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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. 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 Scheme Syllabus Paper GCE A LEVEL October/vember 9 Scheme tes s 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). 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. or 9. istead of. Cambridge Iteratioal Examiatios

3 Page 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. Cambridge Iteratioal Examiatios

4 Page Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio s Part Re-writes equatio ad uses compoud agle formula. r cos θ π cosθ siθ r M A Chages to cartesia. r cos θ r siθ x y or y x. A Sketches graph. Straight lie at π to the iitial lie. B Poit (,) clearly idicated. B [5] (i) (ii) Uses formula for mea value. Itegrates Uses formula for y- coordiate. Itegrates. x dx x xdx x [ x ] x dx M MA M MA [] Solves AQE. Fids CF. m m m ± i e t CF: ( Acost Bsi t) M A Form for PI ad differetiates. PI: x pt qt r x& pt q & x p M Compares cefficiets ad solves. p p p q q p q r r MA t GS: x e ( Acost Bsi t) t t A [] Cambridge Iteratioal Examiatios

5 Page 5 Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio s Part Verifies give result. Uses method of differeces to sum first series. r ( r )( r ) ( r ) r( r ) r( r )( r r ) r ( r ) (AG) r r( r ) {[ f ( ) f ( ) ] [ f ( ) f ( ) ]... [ f () f (]} ( )( ) (AG) (Award B if ot hece.) B M A Subtracts r r to obtai sum of secod series. r r r ) r r r ( )( ) ( ) ( r ( )( ) ( )( ) (AG) M A Splits series ito two series. Applies sum of squares formula to obtai result. (... ) (... ( ) ) ( )( ) ( ) M MA [] 5 Itegrates by parts to obtai reductio formula. x e x dx x I e x (AG) x -x e dx M A (States propositio.) Proves base case. P : I! -x -x I e dx e! I P true. B B Shows P P k k. k! P k : I k k for some iteger k. k k! ( k )! I k k k P P k k B MA States coclusio. Hece by PMI P is true for all positive itegers. A [] Cambridge Iteratioal Examiatios

6 Page Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio s Part Proves iitial result. (cosθ isiθ ) cos θ isi θ c c (is) c (is) c(is) (is) cosθ c c ( c ) ( c ) cos θ cos θ (AG) M M A Verifies ay two cases (B) Verifies remaiig two cases (B) cos π cosπ π cos π cos π cos π cosπ π cos π cos π cos π cos π cos π cos π ( ) cosπ B B Shows roots of equatio to be as give. States sum of roots of equatio. cos θ cos θ ( cos θ cosθ ) c c c c (*) 5 cos π, cos π, cos π, - are the roots. (AG) Sum of roots of (*) are Result (AG) M A MA [9] 7 Fids asymptotes to C. Differetiates ad equates to. Vertical asymptote x. y λx y λ x is oblique asymptote. x y λ ( x ) for turig poits. λ > ( ) x o turig poits if λ <. B MA MA A Or y λx λx λ Uses discrimiat to show λ < o T.P.s. (MA) (A) Sketch of graph. Deduct mark for poor forms at ifiity. Deduct mark if itersectios with axes ot show. Axes ad asymptotes. LH brach. RH brach. (Idicatig itersectios with axes at (,) ad (,).) B BB [9] Cambridge Iteratioal Examiatios

7 Page 7 Scheme Syllabus Paper GCE A LEVEL October/vember 9 Differetiates. Commetary Solutio s Part Squares ad adds. Uses arc legth formula. Itegrates. Obtais result. & & x t t ds dt s t t y t t d t t t t t t lt 9 l l ( 9.77) B MA M A A Uses surface area formula. Itegrates. Iserts limits. Obtais result. 7 S π t t dt t t dt t π 9 π t t 9 π [ ] 9 π ( or 9.π ) 7 M A M A [] Cambridge Iteratioal Examiatios

8 Page Scheme Syllabus Paper GCE A LEVEL October/vember 9 Commetary Solutio s Part 9 Fids vector ormal i j k to Π. i j 7k MA Dot product of this with geeral poit o l. Deduces result. t 5t or 5 7 t 7 Idepedet of t parallel, or parallel. A A Cartesia equatio of Π. Substitutes geeral poit of l. Fids value of parameter. Fids p.v. of itersectio. Π: x y 7z Sub. x 5 s, y s, z 7 s s ad lie meets Π at poit with p.v. i j 5k Take (9,,) as A, (,,) as B ad let C be foot of perpedicular from A to l. AB 5 B M A A B Fids distace from poit to kow poit o l. BC ( 5 )(i 5jk)(i 5jk) M A [] Fids distace alog l from kow poit to foot of perpedicular from give poit to l. F.t. o ohypoteuse side (must be real). Writes a set of three equatios i three ukows for the itersectio of l with Π. Solves the set of equatios. Fids p.v. of itersectio. AC 5 or (.) Alteratively: 5 s λ µ s λ µ 7 s λ µ s, λ, µ ad lie meets Π at poit with p.v. i j 5k A (B) (MA) (A) Cambridge Iteratioal Examiatios

9 Page 9 Scheme Syllabus Paper GCE A LEVEL October/vember 9 9 Commetary Solutio s Part Fids vector BA. BA i 5j j B 5 i j 5 5 k MA or (.) 5 A () Alteratively: (A) Fids distace from poit to kow poit o l. Fids distace alog l from kow poit to foot of perpedicular from give poit to l. F.t. o o-hypoteuse side (must be real). Take (9,,) as A, (,,) as B ad let C be foot of perpedicular from A to l. AB 5 BC (i 5j k)(i 5j k) AC 5 or (.) (B) (M) (A) (A ) () Fids vector. AC (B) AC t 5t 5 t (B) Uses AC perpedicular to l to fid t. t 5t 5. 5 t t (MA) Fids legth AC. AC (A) () [] Cambridge Iteratioal Examiatios

10 Page Scheme Syllabus Paper GCE A LEVEL October/vember 9 Cambridge Iteratioal Examiatios Commetary Solutio s Part States eigevalues. Fids eigevectors. States P ad D. Fids iverse of P. Fids A. States required limit. Eigevalues are,,. λ e k j i ~ λ e k j i λ e k j i ~ P D Det P Adj P P - A PDP -. P or P.. ]. [ ].. [ ]. [ A as. B MA A A B B MA MA A B 5 []

11 Page Scheme Syllabus Paper GCE A LEVEL October/vember 9 (i) Commetary Solutio s Part EITHER Substitute α ito equatio. α is a root α α 5α Multiply byα. Obtai result. α α 5α α Repeat for β, γ, δ ad sum M S S 5S S (AG) A Uses ( α ) α αβ Fids S from formula. S ( ) S 5 B MA (ii) S αβγ αβγδ Fids S from formula. Fids S 5 from formula. S S S ( 5) MA MA MA α β SS S5 ( 5) ( 75) 5 M MA [] Cambridge Iteratioal Examiatios

12 Page Scheme Syllabus Paper GCE A LEVEL October/vember 9 Cambridge Iteratioal Examiatios Commetary Solutio s Part (i) (ii) (iii) OR Reduces M to echelo form. Uses dimesio theorem. States basis for R. Fids cartesia equatio for R. Fids basis for ull space. Evaluates k. Fids a particular solutio. Fids geeral solutio Dim(M) Basis for R is,. (OE) λ µ x λ µ y z y x λ µ z t z y x t z y λ t ad µ z λ µ y ad µ λ x Basis of ull space is -, (OE) 9 7 k k (OE) (via equatios) x 5 µ λ. MA A B MA M AA B MA MA 9 5 []

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,