www.onlineamhlp.com www.onlineamhlp.com UIVERSITY OF CAMBRIDGE ITERATIOAL EXAMIATIOS GCE Advancd Lvl MARK SCHEME for th Octobr/ovmbr qustion papr for th guidanc of tachrs 9 FURTHER MATHEMATICS 9/ Papr, maimum raw mark This mark schm is publishd as an aid to tachrs and candidats, to indicat th rquirmnts of th amination. It shows th basis on which Eaminrs wr instructd to award marks. It dos not indicat th dtails of th discussions that took plac at an Eaminrs mting bfor marking bgan, which would hav considrd th accptability of altrnativ answrs. Mark schms must b rad in conjunction with th qustion paprs and th rport on th amination. CIE will not ntr into discussions or corrspondnc in connction with ths mark schms. CIE is publishing th mark schms for th Octobr/ovmbr qustion paprs for most IGCSE, GCE Advancd Lvl and Advancd Subsidiary Lvl syllabuss and som Ordinary Lvl syllabuss. www.onlineamhlp.com

Pag Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 Mark Schm ots Marks ar of th following thr typs: M A B Mthod mark, awardd for a valid mthod applid to th problm. Mthod marks ar not lost for numrical rrors, algbraic slips or rrors in units. Howvr, it is not usually sufficint for a candidat just to indicat an intntion of using som mthod or just to quot a formula; th formula or ida must b applid to th spcific problm in hand,.g. by substituting th rlvant quantitis into th formula. Corrct application of a formula without th formula bing quotd obviously arns th M mark and in som cass an M mark can b implid from a corrct answr. Accuracy mark, awardd for a corrct answr or intrmdiat stp corrctly obtaind. Accuracy marks cannot b givn unlss th associatd mthod mark is arnd (or implid). Mark for a corrct rsult or statmnt indpndnt of mthod marks. Whn a part of a qustion has two or mor mthod stps, th M marks ar gnrally indpndnt unlss th schm spcifically says othrwis; and similarly whn thr ar svral B marks allocatd. Th notation DM or DB (or dp*) is usd to indicat that a particular M or B mark is dpndnt on an arlir M or B (astriskd) mark in th schm. Whn two or mor stps ar run togthr by th candidat, th arlir marks ar implid and full crdit is givn. Th symbol implis that th A or B mark indicatd is allowd for work corrctly following on from prviously incorrct rsults. Othrwis, A or B marks ar givn for corrct work only. A and B marks ar not givn for fortuitously corrct answrs or rsults obtaind from incorrct working. ot: B or A mans that th candidat can arn or. B// mans that th candidat can arn anything from to. Th marks indicatd in th schm may not b subdividd. If thr is gnuin doubt whthr a candidat has arnd a mark, allow th candidat th bnfit of th doubt. Unlss othrwis indicatd, marks onc gaind cannot subsquntly b lost,.g. wrong working following a corrct form of answr is ignord. Wrong or missing units in an answr should not lad to th loss of a mark unlss th schm spcifically indicats othrwis. For a numrical answr, allow th A or B mark if a valu is obtaind which is corrct to s.f., or which would b corrct to s.f. if roundd ( d.p. in th cas of an angl). As statd abov, an A or B mark is not givn if a corrct numrical answr ariss fortuitously from incorrct working. For Mchanics qustions, allow A or B marks for corrct answrs which aris from taking g qual to 9.8 or 9.8 instad of. UCLES

Pag Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 Th following abbrviations may b usd in a mark schm or usd on th scripts: AEF AG BOD CAO CWO ISW MR PA SOS SR Any Equivalnt Form (of answr is qually accptabl) Answr Givn on th qustion papr (so tra chcking is ndd to nsur that th dtaild working lading to th rsult is valid) Bnfit of Doubt (allowd whn th validity of a solution may not b absolutly clar) Corrct Answr Only (mphasising that no follow through from a prvious rror is allowd) Corrct Working Only oftn writtn by a fortuitous' answr Ignor Subsqunt Working Misrad Prmatur Approimation (rsulting in basically corrct work that is insufficintly accurat) S Othr Solution (th candidat maks a bttr attmpt at th sam qustion) Spcial Ruling (dtailing th mark to b givn for a spcific wrong solution, or a cas whr som standard marking practic is to b varid in th light of a particular circumstanc) Pnaltis MR PA A pnalty of MR is dductd from A or B marks whn th data of a qustion or part qustion ar gnuinly misrad and th objct and difficulty of th qustion rmain unaltrd. In this cas all A and B marks thn bcom follow through marks. MR is not applid whn th candidat misrads his own figurs this is rgardd as an rror in accuracy. An MR pnalty may b applid in particular cass if agrd at th coordination mting. This is dductd from A or B marks in th cas of prmatur approimation. Th PA pnalty is usually discussd at th mting. UCLES

Pag Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 UCLES ) ( )) ( ( d d y MA prssion simplifid Lngth ( ) ( ) [ ] d M intgrat ( ) ( ) AG A cao [] nth trm is n n MA... S M sum of trms A aftr cancllation [] Limit ¾ B [] Ara 8 / d B A A y ln )d ( M us of A y d M intgrat A corrct Final answr: ln 8 or ln or 9 ln 8 tc (ACF) A []

Pag Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 n : 7 which is divisibl by B Assum 7 k k is divisibl by B Considr 7 (k ) (k ) 7 7 k. k M (k ) th trm 9(7 k k ). k M in appropriat form which is divisibl by A convincing argumnt [] Altrnativ solution for final thr marks: Considr (7 k k ) (7 k k ) M 8(7 k k ). k M in appropriat form which is divisibl by A convincing argumnt I n [ ( ) n n cos] ( n )( ) cos d MA ( ( n )) ( n )[(( ) n sin ) ( ) n sin d] M intgrat by parts again I n (n )(n ) I n AG A [] I I ; I I ; I I M I sin d cos B I ( ( I )).77 MA [] OR I cos B I cos M (us of RF) I cos A I.77 A cao Accpt dcimal vrsions α α α α α α α MA Dim α AG A [] a b c a b c Show a b c M attmpt to solv a b c b c Linarly indpndnt and dim R(T) not : basis A [] a b c p a b c Attmpt to find a, b, c in trms of q or p a b c b c q MA p q A [] Altrnativ solution: Us row oprations as in (i) p p Final column p p q p q M A A UCLES

Pag Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 7 y y y M us in givn cubic quation Givs y 7y y AG A [] n : givn prssion sum of roots 7/ B n : " αβ" ( ) ( ) α α B [] From cubic in y, 7 7. α 7/ A [] α LHS ( β )( γ )( α M ( α ) 7 M rcognis product of roots 7/ AG A [] M 8 (i) sinθ sinθ sinθ M, and, A (both) [] (ii) B B B circl cardioid bhaviour at origin cardioid closd and symmtry [] (iii) Subtract intgrands M ( cos θ sinθ )dθ M [ θ sin θ cosθ ] MA AG A [] Altrnativ: Ara insid C : 9 9sin θ dθ θ sin θ M 9 A UCLES

Pag 7 Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 Ara insid C : sinθ ( cos θ )dθ θ cosθ sin θ M 9 8 (A if not arnd arlir) Subtraction M Rquird ara AG A [] 9 ( )[( λ)( λ) ] ( ( λ)) λ M charactristic quation ( λ )( λ )( λ ) M factoris λ,, A λ λ y λ z Solv for λ : (,, ) MA Solv for λ : (,, ) A Solv for λ : (,, ) A [7] M B ignvctors as columns (cpt ) D 8 MA ft on ignvalus [] cosθ c c s cs MA us of d Moivr for (c is) sin θ c s c s s A t t t tan θ AG MA intrmdiat stp ndd [] t t n tan θ θ M Solutions tan for n,,, A justify valus of n [] Roots ± tan, ± tan B Product of ths roots M tan tan A [] UCLES

Pag 8 Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 z y y B z y y B Obtain rsult B [] Auiliary quation: m : m ±i M CF: Acos Bsin A PI: z a b c Diffrntiat twic and substitut M a, b, c A GS: z Acos Bsin A thir CF thir PI y, : (z ) givs A B z Asin Bcos M y, : (z ) givs B A y cos sin A [9] EITHER (i) y ( λ)( λ) ( λ)( ) M... (λ ) λ λ A Hnc at most valus of and at most stationary points A [] (ii) For ral distinct roots, λ > (λ )( λ ) M us of discriminant λ ( λ) > λ > AG A [] (iii) Vrt. asymptots whn λ M b ac > λ > For two vrt. asymp. λ < A [] (iv) (a) y λ M or λ A (both) λ (b) y : λ B [] (v) (a) λ < : no stat points: vrt. asymp B B branchs compltly corrct shap (b) λ < : stats points: no vrt. asymp B ma, min, horiz asymp B corrct shap [] UCLES

Pag 9 Mark Schm: Tachrs vrsion Syllabus Papr GCE A LEVEL Octobr/ovmbr 9 OR ormal to plan: (,, ) (,, ) (,, ) MA r.(,, ) d and point (,, ) M substitut point into plan qn d y z A [] Altrnativ: λ µ y λ z λ µ } z λ MA z ( y ) M y z A y z y z Solv by liminating on variabl M Us paramtr and prss all variabls in trms of it M.g. t, y t, z t r (,, ) t (,, ) A or quivalnt [] Altrnativ: Dirction of lin t MA Find any point on lin.g., tc. r t B Lin l: r (a, a, ) α(c,, c) Plan: y z Distanc A to plan: a (a ) M a 9 M corrct us of modulus sign a A c c sinθ MA 9c 9 c c M solv for c 9 c c c : c A (Pnalis only onc for ngativ valus.) [7] UCLES