GCE AS/A LEVEL WJEC GCE AS/A Level in FURTHER MATHEMATICS APPROVED BY QUALIFICATIONS WALES SPECIFICATION Teching from 07 For wrd from 08 (AS) For wrd from 09 (A level) This Qulifictions Wles regulted qulifiction is not vilble to centres in Englnd.
GCE AS nd A LEVEL FURTHER MATHEMATICS WJEC GCE AS nd A LEVEL in FURTHER MATHEMATICS For teching from 07 For AS wrd from 08 For A level wrd from 09 This specifiction meets the Approvl Criteri for GCE AS nd A Level Further Mthemtics nd the GCE AS nd A Level Qulifiction Principles which set out the requirements for ll new or revised GCE specifictions developed to be tught in Wles from September 07. Summry of ssessment. Introduction 5. Aims nd objectives 5. Prior lerning nd progression 6.3 Equlity nd fir ccess 6.4 Welsh Bcclurete 7.5 Welsh perspective 7. Subject content 8. AS Unit. AS Unit 5.3 AS Unit 3 8.4 A Unit 4 0.5 A Unit 5 6.6 A Unit 6 9 3. Assessment 3 3. Assessment objectives nd weightings 3 4. Technicl informtion 33 4. Mking entries 33 4. Grding, wrding nd reporting 34 Appendix A Mthemticl nottion 35 Appendix B Mthemticl formule nd identities 4 Pge
GCE AS nd A LEVEL FURTHER MATHEMATICS GCE AS nd A LEVEL FURTHER MATHEMATICS (Wles) SUMMARY OF ASSESSMENT This specifiction is divided into totl of 5 units, 3 AS units nd A units. Weightings noted below re expressed in terms of the full A level qulifiction. All AS units nd A Unit 4 re compulsory. AS (3 units) AS Unit : Further Pure Mthemtics A Written exmintion: hour 30 minutes 3% of qulifiction 70 mrks The pper will comprise number of short nd longer, both structured nd unstructured questions, which my be set on ny prt of the subject content of the unit. A number of questions will ssess lerners' understnding of more thn one topic from the subject content. A clcultor will be llowed in this exmintion. AS Unit : Further Sttistics A Written exmintion: hour 30 minutes 3% of qulifiction 70 mrks The pper will comprise number of short nd longer, both structured nd unstructured questions, which my be set on ny prt of the subject content of the unit. A number of questions will ssess lerners' understnding of more thn one topic from the subject content. A clcultor will be llowed in this exmintion. AS Unit 3: Further Mechnics A Written exmintion: hour 30 minutes 3% of qulifiction 70 mrks The pper will comprise number of short nd longer, both structured nd unstructured questions, which my be set on ny prt of the subject content of the unit. A number of questions will ssess lerners' understnding of more thn one topic from the subject content. A clcultor will be llowed in this exmintion.
GCE AS nd A LEVEL FURTHER MATHEMATICS 3 A Level (the bove plus further units) Cndidtes must tke Unit 4 nd either Unit 5 or Unit 6. A Unit 4: Further Pure Mthemtics B Written exmintion: hours 30 minutes 35% of qulifiction 0 mrks This unit is compulsory. The pper will comprise number of short nd longer, both structured nd unstructured questions, which my be set on ny prt of the subject content of the unit. A number of questions will ssess lerners' understnding of more thn one topic from the subject content. A clcultor will be llowed in this exmintion A Unit 5: Further Sttistics B Written exmintion: hour 45 minutes 5% of qulifiction 80 mrks Lerners will sit either Unit 5 or Unit 6. The pper will comprise number of short nd longer, both structured nd unstructured questions, which my be set on ny prt of the subject content of the unit. A number of questions will ssess lerners' understnding of more thn one topic from the subject content. A clcultor will be llowed in this exmintion A Unit 6: Further Mechnics B Written exmintion: hour 45 minutes 5% of qulifiction 80 mrks Lerners will sit either Unit 5 or Unit 6. The pper will comprise number of short nd longer, both structured nd unstructured questions, which my be set on ny prt of the subject content of the unit. A number of questions will ssess lerners' understnding of more thn one topic from the subject content. A clcultor will be llowed in this exmintion This is unitised specifiction which llows for n element of stged ssessment. Assessment opportunities will be vilble in the summer ssessment period ech yer, until the end of the life of the specifiction. Unit, Unit nd Unit 3 will be vilble in 08 (nd ech yer therefter) nd the AS qulifiction will be wrded for the first time in summer 08. Unit 4, Unit 5 nd Unit 6 will be vilble in 09 (nd ech yer therefter) nd the A level qulifiction will be wrded for the first time in summer 09. Qulifiction Number listed on The Register: GCE AS: 603/984/7 GCE A level: 603/980/X Qulifictions Wles Approvl Number listed on QiW: GCE AS: C00/73/8 GCE A level: C00/53/7
GCE AS nd A LEVEL FURTHER MATHEMATICS 5 GCE AS AND A LEVEL FURTHER MATHEMATICS INTRODUCTION. Aims nd objectives This WJEC GCE AS nd A Level in Further Mthemtics provides brod, coherent, stisfying nd worthwhile course of study. It encourges lerners to develop confidence in, nd positive ttitude towrds, mthemtics nd to recognise its importnce in their own lives nd to society. The specifiction hs been designed to respond to the proposls set out in the report of the ALCAB pnel on mthemtics nd further mthemtics. The WJEC GCE AS nd A level in Further Mthemtics encourges lerners to: develop their understnding of mthemtics nd mthemticl processes in wy tht promotes confidence nd fosters enjoyment; develop bilities to reson logiclly nd recognise incorrect resoning, to generlise nd to construct mthemticl proofs; extend their rnge of mthemticl skills nd techniques nd use them in more difficult, unstructured problems; develop n understnding of coherence nd progression in mthemtics nd of how different res of mthemtics cn be connected; 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; use mthemtics s n effective mens of communiction; red nd comprehend mthemticl rguments nd rticles concerning pplictions of mthemtics; 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; develop n wreness of the relevnce of mthemtics to other fields of study, to the world of work nd to society in generl; tke incresing responsibility for their own lerning nd the evlution of their own mthemticl development.
GCE AS nd A LEVEL FURTHER MATHEMATICS 6. Prior lerning nd progression Any requirements set for entry to course following this specifiction re t the discretion of centres. It is resonble to ssume tht mny lerners will hve chieved qulifictions equivlent to Level t KS4. Skills in Numercy/Mthemtics, Litercy/English nd Informtion nd Communiction Technology will provide good bsis for progression to this Level 3 qulifiction. Cndidtes my be expected to hve obtined (or to be obtining concurrently) n Advnced GCE in Mthemtics. This specifiction builds on the knowledge, understnding nd skills estblished t GCSE. This specifiction provides suitble foundtion for the study of mthemtics or relted re through rnge of higher eduction courses, progression to the next level of voctionl qulifictions or employment. In ddition, the specifiction provides coherent, stisfying nd worthwhile course of study for lerners who do not progress to further study in this subject. This specifiction is not ge specific nd, s such, provides opportunities for lerners to extend their life-long lerning..3 Equlity nd fir ccess This specifiction my be followed by ny lerner, irrespective of gender, ethnic, religious or culturl bckground. It hs been designed to void, where possible, fetures tht could, without justifiction, mke it more difficult for lerner to chieve becuse they hve prticulr protected chrcteristic. The protected chrcteristics under the Equlity Act 00 re ge, disbility, gender ressignment, pregnncy nd mternity, rce, religion or belief, sex nd sexul orienttion. The specifiction hs been discussed with groups who represent the interests of diverse rnge of lerners, nd the specifiction will be kept under review. Resonble djustments re mde for certin lerners in order to enble them to ccess the ssessments (e.g. cndidtes re llowed ccess to Sign Lnguge Interpreter, using British Sign Lnguge). Informtion on resonble djustments is found in the following document from the Joint Council for Qulifictions (JCQ): Access Arrngements nd Resonble Adjustments: Generl nd Voctionl Qulifictions. This document is vilble on the JCQ website (www.jcq.org.uk). As consequence of provision for resonble djustments, very few lerners will hve complete brrier to ny prt of the ssessment.
GCE AS nd A LEVEL FURTHER MATHEMATICS 7.4 Welsh Bcclurete In following this specifiction, lerners should be given opportunities, where pproprite, to develop the skills tht re being ssessed through the Skills Chllenge Certificte within the Welsh Bcclurete: Litercy Numercy Digitl Litercy Criticl Thinking nd Problem Solving Plnning nd Orgnistion Cretivity nd Innovtion Personl Effectiveness..5 Welsh perspective In following this specifiction, lerners should be given opportunities, where pproprite, to consider Welsh perspective if the opportunity rises nturlly from the subject mtter nd if its inclusion would enrich lerners understnding of the world round them s citizens of Wles s well s the UK, Europe nd the world.
GCE AS nd A LEVEL FURTHER MATHEMATICS 8 SUBJECT CONTENT Mthemtics is, inherently, sequentil subject. There is progression of mteril through ll levels t which the subject is studied. The specifiction content therefore builds on the skills, knowledge nd understnding set out in the whole GCSE subject content for Mthemtics nd Mthemtics-Numercy for first teching from 05. It lso builds upon the skills, knowledge nd understnding in AS nd A level Mthemtics. Overrching themes This GCE AS nd A Level specifiction in Further Mthemtics requires lerners to demonstrte the following overrching knowledge nd skills. These must be pplied, long with ssocited mthemticl thinking nd understnding, cross the whole of the detiled content set out below. The knowledge nd skills required for AS Further Mthemtics re shown in bold text. The text in stndrd type pplies to A only. Mthemticl rgument, lnguge nd proof GCE AS nd A Level Further Mthemtics specifictions must use the mthemticl nottion set out in Appendix A nd must require lerners to recll the mthemticl formule nd identities set out in Appendix B. Knowledge/Skill Construct nd present mthemticl rguments through pproprite use of digrms; sketching grphs; logicl deduction; precise sttements involving correct use of symbols nd connecting lnguge, including: constnt, coefficient, expression, eqution, function, identity, index, term, vrible Understnd nd use mthemticl lnguge nd syntx s set out in the content Understnd nd use lnguge nd symbols ssocited with set theory, s set out in the content Understnd nd use the definition of function; domin nd rnge of functions Comprehend nd critique mthemticl rguments, proofs nd justifictions of methods nd formule, including those relting to pplictions of mthemtics
GCE AS nd A LEVEL FURTHER MATHEMATICS 9 Mthemticl problem solving Knowledge/Skill Recognise the underlying mthemticl structure in sitution nd simplify nd bstrct ppropritely to enble problems to be solved Construct extended rguments to solve problems presented in n unstructured form, including problems in context Interpret nd communicte solutions in the context of the originl problem Understnd the concept of mthemticl problem solving cycle, including specifying the problem, collecting informtion, processing nd representing informtion nd interpreting results, which my identify the need to repet the cycle Understnd, interpret nd extrct informtion from digrms nd construct mthemticl digrms to solve problems Mthemticl modelling Knowledge/Skill Trnslte sitution in context into mthemticl model, mking simplifying ssumptions Use mthemticl model with suitble inputs to engge with nd explore situtions (for given model or model constructed or selected by the lerner) Interpret the outputs of mthemticl model in the context of the originl sitution (for given model or model constructed or selected by the lerner) Understnd tht mthemticl model cn be refined by considering its outputs nd simplifying ssumptions; evlute whether the model is pproprite Understnd nd use modelling ssumptions
GCE AS nd A LEVEL FURTHER MATHEMATICS 0 Use of dt in sttistics This specifiction requires lerners, during the course of their study, to: develop skills relevnt to exploring nd nlysing lrge dt sets (these dt must be rel nd sufficiently rich to enble the concepts nd skills of dt presenttion nd interprettion in the specifiction to be explored); use technology such s spredsheets or specilist sttisticl pckges to explore dt sets; interpret rel dt presented in summry or grphicl form; use dt to investigte questions rising in rel contexts. Lerners should be ble to demonstrte the bility to explore lrge dt sets, nd ssocited contexts, during their course of study to enble them to perform tsks, nd understnd wys in which technology cn help explore the dt. Lerners should be ble to demonstrte the bility to nlyse subset or fetures of the dt using clcultor with stndrd sttisticl functions.
GCE AS nd A LEVEL FURTHER MATHEMATICS. AS UNIT Unit : Further Pure Mthemtics A Written exmintion: hour 30 minutes 3% of A level qulifiction (33% of AS qulifiction) 70 mrks The subject content is set out on the following pges. There is no hierrchy implied by the order in which the content is presented, nor should the length of the vrious sections be tken to imply ny view of their reltive importnce. Cndidtes will be expected to be fmilir with the knowledge, skills nd understnding implicit in AS Mthemtics. Where specific content requires knowledge of concepts or results from A Mthemtics, this will be mde explicit in the Guidnce section of the content. Topics.. Proof Construct proofs using mthemticl induction. Contexts include sums of series, powers of mtrices nd divisibility. Guidnce Including ppliction to the proof of the binomil theorem for positive integrl power. eg. the proof of the divisibility of 5 n by 4. Knowledge of the nottion is ssumed... Complex Numbers Solve ny qudrtic eqution with rel coefficients. Solve cubic or qurtic equtions with rel coefficients (given sufficient informtion to deduce t lest one root for cubics or t lest one complex root or qudrtic fctor for qurtics). Add, subtrct, multiply nd divide complex numbers in the form x + iy, with x nd y rel. Understnd nd use the terms rel prt nd imginry prt.
GCE AS nd A LEVEL FURTHER MATHEMATICS Topics Understnd nd use the complex conjugte. Guidnce The complex conjugte of z will be denoted by z. Know tht non-rel roots of polynomil equtions with rel coefficients occur in conjugte pirs Equte the rel nd imginry prts of complex number. Including the solution of equtions such s i zz. i Use nd interpret Argnd digrms Understnd nd use the Crtesin (lgebric) nd modulusrgument (trigonometric) forms of complex number. Convert between the Crtesin form nd modulus-rgument form of complex number. Multiply nd divide complex numbers in modulus-rgument form. Construct nd interpret simple loci in n Argnd digrm, such s z r nd rg( z). Simple cses of trnsformtions of lines nd curves defined by w = f (z). Includes representing complex numbers by points in n Argnd digrm. z = x + iy nd z = r(cos + isin) where = rg(z) my be tken to be in either [0, ) or (, ] or [0, 360 o ) or (-80 o, 80 o ]. Knowledge of rdins is ssumed. Knowledge of rdins nd compound ngle formule is ssumed. For exmple, z z i. Knowledge of rdins is ssumed. For exmple, the imge of the line x + y = under the trnsformtion defined by w z.
GCE AS nd A LEVEL FURTHER MATHEMATICS 3 Topics..3 Mtrices Add, subtrct nd multiply conformble mtrices. Multiply mtrix by sclr. Guidnce Understnd nd use zero nd identity mtrices. Understnd nd use the trnspose of x mtrix. Use mtrices to represent liner nd non-liner trnsformtions in -D, involving x nd 3 x 3 mtrices, successive trnsformtions, single trnsformtions in 3-D (3-D trnsformtions confined to reflection in one of x = 0, y = 0, z = 0 or rottion bout one of the coordinte xes). Trnsformtions to only include trnsltion, rottion nd reflection, using x nd/or 3 x 3 mtrices. Knowledge tht the trnsformtion represented by AB is the trnsformtion represented by B followed by the trnsformtion represented by A. Knowledge of 3-D vectors is ssumed. Find invrint points nd lines for liner nd non-liner trnsformtions. Clculte determinnts of x mtrices. Understnd nd use singulr nd non-singulr mtrices. Understnd nd use properties of inverse mtrices. Clculte nd use the inverse of non-singulr x mtrices. Use nd understnd the nottion M or det M or b c d or...4 Further Algebr nd Functions Understnd nd use the reltionship between roots nd coefficients of polynomil equtions up to qurtic equtions. Form polynomil eqution whose roots re liner trnsformtion of the roots of given polynomil eqution (of t lest cubic degree).
GCE AS nd A LEVEL FURTHER MATHEMATICS 4 Topics Understnd nd use formule for the sums of integers, squres nd cubes nd use these to sum other series. Understnd nd use the method of differences for summtion of series, including the use of prtil frctions. Guidnce Summtion of finite series. Use of formule for r n n r, r 3 nd r. r r Including mthemticl induction (see section on Proof) nd difference methods. Summtion of series such s n n r r( r ) nd 3 (r ). r n Knowledge of the nottion nd prtil frctions is ssumed...5 Further Vectors Understnd nd use the vector nd Crtesin forms of n eqution of stright line in 3-D. r b nd x y z b b b 3 3 Knowledge of 3-D vectors is ssumed. Understnd nd use the vector nd Crtesin forms of the eqution of plne. Clculte the sclr product nd use it to express the eqution of plne, nd to clculte the ngle between two lines, the ngle between two plnes nd the ngle between line nd plne. b. b cos The form rn. k for plne. Use the sclr product to check whether vectors re perpendiculr. Find the intersection of line nd plne. Clculte the perpendiculr distnce between two lines, from point to line nd point to plne.
GCE AS nd A LEVEL FURTHER MATHEMATICS 5. AS UNIT Unit : Further Sttistics A Written exmintion: hour 30 minutes 3% of A level qulifiction (33% of AS qulifiction) 70 mrks The subject content is set out on the following pges. There is no hierrchy implied by the order in which the content is presented, nor should the length of the vrious sections be tken to imply ny view of their reltive importnce. Cndidtes will be expected to be fmilir with the knowledge, skills nd understnding implicit in AS Mthemtics. Where specific content requires knowledge of concepts or results from A Mthemtics, this will be mde explicit in the Guidnce section of the content. Topics.. Rndom Vribles nd the Poisson Process Understnd nd use the men nd vrince of liner combintions of independent rndom vribles. ie. use of results: E(X + b) = E(X) + b Vr(X + b) = Vr(X) E(X + by) = E(X) + be(y) Guidnce For discrete nd continuous rndom vribles. For independent X nd Y, use the results: E(XY) = E(X) E(Y) Vr(X + by) = Vr(X) + b Vr(Y) Probbility: Discrete probbility distributions. Find the men nd vrince of simple discrete probbility distributions. Use of E( X ) xp( X x) nd Vr( X ) x P( X x)
GCE AS nd A LEVEL FURTHER MATHEMATICS 6 Topics Probbility: Continuous probbility distributions. Understnd nd use probbility density nd cumultive distribution functions nd their reltionships. Guidnce Use of the results f ( x) F( x) nd F( x) f ( t)dt. x Find nd use the medin, qurtiles nd percentiles. Find nd use the men, vrince nd stndrd devition. Understnd nd use the expected vlue of function of continuous rndom vrible. Sttisticl distributions: Poisson nd exponentil distributions. E[g (X)] = g( x) f ( x) dx Simple functions only, e.g. X nd X. Use of formul nd tbles/clcultor for Poisson distribution. Find nd use the men nd vrince of Poisson distribution nd n exponentil distribution. Knowledge nd use of: If X ~ Po() then E(X) = nd Vr(X) = Understnd nd use Poisson s n pproximtion to the binomil distribution. Apply the result tht the sum of independent Poisson rndom vribles hs Poisson distribution. If Y ~ Exp( Y ) then E( Y) nd Vr( Y) Use of the exponentil distribution s model for intervls between events. Lerners will be expected to know tht d e kx e kx k dx
GCE AS nd A LEVEL FURTHER MATHEMATICS 7 Topics.. Exploring reltionships between vribles nd goodness of fit of model Understnd nd use correltion nd liner regression: Guidnce Explore the reltionships between severl vribles. Clculte nd interpret Spermn s rnk correltion coefficient Person s product-moment correltion coefficient. Clculte nd interpret the coefficients for lest squres regression line in context; interpoltion nd extrpoltion. Understnd nd use the Chi-squred distribution: ( O E) Conduct goodness of fit test using, or equivlent form, E s n pproximte sttistic (for use with ctegoricl dt). Use test to test for ssocition in contingency tble nd interpret results To include tests for significnce. Excludes tied rnks. Use of tbles for Spermn's nd Person's product moment correltion coefficients. Be ble to choose between Spermn s rnk correltion coefficient nd Person s product-moment correltion coefficient for given context. Including from summry sttistics. For use with binomil, discrete uniform nd Poisson distributions, for known prmeters only. To include pooling. Not including Ytes continuity correction.
GCE AS nd A LEVEL FURTHER MATHEMATICS 8.3 AS UNIT 3 Unit 3: Further Mechnics A Written exmintion: hour 30 minutes 3% of A level qulifiction (33% of AS qulifiction) 70 mrks The subject content is set out on the following pges. There is no hierrchy implied by the order in which the content is presented, nor should the length of the vrious sections be tken to imply ny view of their reltive importnce. Cndidtes will be expected to be fmilir with the knowledge, skills nd understnding implicit in AS Mthemtics. Where specific content requires knowledge of concepts or results from A Mthemtics, this will be mde explicit in the Guidnce section of the content. Topics.3. Momentum nd Impulse Understnd nd use momentum nd impulse. Understnd nd use conservtion of momentum. Guidnce Problems will be restricted to the one-dimensionl cse. Understnd nd use Newton s Experimentl Lw for (i) the direct impct of two bodies moving in the sme stright line, (ii) the impct of body moving t right-ngles to plne..3. Hooke's Lw, Work, Energy nd Power Solve problems involving light strings nd springs obeying Hooke s Lw. Understnd nd use work, energy nd power. Clcultion of work done by using chnge of energy. Understnd nd use grvittionl potentil energy, kinetic energy, elstic energy. Understnd nd use conservtion of energy. Understnd nd use the Work-energy Principle.
GCE AS nd A LEVEL FURTHER MATHEMATICS 9 Topics.3.3 Circulr Motion Understnd nd use circulr motion. Guidnce Angulr speed nd the use of v = r. Rdil ccelertion in circulr motion in the form r nd r v. Understnd nd use the motion of prticle in horizontl circle with uniform ngulr speed. Understnd nd use the motion in verticl circle..3.4 Differentition nd Integrtion of Vectors Differentite nd integrte vectors in component form with respect to sclr vrible. Understnd nd use vector quntities including displcement, velocity, ccelertion, force nd momentum. Problems on bnked trcks including the condition for no side slip. The conicl pendulum. The motion of prticle in horizontl circle where the prticle is (i) constrined by two strings, (ii) threded on one string, (iii) constrined by one string nd smooth horizontl surfce. Knowledge of resolution of forces in ny given direction is ssumed. To include the determintion of points where the circulr motion breks down (e.g. loss of contct with surfce or string becoming slck). The condition for prticle to move in complete verticl circles when (i) it is ttched to light string, (ii) it is ttched to light rigid rod, (iii) it moves on the inside surfce of sphere. The tngentil component of the ccelertion is not required. Knowledge of resolution of forces in ny given direction is ssumed. Extends to vectors in 3 dimensions. Resultnts of vector quntities. Simple pplictions including the reltive motion of two objects nd the determintion of the shortest distnce between them. Knowledge of 3-D vectors is ssumed.
GCE AS nd A LEVEL FURTHER MATHEMATICS 0.4 A UNIT 4 Unit 4: Further Pure Mthemtics B Written exmintion: hours 30 minutes 35% of A level qulifiction 0 mrks This unit is compulsory. The subject content is set out on the following pges. There is no hierrchy implied by the order in which the content is presented, nor should the length of the vrious sections be tken to imply ny view of their reltive importnce. Cndidtes will be expected to be fmilir with the knowledge, skills nd understnding implicit in A level Mthemtics. Topics.4. Complex Numbers Understnd de Moivre s theorem nd use it to find multiple ngle formule nd sums of series. Guidnce To include proof by induction of de Moivre s Theorem for positive integer vlues of n. 4 For exmple, showing tht cos 4 8cos 8cos nd cos (cos 4 4cos 3). 8 4 Know nd use the definition i z re. e i cos isin nd the form i Find the n distinct nth roots for r e for r 0 nd know tht they form the vertices of regulr n-gon in the Argnd digrm. Use complex roots of unity to solve geometric problems.
GCE AS nd A LEVEL FURTHER MATHEMATICS.4. Further Trigonometry Solve trigonometric equtions. Topics Use the formule for sin A sin B, cos A cos B nd for sinx, cosx nd tnx in terms of t, where t = tn. x Guidnce Questions imed solely t proving identities will not be set. For exmple, cos cos cos 3 0 nd sin x tn x 0. Find the generl solution of trigonometric equtions..4.3 Mtrices Clculte determinnts up to 3 x 3 mtrices nd interpret s scle fctors, including the effect on orienttion. Clculte nd use the inverse of non-singulr 3 x 3 mtrices. Solve three liner simultneous equtions in three vribles by use of the inverse mtrix nd by reduction to echelon form. Understnd nd use the determinntl condition for the solution of simultneous equtions which hve unique solution. Including knowledge of the term djugte mtrix. To include equtions which () hve unique solution, (b) hve non-unique solutions, (c) re not consistent. Interpret geometriclly the solution nd filure of three simultneous liner equtions.
GCE AS nd A LEVEL FURTHER MATHEMATICS Topics.4.4 Further Algebr nd Functions Find the Mclurin series of function (including the generl term) Guidnce Recognise nd use the Mclurin series for cos x nd n ( x), x e, ln( x), sin x, nd be wre of the rnge of vlues of x for which they re vlid. Proof not required. Understnd nd use prtil frctions with denomintors of the form ( x b)( cx d)..4.5 Further Clculus Evlute improper integrls, where either the integrnd is undefined t vlue in the rnge of integrtion or the rnge of integrtion extends to infinity. Derive formule for nd clculte volumes of revolution. Rottion my be bout the x-xis or the y-xis. Understnd nd evlute the men vlue of function. Integrte using prtil frctions (extend to qudrtic fctors ( x c) in the denomintor). Differentite inverse trigonometric functions. Integrte functions of the form nd nd be x x ble to choose trigonometric substitutions to integrte ssocited functions. Men vlue of function f( x ) = b b f ( x)dx
GCE AS nd A LEVEL FURTHER MATHEMATICS 3 Topics.4.6 Polr Coordintes Understnd nd use polr coordintes nd be ble to convert between polr nd Crtesin coordintes. Sketch curves with r given s function of θ, including the use of trigonometric functions. Guidnce Where r 0 nd the vlue of my be tken to be in either [0, ) or (, ]. Cndidtes will be expected to sketch simple curves such s r = (b + ccos) nd r = cosn. Includes the loction of points t which tngents re prllel to, or perpendiculr to, the initil line. Find the re enclosed by polr curve..4.7 Hyperbolic functions Understnd the definitions of hyperbolic functions, sinh x, cosh x nd tnh x, including their domins nd rnges, nd be ble to sketch their grphs. Excludes the intersection of curves. cosh x e e tnh x x x x x, sinh x e e x x sinh x e e x x cosh x e e Know nd use the formule for sinh( A B), cosh( A B), tnh( A B), sinh A, cosh A nd tnh A. Knowledge nd use of the identity equivlents. cosh Asinh A nd its Differentite nd integrte hyperbolic functions. eg. Differentite sinh x, xcosh x Understnd nd be ble to use the definitions of the inverse hyperbolic functions nd their domins nd rnges. sinh x ln x x cosh x ln x x, x x tnh x ln, x x
GCE AS nd A LEVEL FURTHER MATHEMATICS 4 Topics Derive nd use the logrithmic forms of the inverse hyperbolic functions. Guidnce Integrte functions of the form x nd x, nd be ble to choose substitutions to integrte ssocited functions..4.8 Differentil equtions Find nd use n integrting fctor to solve differentil equtions of dy the form P( x) y Q( x) nd recognise when it is pproprite to dx do so. Find both generl nd prticulr solutions to differentil equtions. Use differentil equtions in modelling in vriety of contexts. Contexts will not include mechnics contexts. Solve differentil equtions of the form y y by 0, where nd b re constnts, by using the uxiliry eqution. Solve differentil equtions of the form y y by f (x), where nd b re constnts, by solving the homogenous cse nd dding prticulr integrl to the complementry function (in cses where f (x) is polynomil, exponentil or trigonometric function). Understnd nd use the reltionship between the cses when the discriminnt of the uxiliry eqution is positive, zero nd negtive nd the form of solution of the differentil eqution. f( x ) will hve one of the forms A Bx, mcosx nsinx. cx dx e, k e qx or
GCE AS nd A LEVEL FURTHER MATHEMATICS 5 Topics Anlyse nd interpret models of situtions with one independent vrible nd two dependent vribles s pir of coupled st order simultneous equtions nd be ble to solve them. Guidnce For exmple, predtor-prey models. Restricted to first order differentil equtions of the form dx x by f () t dt dy cx dy g() t dt
GCE AS nd A LEVEL FURTHER MATHEMATICS 6.5 A UNIT 5 Unit 5: Further Sttistics B Written exmintion: hour 45 minutes 5% of A level qulifiction 80 mrks Cndidtes will choose either Unit 5 or Unit 6. The subject content is set out on the following pges. There is no hierrchy implied by the order in which the content is presented, nor should the length of the vrious sections be tken to imply ny view of their reltive importnce. Cndidtes will be expected to be fmilir with the knowledge, skills nd understnding implicit in A Level Mthemtics. Topics.5. Smples nd Popultions Understnd nd use unbised estimtors: Guidnce Understnd nd use the vrince criterion for choosing between unbised estimtors. Understnd nd use unbised estimtors of probbility nd of popultion men nd their stndrd errors. Understnd nd use n unbised estimtor of popultion vrince. Use of s x i x. n
GCE AS nd A LEVEL FURTHER MATHEMATICS 7 Topics.5. Sttisticl Distributions Understnd nd use the result tht liner combintion of independent normlly distributed rndom vribles hs norml distribution. Guidnce Understnd nd use the fct tht the distribution of the men of rndom smple from norml distribution with known men nd vrince is lso norml. Know nd use the Centrl Limit Theorem: Understnd nd use the fct tht the distribution of the men of lrge rndom smple from ny distribution with known men nd vrince is pproximtely normlly distributed..5.3 Hypothesis Testing Understnd nd use tests for: () specified men of ny distribution whose vrince is estimted from lrge smple. (b) difference of two mens for two independent norml distributions with known vrinces. (c) specified men of norml distribution with unknown vrince. For popultion with men nd vrince X ~ N, n Using the Centrl Limit Theorem., for lrge n The specified difference my be different from zero. To include estimting the vrince from the dt nd using the Student s t-distribution. The significnce level will be given nd questions involving the Student s t-distribution will not require the clcultion of p-vlues. Interpret results for these tests in context. Non-prmetric tests: Understnd nd use Mnn-Whitney nd Wilcoxon signed-rnk tests, understnding pproprite test selection nd interpreting the results in context. Alterntive tests for when distributionl model cnnot be ssumed. Excludes tied rnks.
GCE AS nd A LEVEL FURTHER MATHEMATICS 8 Topics.5.4 Estimtion Understnd nd use confidence intervls: Understnd nd use confidence limits for () the men of norml distribution with (i) known vrince nd (ii) unknown vrince, Guidnce Cndidtes will be expected to be fmilir with the term confidence intervl, including its interprettion. Estimting the vrince from the dt nd using the Student s t-distribution. (b) the difference between the mens of two norml distributions whose vrinces re known. Understnd nd use pproximte confidence limits, given lrge smples, for probbility or proportion. Using norml pproximtion. Interpret results in prcticl contexts.
GCE AS nd A LEVEL FURTHER MATHEMATICS 9.6 A UNIT 6 Unit 6: Further Mechnics B Written exmintion: hour 45 minutes 5% of A level qulifiction 80 mrks Cndidtes will choose either Unit 5 or Unit 6. The subject content is set out on the following pges. There is no hierrchy implied by the order in which the content is presented, nor should the length of the vrious sections be tken to imply ny view of their reltive importnce. Cndidtes will be expected to be fmilir with the knowledge, skills nd understnding implicit in A Level Mthemtics..6. Rectiliner motion Topics Guidnce Form nd solve simple equtions of motion in which (i) ccelertion is given s function of time, displcement or velocity, (ii) velocity is given s function of time or displcement..6. Momentum nd Impulse Understnd nd use momentum nd impulse in two dimensions, using vectors. To include use of d x dv dv v. dt dt dx
GCE AS nd A LEVEL FURTHER MATHEMATICS 30 Topics.6.3 Moments nd Centre of Mss Understnd nd use the centre of mss of coplnr system of prticles. Guidnce Cndidtes will be expected to be fmilir with the term centre of grvity. Understnd nd use the centre of mss of uniform lmine: tringles, rectngles, circles, semicircles, qurter-circles nd composite shpes. Solve problems involving simple cses of equilibrium of plne lmin nd/or coplnr system of prticles connected by light rods. Understnd nd use the centre of mss of uniform rigid bodies nd composite bodies..6.4 Equilibrium of Rigid Bodies Understnd nd use the equilibrium of single rigid body under the ction of coplnr forces where the forces re not ll prllel..6.5 Differentil Equtions Use differentil equtions in modelling in kinemtics. Understnd nd use simple hrmonic motion. The lmin or system of prticles my be suspended from fixed point. The use of symmetry nd/or integrtion to determine the centre of mss of uniform body. Problems my include rods resting ginst rough or smooth wlls nd on rough ground. Considertion of jointed rods is not required. Questions involving toppling will not be set. To include the use of first nd second order differentil equtions. Cndidtes will be expected to set up the differentil eqution of motion, identify the period, mplitude nd pproprite forms of solution. Cndidtes my quote formule in problems unless the question specificlly requires otherwise. Questions my involve light elstic strings or springs. Questions my require the refinement of the mthemticl model to include dmping. Angulr S.H.M. is not included.
GCE AS nd A LEVEL FURTHER MATHEMATICS 3 3 ASSESSMENT 3. Assessment objectives nd weightings Below re the ssessment objectives for this specifiction. Lerners must: AO Use nd pply stndrd techniques Lerners should be ble to: select nd correctly crry out routine procedures; nd ccurtely recll fcts, terminology nd definitions AO Reson, interpret nd communicte mthemticlly Lerners should be ble to: construct rigorous mthemticl rguments (including proofs); mke deductions nd inference; ssess the vlidity of mthemticl rguments; explin their resoning; nd use mthemticl lnguge nd nottion correctly. AO3 Solve problems within mthemtics nd in other contexts Lerners should be ble to: trnslte problems in mthemticl nd non-mthemticl contexts into mthemticl processes; interpret solutions to problems in their originl context, nd, where pproprite, evlute their ccurcy nd limittions; trnslte situtions in context into mthemticl models; use mthemticl models; nd evlute the outcomes of modelling in context, recognise the limittions of models nd, where pproprite, explin how to refine them. Approximte ssessment objective weightings re shown below s percentge of the full A level, with AS weightings in brckets. AO AO AO3 Totl AS Unit 8% (0%) % (6%) % (6%) 3% (33%) AS Unit 6% (5%) 3% (9%) 3% (9%) 3% (33%) AS Unit 3 6% (5%) 3% (9%) 3% (9%) 3% (33%) Totl for AS units only 0% 0% 0% 40% A Unit 4 0% 7% 7% 35% A Unit 5 (option) A Unit 6 (option) 0% 7% 7% 5% Totl for A units only 30% 5% 5% 60% Finl Totl A Level 50% 5% 5% 00%
GCE AS nd A LEVEL FURTHER MATHEMATICS 3 Use of technology The use of technology, in prticulr mthemticl nd sttisticl grphing tools nd spredsheets, permetes the study of GCE AS nd A Level Further Mthemtics. A clcultor is required for use in ll ssessments in this specifiction. Clcultors used must include the following fetures: n itertive function; the bility to compute summry sttistics nd ccess probbilities from stndrd sttisticl distributions. Clcultors must lso meet the regultions set out below. Clcultors must be: Clcultors must not: of size suitble for use on the desk; either bttery or solr powered; free of lids, cses nd covers which hve printed instructions or formuls. The cndidte is responsible for the following: the clcultor s power supply; the clcultor s working condition; clering nything stored in the clcultor. be designed or dpted to offer ny of these fcilities: - lnguge trnsltors; symbolic lgebr mnipultion; symbolic differentition or integrtion; communiction with other mchines or the internet; be borrowed from nother cndidte during n exmintion for ny reson;* hve retrievble informtion stored in them - this includes: dtbnks; dictionries; mthemticl formuls; text. * An invigiltor my give cndidte replcement clcultor. Formul Booklet A formul booklet will be required in ll exmintions. This will exclude ny formule listed in Appendix B. Copies of the formul booklet my be obtined from the WJEC. Sttisticl Tbles Cndidtes my use book of sttisticl tbles for Unit nd Unit 5. The following book of sttisticl tbles is llowed in the exmintions: Elementry Sttisticl Tbles (RND/WJEC Publictions).
GCE AS nd A LEVEL FURTHER MATHEMATICS 33 4 TECHNICAL INFORMATION 4. Mking entries This is unitised specifiction which llows for n element of stged ssessment. Assessment opportunities will be vilble in the summer ssessment period ech yer, until the end of the life of the specifiction. Unit, Unit nd Unit 3 will be vilble in 08 (nd ech yer therefter) nd the AS qulifiction will be wrded for the first time in summer 08. Unit 4, Unit 5 nd Unit 6 will be vilble in 09 (nd ech yer therefter) nd the A level qulifiction will be wrded for the first time in summer 09. Cndidtes my resit n individul unit ONCE only. The better uniform mrk score from the two ttempts will be used in clculting the finl overll qulifiction grde(s). A qulifiction my be tken more thn once. However, if ll units hve been ttempted twice, cndidtes will hve to mke fresh strt by entering ll units nd the pproprite csh-in(s). No result from units tken prior to the fresh strt cn be used in ggregting the new grde(s). The entry codes pper below. Title Entry codes English-medium Welsh-medium AS Unit Further Pure Mthemtics A 305U 305N AS Unit Further Sttistics A 305U 305N AS Unit 3 Further Mechnics A 305U3 305N3 A Unit 4 Further Pure Mthemtics B 305U4 305N4 A Unit 5 Further Sttistics B 305U5 305N5 A Unit 6 Further Mechnics B 305U6 305N6 AS Qulifiction csh-in 305QS 305CS A level Qulifiction csh-in 305QS 305CS The current edition of our Entry Procedures nd Coding Informtion gives up-to-dte entry procedures. There is no restriction on entry for this specifiction with ny other WJEC AS or A level specifiction.
GCE AS nd A LEVEL FURTHER MATHEMATICS 34 4. Grding, wrding nd reporting The overll grdes for the GCE AS qulifiction will be recorded s grde on scle A to E. The overll grdes for the GCE A level qulifiction will be recorded s grde on scle A* to E. Results not ttining the minimum stndrd for the wrd will be reported s U (unclssified). Unit grdes will be reported s lower cse letter to e on results slips but not on certifictes. The Uniform Mrk Scle (UMS) is used in unitised specifictions s device for reporting, recording nd ggregting cndidtes' unit ssessment outcomes. The UMS is used so tht cndidtes who chieve the sme stndrd will hve the sme uniform mrk, irrespective of when the unit ws tken. Individul unit results nd the overll subject wrd will be expressed s uniform mrk on scle common to ll GCE qulifictions. An AS GCE hs totl of 40 uniform mrks nd n A level GCE hs totl of 600 uniform mrks. The mximum uniform mrk for ny unit depends on tht unit s weighting in the specifiction. Uniform mrks correspond to unit grdes s follows: Unit grde Unit weightings Mximum unit uniform mrk b c d e Unit (3%) 80 (rw mrk mx=70) 64 56 48 40 3 Unit (3%) 80 (rw mrk mx=70) 64 56 48 40 3 Unit 3 (3%) 80 (rw mrk mx=70) 64 56 48 40 3 Unit 4 (35%) 0 (rw mrk mx=0) 68 47 6 05 84 Unit 5 (5%) Unit 6 (5%) 50 (rw mrk mx=80) 0 05 90 75 60 The uniform mrks obtined for ech unit re dded up nd the subject grde is bsed on this totl. Qulifiction grde Mximum uniform mrks A B C D E GCE AS 40 9 68 44 0 96 GCE A level 600 480 40 360 300 40 At A level, Grde A* will be wrded to cndidtes who hve chieved Grde A (480 uniform mrks) in the overll A level qulifiction nd t lest 90% of the totl uniform mrks for the A units (34 uniform mrks).
GCE AS nd A LEVEL FURTHER MATHEMATICS 35 APPENDIX A Mthemticl Nottion The tbles below set out the nottion tht must be used in the WJEC GCE AS nd A Level Further Mthemtics specifiction. Lerners will be expected to understnd this nottion without the need for further explntion. AS lerners will be expected to understnd nottion tht reltes to AS content, nd will not be expected to understnd nottion tht reltes only to A Level content. Set Nottion. is n element of. is not n element of.3 is subset of.4 is proper subset of.5 {x, x, } the set with elements x, x,.6 {x: } the set of ll x such tht.7 n(a) the number of elements in set A.8 the empty set.9 ε the universl set.0 Α the complement of the set A. N the set of nturl numbers, {,, 3, }. Z the set of integers, {0, ±, ±, ±3, }.3 Z + the set of positive integers, {,, 3, } + the set of non-negtive integers,.4 Z 0 {0,,, 3, }.5 R the set of rel numbers the set of rtionl numbers.6 Q { p q p Z, q Z+ }.7 union.8 intersection.9 (x, y) the ordered pir x, y.0 [, b] the closed intervl {x R: x b}. [, b) the intervl {x R: x < b}. (, b] the intervl {x R: < x b}.3 (, b) the open intervl {x R: < x < b}.4 C the set of complex numbers
GCE AS nd A LEVEL FURTHER MATHEMATICS 36 Miscellneous Symbols. = is equl to. is not equl to.3 is identicl to or congruent to.4 is pproximtely equl to.5 infinity.6 is proportionl to.7 therefore.8 becuse.9 < is less thn.0,. > is greter thn., is less thn or equl to, is not greter thn is greter thn or equl to, is not less thn.3 p q p implies q (if p then q).4 p q p is implied by q (if q then p).5 p q p implies nd is implied by q (p is equivlent to q).6 first term for n rithmetic or geometric sequence.7 l lst term for n rithmetic sequence.8 d.9 r common difference for n rithmetic sequence common rtio for geometric sequence.0 S n sum to n terms of sequence. S sum to infinity of sequence 3 Opertions 3. + b plus b 3. - b minus b 3.3 b, b,.b multiplied by b 3.4 b, b n divided by b 3.5 i i= n + + + n 3.6 i i=,, n 3.7 the non-negtive squre root of 3.8 the modulus of 3.9 n! n fctoril: n! = n (n ), n N; 0! =
GCE AS nd A LEVEL FURTHER MATHEMATICS 37 3.0 ( n ), n Cr, C r n r the binomil coefficient n! for n, r Z + r!(n r)! 0, r n or n(n ) (n r+) r! for n Q, r Z 0 + 4 Functions 4. f(x) the vlue of the function f t x 4. f : x y the function f mps the element x to the element y 4.3 f the inverse function of the function f 4.4 gf the composite function of f nd g which is defined by gf(x) = g(f(x)) 4.5 lim x f(x) the limit of f(x) s x tends to 4.6 x, δx n increment of x 4.7 4.8 dy dx d n y dx n 4.9 f'(x), f''(x),, f n (x) 4.0 x, x, 4. y dx b 4. y dx the derivtive of y with respect to x the n th derivtive of y with respect to x the first, second, n th derivtives of f(x) with respect to x the first, second, derivtives of x with respect to t the indefinite integrl of y with respect to x the definite integrl of y with respect to x between the limits x = nd x = b 5 Exponentil nd Logrithmic Functions 5. e bse of nturl logrithms 5. e x, exp x exponentil function of x 5.3 log x logrithm to the bse of x 5.4 ln x, log e x nturl logrithm of x 6 Trigonometric Functions 6. sin, cos, tn, cosec, sec, cot } the trigonometric functions 6. sin, cos, tn rcsin, rccos, rctn } the inverse trigonometric functions 6.3 degrees 6.4 rd rdins 6.5 cosec, sec, cot rccosec, rcsec, rccot } the inverse trigonometric functions
GCE AS nd A LEVEL FURTHER MATHEMATICS 38 6.6 6.7 sinh, cosh, tnh, cosech, sech, coth } sinh, cosh, tnh, cosech, sech, coth } rsinh, rcosh, rtnh, rcosech, rsech, rcoth } the hyperbolic functions the inverse hyperbolic functions 7 Complex Numbers 7. i, j squre root of - 7. x + iy 7.3 r(cos θ+ isinθ) 7.4 z complex number with rel prt x nd imginry prt y modulus rgument form of complex number with modulus r nd rgument θ complex number, z = x + iy = r(cosθ + isinθ) 7.5 Re(z) the rel prt of z, Re (z) = x 7.6 Im(z) the imginry prt of z, Im(z)=y 7.7 z the modulus of z, z = x + y 7.8 rg(z) the rgument of z, rg(z) = θ, -π < θ π 7.9 z * or z the complex conjugte of z, x - iy 8 Mtrices 8. M mtrix M 8. 0 zero mtrix 8.3 I identity mtrix 8.4 M the inverse of the mtrix M 8.5 M T the trnspose of the mtrix M 8.6, det M or M 8.7 Mr 9 Vectors 9.,, 9. AB the determinnt of the squre mtrix M imge of column vector r under the trnsformtion ssocited with the mtrix M the vector,, ; these lterntives pply throughout section 9 the vector represented in mgnitude nd direction by the directed line segment AB 9.3 unit vector in the direction of 9.4 i, j, k unit vectors in the directions of the crtesin coordinte xes 9.5, the mgnitude of
GCE AS nd A LEVEL FURTHER MATHEMATICS 39 9.6 AB, AB the mgnitude of AB 9.7 ( b ), i + bj column vector nd corresponding unit vector nottion 9.8 r position vector 9.9 s displcement vector 9.0 v velocity vector 9. ccelertion vector 9..b the sclr product of nd b 0 Differentil Equtions 0. ω ngulr speed Probbility nd Sttistics. A, B, C, etc. events. A B union of the events A nd B.3 A B intersection of the events A nd B.4 P(A) probbility of the event A.5 A' complement of the event A.6 P(A B).7 X, Y, R, etc. rndom vribles.8 x, y, r, etc. probbility of the event A conditionl on the event B vlues of the rndom vribles X, Y, R etc.9 x, x, vlues of observtions.0 f, f,. p(x), P(X=x). p, p, frequencies with which the observtions x, x, occur probbility function of the discrete rndom vrible X probbilities of the vlues x, x, of the discrete rndom vrible X.3 E(X) expecttion of the rndom vrible X.4 Vr(X) vrince of the rndom vrible X.5 ~ hs the distribution.6 B(n, p) binomil distribution with prmeters n nd p, where n is the number of trils nd p is the probbility of success in tril.7 q q = p for binomil distribution.8 N(μ, σ ) Norml distribution with men μ nd vrince σ.9 Z ~ N(0,) stndrd Norml distribution.0 φ probbility density function of the stndrdised Norml vrible with distribution N(0,)
GCE AS nd A LEVEL FURTHER MATHEMATICS 40. Φ corresponding cumultive distribution function. μ popultion men.3 σ popultion vrince.4 σ popultion stndrd devition.5 x smple men.6 s smple vrince.7 s smple stndrd devition.8 H 0 null hypothesis.9 H lterntive hypothesis.30 r product moment correltion coefficient for smple.3 ρ product moment correltion coefficient for popultion Mechnics. kg kilogrms. m metres.3 km kilometres.4 m/s, m s metres per second (velocity).5 m/s, m s metres per second per second (ccelertion).6 F force or resultnt force.7 N newton.8 N m newton metre (moment of force).9 t time.0 s displcement. u initil velocity. v velocity or finl velocity.3 ccelertion.4 g ccelertion due to grvity.5 μ coefficient or friction
GCE AS nd A LEVEL FURTHER MATHEMATICS 4 APPENDIX B Mthemticl Formule nd Identities Lerners must be ble to use the following formule nd identities for GCE AS nd A Level Further Mthemtics, without these formule nd identities being provided, either in these forms or in equivlent forms. These formule nd identities my only be provided where they re the strting point for proof or s result to be proved. Pure Mthemtics Qudrtic Equtions x bx c 0 hs roots Lws of Indices x y x y x y x y xy ( ) xy b b 4c Lws of Logrithms n x n log x for 0 nd x 0 log log x log x log k log x log y log y log x k ( xy) Coordinte Geometry x y A stright line grph, grdient m pssing through x y y x m( x ), y hs eqution Stright lines with grdients m nd m re perpendiculr when m m Sequences Generl term of rithmetic progression: u n ( n ) d Generl term of geometric progression: n u r n
GCE AS nd A LEVEL FURTHER MATHEMATICS 4 Trigonometry In the tringle ABC b Sine rule: sin A sin B c sin C Cosine rule: b c b cos A Are = bsin C cos A sin A sec A tn A cosec A cot A sin A sin Acos A cos A cos A sin tn tn A tn A A Mensurtion A Circumference nd Are of circle, rdius r nd dimeter d : C r = d A r Pythgors' Theorem: In ny right-ngled tringle where, b nd c re the lengths of the sides nd c is the hypotenuse: c b Are of trpezium = ( b) h, where nd b re the lengths of the prllel sides nd h is their perpendiculr seprtion. Volume of prism = re of cross section length For circle of rdius, r, where n ngle t the centre of rdins subtends n rc of length s nd encloses n ssocited sector of re A : s r A r Complex Numbers i i For two complex numbers z r e nd z re : i( ) z z r r e z r i( ) e z r