Math as, A2 Syllabus Gr12

6SHFL¿FDWLRQGCE Mathematics

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Mathematics (8371)

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (8372)

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Pure Mathematics (8373)

Pearson Edexcel Level 3 Advanced Subsidiary GCE in Further Mathematics (Additional) (8374)First examination 2014

Pearson Edexcel Level 3 Advanced GCE in Mathematics (9371)

Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9372)

Pearson Edexcel Level 3 Advanced GCE in Pure Mathematics (9373)

Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (Additional) (9374)First examination 2014

Issue 3

Pearson Edexcel Level 3 GCE in Mathematics © Pearson Education Limited 2013 Introduction 1

About this specification

Supporting you

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Edexcel GCE in Mathematics, Further Mathematics, Pure Mathematics and Further Mathematics (Additional) is GHVLJQHG�IRU�XVH�LQ�VFKRRO�DQG�FROOHJHV��,W�LV�SDUW�RI�D�VXLWH�RI�*&(�TXDOL¿FDWLRQV�RIIHUHG�E\�(GH[FHO�

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� pathways leading to full Advanced Subsidiary and Advanced GCE in Mathematics, Further Mathematics, Pure Mathematics and Further Mathematics (Additional)

� updated Decision Mathematics 1 and Decision Mathematics 2 units, for a more balanced approach to content

� XSGDWHG�)XUWKHU�3XUH�0DWKHPDWLFV��XQLW�IRU�WHDFKLQJ�LQ�WKH�¿UVW�\HDU�RI�VWXG\

� updated Further Pure Mathematics 2 and Further Pure Mathematics 3 units to allow a coherent curriculum in further mathematics

� no change to Core, Mechanics or Statistics units content

� substantial Professional development and training programme

� past papers, specimen papers, examiner reports and further support materials available

� 18 units tested fully by written examination

� YDULHW\�RI�HQGRUVHG�HOHFWURQLF�VXSSRUW�PDWHULDO��LQFOXGLQJ�([DP�:L]DUG��7RSLF�7XWRU�DQG�([DP�7XWRU

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Contents © Pearson Education Limited 2013 Pearson Edexcel Level 3 GCE in Mathematics2

Contents

A Specification at a glance 4

Summary of unit content 4

B Specification overview 7

Summary of awards 7

Summary of assessment requirements 10

Assessment objectives and weightings 12

Relationship of assessment objectives to units 13

Qualification summary 13

C Mathematics unit content 15

Structure of qualification 16

Unit C1 Core Mathematics 1 19




Unit FP1 Further Pure Mathematics 1 43



Unit M1 Mechanics 1 59





Unit S1 Statistics 1 79

Contents

Pearson Edexcel Level 3 GCE in Mathematics © Pearson Education Limited 2013 Contents 3




Unit D1 Decision Mathematics 1 97


D Assessment and additional information 107

Assessment information 107

Assessment objectives and weighting 109

Additional information 110

E Resources, support and training 115

Resources to support the specification 115

Edexcel’s own published resources 115

Edexcel publications 116

Edexcel support services 117

Training 118

F Appendices 119

Appendix A Grade descriptions 121

Appendix B Notation 125

Appendix C Codes 133

Section A © Pearson Education Limited 2013 Pearson Edexcel Level 3 GCE in Mathematics4

A Specification at a glance

Students study a variety of units, following pathways to their GHVLUHG�TXDOL¿FDWLRQ�

6WXGHQWV�PD\�VWXG\�XQLWV�OHDGLQJ�WR�WKH�IROORZLQJ�DZDUGV�

� Advanced GCE in Mathematics

� Advanced GCE in Further Mathematics

� Advanced GCE in Pure Mathematics

� Advanced GCE In Further Mathematics (Additional)

� Advanced Subsidiary GCE in Mathematics

� Advanced Subsidiary GCE in Further Mathematics

� Advanced Subsidiary GCE in Pure Mathematics

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Summary of unit content

Core Mathematics

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C1 Algebra and functions; coordinate geometry in the (x, y) plane; sequences and VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�

C2 Algebra and functions; coordinate geometry in the (x, y) plane; sequences and VHULHV��WULJRQRPHWU\��H[SRQHQWLDOV�DQG�ORJDULWKPV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�

C3 Algebra and functions; trigonometry; exponentials and logarithms; GLIIHUHQWLDWLRQ��QXPHULFDO�PHWKRGV�

C4 Algebra and functions; coordinate geometry in the (x, y) plane; sequences and VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ��YHFWRUV�

Further Pure Mathematics


FP1 Series; complex numbers; numerical solution of equations; coordinate V\VWHPV��PDWUL[�DOJHEUD��SURRI�

FP2 ,QHTXDOLWLHV��VHULHV��¿UVW�RUGHU�GLIIHUHQWLDO�HTXDWLRQV��VHFRQG�RUGHU�GLIIHUHQWLDO�HTXDWLRQV��IXUWKHU�FRPSOH[�QXPEHUV��0DFODXULQ�DQG�7D\ORU�VHULHV�

FP3 Further matrix algebra; vectors, hyperbolic functions; differentiation; LQWHJUDWLRQ��IXUWKHU�FRRUGLQDWH�V\VWHPV�

Specification at a glance A

Pearson Edexcel Level 3 GCE in Mathematics © Pearson Education Limited 2013 Section A 5

Mechanics


M1 0DWKHPDWLFDO�PRGHOV�LQ�PHFKDQLFV��YHFWRUV�LQ�PHFKDQLFV��NLQHPDWLFV�RI�D�particle moving in a straight line; dynamics of a particle moving in a straight OLQH�RU�SODQH��VWDWLFV�RI�D�SDUWLFOH��PRPHQWV�

M2 Kinematics of a particle moving in a straight line or plane; centres of mass; ZRUN�DQG�HQHUJ\��FROOLVLRQV��VWDWLFV�RI�ULJLG�ERGLHV�

M3 )XUWKHU�NLQHPDWLFV��HODVWLF�VWULQJV�DQG�VSULQJV��IXUWKHU�G\QDPLFV��PRWLRQ�LQ�D�FLUFOH��VWDWLFV�RI�ULJLG�ERGLHV�

M4 Relative motion; elastic collisions in two dimensions; further motion of SDUWLFOHV�LQ�RQH�GLPHQVLRQ��VWDELOLW\�

M5 Applications of vectors in mechanics; variable mass; moments of inertia of a ULJLG�ERG\��URWDWLRQ�RI�D�ULJLG�ERG\�DERXW�D�¿[HG�VPRRWK�D[LV�

Statistics


S1 Mathematical models in probability and statistics; representation and summary of data; probability; correlation and regression; discrete random YDULDEOHV��GLVFUHWH�GLVWULEXWLRQV��WKH�1RUPDO�GLVWULEXWLRQ�

S2 The Binomial and Poisson distributions; continuous random variables; FRQWLQXRXV�GLVWULEXWLRQV��VDPSOHV��K\SRWKHVLV�WHVWV�

S3 &RPELQDWLRQV�RI�UDQGRP�YDULDEOHV��VDPSOLQJ��HVWLPDWLRQ��FRQ¿GHQFH�LQWHUYDOV�DQG�WHVWV��JRRGQHVV�RI�¿W�DQG�FRQWLQJHQF\�WDEOHV��UHJUHVVLRQ�DQG�FRUUHODWLRQ�

S4 Quality of tests and estimators; one-sample procedures; two-sample SURFHGXUHV�

Decision Mathematics


D1 Algorithms; algorithms on graphs; the route inspection problem; critical path DQDO\VLV��OLQHDU�SURJUDPPLQJ��PDWFKLQJV�

D2 Transportation problems; allocation (assignment) problems; the travelling salesman; game theory; further linear programming, dynamic programming; ÀRZV�LQ�QHWZRUNV�

A Specification at a glance

Section A © Pearson Education Limited 2013 Pearson Edexcel Level 3 GCE in Mathematics6

Pearson Edexcel Level 3 GCE in Mathematics © Pearson Education Limited 2013 Section B 7

B Specification overview

Summary of awards

Advanced Subsidiary

Advanced subsidiary awards comprise three teaching units per DZDUG�

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2210 8371 GCE AS Mathematics C1 and C2 M1, S1 or D1

2230 8372 GCE AS Further Mathematics

FP1 Any *

2230 8373 GCE AS Pure Mathematics C1, C2, C3

2230 8374 GCE AS Further Mathematics (Additional)

Any which have not been used for a previous TXDOL¿FDWLRQ

*For GCE AS Further Mathematics, excluded units are C1, C2, C3, C4

**See Appendix C for description of this code and all other codes relevant to this TXDOL¿FDWLRQ

Advanced Subsidiary combinations

Combinations leading to an award in Advanced Subsidiary 0DWKHPDWLFV�FRPSULVH�WKUHH�$6�XQLWV��&RPELQDWLRQV�OHDGLQJ�WR�DQ�award in Advanced Subsidiary Further Mathematics comprise three XQLWV��7KH�FRPELQDWLRQ�OHDGLQJ�WR�DQ�DZDUG�LQ�$GYDQFHG�6XEVLGLDU\�3XUH�0DWKHPDWLFV�FRPSULVHV�XQLWV�&��&��DQG�&��

8371 Advanced Subsidiary Mathematics

Core Mathematics units C1 and C2 plus one of the Applications XQLWV�0��6��RU�'��

8372 Advanced Subsidiary Further Mathematics

Further Pure Mathematics unit FP1 plus two other units (excluding &�±&��6WXGHQWV�ZKR�DUH�DZDUGHG�FHUWL¿FDWHV�LQ�ERWK�$GYDQFHG�GCE Mathematics and AS Further Mathematics must use unit results IURP�QLQH�GLIIHUHQW�WHDFKLQJ�PRGXOHV�


Section B © Pearson Education Limited 2013 Pearson Edexcel Level 3 GCE in Mathematics8

8373 Advanced Subsidiary Pure Mathematics

&RUH�0DWKHPDWLFV�XQLWV�&��&��DQG�&��

8374 Additional Qualification in Advanced Subsidiary Further Mathematics (Additional)

6WXGHQWV�ZKR�FRPSOHWH�¿IWHHQ�XQLWV�ZLOO�KDYH�DFKLHYHG�WKH�equivalent of the standard of Advanced Subsidiary GCE Further 0DWKHPDWLFV�LQ�WKHLU�DGGLWLRQDO�XQLWV��6XFK�VWXGHQWV�ZLOO�EH�HOLJLEOH�for the award of Advanced Subsidiary GCE Further Mathematics (Additional) in addition to the awards of Advanced GCE Mathematics DQG�$GYDQFHG�*&(�)XUWKHU�0DWKHPDWLFV�

Advanced Level

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2210 9371 GCE Mathematics C1, C2, C3, C4 M1 and S1 or M1 and D1 or S1 and D1 or S1 and S2 or M1 and M2 or D1 and D2

2230 9372 GCE Further Mathematics FP1 and either FP2 or FP3 Any *

2230 9373 GCE Pure Mathematics C1, C2, C3, C4, FP1 FP2 or FP3

2230 9374 GCE Further Mathematics (Additional)

Any which have not been used for a previous TXDOL¿FDWLRQ

*For GCE Further Mathematics, excluded units are C1, C2, C3, C4

** See Appendix C for description of this code and all other codes relevant to these TXDOL¿FDWLRQV�

Specification overview B


Advanced GCE combinations

Combinations leading to an award in mathematics must comprise VL[�XQLWV��LQFOXGLQJ�DW�OHDVW�WZR�$��XQLWV��&RPELQDWLRQV�OHDGLQJ�WR�an award in Further Mathematics must comprise six units, including DW�OHDVW�WKUHH�$��XQLWV��&RPELQDWLRQV�OHDGLQJ�WR�DQ�DZDUG�LQ�3XUH�0DWKHPDWLFV�FRPSULVH�XQLWV�&�±&��)3��DQG�RQH�RI�)3��RU�)3��

9371 Advanced GCE Mathematics

Core Mathematics units C1, C2, C3 and C4 plus two Applications XQLWV�IURP�WKH�IROORZLQJ�VL[�FRPELQDWLRQV��0��DQG�0��6��DQG�6��'��DQG�'��0��DQG�6��6��DQG�'��0��DQG�'��

9372 Advanced GCE Further Mathematics

Further Pure Mathematics units FP1, FP2, FP3 and a further three $SSOLFDWLRQV�XQLWV��H[FOXGLQJ�&�±&��WR�PDNH�D�WRWDO�RI�VL[�XQLWV��RU�FP1, either FP2 or FP3 and a further four Applications units �H[FOXGLQJ�&�±&��WR�PDNH�D�WRWDO�RI�VL[�XQLWV��6WXGHQWV�ZKR�DUH�DZDUGHG�FHUWL¿FDWHV�LQ�ERWK�$GYDQFHG�*&(�0DWKHPDWLFV�DQG�Advanced GCE Further Mathematics must use unit results from 12 GLIIHUHQW�WHDFKLQJ�PRGXOHV�

9373 Advanced GCE Pure Mathematics

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9374 Additional Qualification in Advanced Further Mathematics (Additional)

Students who complete eighteen units will have achieved the equivalent of the standard of Advanced GCE Further Mathematics �$GGLWLRQDO��LQ�WKHLU�DGGLWLRQDO�XQLWV��6XFK�VWXGHQWV�ZLOO�EH�HOLJLEOH�for the award of Advanced GCE Further Mathematics (Additional) in addition to the awards of Advanced GCE Mathematics and Advanced *&(�)XUWKHU�0DWKHPDWLFV�



Summary of assessment requirements

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C1 Core Mathematics 1

6663 AS 1 written paper

June ��RI�AS

��RI�Advanced GCE



June ��RI�AS



6665 A2 1 written paper

June ��RI�AS




June ��RI�AS


FP1 Further Pure Mathematics 1


June ��RI�AS




June ��RI�AS




June ��RI�AS


M1 Mechanics 1 6677 AS 1 written paper

June ��RI�AS


M2 Mechanics 2 6678 A2 1 written paper

June ��RI�AS



June ��RI�AS



June ��RI�AS



June ��RI�AS




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S1 Statistics 1 6683 AS 1 written paper

June ��RI�AS


S2 Statistics 2 6684 A2 1 written paper

June ��RI�AS


S3Statistics 3 6691 A2

1 written paper

June ��RI�AS


S4Statistics 4 6686 A2

1 written paper

June ��RI�AS


D1 Decision Mathematics 1


June ��RI�AS


D2 Decision Mathematics 2


June ��RI�AS


*See Appendix C for a description of this code and all other codes relevant to this TXDOL¿FDWLRQ�

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Assessment objectives and weightings

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AO1 UHFDOO��VHOHFW�DQG�XVH�WKHLU�NQRZOHGJH�RI�PDWKHPDWLFDO�IDFWV��FRQFHSWV�DQG�techniques in a variety of contexts

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AO2 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

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AO3 UHFDOO��VHOHFW�DQG�XVH�WKHLU�NQRZOHGJH�RI�VWDQGDUG�PDWKHPDWLFDO�PRGHOV�WR�UHSUHVHQW�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 UH¿QHPHQW�RI�VXFK�PRGHOV

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AO4 comprehend translations of common realistic contexts into mathematics; use the UHVXOWV�RI�FDOFXODWLRQV�WR�PDNH�SUHGLFWLRQV��RU�FRPPHQW�RQ�WKH�FRQWH[W��DQG��ZKHUH�appropriate, read critically and comprehend longer mathematical arguments or examples of applications

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AO5 use contemporary calculator technology and other permitted resources (such DV�IRUPXODH�ERRNOHWV�RU�VWDWLVWLFDO�WDEOHV��DFFXUDWHO\�DQG�HI¿FLHQWO\��XQGHUVWDQG�ZKHQ�QRW�WR�XVH�VXFK�WHFKQRORJ\��DQG�LWV�OLPLWDWLRQV��*LYH�DQVZHUV�WR�DSSURSULDWH�DFFXUDF\�

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$VVHVVPHQW�REMHFWLYH AO1 AO2 AO3 AO4 AO5

Core Mathematics C1 30–35 25–30 5–15 5–10 1–5




Further Pure Mathematics FP1 25–30 25–30 0–5 5–10 5–10



Mechanics M1 20–25 20–25 15–20 6–11 4–9

Mechanics M2 20–25 20–25 10–15 7–12 5–10

Mechanics M3 20–25 25–30 10–15 5–10 5–10

Mechanics M4 20–25 20–25 15–20 5–10 5–10

Mechanics M5 20–25 20–25 15–20 5–10 5–10

Statistics S1 20–25 20–25 15–20 5–10 5–10

Statistics S2 25–30 20–25 10–15 5–10 5–10

Statistics S3 25–30 20–25 10–15 5–10 5–10

Statistics S4 15–20 15–20 15–20 5–10 10-15

Decision Mathematics D1 20–25 20–25 15–20 5–10 5–10

Decision Mathematics D2 25–30 20–25 10–15 7–13 0–5

Qualification summary

Subject criteria 7KH�*HQHUDO�&HUWL¿FDWH�RI�(GXFDWLRQ�LV�SDUW�RI�WKH�/HYHO��SURYLVLRQ��7KLV�VSHFL¿FDWLRQ�LV�EDVHG�RQ�WKH�*&(�$6�DQG�$�/HYHO�6XEMHFW�criteria for Mathematics, which is prescribed by the regulatory DXWKRULWLHV�DQG�LV�PDQGDWRU\�IRU�DOO�DZDUGLQJ�ERGLHV��

7KH�*&(�LQ�0DWKHPDWLFV�HQDEOHV�VWXGHQWV�WR�IROORZ�D�ÀH[LEOH�course in mathematics, to better tailor a course to suit the LQGLYLGXDO�QHHGV�DQG�JRDOV�



Aims The 18 units have been designed for schools and colleges to SURGXFH�FRXUVHV�ZKLFK�ZLOO�HQFRXUDJH�VWXGHQWV�WR�

� develop their understanding of mathematics and mathematical SURFHVVHV�LQ�D�ZD\�WKDW�SURPRWHV�FRQ¿GHQFH�DQG�IRVWHUV�HQMR\PHQW

� develop abilities to reason logically and recognise incorrect reasoning, to generalise and to construct mathematical proofs

� H[WHQG�WKHLU�UDQJH�RI�PDWKHPDWLFDO�VNLOOV�DQG�WHFKQLTXHV�DQG�XVH�WKHP�LQ�PRUH�GLI¿FXOW��XQVWUXFWXUHG�SUREOHPV

� develop an understanding of coherence and progression in mathematics and of how different areas of mathematics can be connected

� 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 EH�UH¿QHG�DQG�LPSURYHG

� use mathematics as an effective means of communication

� read and comprehend mathematical arguments and articles concerning applications of mathematics

� DFTXLUH�WKH�VNLOOV�QHHGHG�WR�XVH�WHFKQRORJ\�VXFK�DV�FDOFXODWRUV�and computers effectively, recognise when such use may be inappropriate and be aware of limitations

� develop an awareness of the relevance of mathematics to other ¿HOGV�RI�VWXG\��WR�WKH�ZRUOG�RI�ZRUN�DQG�WR�VRFLHW\�LQ�JHQHUDO

� WDNH�LQFUHDVLQJ�UHVSRQVLELOLW\�IRU�WKHLU�RZQ�OHDUQLQJ�DQG�WKH�HYDOXDWLRQ�RI�WKHLU�RZQ�PDWKHPDWLFDO�GHYHORSPHQW�

AS/A2 knowledge and understanding and skills

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Pearson Edexcel Level 3 GCE in Mathematics © Pearson Education Limited 2013 Section C 15

C Mathematics, Further Mathematics, Pure Mathematics and Further Mathematics (Additional) unit content

Structure of qualification 16



















C Mathematics unit content

Section C © Pearson Education Limited 2013 Pearson Edexcel Level 3 GCE in Mathematics16

Structure of qualification

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GCE AS Mathematics

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GCE AS Further Mathematics

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GCE AS Pure Mathematics

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GCE AS Further Mathematics (Additional)

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Mathematics unit content C


GCE A Level Mathematics

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GCE A Level Further Mathematics

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GCE A Level Pure Mathematics

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GCE A Level Further Mathematics (Additional)

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Unit C1 Core Mathematics 1 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics

C1.1 Unit description

Algebra and functions; coordinate geometry in the (x, y) plane; VHTXHQFHV�DQG�VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�

C1.2 Assessment information

Preamble Construction and presentation of rigorous mathematical arguments through appropriate use of precise statements and logical deduction, involving correct use of symbols and appropriate FRQQHFWLQJ�ODQJXDJH�LV�UHTXLUHG��6WXGHQWV�DUH�H[SHFWHG�WR�exhibit 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’, µVXI¿FLHQW¶��DQG�QRWDWLRQ�VXFK�DV�∴, ⇒, ⇐ and ⇔�

Examination 7KH�H[DPLQDWLRQ�ZLOO�FRQVLVW�RI�RQH��í�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�WHQ�TXHVWLRQV�RI�YDU\LQJ�OHQJWK��7KH�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�VKRXOG�EH�DWWHPSWHG�

For this unit, students may QRW have access to any calculating aids, LQFOXGLQJ�ORJ�WDEOHV�DQG�VOLGH�UXOHV�

Formulae )RUPXODH�ZKLFK�VWXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DUH�JLYHQ�EHORZ�DQG�WKHVH�ZLOO�QRW�DSSHDU�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, which will be provided for XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�

This section lists formulae that students are expected to remember DQG�WKDW�PD\�QRW�EH�LQFOXGHG�LQ�IRUPXODH�ERRNOHWV�

Quadratic equations

ax2 + bx + c = 0 has roots

− ± −b b aca

2 42

Unit C1 Core Mathematics C1 (AS)


Differentiation

function

xn

derivative

nxn – 1

Integration

function

xn

integral

11n + xn + 1

+ c, n��±�

1 Algebra and functions

What students need to learn:

/DZV�RI�LQGLFHV�IRU�DOO�UDWLRQDO�H[SRQHQWV� The equivalence of am/n and n√am

VKRXOG�EH�NQRZQ�

8VH�DQG�PDQLSXODWLRQ�RI�VXUGV� Students should be able to UDWLRQDOLVH�GHQRPLQDWRUV�

4XDGUDWLF�IXQFWLRQV�DQG�WKHLU�JUDSKV�

7KH�GLVFULPLQDQW�RI�D�TXDGUDWLF�IXQFWLRQ�

&RPSOHWLQJ�WKH�VTXDUH��6ROXWLRQ�RI�TXDGUDWLF�HTXDWLRQV�

Solution of quadratic equations by factorisation, use of the formula DQG�FRPSOHWLQJ�WKH�VTXDUH�

6LPXOWDQHRXV�HTXDWLRQV��DQDO\WLFDO�VROXWLRQ�E\�VXEVWLWXWLRQ�

For example, where one equation is linear and one equation is TXDGUDWLF�

6ROXWLRQ�RI�OLQHDU�DQG�TXDGUDWLF�LQHTXDOLWLHV�� For example, ax + b > cx + d,

px2 + qx + r ≥ 0, px2

+ qx + r < ax + b.

Core Mathematics C1 (AS) Unit C1


Algebraic manipulation of polynomials, including H[SDQGLQJ�EUDFNHWV�DQG�FROOHFWLQJ�OLNH�WHUPV��IDFWRULVDWLRQ�

Students should be able to XVH�EUDFNHWV��)DFWRULVDWLRQ�RI�polynomials of degree n, n ≤ 3,

eg x3 + 4x2

+ 3x. The notation f(x) PD\�EH�XVHG��8VH�RI�WKH�IDFWRU�theorem is not UHTXLUHG��

*UDSKV�RI�IXQFWLRQV��VNHWFKLQJ�FXUYHV�GH¿QHG�E\�VLPSOH�HTXDWLRQV��*HRPHWULFDO�LQWHUSUHWDWLRQ�RI�DOJHEUDLF�VROXWLRQ�RI�HTXDWLRQV��8VH�RI�LQWHUVHFWLRQ�SRLQWV�RI�JUDSKV�RI�IXQFWLRQV�WR�VROYH�HTXDWLRQV�

Functions to include simple cubic functions and the reciprocal

function y = k

x with x ≠ 0.

Knowledge of the term asymptote LV�H[SHFWHG�

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).

Students should be able to apply one of these transformations to any of the above functions (quadratics, cubics, reciprocal) DQG�VNHWFK�WKH�UHVXOWLQJ�JUDSK��

Given the graph of any function y = f(x) students should be able to VNHWFK�WKH�JUDSK�UHVXOWLQJ�IURP�RQH�RI�WKHVH�WUDQVIRUPDWLRQV�



2 Coordinate geometry in the (x, y) plane


Equation of a straight line, including the forms y – y

1 = m(x – x

1) and ax + by + c = 0.

7R�LQFOXGH�

(i) the equation of a line through two given points

(ii) the equation of a line parallel (or perpendicular) to a given line through a given SRLQW��)RU�H[DPSOH��WKH�OLQH�perpendicular to the line 3x + 4y = 18 through the point (2, 3) has equation

y – 3 = (x – 2).

Conditions for two straight lines to be parallel or SHUSHQGLFXODU�WR�HDFK�RWKHU�

3 Sequences and series


Sequences, including those given by a formula for the nth term and those generated by a simple relation of the form x

n+1 = f(x

n).

Arithmetic series, including the formula for the sum RI�WKH�¿UVW�n�QDWXUDO�QXPEHUV�

The general term and the sum to n�WHUPV�RI�WKH�VHULHV�DUH�UHTXLUHG��The proof of the sum formula VKRXOG�EH�NQRZQ��

Understanding of ∑ notation will be H[SHFWHG��

4

3



4 Differentiation


The derivative of f(x) as the gradient of the tangent to the graph of y = f (x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change; VHFRQG�RUGHU�GHULYDWLYHV�

For example,�NQRZOHGJH�WKDW�

is

the rate of change of y with respect to x��.QRZOHGJH�RI�WKH�FKDLQ�UXOH�LV�QRW�UHTXLUHG��

The notation f ′(x) PD\�EH�XVHG�

Differentiation of xn, and related sums and GLIIHUHQFHV�

For example, for n ≠ 1, the ability to differentiate expressions such

as (2x + 5)(x − 1) and 2 5 3

3 1 2x + x

x−

/ is

expectHG�

Applications of differentiation to gradients, tangents DQG�QRUPDOV�

8VH�RI�GLIIHUHQWLDWLRQ�WR�¿QG�equations of tangents and normals DW�VSHFL¿F�SRLQWV�RQ�D�FXUYH�

5 Integration


,QGH¿QLWH�LQWHJUDWLRQ�DV�WKH�UHYHUVH�RI�GLIIHUHQWLDWLRQ�

6WXGHQWV�VKRXOG�NQRZ�WKDW�D�FRQVWDQW�RI�LQWHJUDWLRQ�LV�UHTXLUHG�

Integration of xn. For example, the ability to integrate expressions such as

12

2 312x x− −

and ( )x

x+ 2 2

12

is

expecteG�

Given f ′(x) and a point on the curve, students should be able to ¿QG�DQ�HTXDWLRQ�RI�WKH�FXUYH�LQ�WKH�form y = f(x).

dd

yx


Unit C2 Core Mathematics 2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics


Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQ�IRU�&��LWV�SUHDPEOH�DQG�LWV�DVVRFLDWHG�IRUPXODH��LV�DVVXPHG�DQG�PD\�EH�WHVWHG�

Examination 7KH�H[DPLQDWLRQ�ZLOO�FRQVLVW�RI�RQH��í�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�QLQH�TXHVWLRQV�RI�YDU\LQJ�OHQJWK��7KH�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�VKRXOG�EH�DWWHPSWHG�

Calculators Students are expected to have available a calculator with at least

WKH�IROORZLQJ�NH\V��+, −, ×, ÷, ʌ, x2, √x,

1

x, xy

, ln x, ex, x!, sine, cosine and

tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG��





Laws of logarithmslog

ax + log

ay = log

a(xy)

log log loga a ax y xy

− ≡⎛

⎝⎜

⎞

⎠⎟

klogax = log

a(xk

)

Trigonometry

In the triangle ABC

aA

bB

cCsin sin sin

= =

area =

Area

area under a curve = y x ya

b d ( 0)∫ ≥

12 ab Csin





Simple algebraic division; use of the Factor Theorem DQG�WKH�5HPDLQGHU�7KHRUHP�

Only division by (x + a) or (x – a) will EH�UHTXLUHG�

6WXGHQWV�VKRXOG�NQRZ�WKDW�LI f(x) = 0 when x = a, then (x – a) is a factor of f(x)�

Students may be required to factorise cubic expressions such as x3

+ 3x2 – 4 and 6x3

+ 11x2 – x – 6�

Students should be familiar with the terms ‘quotient’ and ‘remainder’ and be able to determine the remainder when the polynomial f(x) is divided by (ax + b)�



Coordinate geometry of the circle using the equation of a circle in the form (x – a)

2 + (y – b)2 = r2 and

LQFOXGLQJ�XVH�RI�WKH�IROORZLQJ�FLUFOH�SURSHUWLHV�

(i) the angle in a semicircle is a right angle;

(ii) the perpendicular from the centre to a chord bisects the chord;

�LLL��WKH�SHUSHQGLFXODULW\�RI�UDGLXV�DQG�WDQJHQW�

6WXGHQWV�VKRXOG�EH�DEOH�WR�¿QG�the radius and the coordinates of the centre of the circle given the equation of the circle, and vice YHUVD�





7KH�VXP�RI�D�¿QLWH�JHRPHWULF�VHULHV��WKH�VXP�WR�LQ¿QLW\�RI�D�FRQYHUJHQW�JHRPHWULF�VHULHV��LQFOXGLQJ�the use of ⏐r⏐ < 1�

The general term and the sum to n WHUPV�DUH�UHTXLUHG��

The proof of the sum formula VKRXOG�EH�NQRZQ�

Binomial expansion of (1 + x)n for positive integer n��

The notations n! and nr⎛

⎝⎜⎞

⎠⎟�

Expansion of (a + bx)n may be

UHTXLUHG�

4 Trigonometry


The sine and cosine rules, and the area of a triangle in the form 1

2 ab sin C.

Radian measure, including use for arc length and DUHD�RI�VHFWRU�

Use of the formulae V� �Uș and A =

12 r2ș�IRU�D�FLUFOH�

6LQH��FRVLQH�DQG�WDQJHQW�IXQFWLRQV��7KHLU�JUDSKV��V\PPHWULHV�DQG�SHULRGLFLW\�

Knowledge of graphs of curves with equations such as

y = 3 sin x, y = sin

x +⎛⎝⎜

⎞⎠⎟

π6

, y = sin 2x is

H[SHFWHG�

Knowledge and use of tan ș = , and sin

2ș + cos2 ș�= 1�

sincos

θθ



Solution of simple trigonometric equations in a given LQWHUYDO�

Students should be able to solve equations such as

sin x +⎛⎝⎜

⎞⎠⎟

π2 =

34 for 0 < x < 2ʌ,

cos (x + 30°) = 12 for −180° < x < 180°,

tan 2x = 1 for 90° < x < 270°,

6 cos2 x + sin x − 5 = 0, 0° ≤ x < 360,

sin2 =

12 for –π ≤ x < π.

5 Exponentials and logarithms


y = ax DQG�LWV�JUDSK�

Laws of logarithms To include

log a

xy = log a x + log

a y,

log a x

y

= log a x − log

a y,

loga xk

= k log a x,

log loga axx1 = −

,

log a a = 1

The solution of equations of the form ax = b� Students may use the change of EDVH�IRUPXOD�

x +⎛⎝⎜

⎞⎠⎟

π6



6 Differentiation


Applications of differentiation to maxima and minima and stationary points, increasing and decreasing IXQFWLRQV�

The notation f″(x) may be used for WKH�VHFRQG�RUGHU�GHULYDWLYH��

To include applications to curve VNHWFKLQJ��0D[LPD�DQG�PLQLPD�problems may be set in the context RI�D�SUDFWLFDO�SUREOHP�

7 Integration


(YDOXDWLRQ�RI�GH¿QLWH�LQWHJUDOV�

,QWHUSUHWDWLRQ�RI�WKH�GH¿QLWH�LQWHJUDO�DV�WKH�DUHD�XQGHU�D�FXUYH�

Students will be expected to be able to evaluate the area of a region bounded by a curve and JLYHQ�VWUDLJKW�OLQHV�

(J�¿QG�WKH�¿QLWH�DUHD�ERXQGHG�E\�the curve y = 6x – x2

and the line y = 2x.

�x dy�ZLOO�QRW�EH�UHTXLUHG�

Approximation of area under a curve using the WUDSH]LXP�UXOH�

For example,

evaluate √ +⌠

⌡

⎮⎮

( )2 10

1

x xd

using the values of √(2x + 1) at x = 0,

0.25, 0.5, 0.75 and 1.


Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics


Algebra and functions; trigonometry; exponentials and logarithms; GLIIHUHQWLDWLRQ��QXPHULFDO�PHWKRGV�


Prerequisites and preamble

Prerequisites

$�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&��DQG�&��WKHLU�SUHDPEOHV��prerequisites and associated formulae, is assumed and may be WHVWHG�

Preamble

Methods of proof, including proof by contradiction and disproof by FRXQWHU�H[DPSOH��DUH�UHTXLUHG��$W�OHDVW�RQH�TXHVWLRQ�RQ�WKH�SDSHU�ZLOO�UHTXLUH�WKH�XVH�RI�SURRI�

Examination The examination will consist of one 1½�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�VHYHQ�TXHVWLRQV�RI�YDU\LQJ�OHQJWK��7KH�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�VKRXOG�EH�DWWHPSWHG�


the folloZLQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x, , xy



1

x

Unit C3 Core Mathematics C3 (A2)




Trigonometrycos

2A + sin2 A�Ł��

sec2A�Ł��WDQ2

A

cosec2A�Ł��FRW2 A

sin 2A�Ł��VLQA cos A

cos2A�Ł�FRV2A – sin2 A

tan 2 2 tan 1 tan2A A

A≡

−

Differentiation

function

sin kx

cos kx

ekx

ln x

f (x) + g (x)

f (x) g (x)

f (g (x))

derivative

k cos kx

–k sin kx

kekx

1x

f ’ (x) + g ’ (x)

f ’ (x) g (x) +f (x) g ’ (x)

f ’ (g (x)) g ’ (x)

Core Mathematics C3 (A2) Unit C3




6LPSOL¿FDWLRQ�RI�UDWLRQDO�H[SUHVVLRQV�LQFOXGLQJ�IDFWRULVLQJ�DQG�FDQFHOOLQJ��DQG�DOJHEUDLF�GLYLVLRQ�

Denominators of rational expressions will be linear or

quadratic, eg 1

ax + b,

ax + bpx + qx + r2 ,

xx

3

211

+− �

'H¿QLWLRQ�RI�D�IXQFWLRQ��'RPDLQ�DQG�UDQJH�RI�IXQFWLRQV��&RPSRVLWLRQ�RI�IXQFWLRQV��,QYHUVH�IXQFWLRQV�DQG�WKHLU�JUDSKV�

The concept of a function as a one-one or many-one mapping from ! (or a subset of !) to !��The notation f : x and f(x) will be XVHG��

6WXGHQWV�VKRXOG�NQRZ�WKDW fg will mean ‘do g�¿UVW��WKHQ f�¶��

6WXGHQWV�VKRXOG�NQRZ�WKDW�LI�f −1

exists, then f −1

f(x) = ff −1

(x) = x.

7KH�PRGXOXV�IXQFWLRQ� 6WXGHQWV�VKRXOG�EH�DEOH�WR�VNHWFK�the graphs of y = ⏐ax + b⏐ and the graphs of y = ⏐f(x)⏐ and y = f(⏐x⏐), given the graph of y = f(x)�

Combinations of the transformations y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax)�

6WXGHQWV�VKRXOG�EH�DEOH�WR�VNHWFK�the graph of, for example, y = 2f(3x), y = f(−x) + 1, given the graph of y = f(x) or the graph of, for example, y = 3 + sin 2x,

y = −cos x +⎛⎝⎜

⎞⎠⎟

π4 �

The graph of y = f(ax + b) will not be UHTXLUHG�



2 Trigonometry


Knowledge of secant, cosecant and cotangent and of DUFVLQ��DUFFRV�DQG�DUFWDQ��7KHLU�UHODWLRQVKLSV�WR�VLQH��FRVLQH�DQG�WDQJHQW��8QGHUVWDQGLQJ�RI�WKHLU�JUDSKV�DQG�DSSURSULDWH�UHVWULFWHG�GRPDLQV�

Angles measured in both degrees DQG�UDGLDQV�

Knowledge and use of sec2�ș = 1 + tan

2�ș and

cosec2 ș = 1 + cot

2 ș.

Knowledge and use of double angle formulae; use of formulae for sin (A ± B), cos (A ± B) and tan (A ± B) and of expressions for a cos�ș��E�sin�ș�in the equivalent forms of r cos (ș�± a) or r sin (ș�± a)�

To include application to half DQJOHV��.QRZOHGJH�RI�WKH t (tan

12θ )

formulae will not�EH�UHTXLUHG�

Students should be able to solve equations such as a cos ș�+ b sin ș = c in a given interval, and to prove simple identities such as cos x cos 2x + sin x sin 2x ≡ cos x�

3 Exponentials and logarithms


The function ex DQG�LWV�JUDSK� To include the graph of y = eax + b + c.

The function ln x and its graph; ln x as the inverse function of ex�

Solution of equations of the form e

ax + b = p and ln (ax + b) = q is H[SHFWHG�



4 Differentiation


Differentiation of ex, ln x, sin x, cos x, tan x and their sums and differences.

Differentiation using the product rule, the quotient UXOH�DQG�WKH�FKDLQ�UXOH��

The use of dd d

d

yx x

y

=⎛

⎝⎜

⎞

⎠⎟

1�

Differentiation of cosec x, cot x and sec x�DUH�UHTXLUHG��6NLOO�ZLOO�EH�expected in the differentiation of functions generated from standard forms using products, quotients and composition, such as 2x4

sin x,

e3x

x , cos x2

and tan2 2x�

(J�¿QGLQJ�ddyx for x = sin 3y�

5 Numerical methods


Location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is FRQWLQXRXV�

Approximate solution of equations using simple iterative methods, including recurrence relations of the form x

n+1 = f(x

n).

Solution of equations by use of iterative procedures for which OHDGV�ZLOO�EH�JLYHQ�


Unit C4 Core Mathematics 4 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics


Algebra and functions; coordinate geometry in the (x, y) plane; VHTXHQFHV�DQG�VHULHV��GLIIHUHQWLDWLRQ��LQWHJUDWLRQ��YHFWRUV�


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&��&��DQG�&��DQG�WKHLU�preambles, prerequisites and associated formulae, is assumed and PD\�EH�WHVWHG�

Examination The examination will consist of one 1½�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�VHYHQ�TXHVWLRQV�RI�YDU\LQJ�OHQJWK��7KH�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�VKRXOG�EH�DWWHPSWHG�


the following�NH\V� +, −, ×, ÷��ʌ��x2, √x,

1

x , xy



Formulae )RUPXODH�ZKLFK�VWXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DUH�JLYHQ�RYHUOHDI�DQG�WKHVH�ZLOO�QRW�DSSHDU�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, which will be provided for XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�

This section lists formulae that students are expected to remember DQG�WKDW�PD\�QRW�EH�LQFOXGHG�LQ�IRUPXODH�ERRNOHWV.



Integration

function

cos kx

sin kx

ekx

1x

f’(x) + g’(x)

f’(g(x)) g’(x)

integral

kx + c

− 1k

kxcos kx + c

1k

kxe + c

ln |x| + c, x��

f (x) + g(x) + c

f (g(x)) + c

Vectors

xyz

abc

xa yb zc⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟•⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= + +



5DWLRQDO�IXQFWLRQV��3DUWLDO�IUDFWLRQV��GHQRPLQDWRUV�QRW�PRUH�FRPSOLFDWHG�WKDQ�UHSHDWHG�OLQHDU�WHUPV��

Partial fractions to include denominators such as (ax + b)(cx + d)(ex + f)

and (ax + b)(cx + d)2.

The degree of the numerator may equal or exceed the degree of WKH�GHQRPLQDWRU��$SSOLFDWLRQV�WR�integration, differentiation and VHULHV�H[SDQVLRQV�

Quadratic factors in the denominator such as (x2 + a), a > 0, are not�UHTXLUHG�

1k

kxsin





Parametric equations of curves and conversion EHWZHHQ�&DUWHVLDQ�DQG�SDUDPHWULF�IRUPV�

6WXGHQWV�VKRXOG�EH�DEOH�WR�¿QG�the area under a curve given its SDUDPHWULF�HTXDWLRQV��6WXGHQWV�will not�EH�H[SHFWHG�WR�VNHWFK�a curve from its parametric HTXDWLRQV�



Binomial series for any rational n�For ⏐x⏐<

b

a , students should be

able to obtain the expansion of (ax + b)

n, and the expansion of rational functions by decomposition LQWR�SDUWLDO�IUDFWLRQV�

4 Differentiation


'LIIHUHQWLDWLRQ�RI�VLPSOH�IXQFWLRQV�GH¿QHG�LPSOLFLWO\�RU�SDUDPHWULFDOO\�

7KH�¿QGLQJ�RI�HTXDWLRQV�RI�tangents and normals to curves given parametrically or implicitly is UHTXLUHG�

([SRQHQWLDO�JURZWK�DQG�GHFD\� Knowledge and use of the result

ddx

(ax) = ax

ln a�LV�H[SHFWHG��

)RUPDWLRQ�RI�VLPSOH�GLIIHUHQWLDO�HTXDWLRQV� Questions involving connected UDWHV�RI�FKDQJH�PD\�EH�VHW�



5 Integration


Integration of ex,

1x , sinx, cosx.

To include integration of standard

functions such as sin 3x, sec2 2x, tan x,

e5x,

12x�

Students should recognise integrals of the form

= ln f(x) + c�

Students are expected to be able to use trigonometric identities to integrate, for example, sin

2 x, tan

2 x,

cos2 3x.

(YDOXDWLRQ�RI�YROXPH�RI�UHYROXWLRQ�π y x2

⌠

⌡

⎮⎮

d is required, but not

π x y2⌠

⌡

⎮⎮

d ��6WXGHQWV�VKRXOG�EH�able

WR�¿QG�D�YROXPH�RI�UHYolution, JLYHQ�SDUDPHWULF�HTXDWLRQV�

Simple cases of integration by substitution and LQWHJUDWLRQ�E\�SDUWV��7KHVH�PHWKRGV�DV�WKH�UHYHUVH�SURFHVVHV�RI�WKH�FKDLQ�DQG�SURGXFW�UXOHV�UHVSHFWLYHO\�

Except in the simplest of cases the VXEVWLWXWLRQ�ZLOO�EH�JLYHQ��

The integral ��ln x dx�LV�UHTXLUHG�

More than one application of integration by parts may be required, for example ��x2 ex

dx�

′⌠

⌡⎮⎮

ff( )

d( )xx

x



6LPSOH�FDVHV�RI�LQWHJUDWLRQ�XVLQJ�SDUWLDO�IUDFWLRQV� Integration of rational expressions such as those arising from partial

fractions, eg 2

3 5x +,

31 2( )x −

�

Note that the integration of other

rational expressions, such as x

x2 5+

and 2

2 1 4( )x − is also required (see

above paragrapKV��

$QDO\WLFDO�VROXWLRQ�RI�VLPSOH�¿UVW�RUGHU�GLIIHUHQWLDO�HTXDWLRQV�ZLWK�VHSDUDEOH�YDULDEOHV�

General and particular solutions ZLOO�EH�UHTXLUHG�

1XPHULFDO�LQWHJUDWLRQ�RI�IXQFWLRQV� $SSOLFDWLRQ�RI�WKH�WUDSH]LXP�UXOH�to functions covered in C3 and &��8VH�RI�LQFUHDVLQJ�QXPEHU�RI�WUDSH]LD�WR�LPSURYH�DFFXUDF\�DQG�HVWLPDWH�HUURU�ZLOO�EH�UHTXLUHG��Questions will not require more WKDQ�WKUHH�LWHUDWLRQV�

Simpson’s Rule is not�UHTXLUHG�

6 Vectors


9HFWRUV�LQ�WZR�DQG�WKUHH�GLPHQVLRQV�

0DJQLWXGH�RI�D�YHFWRU� 6WXGHQWV�VKRXOG�EH�DEOH�WR�¿QG�D�unit vector in the direction of a, and be familiar with ⏐a⏐�

Algebraic operations of vector addition and multiplication by scalars, and their geometrical LQWHUSUHWDWLRQV�



3RVLWLRQ�YHFWRUV��

7KH�GLVWDQFH�EHWZHHQ�WZR�SRLQWV��

OB OA AB→→→

− = = −b a �

The distance d between two points (x

1 , y

1 , z

1) and (x

2 , y

2 , z

2) is given by

d2 = (x

1 – x

2)

2 + (y

1 – y

2)

2 + (z

1 – z

2)

2 ��

9HFWRU�HTXDWLRQV�RI�OLQHV� To include the forms r = a + tb�and r = c + t(d – c)�

Intersection, or otherwise, of two OLQHV�

7KH�VFDODU�SURGXFW��,WV�XVH�IRU�FDOFXODWLQJ�WKH�DQJOH�EHWZHHQ�WZR�OLQHV�

6WXGHQWV�VKRXOG�NQRZ�WKDW�IRU�

OA→

�= a = a1i + a

2j + a

3k and

OB→

= b = b1i + b

2j + b

3k then

a . b = a1b

1 + a

2b

2 + a

3b

3 and

cos ∠AOB ��

6WXGHQWV�VKRXOG�NQRZ�WKDW�LI�a�.�b�= 0, and a�and b are non-]HUR�YHFWRUV��WKHQ�a and b are SHUSHQGLFXODU�

a ba b

.


Unit FP1 Further Pure Mathematics 1 GCE AS and GCE Further Mathematics and GCE Pure Mathematics compulsory unit

FP1.1 Unit description

Series; complex numbers; numerical solution of equations; &RRUGLQDWH�V\VWHPV��PDWUL[�DOJHEUD��SURRI�

FP1.2 Assessment information

Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&��DQG�&��WKHLU�SUHUHTXLVLWHV��SUHDPEOHV�DQG�DVVRFLDWHG�IRUPXODH�LV�DVVXPHG�DQG�PD\�EH�WHVWHG�

Examination The examination will consist of one 1½�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�QLQH�TXHVWLRQV�ZLWK�YDU\LQJ�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�PD\�EH�DWWHPSWHG�

4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�


the followiQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x,

1

x , xy


tangent and their inverses in degrees and decimals of a degree, and LQ�UDGLDQV��PHPRU\��&DOFXODWRUV�ZLWK�D�IDFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�


Unit FP1 Further Pure Mathematics FP1 (AS)

1 Complex numbers


'H¿QLWLRQ�RI�FRPSOH[�QXPEHUV�LQ�WKH�IRUP�a + ib and r cos θ + i r sin θ��

7KH�PHDQLQJ�RI�FRQMXJDWH��modulus, argument, real part, imaginary part and equality of complex numbers should be NQRZQ�

6XP��SURGXFW�DQG�TXRWLHQW�RI�FRPSOH[�QXPEHUV� ⏐z1 z

2⏐ = ⏐z

1⏐ ⏐z

2⏐

Knowledge of the result arg (z

1 z

2) = arg z

1 + arg z

2 is not

UHTXLUHG�

Geometrical representation of complex numbers in WKH�$UJDQG�GLDJUDP��

Geometrical representation of sums, products and TXRWLHQWV�RI�FRPSOH[�QXPEHUV�

Complex solutions of quadratic equations with real FRHI¿FLHQWV�

&RQMXJDWH�FRPSOH[�URRWV�RI�SRO\QRPLDO�HTXDWLRQV�ZLWK�UHDO�FRHI¿FLHQWV�

Knowledge that if z1 is a root of

f(z) = 0 then z1

*�LV�DOVR�D�URRW�

2 Numerical solution of equations


Equations of the form f(x) = 0�VROYHG�QXPHULFDOO\�E\�

(i) interval bisection,

(ii) linear interpolation,

�LLL��WKH�1HZWRQ�5DSKVRQ�SURFHVV�

f(x) will only involve functions used LQ�&��DQG�&��

For the Newton-Raphson process, the only differentiation required ZLOO�EH�DV�GH¿QHG�LQ�XQLW�&��


Further Pure Mathematics FP1 (AS) Unit FP1

3 Coordinate systems


Cartesian equations for the parabola and rectangular K\SHUEROD��

Students should be familiar with WKH�HTXDWLRQV� y2

= 4ax or x = at2, y = 2at and

xy = c2 or x ct y ct

= =, ��

Idea of parametric equation for parabola and UHFWDQJXODU�K\SHUEROD�

The idea of (at2, 2at) as a general

point on the parabola is all that is UHTXLUHG�

7KH�IRFXV�GLUHFWUL[�SURSHUW\�RI�WKH�SDUDEROD� Concept of focus and directrix and parabola as locus of points equidistant from focus and GLUHFWUL[�

7DQJHQWV�DQG�QRUPDOV�WR�WKHVH�FXUYHV� Differentiation of

y = 2

12

12a x , y =

cx

2

.

Parametric differentiation is not UHTXLUHG�

4 Matrix Algebra


Linear transformations of column vectors in two GLPHQVLRQV�DQG�WKHLU�PDWUL[�UHSUHVHQWDWLRQ�

The transformation represented by AB is the transformation represented by B followed by the transformation represented by A��

$GGLWLRQ�DQG�VXEWUDFWLRQ�RI�PDWULFHV�

0XOWLSOLFDWLRQ�RI�D�PDWUL[�E\�D�VFDODU�

3URGXFWV�RI�PDWULFHV�


Unit FP1 Further Pure Mathematics FP1 (AS)

(YDOXDWLRQ�RI��é��GHWHUPLQDQWV� Singular and non-singular PDWULFHV�

Inverse of 2 × 2 matrices. Use of the relation (AB)

–1 = B–1A–1

.

&RPELQDWLRQV�RI�WUDQVIRUPDWLRQV� Applications of matrices to JHRPHWULFDO�WUDQVIRUPDWLRQV�

,GHQWL¿FDWLRQ�DQG�XVH�RI�WKH�matrix representation of single and combined transformations IURP��UHÀHFWLRQ�LQ�FRRUGLQDWH�axes and lines y = x, rotation of multiples of 45° about (0, 0) and enlargement about centre (0, 0), with scale factor, (k ��), where k ∈ !.

The inverse (when it exists) of a given WUDQVIRUPDWLRQ�RU�FRPELQDWLRQ�RI�WUDQVIRUPDWLRQV�

Idea of the determinant as an area scale factor in WUDQVIRUPDWLRQV�

5 Series


6XPPDWLRQ�RI�VLPSOH�¿QLWH�VHULHV�� Students should be able to sum series such as

r=

n

r=

n

r=

n

r r r r + 1 1 1

2 2∑ ∑ ∑, , ( )2 �

The method of differences is not UHTXLUHG�

+ –


Further Pure Mathematics FP1 (AS) Unit FP1

6 Proof


3URRI�E\�PDWKHPDWLFDO�LQGXFWLRQ� To include induction proofs for

(i) summation of series

eg show r n nr

n3 1

42 2

11= +

=∑ ( ) or

r r n n n

r

n( ) ( )( )+ = + +

=∑ 1 1 2

31

(ii) divisibility

eg show 32n

+ 11 is divisible E\��

�LLL��¿QGLQJ�JHQHUDO�WHUPV�LQ�D�sequence

eg if un+1

= 3un + 4 with

u

1 = 1, prove that u

n = 3

n – 2�

(iv) matrix products

eg show

− −⎛

⎝⎜

⎞

⎠⎟ =

− −+

⎛

⎝⎜

⎞

⎠⎟

2 19 4

1 39 3 1

n n nn n

�


Unit FP2 Further Pure Mathematics 2 GCE Further Mathematics and GCE Pure Mathematics A2 optional unit


,QHTXDOLWLHV��VHULHV��¿UVW�RUGHU�GLIIHUHQWLDO�HTXDWLRQV��VHFRQG�RUGHU�differential equations; further complex numbers, Maclaurin and 7D\ORU�VHULHV�


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&��&��&��&��DQG�)3��WKHLU�prerequisites, preambles and associated formulae is assumed and PD\�EH�WHVWHG�

Examination 7KH�H[DPLQDWLRQ�ZLOO�FRQVLVW�RI�RQH��í�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�HLJKW�TXHVWLRQV�ZLWK�YDU\LQJ�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�PD\�EH�DWWHPSWHG�



WKH�IROORZLQJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,

1

x, xy



1 Inequalities


The manipulation and solution of algebraic inequalities and inequations, including those LQYROYLQJ�WKH�PRGXOXV�VLJQ�

The solution of inequalities such as

1x a

> xx b− −

⏐x2 – 1⏐ > 2(x + 1).


Unit FP2 Further Pure Mathematics FP2 (A2)

2 Series


6XPPDWLRQ�RI�VLPSOH�¿QLWH�VHULHV�XVLQJ�WKH�PHWKRG�RI�GLIIHUHQFHV�

Students should be able to sum

series such as 1

11 r rr

n

( )+=∑ by using

partial fractions such as

11

1 11r r r r( )+

= −+

�

3 Further Complex Numbers


Euler’s relation eiș = cos ș + i sin ș. Students should be familiar with

cos θ = 12

(eiθ + e

−iθ ) and

sin ș = 12i

(eiθ − e

−iθ ).

De Moivre’s theorem and its application to trigonometric identities and to roots of a complex QXPEHU�

7R�LQFOXGH�¿QGLQJ�cos Qș and sin Pș in terms of powers of sin ș�and cos�ș�and also powers of sin ș�and cos�ș in terms of multiple DQJOHV��6WXGHQWV�VKRXOG�EH�DEOH�WR�prove De Moivre’s theorem for any integer n�

/RFL�DQG�UHJLRQV�LQ�WKH�$UJDQG�GLDJUDP� Loci such as⏐z − a⏐ = b,

⏐z�í�a⏐ = k⏐z�í�b⏐,

arg (z − a) = ȕ, arg z az b

= −−

β and

regions such as ⏐z − a⏐ ≤ ⏐z − b⏐,

⏐z − a⏐ ≤ b.

Elementary transformations from the z-plane to the w�SODQH��

Transformations such as w = z2 and

w =

az + bcz + d , where a, b, c, d ∈ ", may

EH�VHW�


Further Pure Mathematics FP2 (A2) Unit FP2

4 First Order Differential Equations


)XUWKHU�VROXWLRQ�RI�¿UVW�RUGHU�GLIIHUHQWLDO�HTXDWLRQV�ZLWK�VHSDUDEOH�YDULDEOHV�

The formation of the differential HTXDWLRQ�PD\�EH�UHTXLUHG��Students will be expected to obtain particular solutions and DOVR�VNHWFK�PHPEHUV�RI�WKH�IDPLO\�RI�VROXWLRQ�FXUYHV�

First order linear differential equations of the form dd

+ = yx

Py Q where P and Q are functions of x�

Differential equations reducible to the above types by PHDQV�RI�D�JLYHQ�VXEVWLWXWLRQ�

The integrating factor e�Pdx may be TXRWHG�ZLWKRXW�SURRI�

5 Second Order Differential Equations


The linear second order differential equation

a yx

b yx

cy x2d

d + d

d + = f( )2 where a, b and c are real

constants and the particular integral can be found by LQVSHFWLRQ�RU�WULDO�

The auxiliary equation may have real distinct, equal or complex URRWV� f(x) will have one of the forms kepx

, A + Bx, p + qx + cx2 or m cos Ȧ[ + n sin Ȧ[.

Students should be familiar with the terms ‘complementary IXQFWLRQ¶�DQG�µSDUWLFXODU�LQWHJUDO¶�

Students should be able to solve equations of the form

d2 y

xd 2 + 4y = sin 2x�

Differential equations reducible to the above types by PHDQV�RI�D�JLYHQ�VXEVWLWXWLRQ�



6 Maclaurin and Taylor series


7KLUG�DQG�KLJKHU�RUGHU�GHULYDWLYHV�

'HULYDWLRQ�DQG�XVH�RI�0DFODXULQ�VHULHV�

'HULYDWLRQ�DQG�XVH�RI�7D\ORU�VHULHV�

The derivation of the series expansion of ex

, sin x, cos x, ln (1 + x)

and other simple functions may be UHTXLUHG�

The derivation, for example, of the expansion of sin x in ascending powers of (x − π) up to and including the term in (x − π)

3.

Use of Taylor series method for series solutions of GLIIHUHQWLDO�HTXDWLRQV�

Students may, for example, be UHTXLUHG�WR�¿QG�WKH�VROXWLRQ�LQ�powers of x as far as the term in x4, of the differential equation

d2 yxd 2

+ x dy

xd + y = 0,

such that y = 1, dy

xd = 0 at x = 0�

7 Polar Coordinates


Polar coordinates (r, ș ), r ≥ 0��

Use of the formula 12

2r dα

β

θ⌠

⌡

⎮⎮

IRU�DUHD�

7KH�VNHWFKLQJ�RI�FXUYHV�VXFK�DV

ș = Į, r = p sec (Į�í�ș), r = a,

r = 2a cos ș, r = Nș, r = a(1 ± cos ș),

r = a(3 + 2 cos ș), r = a cos 2ș and

r2 = a2

cos 2ș PD\�EH�VHW�

7KH�DELOLW\�WR�¿QG�WDQJHQWV�SDUDOOHO�to, or at right angles to, the initial OLQH�LV�H[SHFWHG�


Unit FP3 Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A2 optional unit


Further matrix algebra; vectors, hyperbolic functions; differentiation; integration, further coordinate systems


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�&��&��&��&��DQG�)3��WKHLU�prerequisites, preambles and associated formulae is assumed and PD\�EH�WHVWHG�

Examination 7KH�H[DPLQDWLRQ�ZLOO�FRQVLVW�RI�RQH��í�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�HLJKW�TXHVWLRQV�ZLWK�YDU\LQJ�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�PD\�EH�DWWHPSWHG�



WKH�IROORZLQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x,

1

x, xy





1 Hyperbolic functions


'H¿QLWLRQ�RI�WKH�VL[�K\SHUEROLF�IXQFWLRQV�LQ�WHUPV�RI�H[SRQHQWLDOV��*UDSKV�DQG�SURSHUWLHV�RI�WKH�K\SHUEROLF�IXQFWLRQV�

For example, cosh x = ½(ex + e

-x),

sech x = 1

cosh x =

2e ex x+ − .

Students should be able to derive and use simple identities such as

cosh2 x – sinh

2 x = 1 and

cosh2 x + sinh

2 x = cosh 2x

and to solve equations such as

a cosh x + b sinh x = c.

Inverse hyperbolic functions, their graphs, properties DQG�ORJDULWKPLF�HTXLYDOHQWV�

Eg arsinh x = ln[x��¥��x2)].

Students may be required to SURYH�WKLV�DQG�VLPLODU�UHVXOWV�

2 Further Coordinate Systems


Cartesian and parametric equations for the ellipse DQG�K\SHUEROD�

([WHQVLRQ�RI�ZRUN�IURP�)3��

Students should be familiar with WKH�HTXDWLRQV��

xa

yb

2

2

2

2 1+ = ; x = a cos t, y = b sin t.

xa

yb

2

2

2

2 1− = ; x = a sec t,

y = b tan t, x = a cosh t, y = b sinh t.

The focus-directrix properties of the ellipse and K\SHUEROD��LQFOXGLQJ�WKH�HFFHQWULFLW\�

For example, students should NQRZ�WKDW��IRU�WKH�HOOLSVH� b2

= a2 (1 – e2

), the foci are (ae, 0) and

(−ae, 0) and the equations of the

directrices are x = + ae

and

x = −

ae

.



7DQJHQWV�DQG�QRUPDOV�WR�WKHVH�FXUYHV� The condition for y = mx + c to be a tangent to these curves is H[SHFWHG�WR�EH�NQRZQ�

6LPSOH�ORFL�SUREOHPV�

3 Differentiation


Differentiation of hyperbolic functions and H[SUHVVLRQV�LQYROYLQJ�WKHP�

For example, tanh 3x, x sinh2 x,

�

Differentiation of inverse functions, including WULJRQRPHWULF�DQG�K\SHUEROLF�IXQFWLRQV�

For example, arcsin x + x√(1 – x2),

12 artanh x2

.

cosh( )

21x

x√ +



4 Integration


Integration of hyperbolic functions and expressions LQYROYLQJ�WKHP�

Integration of inverse trigonometric and hyperbolic IXQFWLRQV�

For example,

�DUVLQK�x dx��DUFWDQ�x dx.

Integration using hyperbolic and trigonometric VXEVWLWXWLRQV�

To include the integrals of 1/(a2

+ x2), 1/√(a2

− x2), 1/√(a2

+ x2),

1/√(x2 – a2

).

Use of substitution for integrals involving quadratic VXUGV�

In more complicated cases, VXEVWLWXWLRQV�ZLOO�EH�JLYHQ�

7KH�GHULYDWLRQ�DQG�XVH�RI�VLPSOH�UHGXFWLRQ�IRUPXODH� Students should be able to derive formulae such as

nIn = (n – 1)I

n – 2 , n ≥ 2,

for In = sin dnx x

0

2π

∫ ,

In + 2

=

2 11

sin ( )n xn

++

+ In

for In =

sinsin

dnxx

x∫ , n > 0.

The calculation of arc length and the area of a VXUIDFH�RI�UHYROXWLRQ�

The equation of the curve may be given in cartesian or parametric IRUP��(TXDWLRQV�LQ�SRODU�IRUP�ZLOO�QRW�EH�VHW�



5 Vectors


The vector product a × b and the triple scalar product a . b × c.

Use of vectors in problems involving points, lines and SODQHV��

The equation of a line in the form (r − a) × b = 0.

The interpretation of ⏐a × b⏐ as an area and a . b × c�DV�D�YROXPH�

Students may be required to use HTXLYDOHQW�FDUWHVLDQ�IRUPV�DOVR�

Applications to include

(i) distance from a point to a plane,

(ii) line of intersection of two planes,

(iii) shortest distance between two VNHZ�OLQHV�

The equation of a plane in the forms

r.n = p, r = a + sb + tc.

Students may be required to use HTXLYDOHQW�FDUWHVLDQ�IRUPV�DOVR�



6 Further Matrix Algebra


Linear transformations of column vectors in two and WKUHH�GLPHQVLRQV�DQG�WKHLU�PDWUL[�UHSUHVHQWDWLRQ�

&RPELQDWLRQ�RI�WUDQVIRUPDWLRQV��3URGXFWV�RI�PDWULFHV�

7UDQVSRVH�RI�D�PDWUL[��

([WHQVLRQ�RI�ZRUN�IURP�)3��WR��GLPHQVLRQV�

The transformation represented by AB is the transformation represented by B followed by the transformation represented by A��

Use of the relation (AB)T = BTAT

.

Evaluation of 3 ×��GHWHUPLQDQWV�� Singular and non-singular PDWULFHV�

Inverse of 3 ×��PDWULFHV� Use of the relation (AB)−1

= B−1A−1.

The inverse (when it exists) of a given WUDQVIRUPDWLRQ�RU�FRPELQDWLRQ�RI�WUDQVIRUPDWLRQV�

Eigenvalues and eigenvectors of 2 × 2 and 3 × 3 PDWULFHV�

Normalised vectors may be UHTXLUHG�

5HGXFWLRQ�RI�V\PPHWULF�PDWULFHV�WR�GLDJRQDO�IRUP� 6WXGHQWV�VKRXOG�EH�DEOH�WR�¿QG�an orthogonal matrix P such that PTAP LV�GLDJRQDO�


Unit M1 Mechanics 1 GCE AS and GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) AS optional unit

M1.1 Unit description

Mathematical models in mechanics; vectors in mechanics; NLQHPDWLFV�RI�D�SDUWLFOH�PRYLQJ�LQ�D�VWUDLJKW�OLQH��G\QDPLFV�RI�D�particle moving in a straight line or plane; statics of a particle; PRPHQWV�

M1.2 Assessment information

Prerequisites 6WXGHQWV�DUH�H[SHFWHG�WR�KDYH�D�NQRZOHGJH�RI�&��LWV�SUHDPEOHV�DQG�DVVRFLDWHG�IRUPXODH�DQG�RI�YHFWRUV�LQ�WZR�GLPHQVLRQV�

Examination 7KH�H[DPLQDWLRQ�ZLOO�FRQVLVW�RI�RQH��í�KRXU�SDSHU��,W�ZLOO�FRQWDLQ�DERXW�VHYHQ�TXHVWLRQV�ZLWK�YDU\LQJ�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�PD\�EH�DWWHPSWHG�


the folORZLQJ�NH\V� +, −, ×, ÷��ʌ� x2, √x,

1

x , xy

, ln x, ex, sine, cosine and



Unit M1 Mechanics M1 (AS)

Formulae 6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�DQ\�RWKHU�IRUPXODH�ZKLFK�PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW�LQFOXGHG�LQ�WKH�ERRNOHW� Mathematical Formulae including Statistical Formulae and Tables��ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�

6WXGHQWV�ZLOO�EH�H[SHFWHG�WR�NQRZ�DQG�EH�DEOH�WR�UHFDOO�DQG�XVH�WKH�IROORZLQJ�IRUPXODH�

Momentum = mv

Impulse = mv – mu

)RU�FRQVWDQW�DFFHOHUDWLRQ�

v = u + at

s = ut + 12 at 2

s = vt – 12 at 2

v 2 = u2

+ 2as

s = 12 (u + v)t

1 Mathematical Models in Mechanics


The basic ideas of mathematical modelling as applied LQ�0HFKDQLFV�

Students should be familiar with WKH�WHUPV��SDUWLFOH��ODPLQD��ULJLG�body, rod (light, uniform, non-uniform), inextensible string, smooth and rough surface, light VPRRWK�SXOOH\��EHDG��ZLUH��SHJ��Students should be familiar with the assumptions made in using WKHVH�PRGHOV�


Mechanics M1 (AS) Unit M1

2 Vectors in Mechanics


0DJQLWXGH�DQG�GLUHFWLRQ�RI�D�YHFWRU��5HVXOWDQW�RI�YHFWRUV�PD\�DOVR�EH�UHTXLUHG�

Students may be required to resolve a vector into two components or use a vector GLDJUDP��4XHVWLRQV�PD\�EH�VHW�involving the unit vectors i and j�

Application of vectors to displacements, velocities, DFFHOHUDWLRQV�DQG�IRUFHV�LQ�D�SODQH�

Use of velocity =

change of displacementtime

in the case of constant velocity, and of

acceleration = change of velocity

time

in the case of constant DFFHOHUDWLRQ��ZLOO�EH�UHTXLUHG�

3 Kinematics of a particle moving in a straight line


0RWLRQ�LQ�D�VWUDLJKW�OLQH�ZLWK�FRQVWDQW�DFFHOHUDWLRQ� Graphical solutions may be required, including displacement-time, velocity-time, speed-time and acceleration-time JUDSKV��.QRZOHGJH�DQG�XVH�RI�formulae for constant acceleration ZLOO�EH�UHTXLUHG�


Unit M1 Mechanics M1 (AS)

4 Dynamics of a particle moving in a straight line or plane


7KH�FRQFHSW�RI�D�IRUFH��1HZWRQ¶V�ODZV�RI�PRWLRQ� Simple problems involving constant acceleration in scalar form or as a vector of the form ai + bj.

Simple applications including the motion of two FRQQHFWHG�SDUWLFOHV�

Problems may include

(i) the motion of two connected particles moving in a straight line or under gravity when the forces on each particle are constant; problems involving VPRRWK�¿[HG�SXOOH\V�DQG�RU�pegs may be set;

(ii) motion under a force which FKDQJHV�IURP�RQH�¿[HG�YDOXH�to another, eg a particle hitting the ground;

(iii) motion directly up or down a smooth or rough inclined SODQH�

0RPHQWXP�DQG�LPSXOVH��7KH�LPSXOVH�PRPHQWXP�SULQFLSOH��7KH�SULQFLSOH�RI�FRQVHUYDWLRQ�RI�PRPHQWXP�DSSOLHG�WR�WZR�SDUWLFOHV�FROOLGLQJ�GLUHFWO\�

Knowledge of Newton’s law RI�UHVWLWXWLRQ�LV�QRW�UHTXLUHG��3UREOHPV�ZLOO�EH�FRQ¿QHG�WR�WKRVH�RI�D�RQH�GLPHQVLRQDO�QDWXUH�

&RHI¿FLHQW�RI�IULFWLRQ� An understanding of F = µR when a SDUWLFOH�LV�PRYLQJ�


Mechanics M1 (AS) Unit M1

5 Statics of a particle


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Only simple cases of the application of the conditions for equilibrium to uncomplicated V\VWHPV�ZLOO�EH�UHTXLUHG�

&RHI¿FLHQW�RI�IULFWLRQ� An understanding of F ≤ µR in a VLWXDWLRQ�RI�HTXLOLEULXP�

6 Moments


0RPHQW�RI�D�IRUFH� Simple problems involving coplanar parallel forces acting on a body and conditions for HTXLOLEULXP�LQ�VXFK�VLWXDWLRQV�


Unit M2 Mechanics 2 GCE Mathematics, GCE AS and GCE Further Mathematics, GCE AS and GCE Further Mathematics (Additional) A2 optional unit


Kinematics of a particle moving in a straight line or plane; centres RI�PDVV��ZRUN�DQG�HQHUJ\��FROOLVLRQV��VWDWLFV�RI�ULJLG�ERGLHV�


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQ�IRU�0��DQG�LWV�SUHUHTXLVLWHV�DQG�DVVRFLDWHG�IRUPXODH��WRJHWKHU�ZLWK�D�NQRZOHGJH�RI�DOJHEUD��WULJRQRPHWU\��GLIIHUHQWLDWLRQ�DQG�LQWHJUDWLRQ��DV�VSHFL¿HG�LQ�&��&��DQG�&��LV�DVVXPHG�DQG�PD\�EH�WHVWHG�



the followiQJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,

1

x, xy





Kinetic energy = 12

mv 2

Potential energy = mgh


Unit M2 Mechanics M2 (A2)

1 Kinematics of a particle moving in a straight line or plane


Motion in a vertical plane with constant acceleration, HJ�XQGHU�JUDYLW\��

6LPSOH�FDVHV�RI�PRWLRQ�RI�D�SURMHFWLOH�

Velocity and acceleration when the displacement is a IXQFWLRQ�RI�WLPH�

The setting up and solution of

equations of the form t

x

d

d = f(t) or

t

v

d

d

= g(t) will be consistent with the

level of calculus in C2.

Differentiation and integration of a vector with UHVSHFW�WR�WLPH�

For example, given that,

r = t2i + t3/2j,�¿QG r and r at a given WLPH�

2 Centres of mass


Centre of mass of a discrete mass distribution in one DQG�WZR�GLPHQVLRQV�

&HQWUH�RI�PDVV�RI�XQLIRUP�SODQH�¿JXUHV��DQG�VLPSOH�FDVHV�RI�FRPSRVLWH�SODQH�¿JXUHV�

The use of an axis of symmetry will be acceptable where DSSURSULDWH��8VH�RI�LQWHJUDWLRQ�LV�QRW�UHTXLUHG��)LJXUHV�PD\�LQFOXGH�the shapes referred to in the IRUPXODH�ERRN��5HVXOWV�JLYHQ�LQ�WKH�IRUPXODH�ERRN�PD\�EH�TXRWHG�ZLWKRXW�SURRI�


Mechanics M2 (A2) Unit M2

6LPSOH�FDVHV�RI�HTXLOLEULXP�RI�D�SODQH�ODPLQD� The lamina may

�L�� EH�VXVSHQGHG�IURP�D�¿[HG�point;

�LL�� IUHH�WR�URWDWH�DERXW�D�¿[HG�KRUL]RQWDO�D[LV�

(iii) be put on an�LQFOLQHG�SODQH��

3 Work and energy


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Problems involving motion under D�FRQVWDQW�UHVLVWDQFH�DQG�RU�XS�and down an inclined plane may EH�VHW�

4 Collisions


0RPHQWXP�DV�D�YHFWRU��7KH�LPSXOVH�PRPHQWXP�SULQFLSOH�LQ�YHFWRU�IRUP��&RQVHUYDWLRQ�RI�OLQHDU�PRPHQWXP�

'LUHFW�LPSDFW�RI�HODVWLF�SDUWLFOHV��1HZWRQ¶V�ODZ�RI�UHVWLWXWLRQ��/RVV�RI�PHFKDQLFDO�HQHUJ\�GXH�WR�LPSDFW�

6WXGHQWV�ZLOO�EH�H[SHFWHG�WR�NQRZ�and use the inequalities 0 ≤ e ≤ 1 (where e�LV�WKH�FRHI¿FLHQW�RI�UHVWLWXWLRQ��

Successive impacts of up to three particles or two SDUWLFOHV�DQG�D�VPRRWK�SODQH�VXUIDFH�

Collision with a plane surface will QRW�LQYROYH�REOLTXH�LPSDFW�



5 Statics of rigid bodies


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(TXLOLEULXP�RI�ULJLG�ERGLHV� Problems involving parallel and QRQ�SDUDOOHO�FRSODQDU�IRUFHV��Problems may include rods or ladders resting against smooth or rough vertical walls and on VPRRWK�RU�URXJK�JURXQG�


Unit M3 Mechanics 3 GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) A2 optional unit


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Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�0��DQG�0��DQG�WKHLU�SUHUHTXLVLWHV�DQG�DVVRFLDWHG�IRUPXODH��WRJHWKHU�ZLWK�D�NQRZOHGJH�RI�GLIIHUHQWLDWLRQ��LQWHJUDWLRQ�DQG�GLIIHUHQWLDO�HTXDWLRQV��DV�VSHFL¿HG�LQ�&��&��&��DQG�&��LV�DVVXPHG�DQG�PD\�EH�WHVWHG�

Examination 7KH�H[DPLQDWLRQ�ZLOO�FRQVLVW�RI�RQH��í�KRXU�SDSHU��7KH�SDSHU�ZLOO�FRQWDLQ�DERXW�VHYHQ�TXHVWLRQV�ZLWK�YDU\LQJ�PDUN�DOORFDWLRQV�SHU�TXHVWLRQ�ZKLFK�ZLOO�EH�VWDWHG�RQ�WKH�SDSHU��$OO�TXHVWLRQV�PD\�EH�DWWHPSWHG�


the folORZLQJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,

1

x, xy







The tension in an elastic string = λxl

The energy stored in an elastic string = λxl

2

2

)RU�6+0�

x = −ω 2x,

x = a cos ω t or x = a sin ω t ,

v 2 = ω 2

(a2 – x 2

),

T = 2πω

1 Further kinematics


Kinematics of a particle moving in a straight line when the acceleration is a function of the displacement (x), or time (t).

The setting up and solution of

equations where t

v

d

d = f(t),

vx

v

d

d = f(x),

dd

= f( )xt

x or

dd

= f( )xt

t

will be consistent with the level of calculus required in units C1, &��DQG�&��



2 Elastic strings and springs


(ODVWLF�VWULQJV�DQG�VSULQJV��+RRNH¶V�ODZ�

(QHUJ\�VWRUHG�LQ�DQ�HODVWLF�VWULQJ�RU�VSULQJ� 6LPSOH�SUREOHPV�XVLQJ�WKH�ZRUN�HQHUJ\�SULQFLSOH�LQYROYLQJ�NLQHWLF�energy, potential energy and HODVWLF�HQHUJ\�

3 Further dynamics


Newton’s laws of motion, for a particle moving in RQH�GLPHQVLRQ��ZKHQ�WKH�DSSOLHG�IRUFH�LV�YDULDEOH�

The solution of the resulting equations will be consistent with the level of calculus in units C2, &��DQG�&��3UREOHPV�PD\�LQYROYH�the law of gravitation, ie the LQYHUVH�VTXDUH�ODZ�

6LPSOH�KDUPRQLF�PRWLRQ� Proof that a particle moves with simple harmonic motion in a given

situation may be required (ie

showing that = −ω 2x x��

Geometric or calculus methods RI�VROXWLRQ�ZLOO�EH�DFFHSWDEOH��Students will be expected to be familiar with standard formulae, which may be quoted without SURRI�

Oscillations of a particle attached to the end of an HODVWLF�VWULQJ�RU�VSULQJ�

Oscillations will be in the direction RI�WKH�VWULQJ�RU�VSULQJ�RQO\�



4 Motion in a circle


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5DGLDO�DFFHOHUDWLRQ�LQ�FLUFXODU�PRWLRQ��7KH�IRUPV�rω 2

and vr

2�DUH�UHTXLUHG�

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Problems involving the ‘conical pendulum’, an elastic string, PRWLRQ�RQ�D�EDQNHG�VXUIDFH��DV�well as other contexts, may be VHW�

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5 Statics of rigid bodies


Centre of mass of uniform rigid bodies and simple FRPSRVLWH�ERGLHV�

7KH�XVH�RI�LQWHJUDWLRQ�DQG�RU�symmetry to determine the centre of mass of a uniform body will be UHTXLUHG�

6LPSOH�FDVHV�RI�HTXLOLEULXP�RI�ULJLG�ERGLHV� To include

(i) suspension of a body from a ¿[HG�SRLQW�

(ii) a rigid body placed on a KRUL]RQWDO�RU�LQFOLQHG�SODQH�


Unit M4 Mechanics 4 GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) A2 optional unit


Relative motion; elastic collisions in two dimensions; further motion RI�SDUWLFOHV�LQ�RQH�GLPHQVLRQ��VWDELOLW\�


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�0��0��DQG�0��DQG�WKHLU�SUHUHTXLVLWHV�DQG�DVVRFLDWHG�IRUPXODH��WRJHWKHU�ZLWK�D�NQRZOHGJH�

of the calculus covered in FP2 and 1

2 2a xx

+⌠

⌡⎮⎮ d , is assumed and

may be WHVWHG��



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1

x, xy




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Av

B = v

A – v

B



1 Relative motion


Relative motion of two particles, including relative GLVSODFHPHQW�DQG�UHODWLYH�YHORFLW\�

Problems may be set in vector form and may involve problems of interception or closest approach including the determination of course required for closest DSSURDFK�

2 Elastic collisions in two dimensions


Oblique impact of smooth elastic spheres and a VPRRWK�VSKHUH�ZLWK�D�¿[HG�VXUIDFH�

3 Further motion of particles in one dimension


Resisted motion of a particle moving in a straight OLQH�

The resisting forces may include the forms a + bv and a + bv2

where a and b are constants and v is the VSHHG�

'DPSHG�DQG�RU�IRUFHG�KDUPRQLF�PRWLRQ� The damping to be proportional to WKH�VSHHG��6ROXWLRQ�RI�WKH�UHOHYDQW�differential equations will be H[SHFWHG�

4 Stability


Finding equilibrium positions of a system from FRQVLGHUDWLRQ�RI�LWV�SRWHQWLDO�HQHUJ\�

Positions of stable and unstable HTXLOLEULXP�RI�D�V\VWHP�


Unit M5 Mechanics 5 GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional)A2 optional unit


Applications of vectors in mechanics; variable mass; moments of LQHUWLD�RI�D�ULJLG�ERG\��URWDWLRQ�RI�D�ULJLG�ERG\�DERXW�D�¿[HG�VPRRWK�D[LV�


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the followLQJ�NH\V� +, −, ×, ÷��ʌ��x2, √x,

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Angular momentum = Iθ

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2

L = Iθ

L tt

td

1

2

∫ = Iω 2 – Iω

1

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ω 1 = ω

0 + α t

ω 1

2 = ω

0

2

+ 2αș

ș�= ω 0t +

2

1 α t2

ș�=

( )ω ω0 1

2+ t



1 Applications of vectors in mechanics


6ROXWLRQ�RI�VLPSOH�YHFWRU�GLIIHUHQWLDO�HTXDWLRQV� Vector differential equations such as

ddvt = kv,

2

22 2d

d + 2

dd

+ ( + ) = r r

rt

kt

k n f(t)

where k and n are constants,

dd

er r it

t+ =4 .

Use of an integrating factor may EH�UHTXLUHG�

:RUN�GRQH�E\�D�FRQVWDQW�IRUFH�XVLQJ�D�VFDODU�SURGXFW�

0RPHQW�RI�D�IRUFH�XVLQJ�D�YHFWRU�SURGXFW� The moment of a force F about O�LV�GH¿QHG�DV�r�× F, where r is the position vector of the point of application of F�

The analysis of simple systems of forces in three GLPHQVLRQV�DFWLQJ�RQ�D�ULJLG�ERG\�

The reduction of a system of forces acting on a body to a single force, single couple or a couple and a force acting through a VWDWHG�SRLQW�

2 Variable mass


0RWLRQ�RI�D�SDUWLFOH�ZLWK�YDU\LQJ�PDVV� Students may be required to derive an equation of motion from ¿UVW�SULQFLSOHV�E\�FRQVLGHULQJ�WKH�change in momentum over a small WLPH�LQWHUYDO�



3 Moments of inertia of a rigid body


Moments of inertia and radius of gyration of standard DQG�FRPSRVLWH�ERGLHV�

Use of integration including the proof of the standard results given LQ�WKH�IRUPXODH�ERRNOHW�ZLOO�EH�UHTXLUHG�

7KH�SDUDOOHO�DQG�SHUSHQGLFXODU�D[HV�WKHRUHPV�

4 Rotation of a rigid body about a fixed smooth axis


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Use of conservation of energy will EH�UHTXLUHG��&DOFXODWLRQ�RI�WKH�IRUFH�RQ�WKH�D[LV�ZLOO�EH�UHTXLUHG�

$QJXODU�PRPHQWXP�

.LQHWLF�HQHUJ\�

&RQVHUYDWLRQ�RI�DQJXODU�PRPHQWXP��7KH�HIIHFW�RI�DQ�impulse on a rigid body which is free to rotate about D�¿[HG�D[LV�

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Unit S1 Statistics 1 GCE AS and GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) AS optional unit

S1.1 Unit description

Mathematical models in probability and statistics; representation and summary of data; probability; correlation and regression; discrete random variables; discrete distributions; the Normal GLVWULEXWLRQ�

S1.2 Assessment information



the followinJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,

1

x, xy




Unit S1 Statistics S1 (AS)

Formulae 6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�IRUPXODH�ZKLFK�PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW�LQFOXGHG�LQ�WKH�ERRNOHW��Mathematical Formulae including Statistical Formulae and Tables, ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�


Mean = x = Σxn

or ΣΣfxf

Standard deviation = √(Variance)

Interquartile range = IQR = Q3 – Q

1

P(A′) = 1 – P(A)

For independent events A and B,

P(B⏐A) = P(B), P(A⏐B) = P(A),

P(A ∩ B) = P(A) P(B)

E(aX + b) = aE(X) + b

Var (aX + b) = a2 Var (X)

&XPXODWLYH�GLVWULEXWLRQ�IXQFWLRQ�IRU�D�GLVFUHWH�UDQGRP�YDULDEOH��

F(x0 ) = P(X ≤ x0) = p( )x

x x≤∑

0

Standardised Normal Random Variable Z = X − µ

σ

where X ∼ N (µ , σ 2)

1 Mathematical models in probability and statistics


The basic ideas of mathematical modelling as applied LQ�SUREDELOLW\�DQG�VWDWLVWLFV�


Statistics S1 (AS) Unit S1

2 Representation and summary of data


+LVWRJUDPV��VWHP�DQG�OHDI�GLDJUDPV��ER[�SORWV� Using histograms, stem and leaf diagrams and box plots to FRPSDUH�GLVWULEXWLRQV��

%DFN�WR�EDFN�VWHP�DQG�OHDI�GLDJUDPV�PD\�EH�UHTXLUHG�

Drawing of histograms, stem and leaf diagrams or box plots will not be the direct focus of examination TXHVWLRQV�

0HDVXUHV�RI�ORFDWLRQ�²�PHDQ��PHGLDQ��PRGH� Calculation of mean, mode and median, range and interquartile range will not be the direct focus RI�H[DPLQDWLRQ�TXHVWLRQV�

Students will be expected to draw simple inferences and give interpretations to measures RI�ORFDWLRQ�DQG�GLVSHUVLRQ��6LJQL¿FDQFH�WHVWV�ZLOO�QRW�EH�H[SHFWHG�

Data may be discrete, continuous, JURXSHG�RU�XQJURXSHG��8QGHUVWDQGLQJ�DQG�XVH�RI�FRGLQJ�

Measures of dispersion — variance, standard GHYLDWLRQ��UDQJH�DQG�LQWHUSHUFHQWLOH�UDQJHV�

Simple interpolation may EH�UHTXLUHG��,QWHUSUHWDWLRQ�of measures of location and GLVSHUVLRQ�

6NHZQHVV��&RQFHSWV�RI�RXWOLHUV� 6WXGHQWV�PD\�EH�DVNHG�WR�illustrate the location of outliers RQ�D�ER[�SORW��$Q\�UXOH�WR�LGHQWLI\�RXWOLHUV�ZLOO�EH�VSHFL¿HG�LQ�WKH�TXHVWLRQ�


Unit S1 Statistics S1 (AS)

3 Probability


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6DPSOH�VSDFH��([FOXVLYH�DQG�FRPSOHPHQWDU\�HYHQWV��&RQGLWLRQDO�SUREDELOLW\�

Understanding and use of

P(A′) = 1 − P(A),

P(A ∪ B) = P(A) + P(B) − P(A ∩ B),

P(A ∩ B) = P(A) P(B⏐A).

,QGHSHQGHQFH�RI�WZR�HYHQWV� P(B⏐A) = P(B), P(A⏐B) = P(A),

P(A ∩ B) = P(A) P(B).

6XP�DQG�SURGXFW�ODZV� Use of tree diagrams and Venn GLDJUDPV��6DPSOLQJ�ZLWK�DQG�ZLWKRXW�UHSODFHPHQW�

4 Correlation and regression


6FDWWHU�GLDJUDPV��/LQHDU�UHJUHVVLRQ� Calculation of the equation of a linear regression line using WKH�PHWKRG�RI�OHDVW�VTXDUHV��Students may be required to draw this regression line on a scatter GLDJUDP�

Explanatory (independent) and response �GHSHQGHQW��YDULDEOHV��$SSOLFDWLRQV�DQG�LQWHUSUHWDWLRQV��

8VH�WR�PDNH�SUHGLFWLRQV�ZLWKLQ�WKH�range of values of the explanatory variable and the dangers of H[WUDSRODWLRQ��'HULYDWLRQV�ZLOO�QRW�EH�UHTXLUHG��9DULDEOHV�RWKHU�WKDQ�x and y�PD\�EH�XVHG��/LQHDU�FKDQJH�RI�YDULDEOH�PD\�EH�UHTXLUHG�

7KH�SURGXFW�PRPHQW�FRUUHODWLRQ�FRHI¿FLHQW��LWV�XVH��LQWHUSUHWDWLRQ�DQG�OLPLWDWLRQV�

Derivations and tests of VLJQL¿FDQFH�ZLOO�QRW�EH�UHTXLUHG�


Statistics S1 (AS) Unit S1

5 Discrete random variables


7KH�FRQFHSW�RI�D�GLVFUHWH�UDQGRP�YDULDEOH�

The probability function and the cumulative GLVWULEXWLRQ�IXQFWLRQ�IRU�D�GLVFUHWH�UDQGRP�YDULDEOH�

Simple uses of the probability function p(x) where p(x) = P(X = x).

Use of the cumulative distribution IXQFWLRQ��

F(x0 ) = P(X��x

0) = p( )x

x x≤∑

0

0HDQ�DQG�YDULDQFH�RI�D�GLVFUHWH�UDQGRP�YDULDEOH� Use of E(X), E(X 2) for calculating

the variance of X.

Knowledge and use of

E(aX + b) = aE(X) + b,

Var (aX + b) = a2 Var (X).

7KH�GLVFUHWH�XQLIRUP�GLVWULEXWLRQ� The mean and variance of this GLVWULEXWLRQ�

6 The Normal distribution


The Normal distribution including the mean, variance and use of tables of the cumulative distribution IXQFWLRQ�

Knowledge of the shape and the symmetry of the distribution LV�UHTXLUHG��.QRZOHGJH�RI�WKH�probability density function is QRW�UHTXLUHG��'HULYDWLRQ�RI�WKH�mean, variance and cumulative distribution function is not UHTXLUHG��,QWHUSRODWLRQ�LV�QRW�QHFHVVDU\��4XHVWLRQV�PD\�LQYROYH�the solution of simultaneous HTXDWLRQV�


Unit S2 Statistics 2 GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) A2 optional unit


The Binomial and Poisson distributions; continuous random YDULDEOHV��FRQWLQXRXV�GLVWULEXWLRQV��VDPSOHV��K\SRWKHVLV�WHVWV�


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQ�IRU�6��DQG�LWV�SUHUHTXLVLWHV�DQG�DVVRFLDWHG�IRUPXODH��WRJHWKHU�ZLWK�D�NQRZOHGJH�RI�GLIIHUHQWLDWLRQ�DQG�LQWHJUDWLRQ�RI�SRO\QRPLDOV��ELQRPLDO�FRHI¿FLHQWV�LQ�FRQQHFWLRQ�with the binomial distribution and the evaluation of the exponential IXQFWLRQ�LV�DVVXPHG�DQG�PD\�EH�WHVWHG�



the follRZLQJ�NH\V� +, −, ×, ÷, ʌ, x2, √x,

1

x , xy, ln x, e

x, x!, sine, cosine and



Unit S2 Statistics S2 (A2)



For the continuous random variable X having probability density function f(x),

P(a < X ≤ b) = f( ) dx xa

b

∫ .

f(x) = .

1 The Binomial and Poisson distributions


7KH�ELQRPLDO�DQG�3RLVVRQ�GLVWULEXWLRQV� Students will be expected to use these distributions to model a real-world situation and to comment critically on their DSSURSULDWHQHVV��&XPXODWLYH�probabilities by calculation or by UHIHUHQFH�WR�WDEOHV��

Students will be expected to use the additive property of the Poisson distribution — eg if the number of events per minute ∼ Po(λ) then the number of events per 5 minutes ∼ Po(5λ).

The mean and variance of the binomial and Poisson GLVWULEXWLRQV�

1R�GHULYDWLRQV�ZLOO�EH�UHTXLUHG�

The use of the Poisson distribution as an DSSUR[LPDWLRQ�WR�WKH�ELQRPLDO�GLVWULEXWLRQ�

dF(d

xx

)


Statistics S2 (A2) Unit S2

2 Continuous random variables


7KH�FRQFHSW�RI�D�FRQWLQXRXV�UDQGRP�YDULDEOH�

The probability density function and the cumulative distribution function for a continuous random YDULDEOH��

Use of the probability density function f(x), where

P(a < X ≤ b) = f( ) dx xa

b

∫ .

Use of the cumulative distribution function

F(x0) = P(X ≤ x

0) = f( ) dx x

x

−∞∫0

.

7KH�IRUPXODH�XVHG�LQ�GH¿QLQJ�f(x) will be restricted to simple polynomials which may be H[SUHVVHG�SLHFHZLVH�

Relationship between density and distribution IXQFWLRQV�

f(x) =

.

0HDQ�DQG�YDULDQFH�RI�FRQWLQXRXV�UDQGRP�YDULDEOHV�

Mode, median and quartiles of continuous random YDULDEOHV�

3 Continuous distributions


7KH�FRQWLQXRXV�XQLIRUP��UHFWDQJXODU��GLVWULEXWLRQ� Including the derivation of the mean, variance and cumulative GLVWULEXWLRQ�IXQFWLRQ�

Use of the Normal distribution as an approximation to the binomial distribution and the Poisson distribution, with the application of the continuity FRUUHFWLRQ�

dF(d

xx

)



4 Hypothesis tests


3RSXODWLRQ��FHQVXV�DQG�VDPSOH��6DPSOLQJ�XQLW��VDPSOLQJ�IUDPH�

6WXGHQWV�ZLOO�EH�H[SHFWHG�WR�NQRZ�the advantages and disadvantages associated with a census and a VDPSOH�VXUYH\�

&RQFHSWV�RI�D�VWDWLVWLF�DQG�LWV�VDPSOLQJ�GLVWULEXWLRQ�

&RQFHSW�DQG�LQWHUSUHWDWLRQ�RI�D�K\SRWKHVLV�WHVW��1XOO�DQG�DOWHUQDWLYH�K\SRWKHVHV�

Use of hypothesis tests for UH¿QHPHQW�RI�PDWKHPDWLFDO�PRGHOV�

&ULWLFDO�UHJLRQ� Use of a statistic as a test VWDWLVWLF�

2QH�WDLOHG�DQG�WZR�WDLOHG�WHVWV�

Hypothesis tests for the parameter p of a binomial distribution and for the mean of a Poisson GLVWULEXWLRQ�

6WXGHQWV�DUH�H[SHFWHG�WR�NQRZ�how to use tables to carry RXW�WKHVH�WHVWV��4XHVWLRQV�may also be set not involving WDEXODU�YDOXHV��7HVWV�RQ�VDPSOH�proportion involving the normal DSSUR[LPDWLRQ�ZLOO�QRW�EH�VHW�


Unit S3 Statistics 3 GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) A2 optional unit


Combinations of random variables; sampling; estimation, FRQ¿GHQFH�LQWHUYDOV�DQG�WHVWV��JRRGQHVV�RI�¿W�DQG�FRQWLQJHQF\�tables; regression and correlation


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQV�IRU�6��DQG�6��DQG�WKHLU�prerequisites and associated formulae is assumed and may be WHVWHG�



the followiQJ�NH\V��+, −, ×, ÷, ʌ, x2, √x,

1

x, xy





aX ± bY ∼ N(aµ

x ± bµ

y , a2ı

x

2 + b2ı

y

2) where X and Y are independent and

X ∼ N(µx , ı

x

2) and Y

∼ N(µ

y , ı

y

2).



1 Combinations of random variables


Distribution of linear combinations of independent 1RUPDO�UDQGRP�YDULDEOHV�

If X ∼ N(µx , ı

x

2) and Y

∼ N(µ

y , ı

y

2)

independently, then

aX ± bY ∼ N(aµ

x ± bµ

y , a2ı

x

2 + b2ı

y

2).

1R�SURRIV�UHTXLUHG�

2 Sampling


0HWKRGV�IRU�FROOHFWLQJ�GDWD��6LPSOH�UDQGRP�VDPSOLQJ��8VH�RI�UDQGRP�QXPEHUV�IRU�VDPSOLQJ�

2WKHU�PHWKRGV�RI�VDPSOLQJ��VWUDWL¿HG��V\VWHPDWLF��TXRWD�

The circumstances in which they PLJKW�EH�XVHG��7KHLU�DGYDQWDJHV�DQG�GLVDGYDQWDJHV�

3 Estimation, confidence intervals and tests


ConcHSWV�RI�VWDQGDUG�HUURU��HVWLPDWRU��ELDV� The sample mean, , and the sample variance,

s 2 =

11 1

2

nx xi

i

n

−−

=∑ ( ) , as

unbiased estimates of the corresponding population SDUDPHWHUV�

The distribution of the sample mean X � X has mean µ and variance σ 2

n.

If X ∼ N(µ, ı2) then X ∼ N µ σ,

2

n⎛

⎝⎜

⎞

⎠⎟.

1R�SURRIV�UHTXLUHG�

x



&RQFHSW�RI�D�FRQ¿GHQFH�LQWHUYDO�DQG�LWV�LQWHUSUHWDWLRQ�

/LQN�ZLWK�K\SRWKHVLV�WHVWV�

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Students will be expected to NQRZ�KRZ�WR�DSSO\�WKH�1RUPDO�distribution and use the standard HUURU�DQG�REWDLQ�FRQ¿GHQFH�intervals for the mean, rather than be concerned with any theoretical GHULYDWLRQV�

Hypothesis tests for the mean of a Normal GLVWULEXWLRQ�ZLWK�YDULDQFH�NQRZQ�

Use of ∼ N(0, 1).

Use of Central Limit theorem to extend hypothesis WHVWV�DQG�FRQ¿GHQFH�LQWHUYDOV�WR�VDPSOHV�IURP� QRQ�1RUPDO�GLVWULEXWLRQV��8VH�RI�ODUJH�VDPSOH�results to extend to the case in which the variance is XQNQRZQ�

can be treated as N(0, 1)

when n LV�ODUJH��$�NQRZOHGJH�RI�the t-GLVWULEXWLRQ�LV�QRW�UHTXLUHG�

Hypothesis test for the difference between the means of two Normal distributions with variances NQRZQ�

Use of

( ) ( )X Y

n n

x y

x

x

y

y

− − −

+

µ µ

σ σ2 2 ∼ N(0, 1).

Use of large sample results to extend to the case in ZKLFK�WKH�SRSXODWLRQ�YDULDQFHV�DUH�XQNQRZQ�

Use of

( ) ( )X Y

Sn

Sn

x y

x

x

y

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− − −

+

µ µ2 2 ∼ N(0, 1).

$�NQRZOHGJH�RI�WKH�t-distribution LV�QRW�UHTXLUHG�

XS n− µ

/

XS n− µ

/



4 Goodness of fit and contingency tables


7KH�QXOO�DQG�DOWHUQDWLYH�K\SRWKHVHV��7KH�XVH�RI ( )O E

Ei i

ii

n −=∑

2

1 as an approximate �VWDWLVWLF�

Applications to include the discrete uniform, binomial, Normal, Poisson and continuous uniform �UHFWDQJXODU��GLVWULEXWLRQV��Lengthy calculations will not be UHTXLUHG�

'HJUHHV�RI�IUHHGRP� Students will be expected to determine the degrees of freedom when one or more parameters are HVWLPDWHG�IURP�WKH�GDWD��&HOOV�should be combined when E

i < 5.

<DWHV¶�FRUUHFWLRQ�LV�QRW�UHTXLUHG�

5 Regression and correlation


6SHDUPDQ¶V�UDQN�FRUUHODWLRQ�FRHI¿FLHQW��LWV�XVH��LQWHUSUHWDWLRQ�DQG�OLPLWDWLRQV�

Numerical questions involving WLHV�ZLOO�QRW�EH�VHW��6RPH�understanding of how to deal with WLHV�ZLOO�EH�H[SHFWHG�

7HVWLQJ�WKH�K\SRWKHVLV�WKDW�D�FRUUHODWLRQ�LV�]HUR� Use of tables for Spearman’s and product moment correlation FRHI¿FLHQWV�

χ2


Unit S4 Statistics 4 GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) A2 optional unit


Quality of tests and estimators; one-sample procedures; WZR�VDPSOH�SURFHGXUHV�


Prerequisites $�NQRZOHGJH�RI�WKH�VSHFL¿FDWLRQ�IRU�6��6��DQG�6��DQG�LWV�prerequisites and associated formulae is assumed and may be WHVWHG��



the followiQJ�NH\V��+, −, ×, ÷��ʌ��x2, √x,

1

x, xy



Formulae 6WXGHQWV� DUH� H[SHFWHG� WR� NQRZ� DQ\� RWKHU� IRUPXODH� ZKLFK� PLJKW�EH�UHTXLUHG�E\�WKH�VSHFL¿FDWLRQ�DQG�ZKLFK�DUH�QRW� LQFOXGHG� LQ� WKH�ERRNOHW� Mathematical Formulae including Statistical Formulae and Tables��ZKLFK�ZLOO�EH�SURYLGHG�IRU�XVH�ZLWK�WKH�SDSHU��4XHVWLRQV�ZLOO�EH�VHW�LQ�6,�XQLWV�DQG�RWKHU�XQLWV�LQ�FRPPRQ�XVDJH�


Type I error: P(reject H0 ⏐H

0 true)

Type II error: P(do not reject H0 ⏐H

0 false)



1 Quality of tests and estimators


7\SH�,�DQG�7\SH�,,�HUURUV�

6L]H�DQG�3RZHU�RI�7HVW�

7KH�SRZHU�WHVW�

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2 One-sample procedures


+\SRWKHVLV�WHVW�DQG�FRQ¿GHQFH�LQWHUYDO�IRU�WKH�PHDQ�RI�D�1RUPDO�GLVWULEXWLRQ�ZLWK�XQNQRZQ�YDULDQFH�

Use of ∼ tn − 1

. Use of t-WDEOHV�

+\SRWKHVLV�WHVW�DQG�FRQ¿GHQFH�LQWHUYDO�IRU�WKH�YDULDQFH�RI�D�1RUPDO�GLVWULEXWLRQ�

Use of ∼ n-1

.

Use of WDEOHV�

XS n− µ

/

( )n S−1 2

2σ

χ2

χ2



3 Two-sample procedures


Hypothesis test that two independent random samples are from Normal populations with equal YDULDQFHV�

Use of ∼ Fn −1, n −1

under H0.

Use of the tables of the

F-GLVWULEXWLRQ�

8VH�RI�WKH�SRROHG�HVWLPDWH�RI�YDULDQFH� s2 =

+\SRWKHVLV�WHVW�DQG�FRQ¿GHQFH�LQWHUYDO�IRU�WKH�difference between two means from independent Normal distributions when the variances are equal EXW�XQNQRZQ�

Use of t-GLVWULEXWLRQ��

X Y

Sn nx y

−

+1 1 ∼ tn nx y+ − 2 , under H

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Paired t-WHVW�

SS

12

22

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1 12

2 22

1 2

1 12

− + −+ −

1 2


Unit D1 Decision Mathematics 1 GCE AS and GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) AS optional unit

D1.1 Unit description

Algorithms; algorithms on graphs; the route inspection problem; FULWLFDO�SDWK�DQDO\VLV��OLQHDU�SURJUDPPLQJ��PDWFKLQJV�

D1.2 Assessment information

Preamble 6WXGHQWV�VKRXOG�EH�IDPLOLDU�ZLWK�WKH�WHUPV�GH¿QHG�LQ�WKH�JORVVDU\�DWWDFKHG�WR�WKLV�VSHFL¿FDWLRQ��6WXGHQWV�VKRXOG�VKRZ�FOHDUO\�KRZ�DQ�DOJRULWKP�KDV�EHHQ�DSSOLHG��0DWUL[�UHSUHVHQWDWLRQ�ZLOO�EH�UHTXLUHG�EXW�PDWUL[�PDQLSXODWLRQ�LV�QRW�UHTXLUHG��6WXGHQWV�ZLOO�EH�UHTXLUHG�WR�PRGHO�DQG�LQWHUSUHW�VLWXDWLRQV��LQFOXGLQJ�FURVV�FKHFNLQJ�EHWZHHQ�PRGHOV�DQG�UHDOLW\�



the folloZLQJ�NH\V��+, −, ×, ÷, π, x2, √x,

1

x, xy DQG�PHPRU\��&DOFXODWRUV�

with a faFLOLW\�IRU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�


Section D © Pearson Education Limited 2013 Pearson Edexcel Level 3 GCE in Mathematics98

Unit D1 Decision Mathematics D1 (AS)

1 Algorithms


The general ideas of algorithms and the LPSOHPHQWDWLRQ�RI�DQ�DOJRULWKP�JLYHQ�E\�D�ÀRZ�FKDUW�RU�WH[W��

The order of an algorithm is not H[SHFWHG�

:KHQHYHU�¿QGLQJ�WKH�PLGGOH�LWHP�RI�DQ\�OLVW��WKH�PHWKRG�GH¿QHG�LQ�WKH�JORVVDU\�PXVW�EH�XVHG�

6WXGHQWV�VKRXOG�EH�IDPLOLDU�ZLWK�ELQ�SDFNLQJ��EXEEOH�VRUW��TXLFN�VRUW��ELQDU\�VHDUFK�

:KHQ�XVLQJ�WKH�TXLFN�VRUW�algorithm, the pivot should be chosen as the middle item of the OLVW�

2 Algorithms on graphs


The minimum spanning tree (minimum connector) SUREOHP��3ULP¶V�DQG�.UXVNDO¶V��JUHHG\��DOJRULWKP�

'LMNVWUD¶V�DOJRULWKP�IRU�¿QGLQJ�WKH�VKRUWHVW�SDWK�

Matrix representation for Prim’s DOJRULWKP�LV�H[SHFWHG��'UDZLQJ�D�QHWZRUN�IURP�D�JLYHQ�PDWUL[�and writing down the matrix DVVRFLDWHG�ZLWK�D�QHWZRUN�ZLOO�EH�LQYROYHG�

3 The route inspection problem


$OJRULWKP�IRU�¿QGLQJ�WKH�VKRUWHVW�URXWH�DURXQG�D�QHWZRUN��WUDYHOOLQJ�DORQJ�HYHU\�HGJH�DW�OHDVW�RQFH�DQG�HQGLQJ�DW�WKH�VWDUW�YHUWH[��7KH�QHWZRUN�ZLOO�KDYH�XS�WR�IRXU�RGG�QRGHV�

$OVR�NQRZQ�DV�WKH�µ&KLQHVH�SRVWPDQ¶�SUREOHP��6WXGHQWV�ZLOO�be expected to use inspection to consider all possible pairings of RGG�QRGHV�

(The application of Floyd’s algorithm to the odd nodes is not UHTXLUHG��

Pearson Edexcel Level 3 GCE in Mathematics © Pearson Education Limited 2013 Section D 99

Decision Mathematics D1 (AS) Unit D1

4 Critical path analysis


0RGHOOLQJ�RI�D�SURMHFW�E\�DQ�DFWLYLW\�QHWZRUN��IURP�D�SUHFHGHQFH�WDEOH��

Completion of the precedence table for a given DFWLYLW\�QHWZRUN�

$OJRULWKP�IRU�¿QGLQJ�WKH�FULWLFDO�SDWK��(DUOLHVW�DQG�ODWHVW�HYHQW�WLPHV��(DUOLHVW�DQG�ODWHVW�VWDUW�DQG�¿QLVK�WLPHV�IRU�DFWLYLWLHV��

7RWDO�ÀRDW��*DQWW��FDVFDGH��FKDUWV��6FKHGXOLQJ�

$FWLYLW\�RQ�DUF�ZLOO�EH�XVHG��7KH�XVH�RI�GXPPLHV�LV�LQFOXGHG��,Q�D�SUHFHGHQFH�QHWZRUN��precedence tables will only show LPPHGLDWH�SUHGHFHVVRUV�

5 Linear programming


)RUPXODWLRQ�RI�SUREOHPV�DV�OLQHDU�SURJUDPV�

Graphical solution of two variable problems using ruler and vertex methods

Consideration of problems where solutions must KDYH�LQWHJHU�YDOXHV�

6 Matchings


8VH�RI�ELSDUWLWH�JUDSKV�IRU�PRGHOOLQJ�PDWFKLQJV��&RPSOHWH�PDWFKLQJV�DQG�PD[LPDO�PDWFKLQJV�

$OJRULWKP�IRU�REWDLQLQJ�D�PD[LPXP�PDWFKLQJ�

Students will be required to use the maximum matching algorithm to improve a matching E\�¿QGLQJ�DOWHUQDWLQJ�SDWKV��1R�consideration of assignment is UHTXLUHG�


Unit D1 Decision Mathematics D1 (AS)

Glossary for D1

1 Algorithms

In a list containing N items the ‘middle’ item has position [12 (N + 1)] if N is odd [

12(N + 2)] if N is even, so

that if N = 9, the middle item is the 5th and if N = 6 it is the 4th.

2 Algorithms on graphs

A graph G consists of points (vertices or nodes) which are connected by lines (edges or arcs).

A subgraph of G is a graph, each of whose vertices belongs to G and each of whose edges belongs to G.

If a graph has a number associated with each edge (usually called its weight) then the graph is called a

weighted graph or network.

The degree or valency of a vertex is the number of edges incident to it. A vertex is odd (even) if it has odd

(even) degree.

A path�LV�D�¿QLWH�VHTXHQFH�RI�HGJHV��VXFK�WKDW�WKH�HQG�YHUWH[�RI�RQH�HGJH�LQ�WKH�VHTXHQFH�LV�WKH�VWDUW�YHUWH[�of the next, and in which no vertex appears more then once.

A cycle (circuit��LV�D�FORVHG�SDWK��LH�WKH�HQG�YHUWH[�RI�WKH�ODVW�HGJH�LV�WKH�VWDUW�YHUWH[�RI�WKH�¿UVW�HGJH�

Two vertices are connected if there is a path between them. A graph is connected if all its vertices are

connected.

If the edges of a graph have a direction associated with them they are known as directed edges and the graph

is known as a digraph.

A tree is a connected graph with no cycles.

A spanning tree of a graph G is a subgraph which includes all the vertices of G and is also a tree.

A minimum spanning tree (MST) is a spanning tree such that the total length of its arcs is as small as

possible. (MST is sometimes called a minimum connector.)

A graph in which each of the n vertices is connected to every other vertex is called a complete graph.

4 Critical path analysis

The WRWDO�ÀRDW F(i, j) of activity (i, j��LV�GH¿QHG�WR�EH�F(i, j) = lj – e

i – duration (i, j), where e

i is the earliest

time for event i and lj is the latest time for event j.

6 Matchings

A bipartite graph consists of two sets of vertices X and Y. The edges only join vertices in X to vertices in Y,

not vertices within a set. (If there are r vertices in X and s vertices in Y then this graph is Kr,s

.)

A matching is the pairing of some or all of the elements of one set, X, with elements of a second set, Y. If

every member of X is paired with a member of Y the matching is said to be a complete matching.


Unit D2 Decision Mathematics 2 GCE Mathematics, GCE AS and GCE Further Mathematics, GCE AS and GCE Further Mathematics (Additional) A2 optional unit

D2.1 Unit description

Transportation problems; allocation (assignment) problems; the travelling salesman; game theory; further linear programming, G\QDPLF�SURJUDPPLQJ��ÀRZV�LQ�QHWZRUNV

D2.2 Assessment information

Prerequisites and Preamble

Prerequisites

Students should be familiar with the material in, and the glossary DWWDFKHG�WR��WKH�'��VSHFL¿FDWLRQ�

Preamble

6WXGHQWV�VKRXOG�EH�IDPLOLDU�ZLWK�WKH�WHUPV�GH¿QHG�LQ�WKH�JORVVDU\�attached WR�WKLV�VSHFL¿FDWLRQ��6WXGHQWV�VKRXOG�VKRZ�FOHDUO\�KRZ�DQ�DOJRULWKP�KDV�EHHQ�DSSOLHG�



the folloZLQJ�NH\V��+, −, ×, ÷, π, x2, √x,

1

x, xy DQG�PHPRU\��&DOFXODWRUV�

with a facility foU�V\PEROLF�DOJHEUD��GLIIHUHQWLDWLRQ�DQG�RU�LQWHJUDWLRQ�DUH�QRW�SHUPLWWHG�



Unit D2 Decision Mathematics D2 (A2)

1 Transportation problems


7KH�QRUWK�ZHVW�FRUQHU�PHWKRG�IRU�¿QGLQJ�DQ�LQLWLDO�EDVLF�IHDVLEOH�VROXWLRQ�

Problems will be restricted to a maximum of four sources and four GHVWLQDWLRQV��7KH�LGHD�RI�GXPP\�locations and degeneracy are UHTXLUHG�

Use of the stepping-stone method for obtaining an LPSURYHG�VROXWLRQ��,PSURYHPHQW�LQGLFHV�

6WXGHQWV�VKRXOG�LGHQWLI\�D�VSHFL¿F�HQWHULQJ�FHOO�DQG�D�VSHFL¿F�H[LWLQJ�FHOO�

)RUPXODWLRQ�DV�D�OLQHDU�SURJUDPPLQJ�SUREOHP�

2 Allocation (assignment) problems


&RVW�PDWUL[�UHGXFWLRQ� 6WXGHQWV�VKRXOG�UHGXFH�URZV�¿UVW�

7KH�VL]H�RI�WKH�FRVW�PDWUL[�ZLOO�EH�at most 5 ×��

8VH�RI�WKH�+XQJDULDQ�DOJRULWKP�WR�¿QG�OHDVW�FRVW�DOORFDWLRQ�

The idea of a dummy location is UHTXLUHG�

0RGL¿FDWLRQ�RI�PHWKRG�WR�GHDO�ZLWK�D�PD[LPXP�SUR¿W�DOORFDWLRQ�

)RUPXODWLRQ�DV�D�OLQHDU�SURJUDPPLQJ�SUREOHP�


Decision Mathematics D2 (A2) Unit D2

3 The travelling salesman problem


7KH�SUDFWLFDO�DQG�FODVVLFDO�SUREOHPV��7KH�FODVVLFDO�problem for complete graphs satisfying the triangle LQHTXDOLW\�

Determination of upper and lower bounds using PLQLPXP�VSDQQLQJ�WUHH�PHWKRGV�

The use of short cuts to improve XSSHU�ERXQG�LV�LQFOXGHG�

7KH�QHDUHVW�QHLJKERXU�DOJRULWKP� 7KH�FRQYHUVLRQ�RI�D�QHWZRUN�LQWR�D�FRPSOHWH�QHWZRUN�RI�VKRUWHVW�µGLVWDQFHV¶�LV�LQFOXGHG�

4 Further linear programming


)RUPXODWLRQ�RI�SUREOHPV�DV�OLQHDU�SURJUDPV�

The Simplex algorithm and tableau for maximising SUREOHPV�

7KH�XVH�DQG�PHDQLQJ�RI�VODFN�YDULDEOHV� Problems will be restricted to those with a maximum of four variables and four constraints, in addition to non-negativity FRQGLWLRQV�



5 Game theory


7ZR�SHUVRQ�]HUR�VXP�JDPHV�DQG�WKH�SD\�RII�PDWUL[��,GHQWL¿FDWLRQ�RI�SOD\�VDIH�VWUDWHJLHV�DQG�VWDEOH�VROXWLRQV��VDGGOH�SRLQWV��

Reduction of pay-off matrices using dominance DUJXPHQWV��2SWLPDO�PL[HG�VWUDWHJLHV�IRU�D�JDPH�ZLWK�QR�VWDEOH�VROXWLRQ�

Use of graphical methods for 2 × n

or n × 2 games where n = 1, 2 or 3.

Conversion of 3 × 2 and 3 × 3 games to linear SURJUDPPLQJ�SUREOHPV�

6 Flows in networks


$OJRULWKP�IRU�¿QGLQJ�D�PD[LPXP�ÀRZ�

&XWV�DQG�WKHLU�FDSDFLW\�

8VH�RI�WKH�PD[�ÀRZ�²�PLQ�FXW�WKHRUHP�WR�YHULI\�WKDW�D�ÀRZ�LV�D�PD[LPXP�ÀRZ��

0XOWLSOH�VRXUFHV�DQG�VLQNV�

Vertex restrictions are not UHTXLUHG��2QO\�QHWZRUNV�ZLWK�GLUHFWHG�DUFV�ZLOO�EH�FRQVLGHUHG��Only problems with upper FDSDFLWLHV�ZLOO�EH�VHW�

7 Dynamic programming


3ULQFLSOHV�RI�G\QDPLF�SURJUDPPLQJ��%HOOPDQ¶V�SULQFLSOH�RI�RSWLPDOLW\�

6WDJH�YDULDEOHV�DQG�6WDWH�YDULDEOHV��8VH�RI�tabulation to solve maximum, minimum, minimax or PD[LPLQ�SUREOHPV�

%RWK�QHWZRUN�DQG�WDEOH�IRUPDWV�DUH�UHTXLUHG�


Decision Mathematics D2 (A2) Unit D2

Glossary for D2

1 Transportation problemsIn the north-west corner method, WKH�XSSHU�OHIW�KDQG�FHOO�LV�FRQVLGHUHG�¿UVW�DQG�DV�PDQ\�XQLWV�DV�SRVVLEOH�sent by this route.

The stepping stone method is an iterative procedure for moving from an initial feasible solution to an

optimal solution.

Degeneracy occurs in a transportation problem, with m rows and n columns, when the number of occupied

cells is less than (m + n – 1).

In the transportation problem:

The shadow costs Ri , for the ith row, and K

j , for the jth column, are obtained by solving R

i + K

j = C

ij for

occupied cells, taking R1 = 0 arbitrarily.

The improvement index Iij for an unoccupied cell�LV�GH¿QHG�E\�I

ij = C

ij – R

i – K

j.

3 The travelling salesman problemThe travelling salesman problem LV�µ¿QG�D�URXWH�RI�PLQLPXP�OHQJWK�ZKLFK�YLVLWV�HYHU\�YHUWH[�LQ�DQ�undirected network’. In the ‘classical’ problem, each vertex is visited once only. In the ‘practical’ problem, a

vertex may be revisited.

For three vertices A, B and C, the triangular inequality is ‘length AB ≤ length AC + length CB’.

A walk�LQ�D�QHWZRUN�LV�D�¿QLWH�VHTXHQFH�RI�HGJHV�VXFK�WKDW�WKH�HQG�YHUWH[�RI�RQH�HGJH�LV�WKH�VWDUW�YHUWH[�RI�the next.

A walk which visits every vertex, returning to its starting vertex, is called a tour.

4 Further linear programmingThe simplex tableau for the linear programming problem:

Maximise P = 14x + 12y + 13z,

Subject to 4x + 5y + 3z��

5x + 4y + 6z��

will be written as

%DVLF�YDULDEOH x y z r s 9DOXH

r 4 5 3 1 0 16

s 5 4 6 0 1 24

P −14 −12 −13 0 0 0

where r and s are slack variables.

5 Game theoryA two-person game is one in which only two parties can play.

A zero-sum game is one in which the sum of the losses for one player is equal to the sum of the gains for the

other player.



6 Flows in networks

A cut, in a network with source S and sink T, is a set of arcs (edges) whose removal separates the network

into two parts X and Y, where X contains at least S and Y contains at least T. The capacity of a cut is the sum

of the capacities of those arcs in the cut which are directed from X to Y.

If a network has several sources S1 , S

2 , . . ., then these can be connected to a single supersource S. The

capacity of the edge joining S to S1 is the sum of the capacities of the edges leaving S

1.

If a network has several sinks T1 , T

2 , . . ., then these can be connected to a supersink T. The capacity of the

edge joining T1 to T is the sum of the capacities of the edges entering T

1.

7 Dynamic programming

Bellman’s principle for dynamic programming is ‘Any part of an optimal path is optimal.’

The minimax route is the one in which the maximum length of the arcs used is as small as possible.

The maximin route is the one in which the minimum length of the arcs used is as large as possible.


D Assessment and additional information

Assessment information

Assessment requirements

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E Resources, support and training

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Training

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F Appendices

Appendix A Grade descriptions 121

Appendix B Notation 125

Appendix C Codes 133


Appendix A Grade descriptions

The following grade descriptions indicate the level of attainment FKDUDFWHULVWLF�RI�JUDGHV�$��&�DQG�(�DW�$GYDQFHG�*&(��7KH\�JLYH�a general indication of the required learning outcomes at the VSHFL¿HG�JUDGHV��7KH�GHVFULSWLRQV�VKRXOG�EH�LQWHUSUHWHG�LQ�UHODWLRQ�WR�WKH�FRQWHQW�RXWOLQHG�LQ�WKH�VSHFL¿FDWLRQ��WKH\�DUH�QRW�GHVLJQHG�WR�GH¿QH�WKDW�FRQWHQW��7KH�JUDGH�DZDUGHG�ZLOO�GHSHQG�LQ�SUDFWLFH�upon the extent to which the candidate has met the assessment REMHFWLYHV�RYHUDOO��6KRUWFRPLQJV�LQ�VRPH�DVSHFWV�RI�WKH�H[DPLQDWLRQ�PD\�EH�EDODQFHG�E\�EHWWHU�SHUIRUPDQFHV�LQ�RWKHUV�

Grade A Students recall or recognise almost all the mathematical facts, concepts and techniques that are needed, and select appropriate RQHV�WR�XVH�LQ�D�ZLGH�YDULHW\�RI�FRQWH[WV�

Students manipulate mathematical expressions and use graphs, VNHWFKHV�DQG�GLDJUDPV��DOO�ZLWK�KLJK�DFFXUDF\�DQG�VNLOO��7KH\�use mathematical language correctly and proceed logically DQG�ULJRURXVO\�WKURXJK�H[WHQGHG�DUJXPHQWV�RU�SURRIV��:KHQ�confronted with unstructured problems they can often devise and LPSOHPHQW�DQ�HIIHFWLYH�VROXWLRQ�VWUDWHJ\��,I�HUURUV�DUH�PDGH�LQ�WKHLU�FDOFXODWLRQV�RU�ORJLF��WKHVH�DUH�VRPHWLPHV�QRWLFHG�DQG�FRUUHFWHG�

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Grade C Students recall or recognise most of the mathematical facts, concepts and techniques that are needed, and usually select DSSURSULDWH�RQHV�WR�XVH�LQ�D�YDULHW\�RI�FRQWH[WV�

Students manipulate mathematical expressions and use graphs, VNHWFKHV�DQG�GLDJUDPV��DOO�ZLWK�D�UHDVRQDEOH�OHYHO�RI�DFFXUDF\�DQG�VNLOO��7KH\�XVH�PDWKHPDWLFDO�ODQJXDJH�ZLWK�VRPH�VNLOO�DQG�VRPHWLPHV�SURFHHG�ORJLFDOO\�WKURXJK�H[WHQGHG�DUJXPHQWV�RU�SURRIV��:KHQ�FRQIURQWHG�ZLWK�XQVWUXFWXUHG�SUREOHPV�WKH\�VRPHWLPHV�GHYLVH�DQG�LPSOHPHQW�DQ�HIIHFWLYH�DQG�HI¿FLHQW�VROXWLRQ�VWUDWHJ\��7KH\�RFFDVLRQDOO\�QRWLFH�DQG�FRUUHFW�HUURUV�LQ�WKHLU�FDOFXODWLRQV�

Students recall or recognise most of the standard models that are needed and usually select appropriate ones to represent a variety RI�VLWXDWLRQV�LQ�WKH�UHDO�ZRUOG��7KH\�RIWHQ�FRUUHFWO\�UHIHU�UHVXOWV�from calculations using the model to the original situation; they sometimes give sensible interpretations of their results in the FRQWH[W�RI�WKH�RULJLQDO�UHDOLVWLF�VLWXDWLRQ��7KH\�VRPHWLPHV�PDNH�intelligent comments on the modelling assumptions and possible UH¿QHPHQWV�WR�WKH�PRGHO�

Students comprehend or understand the meaning of most WUDQVODWLRQV�LQWR�PDWKHPDWLFV�RI�FRPPRQ�UHDOLVWLF�FRQWH[WV��7KH\�RIWHQ�FRUUHFWO\�UHIHU�WKH�UHVXOWV�RI�FDOFXODWLRQV�EDFN�WR�WKH�JLYHQ�FRQWH[W�DQG�VRPHWLPHV�PDNH�VHQVLEOH�FRPPHQWV�RU�SUHGLFWLRQV��They distil much of the essential mathematical information from H[WHQGHG�SLHFHV�RI�SURVH�KDYLQJ�PDWKHPDWLFDO�FRQWHQW��7KH\�JLYH�VRPH�XVHIXO�FRPPHQWV�RQ�WKLV�PDWKHPDWLFDO�LQIRUPDWLRQ�

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Grade descriptions Appendix A


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Students recall or recognise some of the standard models that are needed and sometimes select appropriate ones to represent D�YDULHW\�RI�VLWXDWLRQV�LQ�WKH�UHDO�ZRUOG��7KH\�VRPHWLPHV�FRUUHFWO\�refer results from calculations using the model to the original situation; they try to interpret their results in the context of the RULJLQDO�UHDOLVWLF�VLWXDWLRQ�

Students sometimes comprehend or understand the meaning of WUDQVODWLRQV�LQ�PDWKHPDWLFV�RI�FRPPRQ�UHDOLVWLF�FRQWH[WV��7KH\�VRPHWLPHV�FRUUHFWO\�UHIHU�WKH�UHVXOWV�RI�FDOFXODWLRQV�EDFN�WR�WKH�JLYHQ�FRQWH[W�DQG�DWWHPSW�WR�JLYH�FRPPHQWV�RU�SUHGLFWLRQV��They distil some of the essential mathematical information from H[WHQGHG�SLHFHV�RI�SURVH�KDYLQJ�PDWKHPDWLFDO�FRQWHQW��7KH\�DWWHPSW�WR�FRPPHQW�RQ�WKLV�PDWKHPDWLFDO�LQIRUPDWLRQ�

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Appendix B Notation

7KH�IROORZLQJ�QRWDWLRQ�ZLOO�EH�XVHG�LQ�DOO�PDWKHPDWLFV�H[DPLQDWLRQV�

1 Set notation

∈ is an element of

∉ is not an element of

x x1 2, , ... { } the set with elements x1, x

2, ...

{ }x:... the set of all x such that ...

n(A) the number of elements in set A

∅ the empty set

the universal set

A’ the complement of the set A

# the set of natural numbers, 1 2 3, , , ... { }

$ the set of integers, 0 1 2 3, , , , ... ± ± ±{ }

$ + the set of positive integers, 1 2 3, , , ... { }

$n the set of integers modulo n, 0 1 2 1, , , ... , n −{ }

% the set of rational numbers, pq

p q: ∈ ∈⎧⎨⎩

⎫⎬⎭

ℤ +ℤ,

% + the set of positive rational numbers, x x∈ >{ }ℚ : 0

%0

+ the set of positive rational numbers and zero, x x∈ ≥{ }ℚ : 0

! the set of real numbers

! + the set of positive real numbers x x∈ >{ }ℝ : 0

!0

+ the set of positive real numbers and zero, x x∈ ≥{ }: 0ℝ

" the set of complex numbers

(x, y) the ordered pair x, y

A × B the cartesian product of sets A and B, ie

A B a b a A b B× = ∈ ∈{ }( ) , : ,⊆ is a subset of

⊂ is a proper subset of

∪ union

∩ intersection

Appendix B Notation


[a, b] the closed interval, x a x b∈ ≤ ≤ }{ ℝ:[a, b), [a, b[ the interval x a x b∈ ≤ <{ }ℝ :(a, b], ]a, b] the interval x a x b∈ < ≤{ }ℝ:(a, b), ]a, b[ the open interval x a x b∈ < <{ }ℝ :

y R x y is related to x by the relation R

y ~ x y is equivalent to x, in the context of some equivalence relation

2 Miscellaneous symbols

= is equal to

�� LV�QRW�HTXDO�WR

Ł� LV�LGHQWLFDO�WR�RU�LV�FRQJUXHQW�WR

§� LV�DSSUR[LPDWHO\�HTXDO�WR�≅ is isomorphic to

∝ is proportional to

< is less than

�� LV�OHVV�WKDQ�RU�HTXDO�WR��LV�QRW�JUHDWHU�WKDQ

> is greater than

�� LV�JUHDWHU�WKDQ�RU�HTXDO�WR��LV�QRW�OHVV�WKDQ

�� LQ¿QLW\p q∧ p and q

p q∨ p or q (or both)

~ p not p

p q⇒ p implies q (if p then q)

p q⇐ p is implied by q (if q then p)

p q⇔ p implies and is implied by q (p is equivalent to q)

∃ there exists

! for all

Notation Appendix B


3 Operations

a + b a plus b

a – b a minus b

a × b, ab, a.b a multiplied by b

a b ab

a b÷ , , a divided by b

aii

n

=∑

1 a

1 + a

2 + … + a

n

aii

n

=∏

1 a

1 × a

2 × … × a

n

√a the positive square root of a

⏐a⏐ the modulus of a

n! n factorial

the�ELQRPLDO�FRHI¿FLHQW�n

r n r!

! ! −( ) for n∈$+

n n n rr

−( ) − +( )1 1 ...!

for n∈%

nr⎛

⎝⎜⎞

⎠⎟

Appendix B Notation


4 Functions

f(x) the value of the function f at x

I��$�ĺ�%� I�LV�D�IXQFWLRQ�XQGHU�ZKLFK�HDFK�HOHPHQW�RI�VHW�A has an image in set

B

f : x�ĺ�y the function f maps the element x to the element y

f –1 the inverse function of the function f

g ƕ�I��JI� WKH�FRPSRVLWH�IXQFWLRQ�RI�I�DQG�J�ZKLFK�LV�GH¿QHG�E\�

(g ƕ f)(x) or gf(x) = g(f(x))

lim f xx a

( )→

the limit of f(x) as x tends to a

¨x��įx an increment of x

ddyx the derivative of y with respect to x

dd

n

ny

x the nth derivative of y with respect to x

f ’(x), f ’’(x),…, f(n) (x�� WKH�¿UVW��VHFRQG��nth derivatives of f(x) with respect to x

�y dx� WKH�LQGH¿QLWH�LQWHJUDO�RI�\�ZLWK�UHVSHFW�WR�x

dy xa

b

∫ the dH¿QLWH�LQWHJUDO�RI�\�ZLWK�UHVSHFW�WR�[�EHWZHHQ�WKH�OLPLWV�

x = a and x = b

∂∂

Vx the partial derivative of V with respect to x

x, ... x, � WKH�¿UVt, second, ... derivatives of x with respect to t

5 Exponential and logarithmic functions

e base of natural logarithms

ex, exp x exponential function of x

loga x logarithm to the base a of x

lin x, loge x natural logarithm of x

lg x, log10

x logarithm of x to base 10

Notation Appendix B


6 Circular and hyperbolic functions

sin, cos, tan,cosec, sec, cot

⎫⎬⎭

the circular functions

arcsin, arccos, arctan,arccosec, arcsec, arccot

⎫⎬⎭

the inverse circular functions

sinh, cosh, tanh,cosech, sech, coth

⎫⎬⎭

the hyperbolic functions

arsinh,arcosh,artanh,arcosech,arsech,arcoth

⎫⎬⎭

the inverse hyperbolic functions

7 Complex numbers

i, j square root of –1

z a complex number, z = x + iy

Re z the real part of z, Re z = x

Im z the imaginary part of z, Im z = y

⏐z⏐ the modulus of z, ⏐z⏐ z x y= +( )2 2

arg z the argument of z, arg z = θ, –ʌ��θ � ʌ

z* the complex conjugate of z, x – iy

8 Matrices

M a matrix M

M-1 the inverse of the matrix M

MT the transpose of the matrix M

det M or⏐M⏐ the determinant of the square matrix M

Appendix B Notation


9 Vectors

a the vector a→AB the vector represented in magnitude and direction by the directed

line segment AB

â a unit vector in the direction of a

i, j, k unit vectors in the directions of the cartesian coordinate axes

⏐a⏐, a the magnitude of a

⏐→AB⏐, AB the magnitude of

→AB

a . b the scalar product of a and b

a × b the vector product of a and b

Notation Appendix B


10 Probability and statistics

A, B, C, etc events

BA ∪ union of the events A and B

BA ∩ intersection of the events A and B

P(A) probability of the event A

A’ complement of the event A

P(A ⏐ B) probability of the event A conditional on the event B

X, Y, R, etc random variables

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

x1, x

2, … observations

f1, f

2, … frequencies with which the observations x

1, x

2, … occur

p(x) probability function P(X = x) of the discrete random variable X

p1, p

2 probabilities of the values x

1, x

2, … of the discrete random variable X

f(x), g(x) the value of the probability density function of a continuous random

variable X

F(x), G(x) the value of the (cumulative) distribution function P(X ��x) of a

continuous random variable X

E(X) expectation of the random variable X

E[g(X)] expectation of g(X)

Var (X) variance of the random variable X

G(t) probability generating function for a random variable which takes

the values 0, 1, 2, ...

%�n, p) binomial distribution with parameters n and p

N(ȝ,�ı2) normal distribution with mean ȝ and variance ı2

ȝ population mean

ı2 population variance

ı population standard deviation

x m, sample mean

s2 2, σ unbiased estimate of population variance from a sample,

sn

x xi2 21

1=

−−( )∑

φ probability density function of the standardised normal variable with

distribution N(0, 1)

Φ corresponding cumulative distribution function

Appendix B Notation


ȡ� SURGXFW�PRPHQW�FRUUHODWLRQ�FRHI¿FLHQW�IRU�D�SRSXODWLRQ

r� SURGXFW�PRPHQW�FRUUHODWLRQ�FRHI¿FLHQW�IRU�D�VDPSOH

Cov(X, Y) covariance of X and Y


Appendix C Codes

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National FODVVL¿FDWLRQ�FRGHV

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7KH�FODVVL¿FDWLRQ�FRGHV�IRU�WKLV�VSHFL¿FDWLRQ�DUH�

2210 Advanced Subsidiary GCE in Mathematics

2330 Advanced Subsidiary GCE in Further Mathematics

2230 Advanced Subsidiary GCE in Pure Mathematics

2230 Advanced Subsidiary GCE in Further Mathematics (Additional)

2210 Advanced GCE in Mathematics

2330 Advanced GCE in Further Mathematics

2230 Advanced GCE in Pure Mathematics

2230 Advanced GCE in Further Mathematics (Additional)

National 4XDOL¿FDWLRQV�)UDPHZRUN��14)��codes

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Publications Code UA035243All the material in this publication is copyright© Pearson Education Limited 2013

Publication code UA035243

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Math as, A2 Syllabus Gr12

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