GCE. Mathematics (MEI) Mark Scheme for January Advanced GCE Unit 4756: Further Methods for Advanced Mathematics PMT

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1 GCE Mathematics (MEI) Advaced GCE Uit 476: Further Methods for Advaced Mathematics Mark Scheme for Jauar Oford Cambridge ad RSA Eamiatios

2 476 Mark Scheme Jauar (a) (i) a d ta a sec M Differetiatig with respect to or d d A For d d asec d d d a a d a d a (a) (ii) 48 4 B A(ag) Completio www with sufficiet detail [] d asec d Or a d d sec 4 4 M Itegral of form a arcta bu or a u d arcta 48 appropriate substitutio arcta A Correct itegral with cosistet limits arcta arcta 4 A Evaluated i terms of π [4] (a) (iii) arcta d M Usig parts with u = arcta ad v = Allow oe other error arcta d A M arcta l a l c A a. Codoe omitted c [4]

3 476 Mark Scheme Jauar (b) (i) r cos r rcos M Usig ad rcos r A A correct cartesia equatio i a form A(ag) Eplaiig that the curve is a circle e.g. writig as OR (b) (ii) (a) (i) rcos cos rsi cos si si M Usig rcos, rsi ad likig cos cos cos A i terms of cos r A(ag) Eplaiig that the curve is a circle e.g. writig as Cetre (, ) B Idepedet Radius B Idepedet [] 4 4 r r 4rsi M Usig r ad rsi For aswer aloe www: r 4si A B for r ksi, B for k = 4 [] j e cos jsi cos jsicos M cos jsicos j Usig e cos jsi ad double agle formulae cos cos jsi A(ag) Completio www M j j -j j OR e e e e cos jsi cos A(ag) Completio www j OR Factorisig ad complete replacemet b trigoometric fuctios e cos jsi cos si jsicos j Usig e cos jsi cos jsicos M ad si cos cos cos jsi A(ag) Completio www [] Allow oe error 6

4 476 Mark Scheme Jauar (a) (ii) (b) (i) C js e e... e j e cos cos jsi cos cos jsi j j4 j M Formig C + js M A M A Recogisig as biomial epasio Applig (i) ad De Moivre o.e. Depedet o MM above C cos cos A(ag) Completio www j Need to see e cos jsi o.e. ad S cos si A [7] j e cos jsi j B Must evaluate trigoometric fuctios [] 4j e j M Award for idea of rotatio b e.g. use of arcta (. rad) (must be ) 4j j (b) (ii) Other two vertices are j AA Ma be give as co-ordiates 4 j -j 4 ad 4je 4je M Award for idea of rotatio b e.g. use of arcta (.96 rad) (must be ) 4j j j If AAAA award SC for awrt AA Ma be give as co-ordiates j ad.46.7j [6] 7

5 476 Mark Scheme Jauar (b) (iii) Legth of ( + 4j) = (i) So legth of side = cos 6 M Complete method A(ag) Completio www [] MI det det M I Alterative: fidig distace betwee (, 4) ad, o.e. M I M Formig A A correct form Sarrus: 9 or e.g. A(ag) Codoe omissio of (ii) [] M Factorisig as far as quadratic Allow oe error 4 A eigevalues are,, 4 A : For a oe of λ =,, 4 M Obtaiig two idepedet equatios z M Obtaiig a o-zero eigevector From which a eigevector could be foud, z A o.e. Allow e.g.,z eigevector is A : z, z A o.e. 8

6 476 Mark Scheme Jauar, z z eigevector is A 4: z, z A o.e., z z eigevector is A [] (iii) E.g. P B Use of eigevectors (ft) as colums M Use of,, 4 (ft) i correct order D 4 A Power [] ot required for M 4 A 9

7 476 Mark Scheme Jauar 4 (i) sih cosh e e d cosh sih B e e d At TPs, d tah d M Cosiderig d d e ; e ad e which has o (real) solutios A(ag) Showig o real roots www tah M Solvig = as far as e or tah etc. e ; cosh ; sih l Attempt to verif M Solvig as far as Award M for substitutig l ad M for clearl attemptig to evaluate l A(ag) Completio www eactl d sih cosh d d sih l cosh l must so B(ag) d be eplaied, e.g. coected with = [7] 4 (ii) B [] For a curve with the followig features: icreasig itersectig the positive -ais (, ) idicated gradiet icreasig with large oe poit of iflectio Award B for a curve lackig oe of these features

8 476 Mark Scheme Jauar sih cosh 4 (iii) 9sih sih cosh 4cosh B 9 cosh 6sih cosh Usig double agle formulae or complete M alterative cosh 6sih A Accept usimplified l Attemptig to itegrate their V d M (igore limits) Codoe sig errors but eed s e e 4 4 sih cosh 4 l 4 4 l A Correct results ad limits c.a.o. Igore omitted π M Substitutig both of their limits M Obtaiig eact values of sihl ad cosh l Give A for oe error, or for all three terms correct ad icorrect limits sih l, coshl (i) OR 8 8 e e l l 4 4 l A Correct results ad limits M Substitutig both of their limits M Obtaiig eact values of A(ag) Completio www [9] e ad e Give A for oe error, or for all three terms correct ad icorrect limits e, e B B [] Three curves of correct shape Correctl idetified Give B for two correct curves a =, a =, a = from left to right

9 476 Mark Scheme Jauar (ii) B B [] (iii) Asmptote B [] (iv) a = : cusp B a = : loop B [] (v) r sec acos rcos acos a r a Curve for a = Curve for a = M Usig rcos M Usig r a a M Makig subject A(ag) Hece asmptote at = B [] (vi) Curve eists for a M Cosiderig If a > the > ad so a M i.e. < + a A(ag) If a < the < ad so a M i.e. + a < A [] Curve with cusp Curve with loop

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.

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. 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