9231 FURTHER MATHEMATICS

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CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advaced Level www.xtremepapers.

1 CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advaced Level MARK SCHEME for the October/vember series 9 FURTHER MATHEMATICS 9/ Paper, maximum raw mark This mark scheme is published as a aid to teachers ad cadidates, to idicate the requiremets of the examiatio. It shows the basis o which Examiers were istructed to award marks. It does ot idicate the details of the discussios that took place at a Examiers meetig before markig bega, which would have cosidered the acceptability of alterative aswers. Mark schemes should be read i cojuctio with the questio paper ad the Pricipal Examier Report for Teachers. Cambridge will ot eter ito discussios about these mark schemes. Cambridge is publishig the mark schemes for the October/vember series for most IGCSE, GCE Advaced Level ad Advaced Subsidiary Level compoets ad some Ordiary Level compoets.

2 Page Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Mark Scheme tes are of the followig three types: M A B Method mark, awarded for a valid method applied to the problem. Method marks are ot lost for umerical errors, algebraic slips or errors i uits. However, it is ot usually sufficiet for a cadidate just to idicate a itetio of usig some method or just to quote a formula; the formula or idea must be applied to the specific problem i had, e.g. by substitutig the relevat quatities ito the formula. Correct applicatio of a formula without the formula beig quoted obviously ears the M mark ad i some cases a M mark ca be implied from a correct aswer. Accuracy mark, awarded for a correct aswer or itermediate step correctly obtaied. Accuracy marks caot be give uless the associated method mark is eared (or implied). Mark for a correct result or statemet idepedet of method marks. Whe a part of a questio has two or more "method" steps, the M marks are geerally idepedet uless the scheme specifically says otherwise; ad similarly whe there are several B marks allocated. The otatio DM or DB (or dep*) is used to idicate that a particular M or B mark is depedet o a earlier M or B (asterisked) mark i the scheme. Whe two or more steps are ru together by the cadidate, the earlier marks are implied ad full credit is give. The symbol implies that the A or B mark idicated is allowed for work correctly followig o from previously icorrect results. Otherwise, A or B marks are give for correct work oly. A ad B marks are ot give for fortuitously "correct" aswers or results obtaied from icorrect workig. te: B or A meas that the cadidate ca ear or. B// meas that the cadidate ca ear aythig from to. The marks idicated i the scheme may ot be subdivided. If there is geuie doubt whether a cadidate has eared a mark, allow the cadidate the beefit of the doubt. Uless otherwise idicated, marks oce gaied caot subsequetly be lost, e.g. wrog workig followig a correct form of aswer is igored. Wrog or missig uits i a aswer should ot lead to the loss of a mark uless the scheme specifically idicates otherwise. For a umerical aswer, allow the A or B mark if a value is obtaied which is correct to s.f., or which would be correct to s.f. if rouded ( d.p. i the case of a agle). As stated above, a A or B mark is ot give if a correct umerical aswer arises fortuitously from icorrect workig. For Mechaics questios, allow A or B marks for correct aswers which arise from takig g equal to 9.8 or 9.8 istead of.

3 Page Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 The followig abbreviatios may be used i a mark scheme or used o the scripts: AEF AG BOD CAO CWO ISW MR PA SOS SR Ay Equivalet Form (of aswer is equally acceptable) Aswer Give o the questio paper (so extra checkig is eeded to esure that the detailed workig leadig to the result is valid) Beefit of Doubt (allowed whe the validity of a solutio may ot be absolutely clear) Correct Aswer Oly (emphasisig that o "follow through" from a previous error is allowed) Correct Workig Oly ofte writte by a fortuitous' aswer Igore Subsequet Workig Misread Premature Approximatio (resultig i basically correct work that is isufficietly accurate) See Other Solutio (the cadidate makes a better attempt at the same questio) Special Rulig (detailig the mark to be give for a specific wrog solutio, or a case where some stadard markig practice is to be varied i the light of a particular circumstace) Pealties MR PA A pealty of MR is deducted from A or B marks whe the data of a questio or part questio are geuiely misread ad the object ad difficulty of the questio remai ualtered. I this case all A ad B marks the become "follow through " marks. MR is ot applied whe the cadidate misreads his ow figures this is regarded as a error i accuracy. A MR pealty may be applied i particular cases if agreed at the coordiatio meetig. This is deducted from A or B marks i the case of premature approximatio. The PA pealty is usually discussed at the meetig.

4 Page Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio Part Fids partial fractios. r - + ( r )( r + ) ( r ) r ( r + ) B () Expresses each term i fractios ( ) ( + ) MA Cacels terms ad sums + ( + ) (OE) MA () Fid sums to ifiity S B () [6] Fids determiat ad commets. Determiat o iverse (AG) MA () Elimiates oe variable. Obtais e.g. x + y MA Obtais geeral solutio x y + t z (OE) M A () [6] Fids complemetary fuctio m + m + m ± i x C.F. e ( Acos x + Bsi x) M A Trial P.I. P.I. y px + qx + r y px + q y p Differetiates twice ad substitutes. p + px + q + px + qx + r x + 8 MA Equates coefficiets ad solves p, q, r MA States G.S. x ( Acos x + Bsi x) + x + y e x A (7) [7] Cambridge Iteratioal Examiatios

5 Page Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio Part Differetiates each equatio wrt θ. x & cosθ y& si θ B Forms quotiet for dy dx dy siθ cosθ dx si θ cotθ (AG) MA States exceptios. ( k aiteger) Exceptforθ k. A () Differetiates wrt x d y d cotθ cosec θ dx dθ x si θ MA Obtais result wheθ. A () [7] Uses sum of roots 6 9 α a a 8 MA Uses product of roots αβγ ( a ) a d 8 M Substitutes for a, ad solves 9 d d 8 d ± 7 M A Roots are 7,, A Uses sum of products i pairs. (Or expads ( x + 7)( x + )( x + ) ad equates coefficiets) 7 αβ k 8 k M A (8) [8] Cambridge Iteratioal Examiatios

6 Page 6 Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio Part 6 (a) Uses formula for MV ad itegrates MV [ l( sec x + ta x) ] sec xdx 6 MA Substitutes limits 6 l ( + ) l + M ad evaluates 6 + l (OE) (.7) A () (b) Differetiates wrt x y l cosx y ta x B Fids ds dx + ( y ) sec x B Obtais arc legth itegral ad itegrates. Obtais result [ l( sec x ta )] s sec x dx + x ( ) l + (.) M A () [8] 7 States vertical asymptote Vertical asymptote is x B Expresses y i a suitable form y x + x ( + ) M States oblique asymptote Oblique asymptote is y x + A () Differetiates wrt x y + x ( + ) MA ad explais result ( + ) x > y > (AG) A () Sketches graph. Axes ad asymptotes B (Deduct mark for poor forms at ifiity) Each brach, showig itersectio with axes. BB () [9] Cambridge Iteratioal Examiatios

7 Page 7 Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio Part 8 Fids ormal AB i + j + k, AC i + j k ad cartesia equatio AB AC 6 i j + k MA 6 x + y z costat + MA () Fids where OD meets plae Equatio of OD: r t ( 6 i + j + 6k) 6 t + 6t t t t E is the poit (,,). B MA A () Obtais agles betwee plaes Usig ( 6i j k ) +. i + j + k siθ MA o θ 9. (.66 rad). A () [] 9 States iductive hypothesis k H k : ( cosθ + i siθ ) cos kθ + i si kθ for some k B Proves H k H k + k+ ( cosθ + isiθ ) ( cosθ + isiθ )( coskθ + isi kθ ) ( cos θ coskθ siθ sikθ ) + i( siθ coskθ + cosθ sikθ ) cos [ + ] θ + si[ k + ]θ H k H k k+ M A A States coclusio H is trivially true, so true for all positive itegers A () Uses de M s Thm. to fid isi θ z z i si θ ( cos θ + i si θ ) ( cos θ i si θ ) B Uses biomial expasio ad groups z z z z ( i siθ ) i si θ z + z + z z z z z + z z z B M A Applies above result ad obtais si i si θ i siθ i siθ + i siθ si θ ( si θ si θ + si θ ) 6 M A (6) [] Cambridge Iteratioal Examiatios

8 Page 8 Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio Part Differetiates wrt θ dr dθ cosθ cos θ M Equates to zero Ad solves equatio c ( c ) c c ( c + )( c ) c or A M A States required poits o C, A () Sketches C. Approximate shape ad locatio Accurate scalig. B B () Uses r dθ + Area si θ ( cosθ cos θ ) dθ M M Obtais a itegrable form cosθ cosθ si θ + [ cosθ ] dθ AA Itegrates θ si θ si θ si θ 6 M Obtais result or. 6 A (6) [] Cambridge Iteratioal Examiatios

9 Page 9 Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio Part EITHER Itegrates by parts ( x ) x ( + x ) dx ( + x )( + x ) dx I x + I I + I MA MA Obtais reductio formula ( + ) + I I (AG) A () Fids I I dx B Uses reductio formula twice I + I I M 6 I + I 8 A Fids I 6 96 I + 8 I (.7) 7 A () Uses reductio formula For - ad - Elimiates I - ad fids I - ( + ) I I I I I + ad I I I + M AA MA () [] Cambridge Iteratioal Examiatios

10 }}}}}}}}} Page Mark Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio Part OR States basic relatioship (twice) Ae λe ad Be µ e B Adds Addig Ae + Be λ e + µ e ( A + B) e ( λ + µ )e M Obtais result λ + µ is a eigevalue (sice e ) A () Fids eigevalues of A A λ A λ 6 A λ MA A A () States eigevalues of M Eigevalues of M are 8,, B () States R R B Recogises that S is the iverse of R S R M Fids S Determiat S MA Fids D 768 D MA (6) [] Cambridge Iteratioal Examiatios

9231 FURTHER MATHEMATICS

9231 FURTHER MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advaced Level MARK SCHEME for the October/vember series 9 FURTHER MATHEMATICS 9/ Paper, maximum raw mark This mark scheme is published as a aid to teachers ad cadidates,