4037 ADDITIONAL MATHEMATICS

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinry Level MARK SCHEME for the October/November 0 series 07 ADDITIONAL MATHEMATICS 07/ Pper, mximum rw mrk 80 This mrk scheme is published s n id to techers nd cndies, to indicte the requirements of the exmintion. It shows the bsis on which Exminers were instructed to wrd mrks. It does not indicte the detils of the discussions tht took plce t n Exminers meeting before mrking begn, which would hve considered the cceptbility of lterntive nswers. Mrk schemes should be red in conjunction with the question pper nd the Principl Exminer Report for Techers. Cmbridge will not enter into discussions bout these mrk schemes. Cmbridge is publishing the mrk schemes for the October/November 0 series for most IGCSE, GCE Advnced Level nd Advnced Subsidiry Level components nd some Ordinry Level components.

2 Pge Mrk Scheme Syllbus Pper GCE O LEVEL October/November 0 07 Mrk Scheme Notes Mrks re of the following three types: M A B Method mrk, wrded for vlid method pplied to the problem. Method mrks re not lost for numericl errors, lgebric slips or errors in units. However, it is not usully sufficient for cndie just to indicte n intention of using some method or just to quote formul; the formul or ide must be pplied to the specific problem in hnd, e.g. by substituting the relevnt quntities into the formul. Correct ppliction of formul without the formul being quoted obviously erns the M mrk nd in some cses n M mrk cn be implied from correct nswer. Accurcy mrk, wrded for correct nswer or intermedite step correctly obtined. Accurcy mrks cnnot be given unless the ssocited method mrk is erned (or implied). Accurcy mrk for correct result or sttement independent of method mrks. When prt of question hs two or more method steps, the M mrks re generlly independent unless the scheme specificlly sys otherwise; nd similrly when there re severl B mrks llocted. The nottion DM or DB (or dep*) is used to indicte tht prticulr M or B mrk is dependent on n erlier M or B (sterisked) mrk in the scheme. When two or more steps re run together by the cndie, the erlier mrks re implied nd full credit is given. The symbol implies tht the A or B mrk indicted is llowed for work correctly following on from previously incorrect results. Otherwise, A or B mrks re given for correct work only. A nd B mrks re not given for fortuitously correct nswers or results obtined from incorrect working. Note: B or A mens tht the cndie cn ern or 0. B,, 0 mens tht the cndie cn ern nything from 0 to. Cmbridge Interntionl Exmintions 0

3 Pge Mrk Scheme Syllbus Pper GCE O LEVEL October/November 0 07 The following bbrevitions my be used in mrk scheme or used on the scripts: AG BOD CAO ISW MR PA SOS Answer Given on the question pper (so extr checking is needed to ensure tht the detiled working leding to the result is vlid) Benefit of Doubt (llowed when the vlidity of solution my not be bsolutely cler) Correct Answer Only (emphsising tht no follow through from previous error is llowed) Ignore Subsequent Working Misred Premture Approximtion (resulting in bsiclly correct work tht is insufficiently ccurte) See Other Solution (the cndie mkes better ttempt t the sme question) Penlties MR A penlty of MR is deducted from A or B mrks when the of question or prt question re genuinely misred nd the object nd difficulty of the question remin unltered. In this cse ll A nd B mrks then become follow through mrks. MR is not pplied when the cndie misreds his own figures this is regrded s n error in ccurcy. OW, This is deducted from A or B mrks when essentil working is omitted. PA S EX This is deducted from A or B mrks in the cse of premture pproximtion. Occsionlly used for persistent slckness usully discussed t meeting. Applied to A or B mrks when extr solutions re offered to prticulr eqution. Agin, this is usully discussed t the meeting. Cmbridge Interntionl Exmintions 0

4 y x Pge Mrk Scheme Syllbus Pper GCE O LEVEL October/November 0 07 () (b) (i) F B, B F, F B nd B F, F B = F or F B = B (ii) S F =, S F = {} or n ( S F ) = 0 [] [] [] (i) or [] (ii) d y sin t sin t = = cos t π sin = = 0. D [] dy dy correct substitution in = o.e. π D for use of their nd substitution of. (i) C = 7 [] 9 (ii) C C = 890, [] for correct method (iii) No women: C 7 = = 99 9 [] 9 for C 7 = for complete, correct method (i), [] for y = tn x y = + sinx for shpe of curve for curve strting t nd finishing t nd going between nd. (ii) π, nd π,, [] for ech or for both x coordintes correct (iii) ft [] Ft from their (i) or correct Cmbridge Interntionl Exmintions 0

5 Pge Mrk Scheme Syllbus Pper GCE O LEVEL October/November 0 07 (i) 80 β 80 α α 0 or 0 for correct tringle Could be implied by subsequent working. β 0 = sin0 80 sinα 80 for complete method (sine rule nd/or cosine rule) to find α or β α =. (or β = 7. ) for α (or β ) Bering = 0. or 0 for bering (ii) v r 0 =, v r = 7. sin 7. sin0 x 0 or = sin0 sin 7. for use of complete method (sine rule nd/or cosine rule) to find vr or x For either v = 7 or x = 9 Time = = or 8. 0 D 0 D for their velocity x or their 0 ( p + x) = p + p x + p x + 0 p x... (i) p = 0 p, p =, for p, for 0 p for correct ttempt to equte (ii) need p ( ) + p ( ) + p ( ) = 80 [] p, p (llow in (i)) for both for ttempt using terms for x identifying nd dding t lest two terms independent of x nd Cmbridge Interntionl Exmintions 0

6 Pge Mrk Scheme Syllbus Pper GCE O LEVEL October/November 0 07 ( t + ) t ( t ) 7 (i) = ( t + ) When = 0, t = so x = D for ttempt to differentite quotient or product ll correct, llow unsimplified D for equting to zero nd ttempt to solve to find t. for x = (ii) d x t = ( ) ( ) ( ) ( + t t t t + ) ( t + ) for ttempt to differentite quotient or product to find ccelertion correct unsimplified When t =, ccelertion = 0. [] 8 (i) f() = p + 8 = 0 p = for use of nd equting to zero, or use of compring coefficients or lgebric long division =, b=, c= B [] for ech of, b nd c (ii) ( x ) ( x ) ( x + ) [] for ttempt to obtin fctors 9 (i) AD = ( 0 )( 0 ) cos π finding AD using cosine rule including squre root. for either rc length 0π 0π Perimeter = + + ( 9. ) = 7.9 D D for correct pln before evlution using correct rc lengths nd AD Awrt 7.9 (ii) Are = π = π + π ( 0 )( 0 ) sin D for re of tringle using the sine rule, or complete correct method for ½ 0 (π/) or ½ 0 (π/) D for correct pln before evlution using correct sector nd tringle res. Awrt Cmbridge Interntionl Exmintions 0

7 Pge 7 Mrk Scheme Syllbus Pper GCE O LEVEL October/November (i) (sec x ) sec x + = 0 sec x (sec x ) = 0 cos x = 0., x = 0, 00 Alt scheme: sin x + = 0 cos x cos x sin x cos x + cos x = 0, cos x = 0., x = 0, 00 (ii) tn y =, tn y = ( ± ) ( or sin y = y ± y = 0.,.7, etc. y = 0.0, 0.907,.9,.9 ( ± ), cos = ( ) ),, for use of correct identity for solution of qudrtic in sec or cos for one correct solution for deling with tn nd sec correctly nd for use of correct identity for solution to obtin cos x for correctly obtining in terms of trig rtio nd squre rooting for deling with correctly for first for others (iii) π sin z + = for deling with nd cosec correctly π z + = 0.,.70, z =.9,.9.9 D, D for deling with π correctly EITHER dy x x (i) = e e dy When x = n, = When x =n, y = 8 Tngent: y 8 = x n y = 0, x = + n.9 When ( ) [] For correct derivtive for grd = from correct working for y = 8 Eqution of tngent using their grdient nd their 8 x x (ii) e + e = 0 for correct integrtion x x [ e e ] = e e = o for correct use of limits e e = 0 [] Answer given so need to see some mnipultion (iii) ( e + ) (e ) = 0 = ln,. or.0 [] for recognising nd deling with qudrtic for correct method of solution to obtin Cmbridge Interntionl Exmintions 0

8 Pge 8 Mrk Scheme Syllbus Pper GCE O LEVEL October/November 0 07 OR (i) dy = e x x x ( + e ) e e x ( e ) x ( + e ) x x ( + e ) A = A,,0 for ttempt to differentite quotient or product ech error For obtined from correct working. (ii) When d y = y = x = 0, y = x ft ft [] for y = for grd = A Ft their y 0 nd A (iii) x x e e = x x ( + e ) ( + e ) = 0. x e 9 = + 0 x ( e ) n 0 ( + c) ft ft for ttempt t reverse differentition Ft on their A, i.e. A for correct sttement for correct use of limits A Ft 0 Cmbridge Interntionl Exmintions 0

Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level. Published

Cambridge International Examinations Cambridge International Advanced Subsidiary and Advanced Level. Published Cmbridge Interntionl Exmintions Cmbridge Interntionl Advnced Subsidiry nd Advnced Level MATHEMATICS 9709/ Pper October/November 06 MARK SCHEME Mximum Mrk: 75 Published This mrk scheme is published s n