GCE. Mathematics. (Amended August 2010) for examination in 2011

Transcription

1 GCE Mthemtics in S P E C I F I C A T I O N (Amended August 00) For teching from September 00 for emintion in 0

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3 FOREWORD This booklet contins CCEA s Advnced Subsidiry (AS) nd Advnced (A level) GCE Mthemtics specifiction for teching from September 004. This specifiction hs been developed to tke ccount of the revised AS nd A level GCE Mthemtics Subject Criteri developed jointly by the Regultory Authorities in Englnd (QCA), Northern Irelnd (CCEA) nd Wles (ACCAC) nd published by QCA in December 00. The AS is the first hlf of the A level nd will be ssessed t stndrd pproprite for cndidtes who hve completed hlf of the A level course. The A level course comprises the AS together with the second hlf of the A level course, referred to s A. A will be ssessed t stndrd pproprite for cndidtes who hve completed the full A level course nd will contin n element of synoptic ssessment. The A level wrd will be bsed on the ggregtion of mrks from the AS (50%) nd A (50%). AS cn be tken s stnd lone qulifiction without progress to A level. The first yer of certifiction of the AS nd A Level is 005. FROM AUTUMN 008

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5 Contents Pge KEY FEATURES 5 SUMMARY OF EXAMINATION INFORMATION 6 INTRODUCTION 8 Rtionle 8 Aims 9 Assessment Objectives 0 Specifiction Structure Key Skills Prohibited Combintions SCHEME OF ASSESSMENT 3 Allowble Combintions 3 The Reltionship between Assessment Units nd Assessment Objectives 5 Nture of Assessment Units 6 The Sequence, Timing nd Resitting of Assessment Units 6 Synoptic Assessment 7 Lnguge of Specifiction nd Assessment Mterils 7 Cndidtes with Prticulr Requirements 7 Awrds nd Certifiction 7 3 SUBJECT CONTENT 8 Module C: AS Core Mthemtics 9 Module C: AS Core Mthemtics Module C3: A Core Mthemtics 3 Module C4: A Core Mthemtics 5 Module FP: Further Pure Mthemtics 7 Module FP: Further Pure Mthemtics 8 Module FP3: Further Pure Mthemtics 3 30 Module M: Mechnics 3 Module M: Mechnics 3 Module M3: Mechnics 3 33 Module M4: Mechnics 4 34 Module S: Sttistics 35 Module S: Sttistics 37 4 GRADE DESCRIPTIONS 39 5 RESOURCE LIST 4 FROM AUTUMN 008 3

6 6 LIST OF FORMULAE WHICH WILL BE GIVEN 43 7 LIST OF FORMULAE WHICH WILL NOT BE GIVEN 54 8 GLOSSARY OF MATHEMATICAL NOTATION 59 APPENDIX 66 Opportunities for developing nd generting evidence for ssessing Key Skills 4 FROM AUTUMN 008

7 KEY FEATURES This suite of syllbuses dheres to the GCE Advnced Subsidiry (AS) nd Advnced (A Level) Subject Criteri for Mthemtics (00). The syllbuses: seek to consolidte nd etend the knowledge, skills nd understnding developed in Key Stge 4; hve structure which will llow cndidtes of ll bilities to hve the opportunity to demonstrte positive chievement; provide suitble foundtion for study of mthemtics nd other subjects in further nd higher eduction nd for rnge of interesting creers; enble schools nd colleges to provide coherent, stisfying nd worthwhile course of study for students who do not progress to further study of mthemtics; give centres the fleibility to decide which scheme of ssessment best cters for the needs of their students. FROM AUTUMN 008 5

8 SUMMARY OF EXAMINATION INFORMATION The specifiction dopts modulr structure. Cndidtes re required to study three teching nd lerning modules for the AS nd si modules for the A level GCE in Mthemtics nd in Further Mthemtics. Teching nd lerning module Module C: AS Core Mthemtics Module C: AS Core Mthemtics Module C3: A Core Mthemtics Module C4: A Core Mthemtics Module FP: Further Pure Mthemtics Module FP: Further Pure Mthemtics Module FP3: Further Pure Mthemtics 3 Assessment Unit Nture of Assessment Assessment weighting (%) C Assessed t AS level. Compulsory for AS nd A level in Mthemtics. C Assessed t AS level. Compulsory for AS nd A level in Mthemtics. C3 Assessed t A level. Compulsory for A level in Mthemtics. C4 Assessed t A level. Compulsory for A level in Mthemtics. FP Assessed t AS level. Compulsory for AS nd A level in Further Mthemtics. FP Assessed t A level. Compulsory for A level in Further Mthemtics. FP3 Assessed t A level. Compulsory for A Level in Further Mthemtics. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. 33 ⅓ % of AS 6 ⅔ % of A Level 33 ⅓ % of AS 6 ⅔ % of A Level Emintion Session Avilbility Summer nd Winter Summer nd Winter 6 ⅔ % of A Level Summer nd Winter 6 ⅔ % of A Level 33 ⅓ % of AS 6 ⅔ % of A Level 33 ⅓ % of AS 6 ⅔ % of A Level 33 ⅓ % of AS 6 ⅔ % of A Level Summer nd Winter Summer nd Winter Summer nd Winter Summer only 6 FROM AUTUMN 008

9 Teching nd lerning module Assessment Unit Nture of Assessment Assessment weighting (%) Emintion Session Avilbility Module M: Mechnics Module M: Mechnics Module M3: Mechnics 3 Module M4: Mechnics 4 Module S: Sttistics Module S: Sttistics M Assessed t AS level. Optionl. M Assessed t A level. Optionl. M3 Assessed t A level. Optionl. M4 Assessed t A level. Optionl. S Assessed t AS level. Optionl. S4 Assessed t A level. Optionl. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. hour 30 minutes eternl emintion pper. Mimum 75 rw mrks. 6 8 questions. 33 ⅓ % of AS 6 ⅔ % of A Level 33 ⅓ % of AS 6 ⅔ % of A Level 33 ⅓ % of AS 6 ⅔ % of A Level Summer nd Winter Summer nd Winter Summer only 6 ⅔ % of A Level Summer only 33 ⅓ % of AS 6 ⅔ % of A Level 33 ⅓ % of AS 6 ⅔ % of A Level Summer nd Winter Summer only NOTE: The Assessment Unit for Module S is designted S4 FROM AUTUMN 008 7

10 INTRODUCTION RATIONALE Mthemtics is inherently sequentil subject. There is progression of mteril through ll levels t which the subject is studied. The content, therefore, builds upon the knowledge, skills nd understnding estblished t GCSE. The core content for AS is subset of the core content for A level. Progression in the subject will etend in nturl wy beyond AS nd A level, into Further Mthemtics or into relted courses in higher eduction. This specifiction dheres to the 00 Subject Criteri for AS nd A level Mthemtics nd hs been designed to conform with the GCE Advnced Subsidiry nd Advnced Level Emintions Qulifiction-Specific Criteri nd Common Criteri estblished jointly by the regultory uthorities in Englnd, Wles nd Northern Irelnd nd published by the Qulifictions nd Curriculum Authority (QCA). In following course bsed on this specifiction students should hve opportunities to: consolidte nd etend the knowledge, skills nd understnding developed in Key Stge 4; demonstrte positive chievement; build suitble foundtion for study of mthemtics nd other subjects in further nd higher eduction; prepre themselves for their economic environment nd for rnge of interesting creers; enjoy coherent, stisfying nd worthwhile course of study. In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. Where pproprite, techers should mke opportunities to ddress spiritul, morl, ethicl, socil nd culturl issues nd promote n wreness of environmentl, helth nd sfety nd Europen issues nd developments. For emple, students tking sttistics module my be given the opportunity to discuss presenttion of dt nd the possible misrepresenttion of informtion to support prticulr point of view. These issues will not be directly ssessed in ny of the ssessment units. This specifiction hs been designed to be s free s possible from ethnic, gender, religious, politicl or other forms of bis. 8 FROM AUTUMN 008

11 AIMS Courses bsed on this specifiction should encourge students to: develop their understnding of mthemtics nd mthemticl processes in wy tht promotes confidence nd fosters enjoyment; b develop bilities to reson logiclly nd recognise incorrect resoning, to generlise nd to construct mthemticl proofs; c etend their rnge of mthemticl skills nd techniques nd use them in more difficult, unstructured problems; d develop n understnding of coherence nd progression in mthemtics nd of how different res of mthemtics cn be connected; e recognise how sitution my be represented mthemticlly nd understnd the reltionship between rel world problems nd stndrd nd other mthemticl models nd how these cn be refined nd improved; f use mthemtics s n effective mens of communiction; g red nd comprehend mthemticl rguments nd rticles concerning pplictions of mthemtics; h cquire the skills needed to use technology such s clcultors nd computers effectively, recognise when such use my be inpproprite nd be wre of limittions; i develop n wreness of the relevnce of mthemtics to other fields of study, to the world of work nd to society in generl; j tke incresing responsibility for their own lerning nd the evlution of their own mthemticl development. FROM AUTUMN 008 9

12 ASSESSMENT OBJECTIVES The ssessment objectives provide n indiction of the skills nd bilities which the ssessment units re designed to ssess, together with the knowledge nd understnding specified in the subject content. It is not lwys possible to mke cler distinction between these different elements in constructing emintion questions nd therefore prticulr question my test more thn one ssessment objective. The ssessment objectives nd the ssocited weightings for AS nd A level re the sme nd re listed below. Cndidtes should be ble to: Assessment Objectives AO AO AO3 AO4 AO5 recll, select nd use their knowledge of mthemticl fcts, concepts nd techniques in vriety of contets; construct rigorous mthemticl rguments nd proofs through use of precise sttements, logicl deduction nd inference nd by the mnipultion of mthemticl epressions, including the construction of etended rguments for hndling substntil problems presented in unstructured form; recll, select nd use their knowledge of stndrd mthemticl models to represent situtions in the rel world; recognise nd understnd given representtions involving stndrd models; present nd interpret results from such models in terms of the originl sitution, including discussion of the ssumptions mde nd refinement of such models; comprehend trnsltions of common relistic contets into mthemtics; use the results of clcultions to mke predictions, or comment on the contet; nd, where pproprite, red criticlly nd comprehend longer mthemticl rguments or emples of pplictions; use contemporry clcultor technology nd other permitted resources (such s formule booklets or sttisticl tbles) ccurtely nd efficiently; understnd when not to use such technology, nd its limittions; give nswers to pproprite ccurcy. Minimum Weighting 30% 30% 0% 5% 5% Not ll ssessment objectives will be ssessed in every pper but the totl ssessment will emine the ssessment objectives s set out bove. The weightings given to ny prticulr objective my vry from yer to yer but will reflect the minimum requirements indicted bove. The weighting given to the ssessment objectives in ech pper will reflect the principles of fitness for purpose nd will tke into ccount the nture of the module being ssessed. 0 FROM AUTUMN 008

13 SPECIFICATION STRUCTURE The specifiction dopts modulr structure. Cndidtes re required to tke three ssessment units for n wrd in AS Mthemtics nd si ssessment units for n wrd in A level Mthemtics. The requirements re the sme for the corresponding wrds in Further mthemtics. The vilble ssessment units re listed below: Assessment Unit Stndrd Requirement C AS Compulsory for AS nd A level Mthemtics C AS Compulsory for AS nd A level Mthemtics C3 A Compulsory for A level Mthemtics C4 A Compulsory for A level Mthemtics F AS Compulsory for AS nd A level Further Mthemtics F A Compulsory for A level Further Mthemtics F3 A Compulsory for A level Further Mthemtics M AS Optionl M A Optionl M3 A Optionl M4 A Optionl S AS Optionl S4 A Optionl The digrm below outlines the dependence of ech ssessment unit on others in the specifiction: M M M3 M4 S S4 C C C3 C4 F F3 F FROM AUTUMN 008

14 KEY SKILLS This specifiction provides opportunities for developing nd generting evidence for ssessing the following ntionlly specified key skills t the levels indicted : Appliction of Number Level 3 Communiction Level 3 Informtion Technology Level 3 Working with Others Level 3 Improving Your Own Lerning nd Performnce Level 3 Problem Solving Level 3 The opportunities provided re referenced to the relevnt key skills specifictions nd eemplified in the Appendi on pge 64. PROHIBITED COMBINATIONS In ny one series of emintions cndidte my not tke emintions on this specifiction together with emintions on nother specifiction of the sme title. It is not permitted to count the wrd from the sme module: in two Advnced GCE subjects; in two AS subjects. Every specifiction is ssigned to ntionl clssifiction code indicting the subject re to which it belongs. Centres should be wre tht cndidtes who enter for more thn one GCE qulifiction with the sme clssifiction code, will hve only one grde (the highest) counted for the purpose of the School nd College Performnce Tbles. The clssifiction codes for this specifiction re: Mthemtics 3 Pure Mthemtics 33 Further Mthemtics FROM AUTUMN 008

15 SCHEME OF ASSESSMENT ALLOWABLE COMBINATIONS From the ssessment units listed in section, cndidtes cn choose si if they wish to be considered for the wrd of n A level GCE grde nd three for the wrd of n AS grde. The llowble combintions re set out below: Specifiction title Entry code Assessment units to be tken AS Mthemtics S C C M AS Mthemtics S C C S A level Mthemtics A C C C3 C4 M M A level Mthemtics A C C C3 C4 S S4 A level Mthemtics A C C C3 C4 M S AS Further Mthemtics S33 F F S AS Further Mthemtics S33 F S M3 AS Further Mthemtics S33 F S S4 AS Further Mthemtics S33 F F M AS Further Mthemtics S33 F M M AS Further Mthemtics S33 F M S4 AS Further Mthemtics AS Further Mthemtics AS Further Mthemtics AS Further Mthemtics AS Further Mthemtics S33 S33 S33 S33 S33 F M M3 F F M F F S4 F F M3 F M3 M4 A level Further Mthemtics A33 F F F3 M3 M4 S A level Further Mthemtics A33 F F F3 M3 S S4 A level Further Mthemtics A33 F F F3 M M M3 A level Further Mthemtics A level Further Mthemtics A33 A33 F F F3 M M3 S4 F F F3 M M3 M4 Units tht contribute to n wrd in AS nd A level Mthemtics my not lso be used for n wrd in Further Mthemtics. To gin certifictes in both Advnced Mthemtics nd AS Further Mthemtics, cndidtes must use 9 different units. To gin certifictes in both Advnced Mthemtics nd Advnced Further Mthemtics, cndidtes must use different units. FROM AUTUMN 008 3

16 Units C, C, M, S nd F re ssessed t AS stndrd; ll other units re ssessed t A stndrd. Centres entering cndidtes for AS/A level Further Mthemtics should be wre of the following implictions of the choice of units for the A level wrd: cndidtes who hve tken single pplictions strnd (either Mechnics or Sttistics) for their A level wrd will hve two AS units (F nd either S or M) vilble for their Further Mthemtics wrd; cndidtes who hve tken both pplictions strnds (Mechnics nd Sttistics) for their A level wrd will hve only one AS unit (F) vilble for their Further Mthemtics wrd. These llowble combintions re consistent with the requirements of the Subject Criteri for Mthemtics (00): AS nd A level specifictions in Mthemtics must ddress t lest one pplictions re; AS Further Mthemtics specifictions must include t lest one unit of pure mthemtics; A Level Further Mthemtics specifictions must include t lest two units of pure mthemtics; A level mthemtics specifictions must include two or three A units; A level Further Mthemtics specifictions must include t lest three A units. Emple of Pthwys to AS nd A level Mthemtics nd to AS nd A level Further Mthemtics AS A AS FM FM C C S C3 C4 M F M S4 F F3 M3 C C M C3 C4 S F M M3 F F3 S4 F S F F3 M3 M4 F3 M3 S4 C C M C3 C4 M F S M3 F F3 M4 F F3 S4 F S S4 F F3 M3 C C S C3 C4 S4 F M M F M F F F3 M3 F3 M M3 Appliction, in writing, must be mde to the Council if centres wish to enter cndidtes for n AS or A level Pure Mthemtics wrd. 4 FROM AUTUMN 008

17 THE RELATIONSHIP BETWEEN ASSESSMENT UNITS AND ASSESSMENT OBJECTIVES TABLE : AS ASSESSMENT WEIGHTINGS Cndidtes must tke three ssessment units to qulify for n AS wrd. Ech of the three units is weighted t 33 /3% of the wrd. The llowble combintions of ssessment units re set out on pge. Assessment Unit AO % AO % Assessment Objectives AO3 % AO4 % C AO5 % C M S F TABLE : A LEVEL ASSESSMENT WEIGHTINGS Cndidtes must tke si ssessment units to qulify for n A level wrd. Ech of the si units is weighted t 6 /3% of the wrd. The llowble combintions of ssessment units re set out in the previous section. Assessment Unit AO % AO % Assessment Objectives AO3 % AO4 % C AO5 % C C C F F F M M M M S S FROM AUTUMN 008 5

18 NATURE OF ASSESSMENT UNITS All ssessment units under this specifiction will be eternl. All ssessment units tke the form of n eternl emintion nd the durtion of ech emintion is shown below: Assessment Unit C ssesses Module C hour 30 minutes Assessment Unit C ssesses Module C hour 30 minutes Assessment Unit C3 ssesses Module C3 hour 30 minutes Assessment Unit C4 ssesses Module C4 hour 30 minutes Assessment Unit F ssesses Module FP hour 30 minutes Assessment Unit F ssesses Module FP hour 30 minutes Assessment Unit F3 ssesses Module FP3 hour 30 minutes Assessment Unit M ssesses Module M hour 30 minutes Assessment Unit M ssesses Module M hour 30 minutes Assessment Unit M3 ssesses Module M3 hour 30 minutes Assessment Unit M4 ssesses Module M4 hour 30 minutes Assessment Unit S ssesses Module S hour 30 minutes Assessment Unit S4 ssesses Module S hour 30 minutes THE SEQUENCE, TIMING AND RESITTING OF ASSESSMENT UNITS In 005, the first yer of certifiction of AS Mthemtics under this specifiction, ssessment units C, C, C3, C4, F, F, M, M, S nd S4 will be offered in the summer emintion series only. No other ssessment units will be offered in 005. Therefter, ssessment units will be offered s follows: Cndidtes my sit the following ssessment units in either the winter or summer emintion sessions: C, C, C3, C4, F, F, M, M, S. Cndidtes my sit the following ssessment units in the summer emintion session only: F3, M3, M4, S4. Assessment units my be retken more thn once. The best result in ech unit must count towrds the finl wrd; other results in these units will be used up. Cndidtes my, however, retke the whole qulifiction more thn once. The results of individul ssessment units, prior to certifiction, will hve shelf life limited only by the shelf life of this specifiction. 6 FROM AUTUMN 008

19 SYNOPTIC ASSESSMENT Synoptic ssessment in mthemtics will ddress cndidtes understnding of the connections between different elements of the subject. It involves the eplicit drwing together of knowledge, understnding nd skills lerned in different prts of the A level course through using nd pplying methods developed t erlier stges of study in solving problems. Mking nd understnding connections in this wy is intrinsic to lerning mthemtics. Synoptic ssessment must form 0% of the totl ssessment for A level mthemtics. The synoptic units in A level mthemtics re C3, C4, M, M, Snd S4. Cndidtes my tke these units in ny order nd in ny emintion series. Cndidtes will utomticlly fulfil the synoptic requirements by tking ny of the llowble combintions shown on pge of this specifiction. In ppers which ddress the A core content, synoptic ssessment requires the use of methods from the AS core content. In ppers which ddress mthemticl content outside the core content, synoptic ssessment requires the use of methods from the core content nd/or methods from erlier stges of the sme spect of mthemtics (pure mthemtics, mechnics or sttistics). LANGUAGE OF SPECIFICATION AND ASSESSMENT MATERIALS The specifiction nd ssocited specimen ssessment mterils re provided in English. CANDIDATES WITH PARTICULAR REQUIREMENTS Detils of rrngements for cndidtes with prticulr ssessment requirements re provided in the Joint Council for Generl Qulifictions GCSE nd GCE Regultions nd Guidnce for Cndidtes with Specil Assessment Needs. AWARDS AND CERTIFICATION The AS nd the Advnced GCE in Mthemtics, Pure Mthemtics nd Further Mthemtics will be wrded on five-grde scle: A, B, C, D nd E. Cndidtes who fil to rech the minimum stndrd for grde E will be recorded s U (unclssified) nd will not receive n AS or A level GCE certificte. The results of individul ssessment units will be reported. Where AS certifiction is not requested, cndidte going on to complete the full A level GCE must nevertheless complete ll the modules nd tke ll the ssessment units required for the AS wrd. CCEA will comply with the grding, wrding nd certifiction requirements of the revised GCE Code of Prctice for courses strting in September 000. FROM AUTUMN 008 7

20 3 SUBJECT CONTENT This specifiction builds on the knowledge, understnding nd skills estblished in GCSE Mthemtics. The core mteril for AS, contined in teching nd lerning modules C nd C, is subset of the core mteril for A level; this is completed in modules C3 nd C4. The subject content is orgnised into thirteen teching nd lerning modules. The content of these modules is set out below. For ech module the mjor topics re listed, together with relted guidnce notes. These notes provide further detil of the content required but they re not intended to be ehustive descriptions of the topics to which they relte. Within the ctegories of Pure Mthemtics, Mechnics nd Sttistics, the modules re set out in the norml sequence in which their ssocited ssessment units would be tken. The content of ech unit should be red in conjunction with the relevnt ims nd ssessment objectives set out in Section of this specifiction. This specifiction for AS nd A level Mthemtics requires: () (b) construction nd presenttion of mthemticl rguments through pproprite use of logicl deduction nd precise sttements involving correct use of symbols nd pproprite connecting lnguge; correct understnding nd use of mthemticl lnguge nd grmmr in respect of terms such s equls, identiclly equls, therefore, becuse, implies, is implied by, necessry, sufficient nd nottion such s,, nd. In ddition, the specifiction for A level mthemtics requires: (c) methods of proof, including proof by contrdiction nd disproof by counter-emple. Cndidtes re epected to mke use of cler, precise nd pproprite mthemticl lnguge in ccordnce with the requirements of Assessment Objective. This will be ssessed by the lloction of mrks ginst Assessment Objective. At lest one re of the ppliction of mthemtics must be ddressed by cndidtes seeking n wrd in AS or A level Mthemtics. For this specifiction, this mens tht cndidtes must study Mechnics, Sttistics or both. The ppliction of mthemtics must count for t lest 30% of the totl credit for the qulifiction. This specifiction for A level Mthemtics stisfies the requirement to include content, minly in the re of pure mthemtics, to study some spect of modelling nd the ppliction of mthemtics. Modelling should be ddressed, s pproprite, in the teching of ll modules in the specifiction. For both AS nd A level Mthemtics, the knowledge, understnding nd skills identified by the respective criteri must ttrct two-thirds of the totl credit for the qulifiction. 8 FROM AUTUMN 008

21 MODULE C AS CORE MATHEMATICS This module covers pproimtely hlf of the core content mteril for AS emintions. The module will be ssessed t AS stndrd nd is compulsory for AS nd A level GCE Mthemtics. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. Cndidtes re not permitted to use ny clculting id in the ssessment unit for this module. Topic Guidnce Notes Lws of indices for ll rtionl indices. Use nd mnipultion of surds. Qudrtic functions nd their grphs; the discriminnt of qudrtic function; completing the squre. Rtionlistion of denomintors Conditions for rel nd equl roots re included. Ecluding reltionship between roots nd coefficients of qudrtic eqution. Solution of qudrtic equtions. Simultneous equtions; nlytic solution by substitution, eg of one liner nd one qudrtic eqution. Solution of liner nd qudrtic inequlities. 3 Algebric mnipultion of polynomils, including epnding brckets nd collecting like terms, fctoristion nd simple lgebric division. Liner with two or three unknowns. Including inequlities reducible to the form f() > 0, where f() is product of liner fctors. Division by liner epressions only. Use of the Fctor Theorem nd the Reminder Theorem. 4 Grphs of functions; sketching curves defined by simple equtions. Geometricl interprettion of lgebric solution of equtions. Use of intersection points of grphs to solve equtions. Knowledge of function nottion. Plotting grphs on grph pper will not be required. Knowledge of the effect of simple trnsformtions on the grph of y = f() s represented by y = f(), y = f() +, y = f( + ), y = f(). 5 Eqution of stright line, including the forms y y = m( ) nd + by + c = 0. Including the mid-point of line segment. Conditions for two stright lines to be prllel or perpendiculr to ech other. FROM AUTUMN 008 9

22 Topic 6 The derivtive of f() s the grdient of the tngent to the grph of y = f() t point; the grdient of the tngent s limit; interprettion s rte of chnge; second order derivtives. Differentition of differences. n nd relted sums nd Applictions of differentition to grdients, tngents nd normls, mim nd minim nd sttionry points, incresing nd decresing functions. Guidnce Notes Questions on differentition from first principles will not be set. Including use of second derivtive. 0 FROM AUTUMN 008

23 MODULE C AS CORE MATHEMATICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. This module, together with module C, contins core content mteril for AS emintions. A knowledge of the content of module C will be ssumed. The module will be ssessed t AS stndrd nd is compulsory for AS nd A level GCE Mthemtics. The ssessment unit for the module is n eternl emintion with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. The use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Co-ordinte geometry of the circle using the eqution of the circle in the form ( ) + (y b) = r, nd including use of the following circle properties: Angle in semicircle is right ngle; Perpendiculr from centre to chord bisects the chord; Perpendiculrity of rdius nd tngent. Sequences, including those given by formul for the nth term nd those generted by simple reltion of the form n+ = f( n ) Arithmetic series, including the formul for the sum of the first n terms. Guidnce Notes Including the form + y + g + fy + c = 0 Behviour of sequences (convergence, divergence nd oscilltion) will be required. Including the use of the nottion formul for the sum of the first n nturl numbers. The sum of finite geometric series; the sum to infinity of convergent geometric series, including the use of r <. Binomil epnsion of ( + ) n for positive n integer n; the nottions n! nd. r 3 Sine nd cosine rules nd the re of tringle in the form bsin C. Rdin mesure, including use for rc length nd re of sector. Cses of ( + b) n reducible to the form k( + ) n my be emined. s = rθ; A r θ Sine, cosine nd tngent functions; their grphs, symmetries nd periodicity. sin θ Knowledge nd use of tn nd cos θ sin θ cos θ. Solution of simple trigonometric equtions in given intervl. For emple: 3 3cos θ sin θ = 0 for π < θ π FROM AUTUMN 008

24 Topic 4 y nd its grph. Lws of logrithms: log log y log ( y) log log y log y k k log log ( ). The solution of equtions of the form = b. Guidnce Notes Knowledge of the effect of simple trnsformtions. Including the chnge of bse rule: logb log log b Ecluding nturl logrithms. Ecluding the use of the logrithmic trnsformtion in chnging eperimentl dt into stright-line form. 5 Indefinite integrtion s the reverse of differentition. Integrtion of n nd relted sums nd differences. Approimtion of re under curve using the trpezium rule. Interprettion of the definite integrl s the re under curve. Evlution of definite integrls. Are defined by curve nd either is or between two curves. FROM AUTUMN 008

25 MODULE C3 A CORE MATHEMATICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. This module covers pproimtely hlf of the core content mteril for A level emintions beyond the AS core mteril contined in modules C nd C. A knowledge of the content of modules C nd C will be ssumed. The module will be ssessed t A stndrd nd is compulsory for A level GCE Mthemtics. The ssessment unit for the module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. The use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Simplifiction of rtionl epressions including fctorizing nd cncelling, nd lgebric division. Guidnce Notes Division by non-liner epressions. Rtionl functions; prtil frctions with denomintors not more complicted thn repeted liner terms. The modulus function. Including - < b Combintions of simple trnsformtions on the grph of y = f() s represented by y = f(), y = f() +, y = f( + ), y = f(). Prmetric equtions of curves; conversion between prmetric nd Crtesin forms. Generl properties of conics re ecluded. 3 Binomil series for ny rtionl vlue of n. 4 Knowledge of secnt, cosecnt nd cotngent nd of rcsin, rccos nd rctn. Their reltionships to sine, cosine nd tngent. Knowledge nd use of the equivlents of sin θ + cos θ =. Use of nottion sin θ sec θ = tn θ + cosec θ = + cot θ etc. Solution of trigonometric equtions in given intervl; for emple, sec θ 5tn θ 5. 5 The function e nd its grph. The function ln nd its grph; ln s the inverse function of e. Eponentil growth nd decy. Knowledge of the effect of simple trnsformtions. Knowledge of the effect of simple trnsformtions. Both discrete nd continuous growth. For emple, the hlf-life of rdioctive element. FROM AUTUMN 008 3

26 Topic Guidnce Notes 6 Differentition of e, ln, sin, cos, tn nd their sums nd differences. For emple, ln3, ln. Differentition using the product rule, the quotient dy rule, the chin rule nd by the use of. d d dy Indefinite integrtion s the reverse of differentition; in prticulr, integrtion of e, For emple: e, sin3,, sin, cos. 7 Loction of roots of f() = 0 by considering chnges of sign of f() in n intervl of in which f() is continuous. For emple, e 3, ln( + ), sin3, tn. Differentition of sec, cosec nd cot. Including second derivtive. Ecluding connected rtes of chnge., sec 3 Approimte solution of equtions using simple itertive methods, including recurrence reltions of the form n = f n Numericl integrtion of functions. Including Newton-Rphson method. Simpson s Rule. 4 FROM AUTUMN 008

27 MODULE C4 A CORE MATHEMATICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. This module, together with modules C, C nd C3, contins the core content mteril for A level emintions. A knowledge of the content of modules C, C nd C3 will ssumed. The module will be ssessed t A stndrd nd is compulsory for A level GCE Mthemtics. The ssessment unit for the module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. The use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Definition of function; domin nd rnge of functions; composition of functions; inverse functions nd their grphs. Guidnce notes Knowledge tht gf() g(f()). f - (). Understnding of the grphs nd pproprite restricted domins of secnt, cosecnt, cotngent, rcsin, rccos nd rctn. 3 Knowledge nd use of double ngle formule; use of formule for sin(a B), cos(a B) nd tn(a B); nd of epressions for cos + bsin in the equivlent forms of rcos( ) or rsin( ). 4 Differentition of simple functions defined implicitly or prmetriclly. Proofs of these my be required. Proofs of these re not required. Solution of trigonometric equtions (ecluding generl solution) including use of the identities listed. Including second derivtive. Formtion of simple differentil equtions. 5 Simple cses of integrtion by substitution nd by prts; these methods s the reverse processes of the chin nd product rules respectively. Simple cses of integrtion using prtil frctions. Evlution of volume of revolution. Anlyticl solution of simple first order differentil equtions with seprble vribles. For emple: sin d; cos 3 d The reltionship with corresponding techniques of differentition should be understood. The t (t = tn /) substitution is ecluded. Integrtion by prts will not require more thn one opertion. Volumes generted by the rottion of the re under single curve bout the -is only. FROM AUTUMN 008 5

28 Topic 6 Vectors in two nd three dimensions. Mgnitude of vector. Algebric opertions of vector ddition nd multipliction by sclrs nd their geometricl interprettions. Guidnce notes Nottion: AB, r, r Unit vectors: i, j, k. Including column vector nottion. Nottion: AB, r, r. Ecluding questions set in geometricl contet. Position vectors; the distnce between two points; vector equtions of lines. The sclr product; its use for clculting the ngle between two lines. Ecluding skew lines..b = bcos where is the ngle between nd b. 6 FROM AUTUMN 008

29 MODULE FP FURTHER PURE MATHEMATICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. A knowledge of the content of modules C nd C will be ssumed. The module will be ssessed t AS stndrd nd is compulsory for AS nd A level GCE Further Mthemtics. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. The use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Guidnce Notes Mtrices: ddition, multipliction, null nd unit mtrices. Solution of liner equtions in nd 3 unknowns. Illustrted by solution of A = y Evlution of inverses of non-singulr mtrices. Liner mppings nd trnsformtions in the plne. Represented by mtrices nd column vectors. To include liner trnsformtions of curves, f(, y) = 0 3 Determinnts; impliction of the zero vlue of the determinnt of: (i) simple trnsformtion mtri; (ii) the coefficient mtri of system of simultneous liner equtions. Determinnts of order nd 3 only. Only order. Order nd 3. 4 Eigenvlues nd eigenvectors of 3 3 mtrices. Including mtrices. Reduction of symmetricl mtrices to digonl form. 5 Binry opertions nd groups; period of n element; cyclic groups, isomorphism between groups. Subgroups. Symmetry groups, permuttion groups, groups of mtrices nd the group of residue clsses mod m re included. Lgrnge s theorem (without proof). 6 Further co-ordinte geometry of circle. Intersection of circles, eqution of common chord, eqution of tngent to circle. 7 Comple numbers; Crtesin nd polr form, modulus, rgument, conjugte. Argnd digrms. Sum, difference, product nd quotient of two comple numbers. Simple loci Representtion on n Argnd digrm of the sum or difference of two comple numbers my be required. For emple, z z = z z ; z - = r. FROM AUTUMN 008 7

30 MODULE FP FURTHER PURE MATHEMATICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. A knowledge of the content of modules C, C, C3, C4 nd FP will ssumed. The module will be ssessed t A stndrd nd is compulsory for A level GCE Further Mthemtics. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. The use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Guidnce Notes Prtil frctions To include qudrtic fctors in the denomintor. Summtion of finite series. Use of r, r nd r 3. To include rithmetic series, geometric series nd telescopic series. 3 Proof by mthemticl induction. 4 Generl solution of trigonometric equtions. 5 Simple tretment of the co-ordinte geometry of the prbol nd ellipse in Crtesin nd prmetric form. Including definition of curves s loci; knowledge of focus, directri nd eccentricity. Chnge of origin without rottion. 6 De Moivre s theorem for generl inde ecluding proof. The n th roots of comple number. Including epnsions of sin n θ nd cos n θ in powers of sin θ nd cos θ nd the reverse process. Etension to epression for tn n θ. The eponentil form of comple number. Comple roots of simple polynomils with rel coefficients. 7 Anlyticl solution of the differentil equtions y + p()y = q() nd y + by + cy = f() where, b nd c re constnts. Solutions stisfying boundry conditions. The form of f() will be restricted to k cos n, k sin n, k n or k e n, where k is constnt nd n is n integer; f() will not be solution of the corresponding homogeneous eqution: y + by + cy = 0 The uiliry eqution my hve comple roots. 8 FROM AUTUMN 008

31 Topic Guidnce Notes 8 Mclurin s theorem. The derivtion of the series epnsion of ( + ) n, e, ln( + ), sin, cos nd tn. To include the derivtion of the series epnsions of simple compound functions. Simple eercises on, pproimtions by nd simple vritions on these epnsions. Summtion of infinite series using series epnsions. sin, cos - ½, tn. FROM AUTUMN 008 9

32 MODULE FP3 FURTHER PURE MATHEMATICS 3 In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. A knowledge of the content of modules C, C, C3, C4, FP nd FP will be ssumed. The module will be ssessed t A stndrd nd is compulsory for A level GCE Further Mthemtics. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. The use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Guidnce notes Differentition of sin, cos, tn. Repeted integrtion by prts. Simple reduction formule. Integrtion by prts will not involve more thn three pplictions. 3 Integrtion of,. Use of pproprite substitutions. 4 The hyperbolic nd inverse hyperbolic functions; their definitions, grphs, derivtives nd integrls. 5 Crtesin eqution of line. Including reltionships between the si hyperbolic functions nd the ln epressions for sinh-nd cosh-. y b z c l m n Including skew lines. 6 Vector product. b s n re..b c s volume. The eqution of line (r ) b = 0. The vector triple product is ecluded. 7 Vector nd Crtesin equtions of plne. n.r = d nd + by + cz = d. Lines norml nd prllel to plne; plnes norml nd prllel to line; ngle between line nd plne; eqution of the line of intersection of two plnes. 30 FROM AUTUMN 008

33 MODULE M MECHANICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. Students should be given opportunities to eplore prcticlly, contets relting to the contents of this module. A knowledge of syllbus C nd C will be ssumed. Modelling nd the ppliction of mthemtics s referred to t the strt of section 3 will be ssessed in this module. The module will be ssessed t AS level nd is optionl. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. Use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Displcement, velocity nd ccelertion; displcement-time grphs; velocity-time grphs. Comment Rectiliner motion only. Equtions for uniform ccelertion. Appliction of differentition nd integrtion to problems in kinemtics set s function of time. Accelertion given s simple function of t. Force s loclised vector. Mgnitude, direction, components nd resultnts. 3 Friction. Modelling ssumptions bout frictionl force. Limiting friction = µn. Angle of friction not required. 4 Equilibrium of prticle. 5 Moment of force bout point. The principle of moments. Prllel forces; couples. Concept of Centre of Grvity. Non-prllel forces. 6 Equilibrium of rigid body. Including ldders nd rods. 7 Mss nd ccelertion. Newton's lws of motion to include motion of connected prticles. Where pulleys re involved they will be smooth nd fied. 8 Impulse nd momentum. Principle of conservtion of liner momentum; direct impct. Rectiliner motion nd direct impcts only (coefficient of restitution not required). FROM AUTUMN 008 3

34 MODULE M MECHANICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. Students should be given opportunities to eplore prcticlly, contets relting to the contents of this module. A knowledge of syllbus C, C, C3, C4 nd M will be ssumed. Modelling nd the ppliction of mthemtics s referred to t the strt of section 3 will be ssessed in this module. The module will be ssessed t A level nd is optionl. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. Use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Displcement, velocity, ccelertion, force etc s vectors. Comment Vectors in the form i + bj + ck. Integrtion nd differentition of vectors. Vectors in the form f(t)i + g(t)j + h(t)k. 3 Vrible ccelertion long stright line. Accelertion s function of time or velocity or displcement. Emples involving constnt power my be set. 4 Projectiles. Motion in verticl plne with constnt ccelertion, ie under grvity. 5 Uniform motion in horizontl circle. Conicl pendulum. Derivtion of the stndrd results for gretest height reched, time of flight, rnge on horizontl plne nd the eqution of the flightpth is required. Emples involving n inclined plne will not be set. = r, = r = / r. Including bnked corners. Ecluding sliding/overturning problems. 6 Grvittionl potentil energy (mgh). Kinetic energy. Work done. Work-energy principle. Principle of conservtion of mechnicl energy. 7 Power treted s rte of doing work (leding to P = Fv) nd rte of increse of energy. Ecluding the clcultion by integrtion of work done by vrible force. Work done = / mv / mu = chnge of kinetic energy. Rectiliner motion only. 3 FROM AUTUMN 008

35 MODULE M3 MECHANICS 3 In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. Students should be given opportunities to eplore prcticlly, contets relting to the contents of this module. A knowledge of syllbus C, C, C3, C4, M nd M will be ssumed. Modelling nd the ppliction of mthemtics s referred to t the strt of section 3 will be ssessed in this module. The module will be ssessed t A level nd is optionl. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. Use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Centre of mss. System of prticles t fied points. Rods. Comment Proof of stndrd results not required. Including use of symmetry. Ecluding use of clculus. Ecluding vrible density. Rectngulr, tringulr nd circulr lmine. Composite lmine. Suspended lmine. Further prticle equilibrium. Including prticle ttched to elstic springs or strings on rough plne. 3 Resultnt velocity. Reltive velocity. 4 Hooke's Lw. Grphicl or vector component method. Including problems involving minimum distnce or interception, but not course for closest pproch. Modelling ssumption: elstic limits. Elstic springs nd strings. 5 Work nd Kinetic energy. Work-energy principle. Energy stored in n elstic spring or string. Simple problems involving kinetic energy, grvittionl potentil energy nd elstic potentil energy. 6 Simple hrmonic motion. Simple pendulum. Oscilltions of prticle ttched to the end of n elstic spring or string. Use of sclr product. Rectiliner motion only. Work = F d mv b mv b = chnge in kinetic energy. Knowledge of definition nd stndrd results. Proof of stndrd results is not required. Oscilltions will be in the direction of the spring or string. FROM AUTUMN

36 MODULE M4 MECHANICS 4 In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. Students should be given opportunities to eplore prcticlly, contets relting to the contents of this module. A knowledge of syllbus C, C, C3, C4, M, M nd M3 will be ssumed. Modelling nd the ppliction of mthemtics s referred to t the strt of section 3 will be ssessed in this module. The module will be ssessed t A level nd is optionl. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. Use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Topic Centre of mss of lmine nd solids. Composite bodies. Suspended bodies. Comment Ecluding vrible density. Including use of clculus. Proof of stndrd results for solid cone nd solid hemisphere only my be required. Tble of stndrd results my be used. Toppling problems. Force systems in two dimensions; generl resultnt of coplnr force systems. Replcement of system by single force, by couple or by single force cting t specific point together with couple. 3 Light pin-jointed frmeworks. Use of Bow s nottion is optionl. Questions my involve identifying forces s tension or thrust s well s clculting their mgnitude. 4 Method of dimensions. Checking of epressions nd equtions for dimensionl consistency. Derivtion of equtions connecting physicl quntities where product reltionship is ssumed. 5 Universl Lw of Grvittion; stellite motion. 6 Further circulr motion on bnked corners. Questions my be set on sliding/overturning problems. 7 Motion in verticl circle. Proof of stndrd results my be required. 8 Direct impct of elstic spheres; Newton's lw of restitution. Questions involving impulsive tensions in strings will not be set. Elstic collisions between smooth sphere nd plne or between smooth spheres. 34 FROM AUTUMN 008

37 MODULE S STATISTICS In following course bsed on this specifiction students should be encourged to mke pproprite use of grphic clcultors nd computers s tools by which the lerning of mthemtics my be enhnced. A knowledge of syllbus C nd C will be ssumed. Modelling nd the ppliction of mthemtics s referred to t the strt of section 3 will be ssessed in this module. The module will be ssessed t AS level nd is optionl. The ssessment unit for this module is n eternl emintion, with mimum of 75 rw mrks, the durtion of which is hour 30 minutes. Use of grphic or scientific clcultor will be permitted in the ssessment unit for this module. Cndidtes should be fmilir with methods of presenting dt, including frequency tbles for ungrouped nd grouped dt, bo plots nd stem-nd-lef digrms. They should lso be fmilir with men, mode nd medium s summry mesures of loction of dt. Questions tht directly test the bility of cndidtes to construct such tbles nd digrms nd clculte such mesures will not be set, but cndidtes will be epected to interpret them nd drw inferences from them. Topic Comment Apprecition of the inherent vribility of dt. Collection, ordering nd presenttion of dt. Clcultion nd interprettion of pproprite summry mesures of the loction nd dispersion of dt. 3 Smple spce: events, mutully eclusive nd ehustive events. Clssicl nd limiting reltive frequency definitions of probbility. 4 Addition Lw; Multipliction Lw; sttisticl dependence nd independence. 5 Probbility functions, men, vrince nd stndrd devition. Cndidtes will be epected to drw inferences bout dt sets nd histogrms of vrying widths nd interpret results t level beyond tht epected t GCSE. Knowledge, understnding nd use of men, medin, mode, inter-qurtile rnge, nd stndrd devition. Computtion of stndrd devition (nd of men) should be by clcultor. Cndidtes should know which key to use when clculting the stndrd devition of dt set. Cndidtes will be epected to drw inferences bout dt sets nd interpret results t level beyond tht epected t GCSE. n (A) P(A) =, where event A is subset of ll n n eqully likely outcomes. f A P(A) = lim, where f A is the frequency of event m m A in m trils. Clcultion of combined probbilities of up to three events using pproprite digrms. Including conditionl probbility. Clcultion of probbilities such s P( X b ) nd of men nd vrince for simple cses. Knowledge nd use of epressions for E( + bx) nd Vr( + bx). FROM AUTUMN

38 Topic 6 Discrete probbility distributions: uniform, binomil, Poisson. 7 Continuous probbility distribution; probbility density function f ; men, vrince nd stndrd devition. 8 Norml distribution; liner trnsformtion of Norml vrible; the stndrd Norml distribution. Comment Knowledge nd use of the probbility functions nd the epressions for the men nd vrince, but not their derivtion. Knowledge of ssumptions for binomil nd Poisson distribution is required. Ability to use the eponentil function to clculte probbilities for Poisson distribution is required. Use of recurrence formule will not be required. Clcultion of probbilities such s P( < X < b ) nd of men nd vrince for simple cses. The pdf will be given s simple function of. Knowledge nd use of epressions for E( + b) nd Vr( + bx). Applictions only. Use of N(0,) tbles. 36 FROM AUTUMN 008