Version.0 klm General Certificate of Eucation June 00 Mathematics MFP Further Pure Mark Scheme
Mark schemes are prepare by the Principal Eaminer an consiere, together with the relevant questions, by a panel of subject teachers. This mark scheme inclues any amenments mae at the stanarisation meeting attene by all eaminers an is the scheme which was use by them in this eamination. The stanarisation meeting ensures that the mark scheme covers the caniates responses to questions an that every eaminer unerstans an applies it in the same correct way. As preparation for the stanarisation meeting each eaminer analyses a number of caniates scripts: alternative answers not alreay covere by the mark scheme are iscusse at the meeting an legislate for. If, after this meeting, eaminers encounter unusual answers which have not been iscusse at the meeting they are require to refer these to the Principal Eaminer. It must be stresse that a mark scheme is a working ocument, in many cases further evelope an epane on the basis of caniates reactions to a particular paper. Assumptions about future mark schemes on the basis of one year s ocument shoul be avoie; whilst the guiing principles of assessment remain constant, etails will change, epening on the content of a particular eamination paper. Further copies of this Mark Scheme are available to ownloa from the AQA Website: www.aqa.org.uk Copyright 00 AQA an its licensors. All rights reserve. COPYRIGHT AQA retains the copyright on all its publications. However, registere centres for AQA are permitte to copy material from this booklet for their own internal use, with the following important eception: AQA cannot give permission to centres to photocopy any material that is acknowlege to a thir party even for internal use within the centre. Set an publishe by the Assessment an Qualifications Alliance. The Assessment an Qualifications Alliance (AQA) is a company limite by guarantee registere in Englan an Wales (company number 67) an a registere charity (registere charity number 07). Registere aress: AQA, Devas Street, Manchester M 6EX
MFP - AQA GCE Mark Scheme 00 June series Key to mark scheme an abbreviations use in marking M m or M A B E mark is for metho mark is epenent on one or more M marks an is for metho mark is epenent on M or m marks an is for accuracy mark is inepenent of M or m marks an is for metho an accuracy mark is for eplanation or ft or F follow through from previous incorrect result MC mis-copy CAO correct answer only MR mis-rea CSO correct solution only RA require accuracy AWFW anything which falls within FW further work AWRT anything which rouns to ISW ignore subsequent work ACF any correct form FIW from incorrect work AG answer given BOD given benefit of oubt SC special case WR work replace by caniate OE or equivalent FB formulae book A, or (or 0) accuracy marks NOS not on scheme EE euct marks for each error G graph NMS no metho shown c caniate PI possibly implie sf significant figure(s) SCA substantially correct approach p ecimal place(s) No Metho Shown Where the question specifically requires a particular metho to be use, we must usually see evience of use of this metho for any marks to be aware. However, there are situations in some units where part marks woul be appropriate, particularly when similar techniques are involve. Your Principal Eaminer will alert you to these an etails will be provie on the mark scheme. Where the answer can be reasonably obtaine without showing working an it is very unlikely that the correct answer can be obtaine by using an incorrect metho, we must awar full marks. However, the obvious penalty to caniates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the caniate to state or write own a result, no metho nee be shown for full marks. Where the permitte calculator has functions which reasonably allow the solution of the question irectly, the correct answer without working earns full marks, unless it is given to less than the egree of accuracy accepte in the mark scheme, when it gains no marks. Otherwise we require evience of a correct metho for any marks to be aware.
MFP - AQA GCE Mark Scheme 00 June series (a) y(.) y() + 0.[ + + sin] MA + 0..87..8(7..).8 to p A Conone > p (b) y(.) y() + (0.){f[., y(.)]} M (a). + (0.){.++sin[.8(7..)]} AF Ft on can s answer to (a).09 to p A CAO Must be.09 Note: If using egrees ma mark is /6 ie MAA0;MAFA0 Total 6 k sin + k sin sin M Substituting into the ifferential equation A k A Accept correct PI (b) (Au. eqn m + 0) m ± i B PI CF: Acos + B sin M AF M0 if m is real OE Ft on incorrect comple values for m For the AF o not accept if left in the form Ae i + Be i (GS: y ) Acos + Bsin sin BF c s CF +c s PI but must have constants Total 7 (a) The interval of integration is infinite E OE (b) e e e M A ke k e for non-zero k e e {+c} AF Conone absence of +c (c) I e a lim a e lim { ae a e a } a e M F(a) F() with an inication of limit a a lim a e 0 M For statement with limit/ limiting a process shown I e A CSO Total 7
MFP - AQA GCE Mark Scheme 00 June series MFP (cont) IF is ep ( ) M an with integration attempte e ln A A PI y ( ) + M A y ( ) LHS. Use of c s IF. PI ( ) 0 + + A m k + A Conone missing A () + A 0 m Use of bounary conitions in attempt to fin constant after intgr. Dep on two M marks, not ep on m A ; (*) A 9 ACF. The A can be aware at line (*) y ( + ) provie a correct earlier eqn in y, an 0 A is seen immeiately before bounary conitions are substitute. Total 9
MFP - AQA GCE Mark Scheme 00 June series MFP (cont) (a) ( ) ( ) M Clear attempt to replace by in cos + epansion of cos conone! missing brackets for the M mark A 8 + (b)(i) y M Chain rule ( e ) A e y ( e )( e ) ( e )( e ) M Quotient rule OE A ACF e y e ( ) ( e ) ( e ) ( e ) ( e ) ( e )( e ) ( e ) m A 6 All necessary rules attempte (ep on previous M marks) (ii) y(0) 0; y (0) ; y (0) ; y (0) 6 M At least three attempte Ln( e ) y(0)+y (0)+ y (0)+ y (0) 6. A CSO AG (The previous 7 marks must have been aware an no ouble errors seen) (c) ln( e )... cos 8 M Using the epansions lim o( ) The notation o( n ) can be Limit 0 replace by a term of the form 8 o( ) k n.. lim o( ) 0 8 o( ) m ACF 8 A CSO Total Division by stage before taking the limit 6
MFP - AQA GCE Mark Scheme 00 June series MFP (cont) 6(a)(i) + y r, r cos θ, y r sin θ B,,0 B for one state or use ( cosθ sinθ ) r r M + y ( y) A ACF (ii) ( ) + ( y + ) M AF Centre (, ); raius AF (b)(i) Area ( + sinθ ) θ M Use of r θ. π (6 8sin sin ) + θ + θ θ B 0 B π Correct epn of [+sinθ ] Correct limits (8 + sinθ + 0.( cos θ)) θ M Attempt to writesin θ in terms of 0 cos θ π 8θ cosθ + θ sin θ 8 0 AF Correct integration ft wrong coefficients 6.π A 6 CSO (ii) For the curves to intersect, the eqn (cosθ sinθ ) + sinθ must have a solution. M Equating rs an simplifying to a suitable form cosθ sinθ R cos( θ + α ), M OE. Forming a relevant eqn from which vali eplanation can be state irectly where R + an cos α R A OE. Correct relevant equation cos( θ + α ) >. Since must have cos X there are no solutions of the equation (cosθ sinθ ) + sinθ so the two curves o not intersect. E Accept other vali eplanations. (iii) Require area answer (b)(i) π ( raius of C ) M 6.π π.π AF Ft on (a)(ii) an (b)(i) Total 9 7
MFP - AQA GCE Mark Scheme 00 June series MFP (cont) 7(a)(i) y y M OE Chain rule t y y so y y t A CSO A.G. (a)(ii) (b) (c) y t M ( f ( t) ) ( f ( t) ) O.E. eg ( g( ) ) ( g( ) ) y + y y t t t m y y t + A CSO A.G. y y t y t y t + y ( 8t + )t y + t y t y 6t + t y t y t Divie by t gives y y + y t M A CSO A.G. Prouct rule O.E. use ep on previous M being aware at some stage Subst. using (a)(i), (a)(ii) into given DE to eliminate all y y Solving + y t (*) Aul. Eqn. m m + 0 (m )(m ) 0 M PI m an A CF Ae t + Be t M Conone for t here; ft c s real values for m For PI try y pt + q M OE p + pt + q t p, q A GS of (*) is y Ae t + Be t CF + PI with arb. constants an both CF + t + BF an PI functions of t only GS of y y (8 + ) + y is e e A 7 y A + B + Total TOTAL 7 8