Further Methods for Advanced Mathematics (FP2) WEDNESDAY 9 JANUARY 2008

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1 ADVANCED GCE 7/ MATHEMATICS (MEI) Furter Metods for Advaced Matematics (F) WEDNESDAY 9 JANUARY 8 Additioal materials: Aswer Booklet (8 pages) Grap paper MEI Eamiatio Formulae ad Tables (MF) Afteroo Time: our miutes INSTRUCTIONS TO CANDIDATES Write your ame i capital letters, your Cetre Number ad Cadidate Number i te spaces provided o te Aswer Booklet. Read eac questio carefully ad make sure you kow wat you ave to do before startig your aswer. Aswer all te questios i Sectio A ad oe questio from Sectio B. You are permitted to use a grapical calculator i tis paper. Fial aswers sould be give to a degree of accuracy appropriate to te cotet. INFORMATION FOR CANDIDATES Te umber of marks is give i brackets [ ] at te ed of eac questio or part questio. Te total umber of marks for tis paper is 7. You are advised tat a aswer may receive o marks uless you sow sufficiet detail of te workig to idicate tat a correct metod is beig used. Tis documet cosists of prited pages. OCR 8 [H//] OCR is a eempt Carity [Tur over

2 Sectio A ( marks) Aswer all te questios (a) Fig. sows te curve wit polar equatio r a( cos θ) for θ π, werea is a positive costat. O Fig. Fid te area of te regio eclosed by te curve. [7] (b) (i) Give tat f() arcta( ),fidf () ad f (). [] (ii) Hece fid te Maclauri series for arcta( ), as far as te term i. [] (iii) Hece sow tat, if is small, arcta( ) d. [] (a) Fid te t roots of j, i te form re were r > adπ < θ π. Illustrate te t roots o a Argad diagram. [] (b) (i) Sow tat ( e )( e ) cosθ. [] Series C ad S are defied by C cosθ cosθ 8cosθ... cos θ, S siθ siθ 8siθ... si θ. (ii) Sow tat C cosθ cos( )θ cos θ, ad fid a similar epressio cosθ for S. [9] OCR 8 7/ Ja8

3 YouaregivetematriM ( 7 ). (i) Fid te eigevalues, ad correspodig eigevectors, of te matri M. [8] (ii) Write dow a matri ad a diagoal matri D suc tat M D. [] (iii) Give tat M ( a b c d ),sowtata, ad fid similar epressios for b, c ad d. [8] Optio : Hyperbolic fuctios Sectio B (8 marks) Aswer oe questio (i) Give tat k adcos k, sowtat ±l(k k ). [] (ii) Fid d, givig te aswer i a eact logaritmic form. [] (iii) Solve te equatio si si, givig te aswers i a eact form, usig logaritms were appropriate. [] (iv) Sow tat tere is o poit o te curve y si si at wic te gradiet is. [] [Questio is prited overleaf.] OCR 8 7/ Ja8

4 Optio : Ivestigatio of curves Tis questio requires te use of a grapical calculator. A curve as parametric equatios t t, y t t, were is a costat. (i) Use your calculator to obtai a sketc of te curve i eac of te cases, ad. Name ay special features of tese curves. [] (ii) By cosiderig te value of we t is large, write dow te equatio of te asymptote. [] Forteremaideroftisquestio,assumetat is positive. (iii) Fid, i terms of, te coordiates of te poit were te curve itersects itself. [] (iv) Sow tat te two poits o te curve were te taget is parallel to te -ais ave coordiates (, ± ). [] 7 Fig. sows a curve wic itersects itself at te poit (, ) ad as asymptote 8. Te statioary poits A ad B ave y-coordiates ad. y A O B 8 Fig. (v) For te curve sketced i Fig., fid parametric equatios of te form at t, y b(t t), were a, ad b aretobedetermied. [] ermissio to reproduce items were tird-party owed material protected by copyrigt is icluded as bee sougt ad cleared were possible. Every reasoable effort as bee made by te publiser (OCR) to trace copyrigt olders, but if ay items requirig clearace ave uwittigly bee icluded, te publiser will be pleased to make ameds at te earliest possible opportuity. OCR is part of te Cambridge Assessmet Group. Cambridge Assessmet is te brad ame of Uiversity of Cambridge Local Eamiatios Sydicate (UCLES), wic is itself a departmet of te Uiversity of Cambridge. OCR 8 7/ Ja8

5 7 Mark Sceme Jauary 8 7 (F) Furter Metods for Advaced Matematics (a) (b)(i) (ii) Area is π a π a ( cos θ ) d a π a f ( ) ( f ( ) f () θ ( cos θ ( cos θ )) [ θ si θ si θ ] ( ) ) ( ( ) ) π 8 π dθ M A B BBB ft A M A M A B For ( cos θ ) dθ Correct itegral epressio icludig limits (may be implied by later work) For cos θ ( cos θ ) Itegratig a b cos θ c cos θ [ Ma B if aswer icorrect ad o mark as previously bee 7 lost ] d Applyig arcta u du u dy or d sec y Applyig cai (or quotiet) rule Stated; or appearig i series Accept. f (), f () 8 arcta( ) π... M AA ft Evaluatig f () or f () For ad ft provided coefficiets are o-zero (iii) ( π ( π π ( π...) d )... ) M A ft A ag Itegratig (award if is missed) for Allow ft from a c provided tat a Codoe a proof wic eglects

6 7 Mark Sceme Jauary 8 (a) π j t roots of j e are r e were r π θ 8 π kπ θ 8 θ 7 π, π, 8 π 8 8 B B M A Accept Implied by at least two correct (ft) furter values or statig k,, (), M A oits at vertices of a square cetre O or correct poits (ft) or poit i eac quadrat (b)(i) ( e )( e ) e e (ii) (e e cosθ OR ( cosθ jsiθ )( cosθ jsiθ ) M ( cosθ ) si θ A cosθ (cos θ si θ ) cosθ A C js e e e e ( ( e e ( ( e ) ) e cosθ C siθ S 8 e e )( e )( e ) ( )... e cosθ cos( ) θ cosθ si( ) θ cosθ ) e e ) si θ cos θ M A A ag M M A M A M A ag A 9 For e e Obtaiig a geometric series Summig (M for sum to ifiity) Give A for two correct terms i umerator Equatig real (or imagiary) parts

7 7 Mark Sceme Jauary 8 (i) Caracteristic equatio is (7 )( ), We, 7 y y 7 y y y y, eigevector is We, 7 y y M AA M M A or y ca be awarded for eiter eigevalue Equatio relatig ad y or ay (o-zero) multiple 7 y y y y, eigevector is M A 8 SR ( M I) ca ear MAAMMAMA (ii) D B ft B ft B if is sigular For B, te order must be cosistet

8 7 Mark Sceme Jauary 8 (iii) D M D M a b c d M M A ft B ft M A ag A 8 May be implied Depedet o MM For or Obtaiig at least oe elemet i a product of tree matrices Give A for oe of b, c, d correct SR If D M is used, ma marks are MMABMAA (d sould be correct) SR If teir is sigular, ma marks are MMABM

9 7 Mark Sceme Jauary 8 (i) (e e ) k e k e e k ± k k ± k M M or cos si e or k± k e l( k k ) or l( k k ) ( k k )( k k ) k ( k ) l( k k ) l( ) l( k k k k ) A M Oe value sufficiet or cos is a eve fuctio (or equivalet) ± l( k k ) A ag (ii) M For arcos or d arcos A A l(...) or ay cos substitutio For arcos or cosu or l( ) or l( ) For or du ( arcos (iii) si si cos cos (or si ) ± l( 8) arcos ) ( l( ) l( ) ) OR e e e (e )(e e ) M B l( ± 8) A M A M M B A Eact umerical logaritmic form Obtaiig a value for or l( ± 8) cos or ( e e )(e e ) (iv) dy cos cos d If d y te cos ( cos ) d cos cos Discrimiat D Sice D < tere are o solutios B M M A Usig cos cos Cosiderig D, or completig square, or cosiderig turig poit

10 7 Mark Sceme Jauary 8 OR Gradiet g cos cos B g si si si ( cos ) we (oly) M g cos 8 cos we M Ma value g we So g is ever equal to A Fial A requires a complete proof sowig tis is te oly turig poit

11 7 Mark Sceme Jauary 8 (i) BBB cusp loop BB (ii) B (iii) Itersects itself we y (iv) ± t ( ± ), 7 d y t dt t ± y ± ( ) ( ) ± ± (v) From asymptote, a 8 a From itersectio poit, From maimum poit, b 7 b 7 M A A M A ag M A ag B M A M A Two differet features (cusp, loop, asymptote) correctly idetified Oe value sufficiet

Further Concepts for Advanced Mathematics (FP1) MONDAY 2 JUNE 2008

Further Concepts for Advanced Mathematics (FP1) MONDAY 2 JUNE 2008 ADVANCED SUBSIDIARY GCE 4755/0 MATHEMATICS (MEI) Further Cocepts for Advaced Mathematics (FP) MONDAY JUNE 008 Additioal materials: Aswer Booklet (8 pages) Graph paper MEI Examiatio Formulae ad Tables (MF)