Level (AS and A) award comprises five compulsory assessment units, and one optional unit chosen from three Further Pure assessment units. For the written papers, each candidate will require a copy of the AQA Booklet of formulae and statistical tables issued for this specification. 9.1 AS Assessment Units Unit MPC /3% of the total AS marks Written Paper Core 1 All questions are compulsory. Calculators are not permitted. Unit MPC 33 1 /3% of the total AS marks Written Paper 1hour 30 minutes 75 marks 1hour 30 minutes 75 marks Core All questions are compulsory. A graphics calculator may be used. Unit MFP /3% of the total AS marks Written Paper 1hour 30 minutes 75 marks Further Pure 1 All questions are compulsory. A graphics calculator may be used. 9. A Compulsory Assessment Units Unit MPC3 16 /3% of the total A Level marks Written Paper 1hour 30 minutes 75 marks Core 3 All questions are compulsory. A graphics calculator may be used. Unit MPC4 16 /3% of the total A Level marks Written Paper 1hour 30 minutes 75 marks Core 4 All questions are compulsory. A graphics calculator may be used. 3

24 Mathematics - Advanced Subsidiary and Advanced, A Optional Assessment Units Unit MFP 16 /3% of the total A Level marks Written Paper 1hour 30 minutes 75 marks Further Pure All questions are compulsory. A graphics calculator may be used. Unit MFP3 16 /3% of the total A Level marks Written Paper 1hour 30 minutes 75 marks Further Pure 3 All questions are compulsory. A graphics calculator may be used. Unit MFP4 16 /3% of the total A Level marks Written Paper 1hour 30 minutes 75 marks Further Pure 4 All questions are compulsory. A graphics calculator may be used. 9.4 Synoptic Assessment The GCE Advanced Subsidiary and Advanced Level Qualificationspecific Criteria state that A Level specifications must include synoptic assessment (representing at least 0% of the total A Level marks). Synoptic assessment in mathematics addresses candidates understanding of the connections between different elements of the subject. It involves the explicit drawing together of knowledge, understanding and skills learned in different parts of the A level course, focusing on the use and application of methods developed at earlier stages of the course to the solution of problems. Making and understanding connections in this way is intrinsic to learning mathematics. The requirement for 0% synoptic assessment is met by synoptic assessment in: Core, Core 3, Core 4. There is no restriction on when synoptic units may be taken. 9.5 Weighting of Assessment Objectives for AS The approximate relationship between the relative percentage weighting of the Assessment Objectives (AOs) and the overall Scheme of Assessment is shown in the following table: Assessment Objectives Unit Weightings (range %) Overall Weighting of MPC1 MPC MFP1 AOs (range %) AO AO AO AO AO Overall Weighting of Units (%) 33 1 / / / (maximum) Candidates marks for each assessment unit are scaled to achieve the correct weightings. 4

25 Advanced Subsidiary and Advanced, Mathematics 9.6 Weighting of Assessment Objectives for A Level The approximate relationship between the relative percentage weighting of the Assessment Objectives (AOs) and the overall Scheme of Assessment is shown in the following table: Unit Weightings (range %) Overall Weighting of Assessment Objectives MPC1 All other units AOs (range %) AO AO AO AO AO Overall Weighting of Units (%) 16 / 3 16 / (maximum) Candidates marks for each assessment unit are scaled to achieve the correct weightings. 5

26 Mathematics - Advanced Subsidiary and Advanced, Scheme of Assessment Further Mathematics Advanced Subsidiary (AS) Advanced Level (AS + A) Candidates for AS and/or A Level Further Mathematics are expected to have already obtained (or to be obtaining concurrently) an AS and/or A Level award in Mathematics. The Advanced Subsidiary (AS) award comprises three units chosen from the full suite of units in this specification, except that the Core units cannot be included. One unit must be chosen from MFP1, MFP, MFP3 and MFP4. All three units can be at AS standard; for example, MFP1, MM1B and MS1A could be chosen. All three units can be in Pure Mathematics; for example, MFP1, MFP and MFP4 could be chosen. The Advanced (A Level) award comprises six units chosen from the full suite of units in this specification, except that the Core units cannot be included. The six units must include at least two units from MFP1, MFP, MFP3 and MFP4. All four of these units could be chosen. At least three of the six units counted towards A Level Further Mathematics must be at A standard. Details of the units which can be used towards AS/A Level Mathematics or AS/A Level Further Mathematics are given in section 8. Details of the additional units available for Further Mathematics, but not Mathematics, are given in sections 10.1 and 10.. Units that contribute to an award in A Level Mathematics may not also be used for an award in Further Mathematics. Candidates who are awarded certificates in both A Level Mathematics and A Level Further Mathematics must use unit results from 1 different teaching modules. Candidates who are awarded certificates in both A Level Mathematics and AS Further Mathematics must use unit results from 9 different teaching modules. Candidates who are awarded certificates in both AS Mathematics and AS Further Mathematics must use unit results from 6 different teaching modules. For the written papers, each candidate will require a copy of the AQA Booklet of formulae and statistical tables issued for this specification. 6

27 Advanced Subsidiary and Advanced, Mathematics 10.1 Further Mathematics Assessment Units (Pure) 10. Further Mathematics Assessment Units (Applied) Unit MFP /3% of the total AS marks 16 /3% of the total A level marks Written Paper 1hour 30 minutes 75 marks Further Pure 1 All questions are compulsory. A graphics calculator may be used. Unit MFP Written Paper 1hour 30 minutes 33 1 /3% of the total 75 marks AS marks 16 /3% of the total A level marks Further Pure All questions are compulsory. A graphics calculator may be used. Unit MFP3 Written Paper 1hour 30 minutes 33 1 /3% of the total 75 marks AS marks 16 /3% of the total A level marks Further Pure 3 All questions are compulsory. A graphics calculator may be used. Unit MFP4 Written Paper 1 hour 30 minutes 33 1 /3% of the total 75 marks AS marks 16 /3% of the total A level marks Further Pure 4 All questions are compulsory. A graphics calculator may be used. Unit MS /3% of the total AS marks 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Statistics 3 All questions are compulsory. A graphics calculator may be used. Unit MS04 Written Paper 1 hour 30 minutes 33 1 /3% of the total 75 marks AS marks 16 /3% of the total A level marks Statistics 4 All questions are compulsory. A graphics calculator may be used. 7

28 Mathematics - Advanced Subsidiary and Advanced, Weighting of Assessment Objectives Unit MM /3% of the total AS marks 16 /3% of the total A level marks Written Paper 1 hour 30 minutes 75 marks Mechanics 3 All questions are compulsory. A graphics calculator may be used. Unit MM04 Written Paper 1 hour 30 minutes 33 1 /3% of the total 75 marks AS marks 16 /3% of the total A level marks Mechanics 4 All questions are compulsory. A graphics calculator may be used. Unit MM05 Written Paper 1 hour 30 minutes 33 1 /3% of the total 75 marks AS marks 16 /3% of the total A level marks Mechanics 5 All questions are compulsory. A graphics calculator may be used. The approximate relationship between the relative percentage weighting of the Assessment Objectives (AOs) and the overall Scheme of Assessment is shown in the following tables: Assessment Objectives Further Mathematics AS Unit Weightings (range %) Further Pure Units Applied Units Overall Weighting of AOs (range %) AO AO AO AO AO Overall Weighting of Units (%) 33 1 / / (maximum) Assessment Objectives Further Mathematics Advanced Unit Weightings (range %) Further Pure Units Applied Units Overall Weighting of AOs (range %) AO AO AO AO AO5 1 1½½ 714 Overall Weighting of Units (%) 16 / 3 16 / (maximum) 8

29 Advanced Subsidiary and Advanced, Mathematics Subject Content 111 Summary of Subject Content 11.1 Pure Core Modules AS MODULE Pure Core 1 Algebra Coordinate Geometry Differentiation Integration AS MODULE Pure Core Algebra and Functions Sequences and Series Trigonometry Exponentials and logarithms Differentiation Integration A MODULE Pure Core 3 Algebra and Functions Trigonometry Exponentials and Logarithms Differentiation Integration Numerical Methods A MODULE Pure Core 4 Algebra and Functions Coordinate Geometry in the (x, y) plane Sequences and Series Trigonometry Exponentials and Logarithms Differentiation and Integration Vectors 11. Further Pure Modules AS MODULE Further Pure 1 Algebra and Graphs Complex Numbers Roots and Coefficients of a quadratic equation Series Calculus Numerical Methods Trigonometry Matrices and Transformations 9

30 Mathematics - Advanced Subsidiary and Advanced, 014 A MODULE Further Pure Roots of Polynomials Complex Numbers De Moivre s Theorem Proof by Induction Finite Series The Calculus of Inverse Trigonometrical Functions Hyperbolic Functions Arc Length and Area of surface of revolution about the x-axis A MODULE - Further Pure 3 Series and Limits Polar Coordinates Differential Equations Differential Equations First Order Differential Equations Second Order A MODULE - Further Pure Statistics AS MODULE - Statistics 1 Vectors and Three-Dimensional Coordinate Geometry Matrix Algebra Solution of Linear Equations Determinants Linear Independence Numerical Measures Probability Binomial Distribution Normal Distribution Estimation Correlation and Regression A MODULE - Statistics Discrete Random Variables Poisson Distribution Continuous Random Variables Estimation Hypothesis Testing Chi-Square ( ) Contingency Table Tests A MODULE - Statistics 3 Further Probability Linear Combinations of Random Variables Distributional Approximations Estimation Hypothesis Testing 30

31 Advanced Subsidiary and Advanced, Mathematics A MODULE - Statistics 4 Geometric and Exponenential Distributions Estimators Estimation Hypothesis Testing Chi-Squared ( ) Goodness of Fit Tests 11.4 Mechanics AS MODULE - Mechanics 1 Mathematical Modelling Kinematics in One and Two Dimensions Statics and Forces Momentum Newton s Laws of Motion Connected Particles Projectiles A MODULE - Mechanics Mathematical Modelling Moments and Centres of Mass Kinematics Newton s Laws of Motion Application of Differential Equations Uniform Circular Motion Work and Energy Vertical Circular Motion A MODULE - Mechanics 3 Relative Motion Dimensional Analysis Collisions in one dimension Collisions in two dimensions Further Projectiles Projectiles on Inclined Planes A MODULE - Mechanics 4 Moments Frameworks Vector Product and Moments Centres of mass by Integration for Uniform Bodies Moments of Inertia Motion of a Rigid Body about a Fixed Axis A MODULE - Mechanics 5 Simple Harmonic Motion Forced and Damped Harmonic Motion Stability Variable Mass Problems Motion in a Plane using Polar Coordinates 31

32 Mathematics - Advanced Subsidiary and Advanced, Decision AS MODULE Decision 1 Simple Ideas of Algorithms Graphs and Networks Spanning Tree Problems Matchings Shortest Paths in Networks Route Inspection Problem Travelling Salesperson Problem Linear Programming Mathematical Modelling A MODULE - Decision Critical Path Analysis Allocation Dynamic Programming Network Flows Linear Programming Game Theory for Zero Sum Games Mathematical Modelling 3

33 Advanced Subsidiary and Advanced, Mathematics 1 AS Module Core 1 Candidates will be required to demonstrate: a. construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language; b. correct understanding and use of mathematical language and grammar in respect of terms such as equals, identically equals, therefore, because, implies, is implied by, necessary, sufficient and notation such as,, and. Candidates are not allowed to use a calculator in the assessment unit for this module. Candidates may use relevant formulae included in the formulae booklet without proof. Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions. Quadratic Equations b b 4ac ax bx c 0 has roots a ab and radius r, has equation Circles A circle, centre, x a y b r Differentiation function derivative n ax n 1 anx x gx fx gx f Integration function integral n a n 1 ax x c n 1 f x gx f g Area Area under a curve b y d x y a x x c 0 n is a whole number n is a whole number 1.1 Algebra Use and manipulation of surds. To include simplification and rationalisation of the denominator of a fraction. Eg ; ;

36 Mathematics - Advanced Subsidiary and Advanced, Integration Indefinite integration as the reverse of differentiation Integration of polynomials. Evaluation of definite integrals. Interpretation of the definite integral as the area under a curve. Integration to determine the area of a region between a curve and the x-axis. To include regions wholly below the x-axis, i.e. knowledge that the integral will give a negative value. Questions involving regions partially above and below the x-axis will not be set. Questions may involve finding the area of a region bounded by a straight line and a curve, or by two curves. 36

37 Advanced Subsidiary and Advanced, Mathematics 13 AS Module Core Trigonometry Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the module Core 1. Candidates will be required to demonstrate: a. Construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language; b. correct understanding and use of mathematical language and grammar in respect of terms such as equals, identically equals, therefore, because, implies, is implied by, necessary, sufficient and notation such as,, and. Candidates may use relevant formulae included in the formulae booklet without proof. Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions. In the triangle ABC a b c sin A sin B sin C 1 area absin C arc length of a circle, l r area of a sector of a circle, sin tan cos sin cos 1 Laws of Logarithms log x log y log xy a a a x log a xlog a ylog a y k k log x log x a a A 1 r Differentiation Function derivative n n 1 ax nax, n is a rational number Integration Function integral n a 1 ax x n, n is a rational number, n 1 n 1 37

38 Mathematics - Advanced Subsidiary and Advanced, Algebra and Functions Laws of indices for all rational exponents. Knowledge of the effect of simple transformations on the graph of y f ( x) as represented by y af ( x), y f ( x) a, y f ( x a), y f ( ax). 13. Sequences and Series Sequences, including those given by a formula for the nth term. Sequences generated by a simple relation of the form x n1 f ( xn). Arithmetic series, including the formula for the sum of the first n natural numbers. The sum of a finite geometric series. The sum to infinity of a convergent ( 1 < r < 1) geometric series. The binomial expansion of (1 + x) n for positive integer n Trigonometry The sine and cosine rules. The area of a triangle in the form 1 absin C. Degree and radian measure. Arc length, area of a sector of a circle. Candidates are expected to use the terms reflection, translation and stretch in the x or y direction in their descriptions of these transformations. 3 Eg graphs of y sin x ycos x 30 ; y x ; y x ; Descriptions involving combinations of more than one transformation will not be tested. To include notation for sums of series. To include their use in finding of a limit L as n by putting L f L. Candidates should be familiar with the notation r <1 in this context. n To include the notations n! and. Use of Pascal's triangle or r formulae to expand a b n will be accepted. 1 Knowledge of the formulae l r, A r. 38

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