Version 1.0: abc. General Certificate of Education. Mathematics MFP2 Further Pure 2. Mark Scheme examination - January series

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1 Versio.0: 008 abc Geeral Certificate of Educatio Mathematics 660 MFP Further Pure Mark Scheme 008 examiatio - Jauary series

2 Mark schemes are prepared by the Pricipal Examier ad cosidered, together with the relevat questios, by a pael of subject teachers. This mark scheme icludes ay amedmets made at the stadardisatio meetig atteded by all examiers ad is the scheme which was used by them i this examiatio. The stadardisatio meetig esures that the mark scheme covers the cadidates resposes to questios ad that every examier uderstads ad applies it i the same correct way. As preparatio for the stadardisatio meetig each examier aalyses a umber of cadidates scripts: alterative aswers ot already covered by the mark scheme are discussed at the meetig ad legislated for. If, after this meetig, examiers ecouter uusual aswers which have ot bee discussed at the meetig they are required to refer these to the Pricipal Examier. It must be stressed that a mark scheme is a workig documet, i may cases further developed ad expaded o the basis of cadidates reactios to a particular paper. Assumptios about future mark schemes o the basis of oe year s documet should be avoided; whilst the guidig priciples of assessmet remai costat, details will chage, depedig o the cotet of a particular examiatio paper. Further copies of this Mark Scheme are available to dowload from the AQA Website: Copyright 008 AQA ad its licesors. All rights reserved. COPYRIGHT AQA retais the copyright o all its publicatios. However, registered cetres for AQA are permitted to copy material from this booklet for their ow iteral use, with the followig importat exceptio: AQA caot give permissio to cetres to photocopy ay material that is ackowledged to a third party eve for iteral use withi the cetre. Set ad published by the Assessmet ad Qualificatios Alliace. The Assessmet ad Qualificatios Alliace (AQA) is a compay limited by guaratee registered i Eglad ad Wales (compay umber 6447) ad a registered charity (registered charity umber 074). Registered address: AQA, Devas Street, Machester M5 6EX Dr Michael Cresswell Director Geeral

3 MFP - AQA GCE Mark Scheme 008 Jauary series Key to mark scheme ad abbreviatios used i markig M m or dm A B E mark is for method mark is depedet o oe or more M marks ad is for method mark is depedet o M or m marks ad is for accuracy mark is idepedet of M or m marks ad is for method ad accuracy mark is for explaatio or ft or F follow through from previous icorrect result MC mis-copy CAO correct aswer oly MR mis-read CSO correct solutio oly RA required accuracy AWFW aythig which falls withi FW further work AWRT aythig which rouds to ISW igore subsequet work ACF ay correct form FIW from icorrect work AG aswer give BOD give beefit of doubt SC special case WR work replaced by cadidate OE or equivalet FB formulae book A, or (or 0) accuracy marks NOS ot o scheme x EE deduct x marks for each error G graph NMS o method show c cadidate PI possibly implied sf sigificat figure(s) SCA substatially correct approach dp decimal place(s) No Method Show Where the questio specifically requires a particular method to be used, we must usually see evidece of use of this method for ay marks to be awarded. However, there are situatios i some uits where part marks would be appropriate, particularly whe similar techiques are ivolved. Your Pricipal Examier will alert you to these ad details will be provided o the mark scheme. Where the aswer ca be reasoably obtaied without showig workig ad it is very ulikely that the correct aswer ca be obtaied by usig a icorrect method, we must award full marks. However, the obvious pealty to cadidates showig o workig is that icorrect aswers, however close, ear o marks. Where a questio asks the cadidate to state or write dow a result, o method eed be show for full marks. Where the permitted calculator has fuctios which reasoably allow the solutio of the questio directly, the correct aswer without workig ears full marks, uless it is give to less tha the degree of accuracy accepted i the mark scheme, whe it gais o marks. Otherwise we require evidece of a correct method for ay marks to be awarded.

4 MFP - AQA GCE Mark Scheme 008 Jauary series MFP (a) (b) Ay method for fidig r or θ M π r 4, θ 4 AA πi 5 4 z 4 e M eeds some referece to a +k πi π i kπ i e 0 + M A for r z 5 icorrect r, θ part (a) AF A for θ AF z π i 9π i 7π i 0 0, 0 7π i 5π i 0 0 e, e e, e, e A,,0 F 5 0 Accept r i ay form eg Correct but some aswers outside rage allow A ft icorrect r, θ i part (a) (a) Attempt to expad ( r ) ( r ) (b) r r ( r + ) or ( ) Total 8 + M r expaded A 4r + A AG MA rows see ( ) ( ) r r + A ( ) r Do ot allow M for ( + ) equal to aythig ot r M M for multiplicatio of bracket or takig r + out as a factor r ( ) r A CAO r ( + )( + ) A 6 AG r 6 Total 9 4

5 MFP - AQA GCE Mark Scheme 008 Jauary series MFP (cot) (a)(i) z i i M A i 4 (ii) Cetre of circle is + i B Do ot accept (,) uless attempt to solve usig trig Substitute ito lie M π arg ( + i) show 6 A (b) Circle: cetre correct 0, through ( ) Half lie: through ( 0, ) B B B through cetre of circle B 4 (c) Shadig iside circle ad below lie BF Bouded by y B Total 4(a)(i) α i B (ii) αβ B (iii) αβγ + i B (b)(i) α ( α ) αβ used M Allow if sig error or missig ( i) AF 7 AF ft errors i (a) (ii) α β ( αβ ) αβ βγ M Allow if sig error i missig ( αβ ) αβγ α A ( )( ) 9 + i i AF ft errors i (a) 7+ i AF 4 ft errors i (a) α β γ + i i M AF ft sig error i αβγ (iii) ( ) z + 7z + 7+ i z i 0 BF (c) ( ) BF Total 4 Correct umbers i correct places Correct sigs 5

6 MFP - AQA GCE Mark Scheme 008 Jauary series MFP (cot) 5 Assume result true for k k + r + r! The ( ) r (( k ) )( k ) k( k ) + + +! + +! MA Takig out ( k + )! as factor m ( k! ) ( k k k) A ( k )( k ) + +! A k show ( + )!! B P P ad k k+ 6(a)(i) ( ) P true E 7 If all 6 marks eared Total 7 cosθ + isi θ cosθ + isiθ M cos θ + icos θsiθ + i cosθsi θ + isiθ A Real parts: cosθ cos θ cosθsi θ A AG (ii) Imagiary parts: si θ cos θ siθ si θ AF (iii) siθ taθ cosθ M Used cos θsiθ si θ cos θ si θcosθ AF Error i si θ taθ ta θ ta θ ta θ taθ ta θ A AG (b)(i) π ta B Used (possibly implied) π x x ta is a root of x M Must be hece x x x+ 0 A (ii) 5π 9π Other roots are ta, ta BB (c) π 5π 9π ta + ta + ta M π 5π ta + ta 4 A Total 4 Must be hece 6

7 MFP - AQA GCE Mark Scheme 008 Jauary series MFP (cot) 7(a) dy dx x tah B x sech B B sih x M cosh x cosh x sih x cosh x sih x cosech x A 6 AG OE ie expressig i sih x ad cosh x M ie use of sih A sih Acosh A Alterative l sih x l cosh x cosh x sih x sih x cosh x cosh x sih x sih x cosh x Use of siha sih A cosh A result (B) (BB) (M) (M) (A) (b)(i) s + cosech x dx M coth x d x A AG (ii) [ l sih ] s x M eeds to be correct l sih l sih A sihcosh l AF must be see sih l ( cosh) A 4 AG Total TOTAL 75 7

MATHEMATICS Unit Further Pure 2

MATHEMATICS Unit Further Pure 2 Geeral Certificate of Educatio Jauary 008 Advaced Level Examiatio MATHEMATICS Uit Further Pure MFP Thursday Jauary 008 9.00 am to 0.0 am For this paper you must have: * a 8-page aswer book * the blue AQA