GCE. Mathematics. Mark Scheme for January Advanced GCE Unit 4727: Further Pure Mathematics 3. physicsandmathstutor.com

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1 GCE Mathematics Advaced GCE Uit 77: Further Pure Mathematics Mark Scheme for Jauary 0 Oford Cambridge ad RSA Eamiatios

2 OCR (Oford Cambridge ad RSA) is a leadig UK awardig body, providig a wide rage of qualificatios to meet the eeds of pupils of all ages ad abilities. OCR qualificatios iclude AS/A Levels, Diplomas, GCSEs, OCR Natioals, Fuctioal Skills, Key Skills, Etry Level qualificatios, NVQs ad vocatioal qualificatios i areas such as IT, busiess, laguages, teachig/traiig, admiistratio ad secretarial skills. It is also resposible for developig ew specificatios to meet atioal requiremets ad the eeds of studets ad teachers. OCR is a ot-for-profit orgaisatio; ay surplus made is ivested back ito the establishmet to help towards the developmet of qualificatios ad support which keep pace with the chagig eeds of today s society. This mark scheme is published as a aid to teachers ad studets, to idicate the requiremets of the eamiatio. It shows the basis o which marks were awarded by Eamiers. It does ot idicate the details of the discussios which took place at a Eamiers meetig before markig commeced. All Eamiers are istructed that alterative correct aswers ad uepected approaches i cadidates scripts must be give marks that fairly reflect the relevat kowledge ad skills demostrated. Mark schemes should be read i cojuctio with the published questio papers ad the Report o the Eamiatio. OCR will ot eter ito ay discussio or correspodece i coectio with this mark scheme. OCR 0 Ay equiries about publicatios should be addressed to: OCR Publicatios PO Bo 5050 Aesley NOTTINGHAM NG5 0DL Telephoe: Facsimile: publicatios@ocr.org.uk

3 77 Mark Scheme Jauary 0 (i) e d Itegratig factor. e B For correct IF d M d ye e For y.their IF e.their IF d d ye e ( c) A For correct itegratio both sides y e e c e ce A For correct solutio AEF as y f( ) (ii) (0, ) c M A e +e y For substitutig (0, ) ito their GS, solvig for c ad obtaiig a solutio of the DE For correct solutio AEF y cosh Allow (i) [,, ] [,, ] M For usig of directio vectors [0, 5, 5] k[,, ] A For correct (,, ) yz A For substitutig (,, ) ad obtaiig AG (Verificatio oly M0) (ii) METHOD M For OR [,, ]. [,, ] [,, ]. [,, ] distace = OR OR ([,, ] [ abc,, ]). [,, ] soi OR ([,, ] [ abc,, ]). [,, ] where ( abc,, ) B For soi is o q 8 A For correct distace AEF METHOD M For formig ad solvig a equatio i t [ t, t, t] o q B For soi ( t) ( t) ( t) t distace A For correct distace AEF METHOD As Method to t (7,0,7) o q M* For fidig poit where ormal meets q distace from (,, ) M For fidig distace from (,, ) (7 ) (0 ) (7 ) 5 (*dep) A For correct distace AEF (i) i i i si e e z or e may be used throughout i B For correct epressio for si soi si z z z z M i i For epadig e e (with at least terms ad biomial coefficiet ) si cos 8cos M For groupig terms ad usig multiple agles si cos cos 8 (ii) si d 8 si si A For aswer obtaied correctly AG M A M A 8 For itegratig (i) to Asi Bsi C For correct itegratio For completig itegratio ad substitutig limits For correct aswer AEF(eact)

4 77 Mark Scheme Jauary 0 (i) EITHER = sum of roots of ( z 0) = 0 OR ( )( ) 0 0 (for ) OR sum of G.P. 0 0 OR OR show o Argad diagram or eplaied i terms of vectors cis cis i + i 0 M A For result show by ay correct method AG (ii) Multiplicatio by ω rotatio through B For correct iterpretatio of by ω (allow 0 ad omissio of, or error i, ) B For idetificatio of vectors soi zz CA, z z BC (igore directio errors) BC rotates through to directio of CA M For likig BC ad CA by rotatio of OR ω Δ ABC has BC = CA, hece result A For statig equal magitudes AG (iii) (ii) z z ( ) z 0 M For usig 0 i (ii) 0 z z z A For obtaiig AG 0 5 (i) Au. equatio m 5m( 0) M 8 For correct auiliary equatio see ad solutio attempted m, A For correct roots CF A For correct CF ( y ) Ae Be f.t. from m with arbitrary costats PI ( y ) p q 5p( pq) M For statig ad substitutig PI of correct form p, q A A For correct value of p, ad of q GS For GS ( y ) Ae Be B 7 f.t. from their CF+PI with arbitrary costats i CF ad oe i PI (ii) 0, 7 AB M 7 i their GS For substitutig 0, ad obtaiig a equatio i A ad B y Ae Be, (0, 0) AB M For fidig y, substitutig (0, 0) ad obtaiig a equatio i A ad B M For solvig their equatios i A ad B A0, B A For correct A ad B CAO ( y ) e B 5 For correct solutio f.t. with their A ad B i their GS (iii) large ( y ) B For correct equatio or fuctio (allow ad ) WWW f.t. from (ii) if valid

5 77 Mark Scheme Jauary 0 (i) a r e a has order, a has order M For cosiderig powers of a e has order A For order of ay oe of a correct a a a, a, a A For all correct r e r has order B For order of r correct (ii) G order Order of elemet () Number of elemets (0) H order Order of elemet () Number of elemets (0) G ad H are the oly o-cyclic groups of order which divides Q has elemet of order, G ad H have, so o o-cyclic subgroups i Q M A A For top lie i either table Allow iclusio of ad respectively (ad other orders if 0 appears below) For order table For order table B For statig that oly G ad H eed be cosidered AEF B 5 For argumet completed by elemets of order AG SR Allow equivalet argumets for B B 9 7 (i) [,, ] [,, ] ( )[, 5, ] M For usig of directio vectors A For correct directio [,, ] [, 5, ] ( )[, 5, ] M A For usig of directio vectors For correct directio [,5,] k [, 5, ] parallel A 5 For argumet completed AG ( k ot essetial) (ii) Lie of itersectio is parallel to l ad m B For correct statemet (iii) METHOD yz 5 e.g. 0 yz M For attempt to fid poits o lies,,0 o l A For a correct poit o oe lie yz e.g. z 0 o m 5y z 7,, 0 A For a correct poit o aother lie yz 5 e.g. z 0 5y z 7,,0 o l Differet poits o commo lie of itersectio A For correct aswer METHOD yz 5 M For fidig (e.g.) y ad z i terms of e.g. z y z, y 7 5 OR elimiatig oe variable A For correct epressios OR equatios LHS of eq = (5 5 ) ( ) A For obtaiig a cotradictio from rd equatio o commo lie of itersectio A For correct aswer METHOD LHS M For attempt to lik equatios RHS 5 A For obtaiig a cotradictio o commo lie of itersectio A For correct aswer SR Variatios o all methods may gai full credit SR f.t. may be allowed from relevat workig 0

6 77 Mark Scheme Jauary 0 8 (i) (( a, b)*( c, d))*( e, f ) ( ac, ad b)*( e, f ) M For distict elemets bracketed ad attempt to epad ( ace, acf ad b) A For correct epressio ( ab, )*(( cd, )*( e, f)) ( ab, )*( cecf, d) ( ace, acf ad b) A For correct epressio agai (ii) ( ab, )*(,) ( aa, b), (, ) * ( ab, ) ( ab, ) M For combiig both ways roud ab b a M For equatig compoets (, b) b (allow from icorrect pairs) A For correct elemets AEF (iii) ( mp, mq ) OR ( pm, p q) (,0) M For either elemet o LHS ( p, q), m m A For correct iverse (iv) ( ab, )*( ab, ) ( a, abb) (,0) OR (, ), b a, ab b M For attempt to fid self-iverses a a self-iverse elemets (, 0) ad (, b) b B A For (, 0). For (, b) AEF (v) (0, y) has o iverse for ay y ot a group B For statig ay oe elemet with o iverse. Allow 0 required, provided referece to iverse is made Some elemets have o iverse B0

7 OCR (Oford Cambridge ad RSA Eamiatios) Hills Road Cambridge CB EU OCR Customer Cotact Cetre 9 Qualificatios (Geeral) Telephoe: Facsimile: geeral.qualificatios@ocr.org.uk For staff traiig purposes ad as part of our quality assurace programme your call may be recorded or moitored Oford Cambridge ad RSA Eamiatios is a Compay Limited by Guaratee Registered i Eglad Registered Office; Hills Road, Cambridge, CB EU Registered Compay Number: 8 OCR is a eempt Charity OCR (Oford Cambridge ad RSA Eamiatios) Head office Telephoe: Facsimile: OCR 0

GCE. Mathematics. Mark Scheme for January Advanced GCE Unit 4726: Further Pure Mathematics 2

GCE. Mathematics. Mark Scheme for January Advanced GCE Unit 4726: Further Pure Mathematics 2 GCE Mathematics Advanced GCE Unit 76: Further Pure Mathematics Mark Scheme for January OCR (Oford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the