MARK SCHEME for the October/November 2014 series 9231 MATHEMATICS. 9231/12 Paper 1, maximum raw mark 100

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambrig Inrnaional Aanc Ll MARK SCHEME for h Ocobr/Nombr sris 9 MATHEMATICS 9/ Papr, maimum raw mar This mar schm is publish as an ai o achrs an canias, o inica h rquirmns of h aminaion. I shows h basis on which Eaminrs wr insruc o awar mars. I os no inica h ails of h iscussions ha oo plac a an Eaminrs ming bfor maring bgan, which woul ha consir h accpabili of alrnai answrs. Mar schms shoul b ra in conjuncion wih h qusion papr an h Principal Eaminr Rpor for Tachrs. Cambrig will no nr ino iscussions abou hs mar schms. Cambrig is publishing h mar schms for h Ocobr/Nombr sris for mos Cambrig IGCSE, Cambrig Inrnaional A an AS Ll componns an som Cambrig O Ll componns. IGCSE is h rgisr ramar of Cambrig Inrnaional Eaminaions.

2 Pag Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 Mar Schm Nos Mars ar of h following hr ps: M A B Mho mar, awar for a ali mho appli o h problm. Mho mars ar no los for numrical rrors, algbraic slips or rrors in unis. Howr, i is no usuall sufficin for a cania jus o inica an innion of using som mho or jus o quo a formula; h formula or ia mus b appli o h spcific problm in han,.g. b subsiuing h rlan quaniis ino h formula. Corrc applicaion of a formula wihou h formula bing quo obiousl arns h M mar an in som cass an M mar can b impli from a corrc answr. Accurac mar, awar for a corrc answr or inrmia sp corrcl obain. Accurac mars canno b gin unlss h associa mho mar is arn (or impli). Mar for a corrc rsul or samn inpnn of mho mars. Whn a par of a qusion has wo or mor "mho" sps, h M mars ar gnrall inpnn unlss h schm spcificall sas ohrwis; an similarl whn hr ar sral B mars alloca. Th noaion DM or DB (or p*) is us o inica ha a paricular M or B mar is pnn on an arlir M or B (asris) mar in h schm. Whn wo or mor sps ar run oghr b h cania, h arlir mars ar impli an full cri is gin. Th smbol implis ha h A or B mar inica is allow for wor corrcl following on from priousl incorrc rsuls. Ohrwis, A or B mars ar gin for corrc wor onl. A an B mars ar no gin for foruiousl "corrc" answrs or rsuls obain from incorrc woring. No: B or A mans ha h cania can arn or. B// mans ha h cania can arn anhing from o. Th mars inica in h schm ma no b subii. If hr is gnuin oub whhr a cania has arn a mar, allow h cania h bnfi of h oub. Unlss ohrwis inica, mars onc gain canno subsqunl b los,.g. wrong woring following a corrc form of answr is ignor. Wrong or missing unis in an answr shoul no la o h loss of a mar unlss h schm spcificall inicas ohrwis. For a numrical answr, allow h A or B mar if a alu is obain which is corrc o s.f., or which woul b corrc o s.f. if roun (.p. in h cas of an angl). As sa abo, an A or B mar is no gin if a corrc numrical answr ariss foruiousl from incorrc woring. For Mchanics qusions, allow A or B mars for corrc answrs which aris from aing g qual o 9.8 or 9.8 insa of. Cambrig Inrnaional Eaminaions

3 Pag Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 Th following abbriaions ma b us in a mar schm or us on h scrips: AEF AG BOD CAO CWO ISW MR PA SOS SR An Equialn Form (of answr is quall accpabl) Answr Gin on h qusion papr (so ra chcing is n o nsur ha h ail woring laing o h rsul is ali) Bnfi of Doub (allow whn h alii of a soluion ma no b absolul clar) Corrc Answr Onl (mphasising ha no "follow hrough" from a prious rror is allow) Corrc Woring Onl ofn wrin b a foruious' answr Ignor Subsqun Woring Misra Prmaur Approimaion (rsuling in basicall corrc wor ha is insufficinl accura) S Ohr Soluion (h cania mas a br amp a h sam qusion) Spcial Ruling (ailing h mar o b gin for a spcific wrong soluion, or a cas whr som sanar maring pracic is o b ari in h ligh of a paricular circumsanc) Pnalis MR PA A pnal of MR is uc from A or B mars whn h aa of a qusion or par qusion ar gnuinl misra an h objc an ifficul of h qusion rmain unalr. In his cas all A an B mars hn bcom "follow hrough " mars. MR is no appli whn h cania misras his own figurs his is rgar as an rror in accurac. An MR pnal ma b appli in paricular cass if agr a h coorinaion ming. This is uc from A or B mars in h cas of prmaur approimaion. Th PA pnal is usuall iscuss a h ming. Cambrig Inrnaional Eaminaions

4 Pag Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 Qn & Par ( ) ( )... ( ) n u r n u r n n Soluion Mars MA MA () B () [] & cos sin & sin cos & & s ( cos sin ) [ ] ( ) (.9)! S,! S,! S,! S (Two corrc B, all four corrc B) S n (n )!! H is ru. H : S ( )! ( )! ( ) ( )! ( )!( ) ([ ] )! Hnc H H So rsul hols for all posii ingrs (b PMI). Vrical asmpo is is h obliqu asmpo. ( ) ( ) ( ) ( ) 8( ) > has ral roos 9 > ( )( 9) > Hnc ( )( 9) < no ral roos. i.. < < 9 no poins on C. (AG) Thumbnail sch, or similar, rquir. MA B M AA () [] B,, () B () B B M A () [7] B MA () MA M A () [7] Cambrig Inrnaional Eaminaions

5 Pag Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 7 a a , or b liminaion mhos, M AA () M z z 9. z.8 (OE) A MA () Plans form a prism, or wors o ha ffc. ( ) ( ) ( ) ( ) ( ) c c is c is c is is i c is cos θ c c s cs c s s s s ( ) ( ) ( s s s s s ) c( s ) c s (AG) n Alrnai: using ( z z ) cosnθ ( cosθ ) cosθ cosθ cosθ cos θ cos θ cos θ cosθ Bu cos θ cos θ cosθ (Can b quo, if nown.) cosθ cos Uss cos θ θ ( cos θ cos θ ) sin θ o obain cos c( s s ) θ 9 cos θ,,, 7, ± ± s 8 Sinc ( ) ( ) ( ) 8 θ (AG) < sin, sin (Or b an ohr ali mho.) sin (AG) Jusificaion rquir for final mar. s B () [8] B MA M A () (M) (B) (A) (MA) B MA A () [9] Cambrig Inrnaional Eaminaions

6 Pag Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 7 I ( ) n n [ ] n( ) n ni n ni n [ ] I I I I I 7 I Or for h las mars: I, I I an I Draws appropria sch, or plains clarl. (Mus b sn.) < Ara blow graph < < < Hnc < < < < 8 Circl sch Carioi corrc locaion an orinaion corrc innaion nar pol. ( a, ) an (, ) a (B for rrs, or ( cos ), a θ θ ( ) cosθ cos θ Ara a ( cosθ ) θ a sn.) a (Half circl Ara of scor) a a θ a a ( cosθ cos θ ) θ (Us of oubl angl formula.) [ ] sinθ sin θ ( ) ( ) a θ a a (Ingraion) a a (AG) MA A () B MA A (MA) (A) () B M A () [] B BB () BB () BM A M M A () [] Cambrig Inrnaional Eaminaions

7 Pag 7 Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 Cambrig Inrnaional Eaminaions 9 ( ) ( ) ( )( ), m m m m m CF: B A : PI GS: ( ) B A B A BB MA () M A M MA MA (7) []

8 Pag 8 Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 Cambrig Inrnaional Eaminaions (i) (ii) (iii) j i ~ j i BA Shors isanc. (.) Alrnai , Shors isanc. (.) ~ j i Carsian quaion of Π: z Disanc of A from Π:. 8 (.8) MA B MA () (M) (A) (MA) (A) MA MA () MA A () []

9 Pag 9 Mar Schm Sllabus Papr Cambrig Inrnaional A Ll Ocobr/Nombr 9 E (i) (ii) (iii) α β γ δ α ( ) β γ δ α β γ δ ( ) B () MA () MA () (i) α βγδ β αγδ γ αβδ δ αβγ α β γ αβγδ δ MA () ( ) ( ) 8 ( ) ( ) 8 7 MA A A O (i) (ii) ( ) ± (wic). ± (wic). (Som inicaion of four roos for final mar.) A ; sinc A is non-singular A ( ). A A A A A A Eignalus of A ar,, ( mar for an on, mars for all hr.) Corrsponing igncors ar:,, (MA for on, A for ach ohr.) (N.B. Ma com from using ignalus of A I.) P Eignalus of A I ar,, (Awar B if obain from A I.) Eignalus of (A I) ar,, D (CAO) A MA (7) [] MA () M A () B,, MA A A B B B B () [] Cambrig Inrnaional Eaminaions

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Lvl MARK SCHEME for th Octobr/Novmbr 0 sris 40 ADDITIONAL MATHEMATICS 40/ Papr, maimum raw mark 80 This mark schm is publishd as an aid to tachrs and candidats,