1(a)(i ) For one loop in correct quadrant(s) For two more loops. Continuous and broken lines Dependent on previous B1B1 M1 A1. For.

Transcription

1 7 Mark Scheme Jauary (a)(i ) (ii) Area is r dθ (b) a a ( θ + a ( + cos θ ) dθ d si θ ) (c) Puttig taθ Itegral is OR a cos θ dθ arcsi arcsi sec θ dθ sec θ cosθ siθ dθ M Puttig siθ A Itegral is θ A M A B B B M A M A B M AA M A M AA M A For oe loop i correct quadrat(s) For two more loops Cotiuous ad broke lies Depedet o previous BB For cos θ dθ For a correct itegral epressio icludig limits (may be implied by later work) For cos θ dθ θ + si θ Accept.a For arcsi For ad Depedet o previous M For ay ta substitutio For (sec θ ) sec θ ad Icludig limits of θ For ay sie substitutio For dθ For chagig to limits of θ Depedet o previous M

2 7 Mark Scheme Jauary (i) w, arg w θ w *, arg w* θ j w, arg jw θ + B B ft BB ft B (ii) jθ jθ jθ jθ ( + w)( + w*) + e + e + ( e )( e ) M + (cosθ + jsi θ ) + (cosθ jsi θ ) + + cosθ jθ (iii) jθ 8 jθ C + js e e + e... e e jθ jθ jθ e + e + jθ jθ e ( + e jθ ( + e )( + ( + + cosθ e e jθ + cosθ cos θ + cosθ C + cosθ si θ siθ S + cosθ jθ jθ ) e ) jθ ) jθ jθ e + e + cosθ A M A (ag) M M A M A M A (ag) A 8 w * ad jw i correct positios relative to their w i first quadrat Give B for at least two poits i correct quadrats jθ for w * e for + correctly obtaied for w (cosθ + jsi θ ) for cos θ correctly obtaied Obtaiig a geometric series Summig a ifiite geometric series Usig comple cojugate of deom Equatig real or imagiary parts Correctly obtaied

3 7 Mark Scheme Jauary (i) ( λ) [ ( λ)( λ) ] [ ( λ) ] + [ ( λ) ] ( λ)( λ + 7λ) (λ ) + (λ + ) λ + λ 9λ (ii) Whe λ, (iii) (iv) (v) ( λ + )( λ + λ ) ( λ + )( λ )( λ + 7) Other eigevalues are, 7 + y + z y + z y + y z z z, + y OR A eigevector is 8 8 M P D 7 8 M M A M 7 (vi) By CHT, M + M 9M I M + M 9I M M M + M I 7 9 M A A (ag) B M A M M A M AA B ft M A ft B M A Evaluatig det( M λ I) Allow oe omissio ad two sig errors det( M λ I) correct Correctly obtaied ( is required) or showig that ( λ + ) is a factor, ad deducig that is a root for (λ + ) quadratic factor At least two equatios Solvig to obtai a eigevector Appropriate vector product Evaluatio of vector product Ay method for verifyig or fidig a eigevector see or implied (ft) (codoe eigevalues i wrog order) Order must be cosistet with P (whe B has bee awarded) Codoe omissio of I Codoe dividig by M

4 7 Mark Scheme Jauary (a) ( e e ) + ( e e (e e )(e + e + ) e ) 8, l, l M M M AA A ft Epoetial form Quadratic i e Solvig to obtai a value of e Eact logarithmic form from positive values of e Depedet o M (b) (c)(i) OR c 8 c c c + M, M c AA ± l, ± l M l, l A e (e e ) d ( [ ] e e ( e ) ( ) + ( ) 9 + ) M M A A B Obtaiig quadratic i c (or s) ( s + s 8 ) Solvig to obtai a value of c (or s) or s, Logarithmic form (icludig ± if c) cao Epoetial form Itegratig to obtai a multiple of e Give B for ay o-zero multiple of this (ii) [ arsih( ) ] ( arsih( ) ) l + + l 9 + arsih( ) ( d ) M A ft B M M A (ag) Itegratio by parts applied to arsih( ) for d Usig both limits (provided both give o-zero values) Logarithmic form for arsih (itermediate step required)

5 7 Mark Scheme Jauary (i), (ii) (iii) k y + Asymptote is dy ( d y k < k > )( (k + ) ( ) ( ) ( ) k )() dy whe d Whe, k + > dy dy < whe <, > whe > d d Hece there is a miimum whe (iv) Curve crosses y whe k ) So curve crosses this asymptote ( k B M A B B B B M A A (ag) M A (ag) M A (ag) Dividig out or B for y stated k < for LH ad RH sectios for cetral sectio, with positive itercepts o both aes k > for LH ad cetral sectios for RH sectio, crossig -ais Usig quotiet rule (or equivalet) Ay correct form Correctly show d y or evaluatig whe d d y or k > whe 8 d

6 7 Mark Scheme Jauary (v) k < B Asymptotes show Itercepts k ad k idicated Miimum o positive y-ais Maimum show Give B for miimum ad maimum o cetral sectio k > B Asymptotes show Itercepts k ad k idicated Miimum o positive y-ais RH sectio crosses y ad approaches it from above Give B for RH sectio approachig both asymptotes correctly

7 7 Mark Scheme Jue (a)(i) B B Correct shape for < θ < icludig maimum i st quadrat Correct form at O ad o etra sectios (ii) (b)(i) (ii) Area is r dθ a ( + cosθ ) d θ a (θ + ( + ) a f ( ) sec ( f ( ) sec a ( + ( + ) cosθ + + cos θ ) dθ siθ + + ) ta( si θ ) f( ), f (), f () f( ) ) OR g ( u ) sec u (where g( u) ta u ) B g ( u) sec u ta u B g( ), g ( ), g ( ) M h h + ) f( ) g(... BAA ( + + ( h h h h ) d h +... ) ( h h h + h h ) M A B BB ft M A B B 7 For itegral of ( + cosθ ) For a correct itegral epressio icludig limits (may be implied by later work) Usig cos θ + cos θ Itegratio of cos θ ad cos θ Evaluatio usig si ( ± ) Ay correct form M Evaluatig f () or f () BAA M A ft A (ag) Codoe sec etc Evaluatig g ( ) or g ( ) Usig series ad itegratig (ft requires three o-zero terms) Correctly show Allow ft from + k + with k

8 7 Mark Scheme Jue (a)(i) z + z cos θ, z jsi θ z BB (ii) z z z z z + z z si θ cos θ + + z z z cos θ cos θ cos θ + si θ cos θ cos θ cos θ cos θ + ( A, B, C, D ) B M A M A ft A Epasio z z Usig z + θ cos with z, or. Allow M if used i partial epasio, or if omitted, etc (b)(i) + j, arg( + j) BB Accept.7;.79, (ii) r θ, 7,, 9, 7 B B Accept,., etc Accept.,.,.,.,. 7 Give B for three correct Give B for oe correct Deduct mark (maimum) if degrees used (,, 9, 8, ) + k ears B; with k,,,, ears B B Give B for four poits correct, or B ft for five poits (iii) e j j p, q j M A Eact evaluatio of a fifth root Give B for correct aswer stated or obtaied by ay other method

9 7 Mark Scheme Jue (i) M k k 8k k k M A M A M A Evaluatig determiat For ( k) must be simplified Fidig at least four cofactors At least siged cofactors correct Trasposig matri of cofactors ad dividig by determiat Fully correct OR Elemetary row operatios applied to M (LHS) ad I (RHS), ad obtaiig at least two zeros i LHS M Obtaiig oe row i LHS cosistig of two zeros ad a multiple of ( k) A or elemetary colum operatios Obtaiig oe row i RHS which is a multiple of a row of the iverse matri A Obtaiig two zeros i every row i LHS M Completig process to fid iverse MA (ii) y z 9 m M M M Substitutig k 7 ito iverse Correct use of iverse Evaluatig matri product m, y m, z m + A ft Give A ft for oe correct Accept usimplified forms or solutio left i matri form OR e.g. elimiatig, y z y z m M y m M m, y m, z m + A Elimiatig oe variable i two differet ways Obtaiig oe of, y, z Give M for ay other valid method leadig to oe of, y, z i terms of m Give A for oe correct (iii) Elimiatig, y + z y + z p For solutios, p M A M Elimiatig oe variable i two differet ways Two correct equatios Depedet o previous M OR Replacig oe colum of matri with colum from RHS, ad evaluatig determiat M determiat + p or p A For solutios, det M Depedet o previous M

10 7 Mark Scheme Jue OR Ay other method leadig to a equatio from which p could be foud M Correct equatio A Let z λ, p λ, y 8 λ, z λ A M (or M) A 7 Obtaiig a lie of solutios Give M whe M for fidig p or + λ, y λ, z 8 λ or λ, y + λ, z λ Accept z, y 8 z or y + z etc

11 7 Mark Scheme Jue (i) + sih + [ + ( e ( e cosh ( e + e e ) + e ) ] ) B B B (ag) For ( e e ) e + e For cosh ( e + e ) For completio (ii) ( + sih ) + sih M Usig (i) sih + sih ( sih )(sih + ) sih, M AA Solvig to obtai a value of sih arsih( ) l( ) l A ft arsih( ) l( + + ) l( ) A ft or l( + ) SR Give A for ± l, ± l( ) (iii) OR e + e e e + M (e )(e + )(e + e ) AA l, l( ) AA ft l (cosh ) d sih l 9 l l (iv) Put cosh u whe, u whe, u ar cosh l 9 d l l 9 sih l (sih u)(sih u du) 9 u du M AA M A (ag) M B A A Obtaiig a liear or quadratic factor For (e ) ad (e + e ) Epressig i itegrable form or ( e + e ) d or e e ) 8 8 ( l l For e 9 ad e 9 M for just statig etc Correctly obtaied Ay cosh substitutio sih( l ) For l Not awarded for arcosh Limits ot required 9

12 7 Mark Scheme Jue (i) B At least two cusps clearly show Give B for at least two arches Has cusps Periodic / Symmetrical i y-ais / Has maima / Is ever below the -ais B B Ay other feature (ii) B At least two miima (zero gradiet) clearly show Give B for geeral shape correct (at least two cycles) The curve has o cusps B For descriptio of ay differece (iii) (A) B At least two loops Give B for geeral shape correct (at least oe cycle) (B) (C) dy siθ d cosθ dy d is ifiite whe cosθ ( si ) Hece width of loop is ( ) θ M A M A M M (iv) k. B A (ag) Correct method of differetiatio Allow M if iverted siθ Allow k cosθ Ay correct value of θ Fidig width of loop Correctly obtaied Codoe egative aswer Give B for a value betwee ad (iclusive)

13 7 Mark Scheme Ja 7 (a)(i) (ii) Area is r dθ (b) (c)(i) (ii) kθ a (e ) d a kθ e k a ( e k + OR k ) d arcta arcta M Puttig taθ A Itegral is dθ A M A f ( ) ta, f () f ( ) sec, f () f ( ) sec ta, f () f ( ) sec + sec ta, f () ta + () +... ( )! h h ta d + (h h + 7h h h ( + h h h ) ( h + ) d 9 h ) θ B B M A M A M AA M A B M A B ft M A ft A ag Correct shape for θ Correct shape for θ Requires decreasig r o at least oe ais Igore other values of θ kθ For (e ) dθ For a correct itegral epressio icludig limits (may be implied by later work) (Codoe reversed limits) Obtaiig a multiple of e kθ as the itegral For arcta For ad Depedet o first M For ay ta substitutio For dθ For chagig to limits of θ Depedet o first M Obtaiig f ( ) For f () ad f () correct ft requires term ad at least oe other to be o-zero Obtaiig a polyomial to itegrate For + 9 ft requires at least two o-zero terms

14 7 Mark Scheme Ja 7 (a)(i) w, arg w z w z, arg z w, arg ( ) ( ) z B BB BB ft Deduct mark if aswers give i form r(cosθ + jsi θ ) but modulus ad argumet ot stated. Accept degrees ad decimal appros (ii) w (cos + jsi ) z + j M A Accept. +. j (b)(i) jθ e jθ + e (cos θ jsi cos θ θ ) + (cos θ + jsi θ ) M A For either bracketed epressio + e jθ e e jθ jθ (e jθ ( cos + e θ ) jθ ) M A ag OR + e jθ + cosθ + jsiθ cos θ + jsi θ cos θ M cos θ (cos θ + jsi θ ) jθ e cos θ A (ii) C + js + e ( + e ) C cos( e jθ θ j jθ cos θ ) cos S si( θ ) cos S si( C cos( θ ) cos + e θ ) cos θ θ θ jθ e jθ θ si( θ ) ta( θ cos( θ ) θ ) M MA M A A B ag Usig (i) to obtai a form from which the real ad imagiary parts ca be writte dow θ j Allow ft from C + js e ay 7 real fuctio of ad θ

15 7 Mark Scheme Ja 7 (i) k k ) ( ) ( det P k k k k P Whe, P k M A M M A ft B ag Evaluatig at least three cofactors Fully correct method for iverse Ft from wrog determiat Correctly obtaied (ii) M M M Eigevalues are,, M AAA For oe evaluatio OR M Eigevalues are,, A A Obtaiig a eigevalue (e.g. by solvig λ λ λ + ) Give A for oe correct Verifyig give eigevectors, likig with eigevalues correctly (iii) M BB MA B ft M A A ag 8 For ad see (for B, these must be cosistet) For SD S (MA if order wrog) or + + Evaluatig product of matrices Ay correct form

16 7 Mark Scheme Ja 7 OR Prove M A+ B by iductio Whe, A+ B M B k k Assumig M A+ B, k k M + AM+ BM MA A k ( B ) AA + A+ k B A True for k True for k + ; hece true for all positive itegers A or k+ k M MA+ MB Depedet o previous 7 marks

17 7 Mark Scheme Ja 7 (i) If arcosh, cosh y y y (e + e y ) y e e + e y ± y M M M ad + must be correct ± y y Sice y, e, so e + A (ii) (iii) OR arcosh y l( +.9. l(. +. (l l ) l ) l( ) d arcosh 9 ( arcosh. arcosh ) l( + 9 ) ) M AA l l sih ( + sih ) l 9 A dy ( + sih )sih (cosh )(cosh ) d ( + sih ) dy d 9 whe 8sih 9 ( + sih ) sih sih + sih, Whe sih, cosh, l( + ) Whe Poit is l( + ), sih, cosh 7, l( + Poit is l( + 7), 7 7) A ag M AA M A M A M M M A ag AA 8 For arcosh (or ay cosh substitutio) For ad (or cosh u ad du ) (or limits of u i logarithmic form) For l( k + k...) Give M for l( k+ k...) For ad l( + 9) Usig quotiet rule Ay correct form 9 (or l( + ) Quadratic i sih (or product of two quadratics i e ) Solvig quadratic to obtai at least oe value of sih (or e ) Obtaiig i logarithmic form (must use a correct formula for arsih) SR BB for verifyig y ad dy whe l( ) d 9 +

18 7 Mark Scheme Ja 7 Alteratives for Q (i) l( + ) l( + ) cosh l( + ) (e + e ) Sice ( + + ) + ( + + ) l( + ) >, arcosh l( + ) M M M A A If y arcosh the l( + ) l(cosh y+ cosh y) l(cosh y+ sih y) sice sih y > y l(e ) y M M A M A 7

19 7 Mark Scheme Ja 7 (i) k B B Geeral shape correct Cusp at O clearly show k. B B Geeral shape correct Dimple correctly show k B (ii) Cusp B (iii) Whe k, there are poits Whe k., there are poits Whe k, there are poits B (iv) (v) (vi) k cos θ + cos θ d k siθ cosθ siθ dθ siθ ( k + cosθ ) whe θ,, or cosθ For just two poits, k d r + ( k + cosθ ) r cosθ k + cos θ ( k + si θ ) Sice cos θ, k d k + + ( k + cosθ ) cosθ Whe k is large, k + k, so d k Curve is very early a circle, with cetre (, ) ad radius k k B B M A M A M A ag M A Give B for two cases correct Allow k > or si θ 8

20 7 Mark Scheme Jue 7 (a)(i) B Must iclude a sharp poit at O ad have ifiite gradiet at θ Give B for r icreasig from zero for < θ <, or decreasig to zero for < θ < (ii) Area is r dθ a ( cosθ ) d θ a a (b) Put siθ ( ) cos θ + ( + cos θ) dθ a si si ) θ θ + θ ( ) Itegral is taθ ( si θ ) cosθ dθ 8cos θ (cosθ ) dθ sec θ dθ M A B BB ft B M A M A ag For itegral of ( cosθ) For a correct itegral epressio icludig limits (may be implied by later work) Usig cos θ ( + cos θ ) Itegratig Accept.78a or cosθ Limits ot required For a + b cosθ ad k cos θ sec θ dθ taθ SR If tahu is used M for sih( l ) A for ( ) (ma / ) 8 (c)(i) f ( ) B Give B for ay o-zero real multiple of this (or for etc) si y (ii) f ( ) ( ( + ) + f ( ) C f () C +...) f ( ) M A M A Biomial epasio ( terms, ) Epasio of ( ) correct (accept usimplified form) Itegratig series for f ( ) Must obtai a o-zero C ot required term 7

21 7 Mark Scheme Jue 7 OR by repeated differetiatio () Fidig f ( ) Evaluatig () f () ( 88) f ( ) +... M M A ft f ( ) +... A Must obtai a o-zero value ft from (c)(i) whe B give 8

22 7 Mark Scheme Jue 7 (a) (cosθ + jsiθ ) c + jc s c s jc s + cs + js Equatig imagiary parts si θ c ( s s c s + s ) s ( s ) s + s M M A M s s siθ si + s s + s θ + si θ + s A ag (b)(i) + j 8, arg( + j) r θ θ, BB B ft B ft M A Accept.8;., (Implies B for 8 ) Oe correct (Implies B for Addig or subtractig ) Accept θ + k, k,, (ii) B Give B for two of B, C, M i the correct quadrats Give B ft for all four poits i the correct quadrats (iii) w arg w ( + ) 7 B ft B Accept.7 Accept.8 (iv) w ( ) 7 8 arg( w ) w 8 j 7 ( cos + jsi ) M A ft A Obtaiig either modulus or argumet Both correct (ft) Allow from arg w etc SR If B, C iterchaged o diagram (ii) B (iii) B B for (iv) MAA 9

23 7 Mark Scheme Jue 7 (i) det( M λ I) ( λ)[( λ)( λ) ] [( λ) + ] + [ ( λ)] ( λ )( + λ + λ ) ( λ) + ( + λ) 8 + 9λ + λ λ λ + λ 8λ + λ λ Characteristic equatio is λ λ 8λ (ii) λ ( λ 8)( λ + ) (iii) (iv) Other eigevalues are 8, Whe λ 8, + y + z 8 ( + y z 8y y z 8z y ad z ; eigevector is Whe λ, + y + z + y z y P M y, z ; eigevector is D 8 M 8M M M + 8M M M (M M + 8M + 8M) + 8M + 9M ) M A M A ag M A M M A M M A B ft M A M M A 8 Obtaiig det( M λ I) Ay correct form Simplificatio Solvig to obtai a o-zero value Two idepedet equatios Obtaiig a o-zero eigevector ( + y+ z 8 etc ca ear MM ) Two idepedet equatios Obtaiig a o-zero eigevector B if P is clearly sigular Order must be cosistet with P whe B has bee eared

24 7 Mark Scheme Jue 7 (a) d arsih 9 + arsih 9 l( + + ) l M A A M A For arsih or for ay sih substitutio For or for sih u For or for du OR M For l( k + k +... ) [ Give M for l( a + b +... ) ] l( ) AA or l( + + ) 9 l 8 l l A (b)(i) sih cosh ( e ( e sih e e ) ) ( e + e ) M A (e e )(e For completio + e ) (e e ) (ii) dy sih sih d For statioary poits, sih sih cosh, y 7 ( ± ) l( sih ( cosh ) sih or cosh + 9 ) l 9 y l, y 9 BB M A A ag A ag B 7 Whe epoetial form used, give B for ay terms correctly differetiated d y Solvig to obtai a value of d sih, cosh or e ( or stated) Correctly obtaied Correctly obtaied The last AA ag ca be replaced by BB ag for a full verificatio (iii) sih sih 8 l l 9 9 BB M A ag Whe epoetial form used, give B for ay terms correctly itegrated Eact evaluatio of sih(l ) ad sih( l )

25 7 Mark Scheme Jue 7 (i) k B B Maimum o LH brach ad miimum o RH brach Crossig aes correctly B B Two braches with positive gradiet Crossig aes correctly k k. B B Maimum o LH brach ad miimum o RH brach Crossig positive y-ais ad miimum i first quadrat (ii) ( + k)( k) + k + k y + k k( k + ) k + + k Straight lie whe k( k + ) k, k M A (ag) BB Workig i either directio For completio (iii)(a) Hyperbola B (B) k y k B B

26 7 Mark Scheme Jue 7 (iv) B B B B B Asymptotes correctly draw Curve approachig asymptotes correctly (both braches) Itercept o y-ais, ad ot crossig the -ais Poits A ad B marked, with miimum poit betwee them Poits A ad B at the same height ( y )

27 7 Mark Scheme Jauary 8 7 (FP) Further Methods for Advaced Mathematics (a) (b)(i) (ii) Area is a a ( cos θ ) d a a f ( ) + ( f ( ) f () θ ( cos θ + ( + cos θ )) [ θ si θ + si θ ] ( + ) + ) ( + ( + ) ) 8 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 has previously bee 7 lost ] d Applyig arcta u du + u dy or d sec y Applyig chai (or quotiet) rule Stated; or appearig i series Accept. f (), f () 8 arcta( + ) M AA ft Evaluatig f () or f () For ad ft provided coefficiets are o-zero (iii) h ( h ( + h h + + h ( h +...) d h h ) +... h h h ) M A ft A ag Itegratig (award if is missed) for Allow ft from a + + c provided that a Codoe a proof which eglects h

28 7 Mark Scheme Jauary 8 (a) j jθ th roots of j e are r e where r θ 8 k θ + 8 θ 7,, B B M A Accept Implied by at least two correct (ft) further values or statig k,, (), M A Poits at vertices of a square cetre O or correct poits (ft) or poit i each quadrat (b)(i) jθ jθ jθ jθ ( e )( e ) e e + (ii) jθ (e + e cosθ jθ OR ( cosθ jsiθ )( cosθ + jsiθ ) M ( cosθ ) + si θ A cosθ + (cos θ + si θ ) cosθ A C + js e e e jθ jθ jθ e ( ( e jθ + e jθ jθ ( ( e ) ) e cosθ C siθ S jθ jθ + 8 e jθ jθ e )( e jθ )( e ) + ( + ) jθ jθ e + cosθ + + cos( + ) θ + cosθ si( + ) θ + cosθ + ) + e + e ) jθ jθ si θ cos θ M A A ag M M A M A M A ag A 9 jθ jθ 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

29 7 Mark Scheme Jauary 8 (i) Characteristic equatio is (7 λ)( λ) + λ λ + λ, Whe λ, 7 y y 7 + y y y y, eigevector is Whe λ, 7 y y M AA M M A or y ca be awarded for either eigevalue Equatio relatig ad y or ay (o-zero) multiple 7 + y y y y, eigevector is M A 8 SR ( M λi) λ ca ear MAAMMAMA (ii) P D B ft B ft B if P is sigular For B, the order must be cosistet

30 7 Mark Scheme Jauary 8 (iii) P P D M + + P P P P D M a + b + c d M M A ft B ft M A ag A 8 May be implied Depedet o MM For P or Obtaiig at least oe elemet i a product of three matrices Give A for oe of b, c, d correct SR If P D P M is used, ma marks are MMABMAA (d should be correct) SR If their P is sigular, ma marks are MMABM

31 7 Mark Scheme Jauary 8 (i) (e + e ) k e k e + e k ± k k ± k M M or cosh + sih 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 cosh is a eve fuctio (or equivalet) ± l( k + k ) A ag (ii) M For arcosh or d arcosh A A l( λ + λ...) or ay cosh substitutio For arcosh or coshu or l( + ) or l( + ) For or du ( arcosh (iii) sih sih cosh cosh (or sih ) ± l( + 8) arcosh ) ( l( + ) l( + ) ) OR e e + e (e )(e e + ) M B l( ± 8) A M A M M B A Eact umerical logarithmic form Obtaiig a value for or l( ± 8) cosh or ( e e )(e + e ) (iv) dy cosh cosh d If d y the cosh ( cosh ) d cosh cosh + Discrimiat D Sice D < there are o solutios B M M A Usig cosh cosh Cosiderig D, or completig square, or cosiderig turig poit

32 7 Mark Scheme Jauary 8 OR Gradiet g cosh cosh B g sih sih sih ( cosh ) whe (oly) M g cosh 8 cosh whe M Ma value g whe So g is ever equal to A Fial A requires a complete proof showig this is the oly turig poit

33 7 Mark Scheme Jauary 8 (i) λ λ λ BBB cusp loop BB (ii) B (iii) Itersects itself whe y (iv) ± t ( ± ) λ λ, + λ λ 7 d y t λ dt λ t ± λ λ + λ + λ λ λ y ± ( ) ( ) λ ± λ ± λ (v) From asymptote, a 8 aλ From itersectio poit, + λ λ From maimum poit, b λ 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

34 7 Mark Scheme Jue 8 7 (FP) Further Methods for Advaced Mathematics (a)(i) r cosθ, y r siθ M (M for cos θ, y siθ ) ( r cos θ + r si θ ) r ( r cosθ )( r siθ ) r cosθ si θ A r cosθ si θ A ag (ii) B Loop i st quadrat B Loop i th quadrat (b) OR Put d arcsi arcsi M siθ A d dθ A MA (c)(i) l( + ) (ii) l( ) + l l( + ) l( ) B M AA M A B B M A Fully correct curve Curve may be draw usig cotiuous or broke lies i ay combiatio For arcsi For ad Eact umerical value Depedet o first M (MA for / ) Ay sie substitutio For dθ M depedet o first M Accept usimplified forms Obtaied from two correct series Terms eed ot be added If M, the B for + +

35 7 Mark Scheme Jue 8 (iii) r (r + ) r l (i) z 8, arg z (ii) z * 8, arg z* z w 8 8 arg( z w) + z w arg( 8 8 z w ) 7 7 ( z cos( ) + jsi( ) w j a b, (iii) r 8 θ k θ + 7, (iv) θ 7 j 7 j ) l w * 8 e, so e k z * 8 e j 8 e j k So e z * j w 8 e ( 7 + ) j j j 8 e j w* j ( 8 e, so e j w k ) +... B B B ag BB B ft B ft B ft B ft B ft M A B ft B M A B ft M A ft M A ft 7 Terms eed ot be added For see or implied Satisfactory completio Must be give separately Remaider may be give i epoetial or r cjsθ form (B for 7 ) (B if left as 8/8) If M, the BB for ad Accept 8 Implied by oe further correct (ft) value Igore values outside the required rage Matchig w* to a cube root with 7 argumet ad k or ft r ft is 8 Matchig z* to a cube root with argumet May be implied r ft is z * Matchig jw to a cube root with argumet May be implied OR M for arg( j w) + arg w (implied by or ) r ft is 8

36 7 Mark Scheme Jue 8 (i) k + Q k k k Whe k, Q 8 (ii) P, D M PDP (iii) Characteristic equatio is ( λ )( λ+ )( λ ) λ λ λ+ M M + MI M M + M M (M + M I) + M M M 9I a, b, c 9 M A M A M A BB B M A B M A M A Evaluatio of determiat (must ivolve k) For ( k ) Fidig at least four cofactors (icludig oe ivolvig k) Si siged cofactors correct (icludig oe ivolvig k) Trasposig ad dividig by det Depedet o previous MM Q correct (i terms of k) ad result for k stated After, SC for Q whe k obtaied correctly with some workig For B, order must be cosistet Give B for M P DP or 8 Good attempt at multiplyig two matrices (o more tha errors), leavig third matri i correct positio 7 Give A for five elemets correct Correct M implies BMA -8 elemets correct implies BMA I ay correct form (Codoe omissio of ) M satisfies the characteristic equatio Correct epaded form (Codoe omissio of I ) 7

37 7 Mark Scheme Jue 8 (i) cosh [ (e + e ) ] (e + + e ) + + sih [ (e e )] (e e ) cosh sih ( ) OR cosh + sih (e + e ) + (e e ) e B cosh sih (e + e ) (e e ) e B B B B ag For completio cosh sih e e B Completio (iii) (ii) ( + sih ) + 9sih sih + 9sih 9 sih, l, l( + ) OR e + 9e e 9e + (e e )(e + e ) M M e, + AA l, l( + ) AA ft dy 8cosh sih+ 9cosh d cosh (8sih + 9) oly whe sih cosh ( ) 9 7 y + 9 ( ) M M AA AA ft B B M A (M for sih ) Obtaiig a value for sih Eact logarithmic form Dep o MM Ma A if ay etra values give Quadratic ad / or liear factors Obtaiig a value for e Igore etra values Depedet o MM Ma A if ay etra values give Just l ears MMAAAA Ay correct form 9 7 or y (sih + ) +... ( ) Correctly showig there is oly oe solutio Eact evaluatio of y or cosh or cosh Give B (replacig MA) for. or better (iv) l ( + cosh + 9sih )d [ sih 9cosh ] l l l+ 8 M A M A ag Epressig i itegrable form Give A for two terms correct sih(l ) ( ) Must see both terms for M Must also see cosh(l ) ( + ) for A 8

38 7 Mark Scheme Jue 8 OR l 9 (e + + e + (e e ))d l 9 9 e e e e M A l M l+ A ag 8 (i) λ. λ λ Epaded epoetial form (M if the is omitted) Give A for three terms correct e l l ad e both see Must also see e ad e for A l l BBB (ii) Ellipse B (iii) y cos( θ ) Maimum y whe θ M A ag or si( θ + ) dy OR siθ + cosθ whe θ M dθ y + A (iv) + cos cos si + si y λ θ θ θ θ λ + cos θ + cosθsiθ + si θ ( λ + )( si θ) + ( + )si θ λ + λ + ( λ )si θ λ θ + y + λ θ + y + λ Whe si, Whe si, Sice si θ, distace from O, + y, is betwee + ad + λ λ M M A ag M M A ag Usig cos θ si θ (v) Whe λ, + y Curve is a circle (cetre O) with radius M A 9

39 7 Mark Scheme Jue 8 (vi) B A, E at maimum distace from O C, G at miimum distace from O B, F are statioary poits D, H are o the -ais Give ½ mark for each poit, the roud dow Special properties must be clear from diagram, or stated Ma if curve is ot the correct shape

40 7 Mark Scheme Jauary 9 7 (FP) Further Methods for Advaced Mathematics (a)(i) f () cos f () M Derivatives cos, si, cos, si, cos f () si f () f () cos f () A Correct sigs f () si f () f () cos f () A Correct values. Dep o previous A A (ag) www cos +... (ii) cos sec E o.e. + ( + a + b ) M Multiply to obtai terms i ad Terms correct i ay form + a + b a+ A (may ot be collected) a, b a+ Correctly obtaied by ay method: a B must ot just be stated b B Correctly obtaied by ay method (b)(i) (ii)(a) (ii)(b) y arcta a a ta y M d dy a sec y A Or sec y (a) ta y ad attempt to differetiate both sides dy d a d dy a( + ta y) A Use sec y + ta y o.e. dy a A (ag) www d a + d arcta + M A A SC: Use derivative of arcta ad Chai Rule (properly show) arcta aloe, or ay ta substitutio ad, or dθ without limits Evaluated i terms of d d + M arcta aloe, or ay ta substitutio + A ad, or dθ without limits ( ) arcta A Evaluated i terms of 9

41 7 Mark Scheme Jauary 9 (i) Modulus B Must be separate B Accept,. Argumet c (ii) B G: A i first quadrat, argumet - j A B G,, a e B arg b ± M j b e, e 7 j j j (iii) ( ) Aft B i secod quadrat, same mod B i fourth quadrat, same mod Symmetry G: poits ad at least of above, or B, B o aes, or BOB straight lie, or BOB refle Must be i required form (accept r, θ /) Rotate by addig (or subtractig) / to (or from) argumet. Must be / Both. Ft value of r for a. Must be i required form, but do t pealise twice z e e M ( ) 8 or see 8 A (ag) www Others are ad θ j re θ where r M Add to argumet more tha oce,,, Correct costat r ad five values of θ., A Accept θ i [, ] or i degrees (iv) - G G poits o vertices of regular heago Correctly positioed ( roots o real ais). Igore scales SC if G ad poits correctly plotted j j j j (iv) w ze e e e M arg w cos + j si + j A Or B G Same modulus as z (v) w j j j j e 8e M Or z e 8 e 8j A cao. Evaluated 8

42 7 Mark Scheme Jauary 9 (a)(i) Regio for (ii) (ii) Area rdθ a ta θ dθ G G G r icreasig with θ Correct for θ / (igore etra) Gradiet less tha at O M Itegral epressio ivolvig ta θ Attempt to epress ta θ i terms of sec a θ d θ M sec θ ta a θ θ A ta θ θ ad limits, a A A if e.g. triagle this aswer G Mark regio o graph [ ] (b)(i) Characteristic equatio is (. λ)(.7 λ). M λ.9λ. λ,. A.8.8 Whe λ,.. y (M λi) M below.8 +.8y,..y M At least oe equatio relatig ad y y, eigevector is o.e. A..8 Whe λ.,..8 y. +.8y M At least oe equatio relatig ad y 8 eigevector is o.e. A 8 (ii) Q B if Q is sigular. Must label Bft correctly D Bft. If order cosistet. Dep o BB B eared 7

43 7 Mark Scheme Jauary 9 (a)(i) cosh ( ) e e + ( e + + e ) sih ( ) e e ( e + e Both epressios (M if o middle ) M term) ad subtractio cosh sih ( + ) A (ag) www OR cosh + sih e cosh sih e Both, ad multiplicatio cosh sih e e A Completio (ii)(a) cosh + sih + ta y Use of cosh + sih ad sih M ta y sec y A tah sih ta y si y cosh sec y (ii)(b) arsih l( + arsih(ta y) l(ta y + l(ta y + sec y) A (ag) www + ) M Attempt to use l form of arsih + ta y ) A A (ag) www e e OR sih ta y ta y e e ta y M Arrage as quadratic ad solve for e e ta y ± ta y + A o.e. l(ta y + sec y) A www (b)(i) y artah tah y M tah y ad attempt to differetiate d dy sech y dy d sech y tah y Or sech y dy d A Or B for www (ii) (iii) Itegral [ ] artah artah A B ( )( ) A( + ) + B( ) A ½, B ½ d + d + M A (ag) M A artah or ay tah substitutio www M Log itegrals + l + l + + cor l + c o.e. A d l + l + l M Correct form of partial fractios ad attempt to evaluate costats www. Codoe omitted modulus sigs ad costat After scored, SC for correct aswer Substitutio of ½ ad ½ see aywhere (or correct use of, ½) artah l artah l A (ag) www 8 7

44 7 Mark Scheme Jauary 9 (i) y - - G G G Symmetry i horizotal ais (, ) to (, ) (, ) to (, ) (ii)(a) a >. B a <. B (ii)(b) Circle: r is costat B Shape ad reaso (ii)(c) The two loops get closer together B The shape becomes more early circular B (ii)(d) Cusp B a. B 7 (iii) + a cos θ cos θ a B Equatio If a >., < < ad there are two values a of θ i [, ], arccos a ad + arccos a These differ by arccos a arccos a arcta a Tagets are y ad y a M A (ag) M A A Relatig arccos to arcta by triagle or ta θ sec θ a Aft Negative of above a is real for a >. if a > E 8 8 8

45 7 Mark Scheme Jue 9 7 (FP) Further Methods for Advaced Mathematics (a)(i) l( + ) Series for l( ) as far as l( )... B s.o.i. + l l( + ) l( ) M Seeig series subtracted A Valid for < < B Iequalities must be strict (ii) + + ( ) + M Correct method of solutio (b)(i) l A M B for ½ stated Substitutig their ito their series i (a) (i), eve if outside rage of validity. Series must have at least two terms SR: if > correct terms see i (i), allow a better aswer to d.p..9 ( d.p.) A Must be decimal places y... G G G r() a, r(/) a/ idicated Symmetry i θ / Correct basic shape: flat at θ /, ot vertical or horizotal at eds, o dimple Igore beyod θ (ii) r + y r + r si θ M Usig y r si θ a r( + si θ) ( + si θ) + si θ a A (ag) r a y + y (a y) M A + y a ay + y ay a y a a A Usig r + y i r + y a Usimplified A correct fial aswer, ot spoiled

46 7 Mark Scheme Jue 9 (i) λ M λi λ λ det(m λi) ( λ)[( λ)( λ)] + [( λ)] M ( λ)(λ ) + ( λ) A λ λ + λ + 7 det M 7 Attempt at det(m λi) with all elemets preset. Allow sig errors Usimplified. Allow sigs reversed. Codoe omissio of B (ii) f (λ) λ λ + λ + 7 f ( ) + 7 eigevalue B Showig satisfies a correct characteristic equatio f (λ) (λ + )(λ λ + 7) M Obtaiig quadratic factor λ λ + 7 (λ ) + so o real roots A www (M λi)s, λ (M λi)s (λ)s M below y z + y z Obtaiig equatios relatig, y M + z ad z z y z z + z z M Obtaiig equatios relatig two variables to a third. Dep. o first M s A Or ay o-zero multiple. M Solutio by ay method, e.g. use y. of multiple of s, but M if s itself z. quoted without further work., y., z. A Give A if ay two correct 9 C-H: a matri satisfies its ow characteristic (iii) equatio B Idea of λ M M M + M + 7I M M M 7I B (ag) Must be derived www. Codoe omitted I M M I 7M M Multiplyig by M M M + M I A o.e. 8 (iv) M 8 M Correct attempt to fid M M Usig their (iii) or A SC for aswer without workig

47 7 Mark Scheme Jue 9 OR Matri of cofactors: 7 M Adjugate matri 7 : det M 7 M Fidig at least four cofactors Trasposig ad dividig by determiat. Dep. o M above 9

48 7 Mark Scheme Jue 9 (a)(i) (ii) G Correct basic shape (positive gradiet, through (, )) y arcsi si y si y ad attempt to diff. both M sides d dy cos y A Or cos y dy d dy d cos y Positive square root because gradiet positive d arcsi (b) C + js jθ jθ jθ e + e + e This is a geometric series with first term a j e θ, commo ratio r jθ a e Sum to ifiity r e e e e e jθ jθ jθ jθ 9e e 9e e + jθ jθ jθ jθ 9cos ( θ + jsiθ) cos ( θ jsiθ) ( cos θ jsi θ) ( cos θ + jsi θ) cosθ + j siθ cos θ cosθ C cos θ jθ A B M A A www. SC if quoted without workig Dep. o graph of a icreasig fuctio arcsi fuctio aloe, or ay sie substitutio, or dθ www without limits Evaluated i terms of M Formig C + js as a series of powers Idetifyig geometric series ad M attemptig sum to ifiity or to terms j e θ A Correct a ad r jθ e ( ) jθ e A Sum to ifiity Multiplyig umerator ad jθ deomiator by e o.e. M* Or writig i terms of trig fuctios ad realisig the deomiator M M A M Multiplyig out umerator ad deomiator. Dep. o M* Valid attempt to epress i terms of trig fuctios. If trig fuctios used from start, M for usig the compoud agle formulae ad Pythagoras Dep. o M* Equatig real ad imagiary parts. Dep. o M* 7

49 7 Mark Scheme Jue 9 cosθ cosθ siθ S cosθ A (ag) A o.e. 9 8

50 7 Mark Scheme Jue 9 u u e + e (i) cosh u cosh u u e + + e u u u B ( ) u u e e + e + + e cosh u u e e u u u + e + e B cosh u cosh u B (ag) Completio www (ii) arsih y sih y e e y Epressig y i epoetial form M (, must be correct) e ye (e y) y (e y) y + e y ± y + e y ± y + Reachig e by quadratic formula M or completig the square. Codoe o ± Take + because e > Or argumet of l must be B positive l(y + y + ) Completio www but A (ag) idepedet of B (iii) sih u d d du cosh u M ad substitutig for all du elemets + sih + cosh A Substitutig for all elemets correctly cosh udu coshu+ du M Simplifyig to a itegrable form sih u + u + c A u u Ay form, e.g. e e + u Codoe omissio of + c throughout sih u cosh u + u + c Usig double agle formula ad + + arsih + c M attempt to epress cosh u i terms of + c A (ag) Completio www (iv) t + t + (t + ) + B Completig the square t + t+ dt ( t+ ) + dt + d + + arsih M A Simplifyig to a itegrable form, by substitutig t + s.o.i. or complete alterative method Correct limits cosistet with their method see aywhere 9

51 7 Mark Scheme Jue arsih M + l( + ) (l( + ) + ) A (ag) Usig (iii) or otherwise reachig the result of itegratio, ad usig limits Completio www. Codoe 8 etc. 8

52 7 Mark Scheme Jue 9 (i) If a, agle OCP so P is ( cos, si ) M P (, ) A (ag) Completio www OR Circle ( ) + y, lie y + ( ) + ( + ) M Complete algebraic method to fid ± ad hece P A Q ( +, ) B (ii) cos OCP a Attempt to fid cos OCP ad si M a + OCP i terms of a si OCP a + A Both correct P is (a a cos OCP, a si OCP) P (a a, a ) a + a + A (ag) Completio www OR Circle ( a) + y a, lie y + a ( a) + + a Complete algebraic method to a M fid (iii) a ± Q (a + a a+ ± + + a a a + a a a + a a +, ad hece P. a +. A A a ) B G G G Gft Usimplified Locus of P ( st & rd quadrats) through (, ) Locus of P termiates at (, ) Locus of P: fully correct shape Locus of Q ( d & th quadrats: dotted) reflectio of locus of P i y-ais As a, P (, ) B Stated separately As a, y co-ordiate of P B Stated a a a + a as a M Attempt to cosider y as a A Completio www 8 (iv) POQ 9 B Agle i semicircle B o.e. Loci cross at 9 B 8

53 7 Mark Scheme Jauary 7 (FP) Further Methods for Advaced Mathematics (a) y arcta u, y arcta u du d dy du u dy d + u + + ( ) OR ta y sec y dy d sec y + ta y + dy + d ( ) MA A M A A Usig Chai Rule Correct derivative i ay form Correct derivative i terms of Rearragig for or ad differetiatig implicitly M Itegral i form k arcta arcta ( ) d + A k arcta arcta A (ag) (b)(i) r cos θ, y r si θ, + y r M Usig at least oe of these + y y + r r cos θ si θ + A LHS A RHS r ½r si θ + r r si θ + r ( si θ) r siθ A (ag) Clearly obtaied SR: r si θ, y r cos θ used MAAA ma. (ii) Ma r is B Occurs whe si θ M Attemptig to solve Both. Accept degrees. θ, A A if etras i rage Mi r B Occurs whe si θ M Attemptig to solve (must be ) θ 7 Both. Accept degrees., A A if etras i rage

54 7 Mark Scheme Jauary (iii) y... G Closed curve, roughly elliptical, with. o poits or dets G Major ais alog y 8 (a) cos θ + j si θ (cos θ + j si θ) M Usig de Moivre cos θ + cos θ j si θ + cos θ j si θ + cos θ j si θ + cos θ j si θ + j si θ M Usig biomial theorem appropriately cos θ cos θ si θ + cos θ si Correct real part. Must evaluate θ + j( ) A powers of j cos θ cos θ cos θ si θ + cos θ si θ M Equatig real parts cos θ cos θ(cos θ) + cos θ(cos θ) M Replacig si θ by cos θ cos θ cos θ + cos θ A a, b, c (b) C + js M Formig series C + js as epoetials (c) ( ) jθ+ j θ+ jθ e + e e A Need ot see whole series This is a G.P. M Attemptig to sum fiite or ifiite G.P. jθ a e, r e j Correct a, r used or stated, ad terms A Must see j Sum e jθ e e e j j jθ j Numerator ( e ) ad j e so sum E Covicig eplaatio that sum C ad S E C S. Dep. o previous E Both E marks dep. o marks above 7 t e + t+ t B Igore terms i higher powers t t M Substitutig Maclauri series t e t+ t A t ( ) + t t+... Suitable maipulatio ad use of M t+ t + t biomial theorem OR t + + t t t t t Hece t t e t OR ( e )( t) ( t t...)( t) t M A A (ag) + + M Substitutig Maclauri series A Correct epressio t + terms i t M Multiplyig out t t t e A Covicig eplaatio 8

55 7 Mark Scheme Jauary M Evaluatig determiat (i) a + a A a M a a M Fidig at least four cofactors a A Si siged cofactors correct M Trasposig ad dividig by det Whe a, (ii) y b z M A M M correct (i terms of a) ad result for a stated SR: After scored, SC for M whe a, obtaied correctly with some workig Attemptig to multiply ( b ) T by give matri (M if wrog order) M Multiplyig out 8, y b, z b A A for oe correct OR + y b y b o.e. M Elimiatig oe ukow i ways Or e.g. + z b, Or e.g. y z b, y z 7 M Solve to obtai oe value. Dep. o M above Oe ukow correct A After M, SC for value of M Fidig the other two ukows 8 y b, z b A Both correct (iii) e.g. y b + M Elimiatig oe ukow i ways y AA Two correct equatios Or e.g. + z b, + z Or e.g. y + z b, y + z 7 Cosistet if b + M Attemptig to fid b b Solutio is a lie A B 7 8

56 7 Mark Scheme Jauary (i) e e sih e + e sih + ( sih e e ) e + e + B e + e e + e cosh B Correct completio sih sih cosh B Both correct derivatives sih sih cosh B Correct completio (ii) cosh + sih ( + sih ) + sih M Usig idetity sih + sih A Correct quadratic ( sih )(sih + ) M Solvig quadratic sih ¼, A Both arsih(¼) l( + 7 ) A Must evaluate arsih() l( + ) A OR e e e e + + ( e e )( e e ) M Use of arsih l( + + ) o.e. Must obtai at least oe value of + MA Factorisig quartic e + ± 7 or ± MA Solvig either quadratic l( + 7 ) or l( + ) MAA Usig l (depedet o first M) 7 (iii) cosh t t t e + e t t e e + M Formig quadratic i e t t t ( e )( e ) M Solvig quadratic t e, t ±l A d arcosh B A (ag) Covicig workig arcosh arcosh M Substitutig limits l A A for ±l OR d l ( + ) B l 8 l M Substitutig limits l A 7 8 7

57 7 Mark Scheme Jauary (i) Horz. projectio of QP k cos θ B Vert. projectio of QP k si θ B Subtract OQ ta θ B Clearly obtaied (ii) k k y y.... k ½ k G G Loop Cusp y y G G (iii)(a) for all k, y ais is a asymptote B Both (B) k B (C) k > B (iv) Crosses itself at (, ) k cos θ ½ θ M Obtaiig a value of θ curve crosses itself at A Accept (v) y 8 si θ ta θ dy dθ 8 cos θ sec θ 8 cos θ at highest poit cos θ cos θ 8 cos θ ± M Complete method givig θ θ at top A y A Both (vi) RHS ( ) k cosθ ( k k cos θ) k cos θ M Epressig oe side i terms of θ ( k cosθ ) k si θ k cos θ ( ) k cosθ ta θ M Usig trig idetities (( k cosθ taθ ) ) ( si θ ta θ) k LHS E 8 8

58 GCE Mathematics (MEI) Advaced GCE 7 Further Methods for Advaced Mathematics (FP) Mark Scheme for Jue Oford Cambridge ad RSA Eamiatios

59 7 Mark Scheme Jue (a)(i) f (t) arcsi t f (t) t f (t) ( ) t t ( ) t ( ) t B Ay form t M Usig Chai Rule (ii) f () arcsi ( + ½) f () arcsi (½) (b) f () ( ) ad f () ( ( ) ) f () f () + f () + 9 A (ag) B (ag) M A (ag) obtaied clearly from f () www Clear substitutio of or t ½ f () + M Evaluatig f () ad dividig by term i is 9 A Accept.8 or better (c) Area rdθ a a dθ ( + θ) a + θ a + a d 9+ G G Complete spiral with r() < r() r() a, r() a/ idicated or r() > r(/) > r() > r(/) > r() Dep. o G above Ma. G if ot fully correct dθ M Itegral epressio ivolvig r ( + θ) A M A d 9 + arcta M AA Correct result of itegratio with correct limits Substitutig limits ito a epressio of the form k. Dep. o M above + θ arcta ad arcta M Substitutig limits. Dep. o M above A Evaluated i terms of 9

60 7 Mark Scheme Jue (a) z + cosθ, z z z jsiθ B Both z z z z + z + M Epadig z z z z z z z z + z z z M Itroducig sies (ad possibly cosies) j si θ j si θ j si θ + j si θ of multiple agles A RHS si θ si θ si θ + si θ 8 Aft Divisio by (j) A 8, B, C (b)(i) th roots of 9j 9 j e j are re θ where r B Accept 9 θ B 8 θ k + M 8 θ 7,, A Implied by at least two correct (ft) further values Or statig k (),,, Allow argumets i rage θ w (ii) Mid-poit of SP has argumet 8 ad modulus of Argumet of w 8 M A B B Poits at vertices of a square cetre O or correct poits (ft) or poit i each quadrat ad modulus 9 M A G Multiplyig argumet by ad modulus raised to power of Both correct w plotted o imag. ais above level of P

61 7 Mark Scheme Jue (a)(i) λ + λ λ + (λ )(λ + λ ) B Substitutig λ or factorisig λ or λ + λ M Obtaiig ad solvig a quadratic (λ )(λ + ) λ ½, λ AA (ii) M B M v v Give B for oe compoet with the B wrog sig M Recogisig that the solutio is a M multiple of the give RHS, y, z A Correct multiple (iii) λ + λ λ + M + M M + I M Usig Cayley-Hamilto Theorem M M + M I M M + M M M Multiplyig by M M ( M + M I) + M M M Substitutig for M M 7 M M + I A, C A 7, B (b) N PDP B Order must be correct where D B ad P B For BB, order must be cosistet P Bft Ft their P N M Attemptig matri product A a c OR Let N b d a c a+ c B Or b d b d + a c a c B Or b d b d a + c, a + c B b + d, b + d B a, c ; b, d MA Solvig both pairs of equatios 9

62 7 Mark Scheme Jue (i) sih cosh e + e e (ii) e e e M Usig epoetial defiitios ad multiplyig or factorisig sih A (ag) Differetiatig, cosh cosh + sih B Oe side correct cosh cosh + sih B Correct completio y (iii) Volume ( cosh ) d M ( cosh ) G cosh cosh + d A cosh cosh + d M Correct shape ad through origi d A correct epaded itegral epressio icludig limits, (may be implied by later work) Attemptig to obtai a itegrable form Dep. o M above sih sih + A Give A for two terms correct sih sih 8.7 A d.p. required. Codoe y cosh + sih dy sih + cosh d B Ay correct form At S.P. sih + cosh sih cosh + cosh M Settig derivative equal to zero ad usig idetity cosh ( sih + ) M Solvig dy to obtai value of sih d cosh (rejected) A Repudiatig cosh sih A 7 M Usig log form of arsih, or settig up l + ad solvig quadratic i e A A if etra roots quoted 7 8

63 7 Mark Scheme Jue (i)(a) Circle (B) B G Sketch of circle, cetre (, ) G Sketch of squarer circle o same aes (C) Square B (D) B Give BB for ot all o-strict or y B uclear (ii)(a) Odd roots eist for all real umbers B Ay equivalet eplaatio (B) Lie B Sketch isufficiet y (C) (D) Asymptote: + y y G B (iii) G G Lie + y outside uit square Lies y ad o uit square (iv)(a) G G if curve beyod (, ) or (, ), y B Accept strict, or idicatio o graph (B) Limit is a plus sig where for y ad vice versa Gft B B Give G for a partial attempt. Ft from (iii) o shape 8

64 GCE Mathematics (MEI) Advaced GCE Uit 7: Further Methods for Advaced Mathematics Mark Scheme for Jauary Oford Cambridge ad RSA Eamiatios

65 7 Mark Scheme Jauary (a)(i) r cos θ, y r si θ, + y r M Usig at least oe of these r (cos θ + si θ) r r(cos θ + si θ) + y + y A (ag) Workig must be covicig + y y ( ) + (y ) which is a circle cetre (, ) radius M Recogise as circle or appropriate algebra leadig to ( a) + (y b) r (ii) Area (b)(i) G G M Attempt at complete circle with cetre i first quadrat A circle with cetre ad radius idicated, or cetre (, ) idicated ad passig through (, ), or (, ) ad (, ) idicated ad passig through (, ) cos si d Itegral epressio ivolvig r i terms of θ cos si cos si d M Multiplyig out si cosd A cos θ + si θ used cos si etc. A Correct result of itegratio with correct limits. Give A for oe error M Substitutig limits. Dep. o both Ms A Mark fial aswer 7 f( ) M Usig Chai Rule A Correct derivative i ay form f( )... M Correctly usig biomial epasio (ii)... 8 A Correct epasio f... c M Itegratig at least two terms A But c because arcta() A Idepedet 9

66 7 Mark Scheme Jauary (a)(i) z + z cos θ B z z j si θ B (ii) (z + z ) z + z + z + + z + z + z M Epadig (z + z ) z + z + (z + z ) + (z + z ) + cos θ cos θ + cos θ + cos θ + cos θ cos cos cos cos θ M Usig z + z cos θ with, or. Allow M if omitted, etc. cos cos cos A (ag) (iii) (z z ) z + z (z + z ) + (z + z ) B si θ cos θ cos θ + cos θ M Usig (i) as i part (ii) A Correct epressio i ay form si θ cos cos cos cos θ si θ cos cos M Attemptig to add or subtract A OR cos θ (cos θ + ) B This used cos θ cos θ + 8 cos θ + M Obtaiig a epressio for cos θ cos θ cos θ + cos θ A Correct epressio i ay form cos θ si θ cos θ cos θ + cos θ cos cos MA Attemptig to add or subtract j (b)(i) z 8e z e j Correctly maipulatig modulus ad M argumet j7 e A 8, 7 or. Codoe r(c + js) j z 9 8e z e j Correctly maipulatig modulus ad M argumet j 9 e A, or. Codoe r(c + js) 9 9 w z z (ii) z z e e j j7 e 7 9 j 8 j 9 e Lies i secod quadrat G G M A A Moduli approimately correct Argumets approimately correct Give GG for two poits approimately correct Correctly maipulatig modulus ad argumet Accept ay equivalet form 9

67 7 Mark Scheme Jauary (i) det(m λi) ( λ)[( λ)( λ) + 8] M Obtaiig det(m λi) + [( λ) ] + [8 + ( λ)] A Ay correct form ( λ)(λ λ + ) + (λ) + ( λ) λ + λ λ + 8λ + λ M Simplificatio λ λ + 8λ A (ag) www, but codoe omissio of (ii) λ λ + 8λ M Factorisig ad obtaiig a quadratic. If M, give B for substitutig λ (λ )(λ λ + ) A Correct quadratic λ λ + b ac 8 M Cosiderig discrimiat o.e. so o other real eigevalues A Coclusio from correct evidece www (iii) λ y z y + z z + y z M Two idepedet equatios z k, y k M Obtaiig a o-zero eigevector eigevector is A eigevector with uit legth is v B Magitude of M v is B Must be a magitude (iv) λ λ + 8λ M M + 8M I M Use of Cayley-Hamilto Theorem M M + 8I M M M Multiplyig by M (M M + 8I) ad rearragig A Must cotai I

68 7 Mark Scheme Jauary (i) sih t + 7 cosh t 8 (et e t ) + 7 (et + e t ) 8 M Substitutig correct epoetial forms e t + e t 8 e t 8e t + M Obtaiig quadratic i e t (e t )(e t ) M Solvig to obtai at least oe value of e t e t or AA Codoe etra values t l( ) or l( ) A These two values o.e. oly. Eact form (ii) dy d sih + cosh or 8e + e B sih + cosh sih + 7 cosh 8 l( ) or l( ) l( ) or l( ) M Complete method to obtai a value A Both co-ordiates i ay eact form l( ) y ( l( ), ) l( ) y ( l( ), ) B Both y co-ordiates dy sih + cosh d tah 7 or e etc. M Ay complete method No solutios because < tah < or e > etc. A (ag) www (,) G G Curve (ot st. lie) with correct geeral shape (positive gradiet throughout) Curve through (, ). Depedet o last G 8 a cosh 7sih d M Attemptig itegratio (iii) 7 a sih cosh 7 sih a cosh a 7 sih a + 7 cosh a 8 A Correct result of itegratio a l( ) or l( ) a l( ) or l( ) M Usig both limits ad a complete method to obtai a value of a a l( ) ( l( ) < ) A Must reject l( ), but reaso eed ot be give 8

69 7 Mark Scheme Jauary (i) a y.. a y G a. G y.. G M Evidece s.o.i. of further ivestigatio (A) Loops whe a > A (B) Cusps whe a A 7 (ii) If, t t M Cosiderig effect o t but y(t) y(t) A (ag) Effect o y Curve is symmetrical i the y-ais B (iii) dy d asi t M Usig Chai Rule acost A dy a si t t ad ± d A Values of t t T.P. is (, a) A t ± T.P. are (±, + a) A Both, i ay form (iv) a : both t ad give the poit (, ) B (ag) Verificatio Gradiets are a ad a (or ad ) Hece agle is arcta( ). radias M A Complete method for agle Accept (or ) 8

70 GCE Mathematics (MEI) Advaced GCE Uit 7: Further Methods for Advaced Mathematics Mark Scheme for Jue Oford Cambridge ad RSA Eamiatios

71 7 Mark Scheme Jue 7 (FP) Further Methods for Advaced Mathematics (a)(i) a d (ii) Area si G G Correct geeral shape icludig symmetry i vertical ais Correct form at O ad o etra sectios. Depedet o first G For a otherwise correct curve with a sharp poit at the bottom, award GG M Itegral epressio ivolvig ( si θ) a si si d M A Epadig Correct itegral epressio, icl. limits (which may be implied by later work) a si cos d si cos Correct result of itegratio. a cos si A Give A for oe error a A Depedet o previous A 7 (b)(i) M arcta aloe, or ay ta substitutio d d arcta A ad A Evaluated i terms of (ii) ta θ M Ay ta substitutio d sec θ dθ sec d sec cos d AA sec, sec Itegratig a cos bθ ad usig cosistet M limits. Depedet o M above si a Aft si b b A 8

72 7 Mark Scheme Jue (a) cos θ + j si θ (cos θ + j si θ) c + c js c s c js + cs + js M Epadig M Separatig real ad imagiary parts. Depedet o first M cos θ c c s + cs A Alterative: c c + c si θ c s c s + s A Alterative: s s + s cscs s ta θ c c s cs tt t M Usig ta θ si ad simplifyig t t cos t t t t t A (ag) (b)(i) arg( ) fifth roots have r B ad θ B No credit for argumets i degrees z j e, j e, e j, 7 j e, M Addig (or subtractig) 9 j e A All correct. Allow θ < (ii) (iii) arg(w) G G B w cos M Aft (iv) w cos e j w cos e j which is real if si B Poits at vertices of regular petago, with oe o egative real ais Poits correctly labelled Attemptig to fid legth F.t. (positive) r from (i) w cos a cos M A Attemptig the th power of his modulus i (iii), or attemptig the modulus of the th power here Accept.9 or better 8

73 7 Mark Scheme Jue (i) det(m) ( ) + ( 8) + k( ) M Obtaiig det(m) i terms of k k A o iverse if k A Accept k after correct determiat Evaluatig at least four cofactors M (icludig oe ivolvig k) k k Si siged cofactors correct A M k k (icludig oe ivolvig k) k 9 Trasposig ad dividig by det(m). M Depedet o previous MM A 7 M Settig k ad multiplyig (ii) A (iii) is a eigevector correspodig to a eigevalue of B B For credit here, / scored i (ii) Accept ivariat poit (iv) + y t, + y, + y t M Elimiatig oe variable i two differet ways (or 9 + 8z t +, + z t, + z ) (or 9y 9z t, y z t, y z ) A Two correct equatios For solutios, t M Validly obtaiig a value of t t A M λ, y λ, z λ A Straight lie B Obtaiig geeral solutio by settig oe ukow λ ad fidig other two i terms of λ (accept ukow istead of λ) Accept sheaf. Idepedet of all previous marks 7 8

74 7 Mark Scheme Jue y y (i) cosh y e e (ii) e y e y y e e y e y l( ) y B Usig correct epoetial defiitio M Obtaiig quadratic i e M M y ±l( ) A (ag) arcosh() l( ) because this is the pricipal value d M arcosh AA arcosh arcosh l d Solvig quadratic A B M 7 Validly attemptig to justify ± i prited aswer Referece to arcosh as a fuctio, or correctly to domais/rages arcosh aloe, or ay cosh substitutio, Substitutig limits ad usig (i) correctly at ay stage (or usig limits of u i logarithmic form). Dep. o first M y l OR l l 8 l M AA A l k k... Give M for l k k..., l o.e. l A (iii) cosh cosh cosh ( cosh ) M Attemptig to epress cosh i terms of cosh cosh cosh + ( cosh )(cosh ) M Solvig quadratic to obtai at least oe real value of cosh cosh (rejected) A Or factor cosh or cosh A l Aft F.t. cosh k, k > l F.t. other value. Ma. AA if additioal Aft real values quoted 8

75 7 Mark Scheme Jue (i) (A) m,. y.. (B) m,. G Negative parabola from (,) to (,), symmetrical about.. y (C) m,. G Bell-shape from (,) to (,), symmetrical about.; flat eds, ad obviously differet to (A). y.. (D) m,. G Skewed curve from (,) to (,), maimum to left of.. y.. (ii) Whe m, the curve is symmetrical Echagig m ad reflects the curve. (iii) If m >, the maimum is to the right of. B o.e. As m icreases relative to, the maimum poit moves further to the right B m dy m m m y d m m dy maimum at m d G Skewed curve from (,) to (,), maimum to right of. B B M A M m A Give BB if the idea is correct but vaguely epressed Usig product rule Ay correct form Settig derivative ad solvig to fid a value of other tha or