6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS


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1 6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes. The emintion pper hs two sections. Section A: Section B: 810 questions, ech worth no more thn 5 mrks. Section Totl: 36 mrks three questions, ech worth bout 1 mrks. Section Totl: 36 mrks Assumed Knowledge Cndidtes re epected to know the content of Intermedite Tier GCSE* nd C1. *See note on pge 34. Subject Criteri The Units C1 nd C re required for Advnced Subsidiry GCE Mthemtics in order to ensure coverge of the subject criteri. The Units C1, C, C3 nd C4 re required for Advnced GCE Mthemtics in order to ensure coverge of the subject criteri. Clcultors In the MEI Structured Mthemtics specifiction, no clcultor is llowed in the emintion for C1. For ll other units, including this one, grphicl clcultor is llowed. MEI/OCR 003 Section C: Generl Informtion 47 Oford, Cmbridge nd RSA Emintions MEI Structured Mthemtics
2 Specifiction Ref. Competence Sttements ALGEBRA Logrithms. C1 Understnd the mening of the word logrithm. Understnd the lws of logrithms nd how to pply them. 3 Know the vlues of log nd log 1. 4 Know how to convert from n inde to logrithmic form nd vice vers. 5 Know the function y nd its grph. Definitions of 6 Be ble to solve n eqution of the form b. 7 n Know how to reduce the equtions y nd y b to liner form nd, using eperimentl dt, to drw grph to find vlues of n, nd b,. SEQUENCES AND SERIES Cs1 Know wht sequence of numbers is nd the mening of finite nd infinite Know tht sequence cn be generted using formul for the k th term, or recurrence reltion of the form k 1 f( k). 3 Know wht series is. 4 Be fmilir with nottion. 5 Know nd be ble to recognise the periodicity of 6 Know the difference between convergent nd divergent Arithmetic series. 7 Know wht is ment by rithmetic series nd 8 Be ble to use the stndrd formule ssocited with rithmetic series nd Geometric series. 9 Know wht is ment by geometric series nd 10 Be ble to use the stndrd formule ssocited with geometric series nd 11 Know the condition for geometric series to be convergent nd be ble to find its sum to infinity. 1 Be ble to solve problems involving rithmetic nd geometric series nd 48 Section C: Generl Informtion MEI/OCR 003 MEI Structured Mthemtics Oford, Cmbridge nd RSA Emintions
3 Ref. Notes Nottion Eclusions C1 y log log ( y) log log y log log log y y k log ( ) klog 3 log 1, lo g n n log 5 For 1. 6 By tking logrithms of both sides. y ALGEBRA Chnge of bse of logrithms. 7 By tking logrithms of both sides nd compring with the eqution y m c. Cs1 SEQUENCES AND SERIES e.g. k 3k ; k 1 k 3 with 1 5. k th term: k 3 With reference to the corresponding sequence. 4 Including the sum of the first n nturl numbers e.g. convergent sequence k 3 k e.g. divergent sequence k 1 k 7 The term rithmetic progression (AP) my lso be used. 1st term, Lst term, l Common difference, d. 8 The nth term, the sum to n terms. Forml tests for convergence. 9 The term geometric progression (GP) my lso be used. 1st term, Common rtio, r. 10 The nth term, the sum to n terms. S n 11 Cndidtes will be epected to be fmilir with the modulus sign in the condition for convergence. S, r 1 1 r 1 These my involve the solution of qudrtic nd simultneous equtions. MEI/OCR 003 Section C: Generl Informtion 49 Oford, Cmbridge nd RSA Emintions MEI Structured Mthemtics
4 Specifiction Ref. Competence Sttements Bsic trigonometry. Ct1 TRIGONOMETRY * Know how to solve rightngled tringles using trigonometry. The sine, cosine Be ble to use the definitions of sin θ nd cos θ for ny ngle. nd tngent 3 Know the grphs of sin θ, cos θ nd tn θ for ll vlues of θ, their symmetries nd functions. periodicities. 4 Know the vlues of sin θ, cos θ nd tn θ when θ is 0, 30, 45, 60, 90 nd 180. Identities. 5 sin Be ble to use tn (for ny ngle). cos 6 Be ble to use the identity sin cos 1. 7 Be ble to solve simple trigonometric equtions in given intervls. Are of tringle. 8 Know nd be ble to use the fct tht the re of tringle is given by ½ bsin C. The sine nd 9 Know nd be ble to use the sine nd cosine rules. cosine rules. Rdins. 10 Understnd the definition of rdin nd be ble to convert between rdins nd degrees. 11 Know nd be ble to find the rc length nd re of sector of circle, when the ngle is given in rdins. 50 Section C: Generl Informtion MEI/OCR 003 MEI Structured Mthemtics Oford, Cmbridge nd RSA Emintions
5 Ref. Notes Nottion Eclusions Ct1 TRIGONOMETRY e.g. by reference to the unit circle. 3 Their use to find ngles outside the first qudrnt. 4 Ect vlues my be epected. 5 e.g. solve sin 3cos for e.g. simple ppliction to solution of equtions. 7 e.g. sin , 150 o in [0 o, 360 o ]. rcsin rccos rctn 8 9 Use of berings my be required. Principl vlues (see C4) Generl solutions The results s r nd 1 A r where is mesured in rdins. MEI/OCR 003 Section C: Generl Informtion 51 Oford, Cmbridge nd RSA Emintions MEI Structured Mthemtics
6 Specifiction Ref. Competence Sttements The bsic process of differentition. CALCULUS Cc1 Know tht the grdient of curve t point is given by the grdient of the tngent t the point. Know tht the grdient of the tngent is given by the limit of the grdient of chord. 3 Know tht the grdient function d y gives the grdient of the curve nd mesures d the rte of chnge of y with respect to. Applictions of 4 n Be ble to differentite y k where k is constnt, nd the sum of such differentition to functions. the grphs of functions. 5 Be ble to find second derivtives. 6 Be ble to use differentition to find sttionry points on curve: mim, minim nd points of inflection. 7 Understnd the terms incresing function nd decresing function. Integrtion s the inverse of differentition. 8 Be ble to find the eqution of tngent nd norml t ny point on curve. 9 Know tht integrtion is the inverse of differentition. 10 n Be ble to integrte functions of the form k where k is constnt nd n 1, nd the sum of such functions. 11 Know wht re ment by indefinite nd definite integrls. 1 Be ble to evlute definite integrls. 13 Be ble to find constnt of integrtion given relevnt informtion. Integrtion to find the re under curve. 14 Know tht the re under grph cn be found s the limit of sum of res of rectngles. 15 Be ble to use integrtion to find the re between grph nd the is. 16 Be ble to find n pproimte vlue of definite integrl using the trpezium rule, nd comment sensibly on its ccurcy. CURVE SKETCHING Sttionry points. CC1 Be ble to use sttionry points when curve sketching. Stretches. Know how to sketch curves of the form y f( ) nd y f( ), given the curve of: y f( ). 5 Section C: Generl Informtion MEI/OCR 003 MEI Structured Mthemtics Oford, Cmbridge nd RSA Emintions
7 Ref. Notes Nottion Eclusions Cc1 CALCULUS 3 The terms incresing function nd decresing function. dy δy Lim d δ 0 δ 4 Simple cses of differentition from first principles. f( h)  f( ) Including rtionl vlues of n. f( )= Lim( ) h 0 h 5 6 d y f ( ) d In reltion to the sign of d y d e.g. (3 5 1)d dy e.g. Find y when given tht 5 d nd y 7 when Generl understnding only. Forml proof. 15 Includes res of regions prtly bove nd prtly below the is. 16 Comments on the error will be restricted to considertion of its direction nd mde with reference to the shpe of the curve. CURVE SKETCHING CC1 Including distinguishing between them. Simple cses only e.g. Given f( ) sin, sketch y sin( ) or y 3sin. Repeted pplictions of the trpezium rule (see C4). Combined trnsformtions (see C3f). MEI/OCR 003 Section C: Generl Informtion 53 Oford, Cmbridge nd RSA Emintions MEI Structured Mthemtics
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