4037 ADDITIONAL MATHEMATICS

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1 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, 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 should b rad in conjunction with th qustion papr and th Principal Eaminr Rport for Tachrs. Cambridg will not ntr into discussions about ths mark schms. Cambridg is publishing th mark schms for th Octobr/Novmbr 0 sris for most IGCSE, GCE Advancd Lvl and Advancd Subsidiary Lvl componnts and som Ordinary Lvl componnts.

2 Pag Mark Schm Syllabus Papr GCE O LEVEL Octobr/Novmbr 0 40 Mark Schm Nots 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). Accuracy 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. Not: B or A mans that th candidat can arn or 0. B,, 0 mans that th candidat can arn anything from 0 to. Cambridg Intrnational Eaminations 0

3 Pag Mark Schm Syllabus Papr GCE O LEVEL Octobr/Novmbr 0 40 ( + 6)( ) Critical valus 6 and 6 < < ( 4 5 ) Multiply top and bottom by OR ( 4 5 ) ( 5 )( p 5 + q) 5 p q + 5( q p) Lading to 5p q 84, q p 6 p = q = [] [4] Attmpt to solv a thr trm quadratic Allow > 6 AND < but not OR or a comma. Mark final answr. Attmpt to pand, allow on rror, must b in th form a + b 5. Must b attmpt to pand top and bottom. Allow for c Must gt to a pair of simultanous quations for this mark (i) dy k dk 4 k = Us y 56p 5 dy with = and d p [] [] on k nds both M marks only for 8kp and must b valuatd 4 (i) (iii) 0 5 log XY log X + log Y p p p [] [] [] Not log p 5 Or log XY p log p XY Do not allow just log p X + log p Y on log p XY Cambridg Intrnational Eaminations 0

4 Pag 4 Mark Schm Syllabus Papr GCE O LEVEL Octobr/Novmbr y 5 o + y 5 o Solv thir linar simultanous quations or y 0. 5 OR from log y y.86 OR from ln y y 5.49 Final follows as bfor, [5] Each in two variabls and not quadratic as far as = or y = 6 (a) (i) (b) 8 or 0 60( ) isw 60( ) (i) + (thir 60) 0 ( ) o [] [] B,,,0 [] ± 40 implis ± 0 or +60 hnc OK if sn in pansion Can b implid Trms must b valuatd (allow 4 0 ) B for 4 trms corrct. for or trms corrct. ISW onc pansion is sn. (i) 500 l L l Substitut for l and corrctly rach 000 L 4 + Dag [] allow l = 500 RHS trms.g. or bttr Dpndnt on both prvious B marks dl d dl Equat to 0 and solv d 0 L 0 d y 4000 and minimum statd 4 d D [5] ithr powr rducd by on both trms corrct n Must gt Both valus Or us of gradint ithr sid of turning point. Cambridg Intrnational Eaminations 0

5 Pag 5 Mark Schm Syllabus Papr GCE O LEVEL Octobr/Novmbr (i) [] Implid by as or valus in a tabl. May b sn in y Plot against with linar scals Must b linar scals y [] At last corrct points plottd and no incorrct points Lin must b ruld and through at last corrct points (iii) (iv) Finds gradint (0.4) a 0.4 ± 0.0 b. ± 0.4 y Rad. 5 [] Condon us of corrct valus from tabl/graph to find gradint and /or quation. Valus rad from graph must b corrct. Obtaining ( ) to 4 from graph or substitut in formula As far as = +v constant 4.8 [] 4. to 4.9 ignor 4.8 or 0 9 Mthod A Taks componnts v sin α 40 v cosα +.8 v cosα 48.4 Solv for v or α α 9.6 v 5. ( ) 0 Mthod B D [8] Allow 0.69 radians.8.6 y D D 6.8 V D V 5. tanα α 9.6 ( 94.56) D [8] 5. or bttr Allow 0.69 radians Cambridg Intrnational Eaminations 0

6 Pag 6 Mark Schm Syllabus Papr GCE O LEVEL Octobr/Novmbr 0 40 Mthod C z ( 80.6) v 40 0 ( 6.) tanδ 4 ( δ 9.4) o V cos 9.4 V 5. sin β sin β 9.8() or 9.8() α β 9.6 Mthod D [8] Or tan( 90 δ ) 4 Allow 0. radians Allow 0.69 radians + ( 80.6) z tanδ 4 ( δ 9.4) o D cos 9.4 V 6.8 / 5. ( ) This mthod has tra stps so not at this point th M mark is for an quation in D but th A mark is for a valu of V. sin β.6 sin β 9.8() or 9.8( ) α β 9.6 [8] Allow 0. radians Allow 0.69 radians Cambridg Intrnational Eaminations 0

7 Pag Mark Schm Syllabus Papr GCE O LEVEL Octobr/Novmbr (i) AB to 5.5 θ π.4( 4.88) Us s rθ ( 58.6) 4. cos.4 [5] AB sin 0. May b implid May b implid 4.9 or bttr o (Sctor) ( π.4)( 5) or π.4 (Triangl) sin.4( 0.9 or ) Ara of major sctor + Ara of triangl 4 or 4 [4] May b implid. May b implid (i) y d d m y ( 9) At Q y = 0, = 6 D [4] For insrtion of = 9 into d y thir. 6. or bttr if corrct. d Using thir valuatd m to find qn y or bttr if corrct. Accpt valu that rounds to 6.0 to sf Ara triangl d.5 or 0. o Uss limits of 0 and 9 in intgratd function. or 5. Ara undr curv subtract ara of triangl.5 or. [6] ± must s both valus insrtd if incorrct answr Condon. if obtaind from Cambridg Intrnational Eaminations 0

8 Pag 8 Mark Schm Syllabus Papr GCE O LEVEL Octobr/Novmbr 0 40 (a) cosc = insrtd into quation sin tan = D [4] On corrct valu. on 80 + ( 64.) Must com from tan = Condon64 and 44 Dduct mark for tras in rang (b) (y ) = 0.9..or.4 Find y using radians (or 0.9 or 0.90).6, 4.04 and 4.8(45) [5] Allow 0.8,. or 45.6 Add thn divid by on a corrct angl On corrct valu Anothr corrct valu Final two valus Dduct mark for tras in rang Cambridg Intrnational Eaminations 0

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level MARK SCHEME for the October/November 0 series 40 ADDITIONAL MATHEMATICS 40/ Paper, maimum raw mark 80 This mark scheme is published as an aid to