9231 FURTHER MATHEMATICS

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Level MARK SCHEME for he May/June series 9 FURTHER MATHEMATICS 9/ Paper, maximum raw mark This mark scheme is published as an aid o eachers and candidaes, o indicae he requiremens of he examinaion. I shows he basis on which Examiners were insruced o award marks. I does no indicae he deails of he discussions ha ook place a an Examiners meeing before marking began, which would have considered he accepabiliy of alernaive answers. Mark schemes should be read in conjuncion wih he quesion paper and he Principal Examiner Repor for Teachers. Cambridge will no ener ino discussions abou hese mark schemes. Cambridge is publishing he mark schemes for he May/June series for mos IGCSE, GCE Advanced Level and Advanced Subsidiary Level componens and some Ordinary Level componens.

2 Page Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 Mark Scheme Noes Marks are of he following hree ypes: M A B Mehod mark, awarded for a valid mehod applied o he problem. Mehod marks are no los for numerical errors, algebraic slips or errors in unis. However, i is no usually sufficien for a candidae jus o indicae an inenion of using some mehod or jus o quoe a formula; he formula or idea mus be applied o he specific problem in hand, e.g. by subsiuing he relevan quaniies ino he formula. Correc applicaion of a formula wihou he formula being quoed obviously earns he M mark and in some cases an M mark can be implied from a correc answer. Accuracy mark, awarded for a correc answer or inermediae sep correcly obained. Accuracy marks canno be given unless he associaed mehod mark is earned (or implied). Mark for a correc resul or saemen independen of mehod marks. When a par of a quesion has wo or more "mehod" seps, he M marks are generally independen unless he scheme specifically says oherwise; and similarly when here are several B marks allocaed. The noaion DM or DB (or dep*) is used o indicae ha a paricular M or B mark is dependen on an earlier M or B (aserisked) mark in he scheme. When wo or more seps are run ogeher by he candidae, he earlier marks are implied and full credi is given. The symbol implies ha he A or B mark indicaed is allowed for work correcly following on from previously incorrec resuls. Oherwise, A or B marks are given for correc work only. A and B marks are no given for foruiously "correc" answers or resuls obained from incorrec working. Noe: B or A means ha he candidae can earn or. B// means ha he candidae can earn anyhing from o. The marks indicaed in he scheme may no be subdivided. If here is genuine doub wheher a candidae has earned a mark, allow he candidae he benefi of he doub. Unless oherwise indicaed, marks once gained canno subsequenly be los, e.g. wrong working following a correc form of answer is ignored. Wrong or missing unis in an answer should no lead o he loss of a mark unless he scheme specifically indicaes oherwise. For a numerical answer, allow he A or B mark if a value is obained which is correc o s.f., or which would be correc o s.f. if rounded ( d.p. in he case of an angle). As saed above, an A or B mark is no given if a correc numerical answer arises foruiously from incorrec working. For Mechanics quesions, allow A or B marks for correc answers which arise from aking g equal o 9.8 or 9.8 insead of. Cambridge Inernaional Examinaions

3 Page Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 The following abbreviaions may be used in a mark scheme or used on he scrips: AEF AG BOD CAO CWO ISW MR PA SOS SR Any Equivalen Form (of answer is equally accepable) Answer Given on he quesion paper (so exra checking is needed o ensure ha he deailed working leading o he resul is valid) Benefi of Doub (allowed when he validiy of a soluion may no be absoluely clear) Correc Answer Only (emphasising ha no "follow hrough" from a previous error is allowed) Correc Working Only ofen wrien by a foruious' answer Ignore Subsequen Working Misread Premaure Approximaion (resuling in basically correc work ha is insufficienly accurae) See Oher Soluion (he candidae makes a beer aemp a he same quesion) Special Ruling (deailing he mark o be given for a specific wrong soluion, or a case where some sandard marking pracice is o be varied in he ligh of a paricular circumsance) Penalies MR PA A penaly of MR is deduced from A or B marks when he daa of a quesion or par quesion are genuinely misread and he objec and difficuly of he quesion remain unalered. In his case all A and B marks hen become "follow hrough " marks. MR is no applied when he candidae misreads his own figures his is regarded as an error in accuracy. An MR penaly may be applied in paricular cases if agreed a he coordinaion meeing. This is deduced from A or B marks in he case of premaure approximaion. The PA penaly is usually discussed a he meeing. Cambridge Inernaional Examinaions

4 Page Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 Qu No Commenary Soluion Marks Par Mark Toal Use of r dθ A ( cosθ cos θ ) dθ M Use of double angle formula and aemp o inegrae. ( cosθ cos θ ) dθ M Inegraes correcly. sin θ θ sinθ A Finds value. (CWO) Accep 8.8 A [] Proves base case. P n : n is divisible by 8. 8 P is rue Saes inducive hypohesis. Assume P k is rue: k 8λ for some k. k k. k. k 8. 8λ B B M Proves inducive sep. P k P k A Saes conclusion. (Since P is rue and P k P k ) P n is for every posiive ineger n (by PMI). A [] Uses b α. a c B Uses subsiuion ( α β c γ ec.) y c x x c y ( y ) ( y) ( y) (heir c) M M o obain required cubic equaion. y y y A Obains equaion whose roos are reciprocals of hose in previous cubic equaion. Uses z y o obain z z z MA Uses α ( α ) αβ ( ) ( α β ) MA [8] Cambridge Inernaional Examinaions

5 Page Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 Inegraes by pars. Rearranges. I n ( x ) x ( x n ) n n n( x ( n ) n ( x ) ( x ( n ) ).x ) MA MA Obains resul. n n (n I n. (AG) A ni ) Uses reducion formula o find I. I I 8 MA Uses reducion formula o find I. I 8 I A [8] Finds parial fracions. ( `r )( r ) r r N r (r )(r ) MA Expresses erms as differences.... N N MA Shows cancellaion. (N ) (AG) A Uses N N N N N M. N (N ) (N ) Applies resul and simplifies. N N N ( N )(N ) A M Deduces inequaliy. < N N.N (AG) 8N A [9] Cambridge Inernaional Examinaions

6 Page Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 Shows e is an eigenvecor of A and gives eigenvalue. Ae e e is an eigenvecor wih eigenvalue. MA Finds characerisic equaion. λ λ λ MA Facorises. ( λ )( λ ) A Saes oher eigenvalues. Oher eigenvalues are and. A Repeas for B. Be e e is an eigenvecor wih eigenvalue B Saes resul for AB. ABe A.e Ae.e e AB has eigenvecor e wih eigenvalue MA [9] 7 Expands and groups. Use of z z and z z. Correcly. ( z ) ( z z ) ( z z ) ( z z ) z MA M Obains resul. ( isinθ ) cosθ cosθ cos θ sin θ (Allow p, q, r, s ) ( cos θ cos θ cos θ ) AA A Inegraes correcly. θ sin θ sin θ sin θ 9 MA Insers limis and evaluaes. 9 8 (SC: f power of consisenly wrong ¾ for nd par.) MA [] Cambridge Inernaional Examinaions

7 Page 7 Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 Cambridge Inernaional Examinaions 8 Reduces marices o echelon form M A A Finds basis for each null space. 9 z z y z y x z y y x MA A Wries x and x appropriaely. λ x and µ x B Finds difference µ λ µ λ µ λ µ λ x x µ λ and 7 µ λ λ and µ M A and solves. ) 7 ( x x T p and q A []

8 Page 8 Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 9 Finds complemenary funcion. m m (m ) m C.F.: x Ae Be M A Finds paricular inegral. P.I.: x ke x& ke & x k 8k k k x e ke M A Adds for general soluion. G.S.: x Ae Be e A Uses iniial condiions o find consans. x ( ) A A e Be Be e x& 7 7 x& ( ) B B B M A Gives paricular soluion. x e e e A 9 Saes limi. lim x B [] Cambridge Inernaional Examinaions

9 Page 9 Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 Saes asympoes. Verical: x and Horizonal: y BB Obains quadraic form in x. yx yx y x x ( y ) x (y ) x ( y ) MA Uses B AC for real roos. For real x (y ) ( y )( y ) y y. M A Finds condiion for y. y ( x x )(x ) (x x )(x ) Solves x 8x 7 ( x 7)( x ) x 7, (since x is verical asympoe). M A Obains saionary poin. Saionary poin is 7, A Skech showing: Axes and asympoes (-.,), (,), (, ) and (,) Lef hand branch. Righ hand branch. B B B B [] Cambridge Inernaional Examinaions

10 Page Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 E Differeniaion. y sec x y sec x an x M Use of sec x an x. ( y ) sec x(sec x ) sec x sec x ( sec ) x M A Subsiue in arc lengh formula. s (sec x ) A Inegrae. Subsiue limis. (i) Use surface area formula Obain correc form. (ii) Differeniaes [ an x x] S sec x(sec ( sec sec x) d sec x sec x sec x ) x (AG) ( secxanx) secxan x sec x ( ) x M A MA A MA and obains prined resul. sec x secx (AG) A Uses resul o deduce S sec x an x surface area. [ ] M A [] Cambridge Inernaional Examinaions

11 Page Mark Scheme Syllabus Paper GCE A LEVEL May/June 9 O Vecor perpendicular o Π i j k MA Obains caresian equaion ( ) ( ) ( ) 7 x y z 7 M A Obains area of riangle ABC. (.9) M A Obains lengh of 9 7 perpendicular from D o riangle ABC. MA Uses Base area Heigh. Eiher MA or riple scalar produc mehod. Or e.g. Obains normal o ABD. i j k 7 MA Uses scalar produc 7 cosθ 7 M o find angle beween normals and hence angle beween Π and Π. 8 cos θ θ.º 7 A [] Cambridge Inernaional Examinaions

Mark Scheme (Results) January 2011

Mark Scheme (Results) January 2011 Mark (Resuls) January 0 GCE GCE Furher Pure Mahemaics FP (6667) Paper Edexcel Limied. Regisered in England and Wales No. 4496750 Regisered Office: One90 High Holborn, London WCV 7BH Edexcel is one of he