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1 PhysicsAnMathsTutor.com

2 Version.0: 006 General Certificate of Eucation abc Mathematics 660 MFP Further Pure Mark Scheme 006 eamination - January series 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. Copyright 006 AQA an its licensors. All rights reserve.

3 MFP AQA GCE Mark Scheme, 006 January 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.

4 AQA GCE Mark Scheme, 006 January series MFP MFP (a) ( m + ) Completing sq or formula m ± i (b)(i) CF is e (A cos + B sin ) {or e A cos( + B) but not Ae ( +i) + Be ( i) } If m is real give M0 On wrong a s an b s but roots must be comple. {P.Int.} try y p + q OE p + ( p + q) p, q On one slip GS y e (Acos + Bsin) + B 6 Their CF + their PI with two arbitrary constants. (ii) 0, y A B Provie an gaine in (b)(i) y () e (Acos + Bsin) + Prouct rule use + e ( Asin + Bcos) + y (0) A+B+ B Slips (a) y e (cos + sin) + Total e e - e Reasonable attempt at parts e e {+c} Conone absence of +c a 0 e ae a e a (0 ) F(a) F(0) ae a e a 5 (b) lim a k a a e 0 B (c) 0 e lim { ae a e a } a 0 0 If this line oe is missing then 0/ On caniate s / in part (a). B must have been earne Total 8

5 MFP AQA GCE Mark Scheme, 006 January series MFP (a) y y () B Accept general cubic. y ( ) + y + Substitution into LHS of DE ( ) + 5 (b) [( ) y] y y + ( ) Completion. If using general cubic all unknown constants must be foun Differentiating ( ) y c wrt y leas to y + ( ) 0 c y is a soln. of y + y 0 Be generous SC Differentiate but not implicitly give ma of / for complete solution (c) c y is a soln with one arb. y constant of 0 + y c y is a CF of the DE GS is CF + PI c y + Total 8 Must be using hence ; CF an PI functions of only CSO Must have eplicitly consiere the link between one arbitrary constant an the GS of a first orer ifferential equation.

6 AQA GCE Mark Scheme, 006 January series MFP MFP (a) ln( )... B sin (b)(i) f () e f(0) B f () cos e f (0) sin sin f () sin e + cos e f (0) sin Prouct rule use Maclaurin f () f(0)+f (0)+ f (0) so st three terms are CSO AG (ii) f () cos (cos sin sin) e + sin +{cos( sin) cos } e f (0) 0 so the coefficient of in the series is zero CSO AG SC for (b): Use of series epansions.ma of /9 (c) sin. B Ignore higher power terms in sin epansion sin + o( ) e + ln( ) Series from (a) & (b) use sin Numerator k (+ ) lim e 0 sin + ln + o( ) Conone if this step is missing + o( ) sin ( ) On caniate s coefficient in (a) provie lower powers cancel Total 5

7 MFP AQA GCE Mark Scheme, 006 January series MFP 5(a)(i) y(.) y() + 0.[ln+/] +0.. (ii) y(.) y() + (0.)[f(., y(.)]. +(0.)[.ln.+(.)/.] On answer to (a)(i) to p CAO (b)(i) IF is e e ln ln e Conone e for M mark AG (be convince) (b)(i) Solutions using the printe answer must be convincing before any marks are aware (ii) y ln y ln ln Integration by parts for k ln y ln + c Conone missing c. y() ln + c m Depenent on at least one of the two previous M marks y c y ln + 6 OE eg ln + (iii) y(.).5. to p B Total 7 6

8 AQA GCE Mark Scheme, 006 January series MFP MFP 6(a) + y y Use of y r sin θ ( r cosθ PI) Use of + y r r r sinθ m r sinθ CSO AG (b) Area (sin + 5) θ θ. Use of r θ. π.. (sin θ + 0sinθ + 5 )θ 0 B B Correct epn. of (sinθ +5) Correct limits π (( cos θ ) + 0sinθ + 5 ) Attempt to writesin θ in terms of 0 cos θ. θ [ 7θ sin θ 0cosθ ] 0 Correct integration ft wrong coeffs 7π. 6 CSO (c) At intersection sinθ sinθ + 5 OE eg r 6( r 5) 5 sin θ 0 OE eg r 6 π Points 6, 6 an 5π 6, 6 OE OPMQ is a rhombus of sie 6 Or two equilateral triangles of sie 6 Etra notes: The SC for Q π Any vali complete metho to fin the Area 6 6 sin oe area (or half area) of quarilateral. 8 6 Accept unsimplifie sur Total 6 Total 75 e sin ! +!!!! for st terms ignoring any higher powers than those shown. 7

9 MFP AQA GCE Mark Scheme, 006 January series for all terms (coul be treate separately ie last term often only comes into (b)(ii) + + (...) + (...) (be convince..ignore any powers of above power ) Coefficient of : + 0 (be convince..ignore any powers of above power ) 6 6 Quite often the n A mark is aware before the st 8

10 Version.0: 0606 abc General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 006 eamination - June series 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. Copyright 006 AQA an its licensors. All rights reserve.

11 MFP AQA GCE Mark Scheme, 006 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.

12 AQA GCE Mark Scheme, 006 June series MFP MFP (a) y + sin y' + cos y'' sin Nee to attempt both y an y sin 5( + cos)+( + sin) 8 0 0cos CSO AG Substitute. an confirm correct (b) Auiliary equation m 5m+ 0 m an CF: A e + B e GS: y A e + B e + + sin B Their CF + + sin (c) 0, y A+ B B Only ft if eponentials in GS 0, y' 0 0 A+ B+ B Only ft if eponentials in GS an ifferentiate four terms at least Solving the simultaneous equations gives A an B y e + e + + sin Total (a) + y (b) k PI ft from (a) k 0. f (.,.5) (...) PI y(.) y() + [ ] m.57 to p 6 If answer not to p withhol this mark Total 9 (a) cot IF is e e ln sin sin AG (b) ( ysin ) sin cos ysin sin Metho to integrate sincos ysin cos + c OE π y when π sin cosπ + c m Depening on at least one M c y sin ( cos ) 6 OE eg y sin sin + Total 9

13 MFP AQA GCE Mark Scheme, 006 June series MFP (cont) (a) Area 6( cos ) θ θ use of r θ π π 0 6( cosθ + cos θ ) θ B B for correct eplanation of [6( cosθ )] for correct limits 9 cos θ + (cosθ + ) θ Attempt to write cos θ in terms of 0 cos θ. π 9 7θ 6sinθ + sin θ Correct integration; only ft if integrating 0 a + bcosθ + ccosθ with non-zero a, b, c. 5π 6 CSO (b)(i) + y 9 r 9 B PI A & B: 6 6cosθ cosθ π Pts of intersection, ; 5π, OE (accept ifferent values of θ not in the given interval) (ii) Length AB r sinθ 5(a) (b) OE eact sur form Total + lim a + 0 a a (+ ) + [ ln(+ ) ln(+ ) ] ln + + a + ln lim ln a a + ln ln ln m 5 CSO Total 7 aln( + ) + bln( + )

14 AQA GCE Mark Scheme, 006 June series MFP MFP (cont) 6(a) y u + y u y + y terms correct LHS of DE u y y LHS: u Substitution into LHS of DE as far as no erivatives of y u + u e CSO AG (b) IF is e e B ue ue + A u e + Ae 5 Alternative : Those using CF+PI Auiliary equation, m + 0 ucf Ae B For u PI try upi ke ke ke + ke { e } LHS k upi e u Ae + e GS (c) y y + Ae Use (b) to reach a st orer DE in y an IF is e e B ye A + ye + A + B y e + A + B 5 Total 5

15 MFP AQA GCE Mark Scheme, 006 June series MFP (cont) 7(a)(i) ( ) + y y+ y... B (ii) sec +... B ; + ; 5 AG be convince Alternative: Those using Maclaurin f() sec f(0) ; f () sec tan; {f (0) 0} (B) f () sec tan + sec ; f (0) () Prouct rule oe f () sec tan + 5tan sec ; (m) Chain rule with prouct rule OE f (iv) () sec tan +8tan sec +5sec 5 f (iv) (0) 5 sec printe result (A) CSO AG (b) f() tan ; f(0) 0; f () sec ; {f (0) } B f () sec(sec tan ); f (0) 0 f () sec tan(sec tan) + sec Chain rule with prouct rule oe f (0) tan ! + CSO AG Alternative: Those using otherwise sin () () cos () (c) tan ( + o( )) B tan + ( ) sec + o ( ) Conone o( k ) missing + o ( ) + o ( ) tan lim 0 sec ft on k after B0 for tan k+ Total TOTAL 75 6

16 Version.0: 007 abc General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 007 eamination - January series

17 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: Copyright 007 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 5 6EX Dr Michael Cresswell Director General

18 MFP - AQA GCE Mark Scheme 007 January 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.

19 MFP - AQA GCE Mark Scheme 007 January series MFP (a) y (.05) [ln ( ) ] (7557..) to p Conone > p (b) k 0.05 ln ( ) 0.077( 75 ) k 0.05 f (.05, ) 0.05 ln ( ) F ( 85 ) F PI y(.05) y( ) + [ k + k] m PI ft caniate s evaluation in (a) Dep on previous two Ms an numerical values for k s 0.69 to p F 6 Must be p ft one slip Total 9 r rsinθ r y B r sinθ y state or use r y+ + y ( y+ ) r + y use + y y + 8y+ 6 F ft one slip 6 y 8 6 Total 6 (a) IF is ep An with integration attempte ln e CSO AG be convince (b) y ( ) + PI ( ) y + + A m ( ) k + Conone missing A (9) + A m Use of bounary conitions to fin constant A ( ) y 6 Any correct form + Total 9

20 MFP - AQA GCE Mark Scheme 007 January series MFP (cont) (a) Integran is not efine at 0 E OE (b) ln ln ln ( c)... k ln ± f ( ), with ( ) involving the original ln + Conone absence of + c f not e ln lim e ln (c) 0 a 0 a lim e a lna a F(b) F(a) a 0 lim But a ln a 0 B Accept a general form e.g. a 0 lim k ln 0 0 e ln So eists an e 0 Total 8 5 Aul. eqn m m+ 0 PI m an PI CF is A e + B e F PI Try y a+ bsin + cc os Conone a missing here y () bcos csin y () bsin ccos F ft can be consistent sign error(s) Substitute into DE gives a B c + b 5 an c b 0 b 0.5, c F F ft a slip ft a slip GS: y A e + B e sin + cos BF y caniate s CF an caniate s PI (must have eactly two arbitrary constants) Total 5

21 MFP - AQA GCE Mark Scheme 007 January series MFP (cont) 6(a)(i) ( ) f (+ ) () (+ ) f ( ) ( ) + F ft a slip f ( ) ( ) 5 + (ii) ( ) ( ) ( ) f + f 0 ; B f (0) ; f (0) ; f (0) F All three attempte ft on k( + ) f () f(0) + f (0) + f (0)+ f (0) CSO AG e + (b) ( ) (+) + ( ) (c) e ( ) ( ) () cos + { o( )} e(+ ) e cos Attempt to epan neee CSO B B lim 0 0 F lim + o ( ) 0 + o ( ) { o( )} + { o ( )} + { o ( )} Series use F ft a slip but must see the intermeiate stage Total 6 6

22 MFP - AQA GCE Mark Scheme 007 January series MFP (cont) 7(a) Area (6 cos ) + θ θ use of r θ π 6 + 8cosθ + 6cos θ θ B for correct epansion of [6 + cosθ )] -π B for limits π 8 + cosθ + (cosθ + ) θ Attempt to write cos θ in terms of -π cos θ [ ] π θ + sinθ + sin θ F correct integration ft wrong coefficients -π π 6 CSO (b) At P, r ; At Q, r ; B PI π P { } r cosθ cos Attempt to use r cosθ Q { } r cosθ cos π Both Since P an Q have same, PQ is vertical so QP is parallel to the vertical π line θ (c)(i) OP ; OS 8; Angle POS π E B B or S (, ) an P (, ) π PS cos oe PS 8 { } Cosine rule use in triangle POS OE PS ( + ) + ( ) (ii) Since 8 ( 8) +, OS OP PS OPS + is a right angle. (Converse of Pythagoras Theorem) E Accept vali equivalents e.g. PR PQ ( ) PS. SRP RSP RPO OPS is a right angle π 6 Total 5 TOTAL 75 7

23 Version abc General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 007 eamination - June series

24 MFP - AQA GCE Mark Scheme 007 June series 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: Copyright 007 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 5 6EX Dr Michael Cresswell Director General

25 MFP - AQA GCE Mark Scheme 007 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.

26 MFP - AQA GCE Mark Scheme 007 June series MFP (a) ypi k e y ke + 5k e Prouct rule to ifferentiate 5 e y e k + 0ke + 0ke +5k e ft e k + 0ke + 5k e ( ) ke + 5k e + 5k e 6e Substitution into ifferential equation k 6 k ft 6 Only ft if e 5 an e 5 terms all cancel out (b) Au. eqn. m 0m+ 5 0 m 5 B PI CF is ( A+ B) e 5 GS ( ) 5 e e 5 y A+ B + Their CF + their/our PI ft ft only on wrong value of k Total 0 (a) y y (.) y (.) to p (b) k PI ft k 0. f (.,.88...) (5...) PI y (.) y () + [ ] m to p 6 Total 9 tan IF is e e ln cos e ln sec Accept either sec ft ft on earlier sign error ( ysec ) sec ysec sec ysec tan + c Conone missing c y when 0 sec c m c ysec tan+ 8 OE; conone solution finishing at c provie no errors Total 8

27 MFP - AQA GCE Mark Scheme 007 June series MFP (cont) (a) ( ) cosθ + sinθ cos θ + sin θ + cosθ sinθ (b) ( + y ) ( + y) + sinθ B AG (be convince) ( r ) ( rcosθ rsinθ) + M,,0 [ for one of + y r OE, rcosθ, y rsinθ use] r 6 r (cos θ + sin θ ) r 6 r ( + sin θ ) Uses (a) OE at any stage r ( + sin θ ) r ( + sin θ ) {r 0} CSO; AG (c)(i) r 0 sin θ θ sin ( ); θ π ; π π π, ft for either (ii) Area ( sin ) + θ θ Use of r θ ( sin sin ) + θ + θ θ B Correct epansion of (+sinθ ) sinθ ( cosθ) θ + + Attempt to write cos θ sin θ in terms of θ cos θ sin θ 6 θ cos θ sin θ 6 π π ft Correct integration ft wrong coefficients only 9π π 6 6 m Using c s values from (c)(i) as limits or the correct limits π 6 CSO Total 5

28 MFP - AQA GCE Mark Scheme 007 June series MFP (cont) 5(a) y u + u y + u ( u ) + ( ) DE ( ) u u 0 Substitution into LHS of DE as far as no ys u u CSO; AG (b) u u Separate variables ln u ln + ln A u A ( ) 5 (c) y + A ( ) Use (b) ( 0) to form DE in y an y A ( ) y A + B ft Total Solution must have two ifferent constants an correct metho use to solve the DE 6

29 MFP - AQA GCE Mark Scheme 007 June series MFP (cont) 6(a)(i) f( ) ln( + e ): f( 0) ln B e Chain rule f ( ) f ( 0) + e (+ e )e e e e f Quotient rule OE ( ) (+ e ) (+ e ) f ( 0) so first three terms are: f( ) ln+ + ln+ + 6 CSO; AG! 8 (+ e ) e e (+ e )e (+ e ) f ( 0) 0 {so coefficient of is zero} (ii) f ( ) (b) 8 ft + B Chain rule with quotient/prouct rule n f ke + e integer n < 0 ft on ( ) ( ) ( ) CSO; AG; All previous ifferentiation correct SC for those not using Maclaurin s theorem: maimum of /9 (c) ln +... B () + e ln + ln +... Uses previous epansions to obtain first non-zero term of the form k sin !! B + sin + o 6 + e ln + ln... ( ) o( ) 6 lim o ( ) CSO Total 5 7

30 MFP - AQA GCE Mark Scheme 007 June series MFP (cont) 7(a) 0 B (b) ( ) u e + u e e Attempts to fin u e ( ) e + u lnu + c u ( ){ c} ln e + + Conone missing c (c) e ( ) + e e + B + e a lim ln( e ) a + a { ( a )} ( ) a { ( a )} ( ) lim ln e + ln e + a ln lim e + ln e + a ( ) ( ) ln ln e + ln e + For using part (b) an F( B) F( A) Total 7 TOTAL 75 For using limiting process 8

31 Version.0: 008 abc General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 008 eamination - January series

32 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: Copyright 008 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 5 6EX Dr Michael Cresswell Director General

33 MFP - AQA GCE Mark Scheme 008 January 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.

34 MFP - AQA GCE Mark Scheme 008 January series MFP (a) y(.) y() + 0.[ ] +0.. (b) y(.) y() + (0.)[f(., y(.))]. +(0.)[.. ] Ft on can s answer to (a) (a) CAO Total 6 Area ( tan ) + θ θ Use of θ r. ( tan tan + θ + θ ) θ B Correct epansion of (+tanθ ) (sec θ + tan θ ) θ + tan θ sec θ use [ θ θ ] π tan + ln(sec ) 0 B Integrating psec θ correctly Integrating qtanθ correctly [( + ln ) 0] ln + 6 Completion. AG CSO be convince (b) OP ; OQ π + tan B Both neee. Accept.7 for OQ Shae area answer (a) sin π OP OQ + ln ( + ) + ln ACF. Conone 0.76 if eact value for area of triangle seen Total 9

35 MFP - AQA GCE Mark Scheme 008 January series MFP (cont) (a) ( m + ) Completing sq or formula m ± i CF is e (A cos + B sin ) {or e A cos( + B) but not Ae ( +i) + Be ( i) } PI try y p 5p 5 PI is y B If m is real give M0 Ft on wrong a s an b s but roots must be comple GS y e (Acos + Bsin) + B 6 Their CF + their PI with two arbitrary constants. (b) 0, y A B Provie previous B aware y () e (Acos+Bsin) + Prouct rule use + e ( Asin+Bcos) y (0) A+B B 5 Ft on one slip y e (cos + 5sin) + Total 0 (a) The interval of integration is infinite E OE (b) e e e Reasonable attempt at parts e 9 e {+c} Conone absence of +c (c) I e lim a lim a { ae a 9 e a } a e e 9 F(a) F() with an inication of limit a lim a ae a 0 For statement with limit/limiting process shown I e 9 Total 7 5

36 MFP - AQA GCE Mark Scheme 008 January series MFP (cont) 5 IF is e + ln( + ) e ( ) ln + e ln + ( ) Ft on e p + ( ) ( y ) ( + ) ( + ) y ( + ) ( + ) ( ) y ( + ) + + c 6 LHS as /(y can s IF) PI an also RHS of form k( +) p Use of suitable substitution to fin RHS or reaching k( +) OE Conone missing c 5 y(0) c 6 m y ( + ) ( + ) 9 Accept other forms of f() eg y + ( ) Total 9 6(a) r sinθcosθ 8 sin θ sinθ cosθ use r cos θ y r sin θ Either one state or use 8 y, y Either OE eg y (b) y O B (c) r secθ is B Sub in y y In cartesian, A(, ) tan θ y π θ r + y 8 Use either tan θ y or r + y π θ ; r 8 r must be given in sur form Altn: Eliminating r to reach eqn. in cosθ Altn: rsin θ (B) π Solving rcosθ an rsin θ an sinθ only () θ () simultaneously () Substitution rsec π (m) tan θ or r + () π θ ; r 8 () nee both r 8 () OE sur Total 8 6

37 MFP - AQA GCE Mark Scheme 008 January series MFP (cont) 7(a)(i) ln ( ) 8 Use of epansion of ln(+) Simplifie numerators. (ii) < B (b)(i) yln cos y () ( sin ) cos y () sec ACF y () sec (sec tan ) Chain rule OE {y () tan ( sec )} Ft a slip accept unsimplifie (ii) y () [sec (sec ) + tan(sec (sec tan))] Prouct rule OE ACF y (0) [() +0] Ft a slip (iii) (c) lncos 0+0+ ( ) ( )! CSO throughout part (b). AG ( ) lim ln + Limit 0 ln cos ( +..) lim 0.. lim o ( ) Limit o ( ) lim o ( ) o ( ) Total 5 Using earlier epansions The notation o( n ) can be replace by a term of the form k n Nee to see stage, ivision by 7

38 MFP - AQA GCE Mark Scheme 008 January series MFP (cont) 8(a)(i) t { } B y y t t Chain rule y t y t Completion. AG (ii) y y t t y y + t t Prouct rule y y.. + t t y y. + t Conone leaving in this form y y y t t AG (b) y y 6 6y 0 y y 7 + 6y 0 Using results in (a) to reach DE of this t t form Aul eqn m 7m (m 6)(m ) 0 m PI m an 6 PI 6t t y Ae + Be Must be solving the correct DE. 6 (Give A0 for y Ae + Be ) 6 y A + B 5 Ft a minor slip only if previous A0 an all three metho marks gaine Total TOTAL 75 8

39 Version.0: 0608 abc General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 008 eamination June series

40 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: Copyright 008 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 5 6EX Dr Michael Cresswell Director General

41 MFP - AQA GCE Mark Scheme 008 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.

42 MFP - AQA GCE Mark Scheme 008 June series MFP k 0. ln ( + ) 0.609(79 ) ( *) PI k 0. f (., + *...) 0. ln( )] ( ) PI k k y(.) y() + [ + ] m Dep on previous two Ms an numerical values for k s to p 6 Must be.65 Total 6 (a) PI: ypi a+ b+ csin + cos y PI b+ ccos sin b+ ccos sin a b csin cos 0sin Substituting into DE b a 0; b ; c 0; c0 Equating coefficients (at least eqns) a ; b ; c ; A, for any two correct ypi + sin cos (b) Au. eqn. m 0 ( y ) e CF A Altn. y y Ae OE OE () ( ygs ) Ae + + sin cos BF (c s CF + c s PI ) with arbitrary constant Total 7 (a) + y y + y + y ( y) B AG (b) + y r Or r cosθ y rsinθ y so + y ( y) r ( rsin θ ) OE eg r cos θ r sinθ PI by the net line r rsinθ or r ( rsin θ ) m Either r( + sin θ ) or r( sin θ ) r > 0 so r + sinθ 5 CSO Total 6

43 MFP - AQA GCE Mark Scheme 008 June series MFP (cont) (a) y u y u u u u u AG Substitution into LHS of DE an completion (b) IF is ep ( ) an with integration attempte ln e or or multiple of u LHS as ifferential of u IF. PI u + A m Must have an arbitrary constant (Dep. on previous only) u + A 6 (c) y A + Replaces u by y integrate an attempts to 5(a) A + + F ft on can s u but solution must have two arbitrary constants Total 0 y B ± ln ln, with f() not involving the original ln ln + c Conone absence of + c 6... k ln f ( ) (b) Integran is not efine at 0 E OE e e (c) { } ln lim ln 0 a 0 a e a a lim ln a F(e) F(a) 6 a 0 6 But lim a ln a 0 B Accept a general form eg So a 0 e ln 0 eists an e CSO 6 Total 7 k lim ln 0 0 5

44 MFP - AQA GCE Mark Scheme 008 June series MFP (cont) 6(a) Au eqn: m m 0 m, PI CF + ( yc ) Ae Be Try ( y ) e PI a ( + b) y ae y ae Substitute into DE gives ae + ae ae b 0e 9 a b B BF 0 (c s CF+c s PI) with arbitrary constants ( y ) e e e GS A B (b) 0, y 7 7 A + B + + BF Only ft if eponentials in GS an two arbitrary constants remain y Ae Be e k As, e 0, y 0 so A 0 B When A 0, B + B BF Must be using A 0 y e + e + CSO Total 6

45 MFP - AQA GCE Mark Scheme 008 June series MFP (cont) sin B! 7(a) ( ) y e (e + ) (b)(i) ( ) y ( + ) ( + ) e e e (e ) Chain rule Prouct rule OE OE y (0) ; y (0) 7 5 CSO (ii) y(0) ; y (0) ; y (0) 7 McC. Thm: y(0) + y (0) + y (0) 7 + e CSO; AG (c) 7 + e sin m Diviing numerator an enominator by to get constant term in each lim 0 + e sin 8 F Total Ft on can s answer to (a) provie of the form a+b 7

46 MFP - AQA GCE Mark Scheme 008 June series MFP (cont) 8(a) θ 0, r 5 + cos0 7 {A lies on C} B θ π, r 5 + cos π {B lies on C} B (b) 5 7 B B Close single loop curve, with (inication of) symmetry Critical values,,5,7 inicate 5 (c) Area (5 + cos θ ) θ Use of π r θ B OE for correct epansion of (5 + cos θ ) ( 5 + 0cosθ + cos θ) θ B π For correct limits π ( 5 + 0cosθ + (cosθ + ) ) θ Attempt to write cos θ in terms of π cos θ [ ] π Correct integration ft wrong non-zero 7 θ + 0sin θ + sin θ π F coefficients in a + bcosθ + ccosθ 7π 6 CSO () Triangle OBQ with OB an angle BOQ α B PI OQ 5 + cos( π + α) OE Area of triangle OQB OB OQsinα m Dep. on correct metho to fin OQ ( cos α) sin α CSO Total TOTAL 75 8

47 Version : klm General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 009 eamination - January series

48 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: Copyright 009 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 5 6EX Dr Michael Cresswell Director General

49 MFP - AQA GCE Mark Scheme 009 January 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.

50 MFP - AQA GCE Mark Scheme 009 January series MFP (a) + y (b) k Bft PI ft from (a) k 0. f (.,.5) ft on (a) (5 ) ft PI conone p y(.) y()+ [ (5...) ] + m (a) (b).57.5 to p ft 5 ft one slip If answer not to p withhol this mark Total 8 IF is e e ln e P ln e y ± AG Be convince LHS as /(y IF) PI y ln + c RHS Conone missing + c here y ln + c Total 7 Area π ( cos ) + θ sin θ θ use of r θ 0 B Correct limits cos + θ 0 ( ) π + Vali metho to reach M k(+cosθ) or acosθ +bcosθ +ccos θ OE {SC: if epans then integrates to get either acosθ + b cosθ OE or c cos θ OE in a vali way} OE eg cosθ cos θ cos θ 6 CSO Total 6

51 MFP - AQA GCE Mark Scheme 009 January series MFP (cont) (a) ln ln Integration by parts ln + c CSO AG (b) 0 ln lim a 0 ln OE a lim {0 [ a ln a a] } F() F(a) OE a 0 But lim a 0 a ln a 0 E Accept a general form eg lim a k ln a 0 a 0 So ln 0 5(a) When θ π, r + cosπ + ( ) Total 6 B Correct verification (b)(i) cos θ + cosθ (ii) Points of intersection Equates r s an attempts to solve. π,, π, Area OMN sin( θ ) sin π M θ N A, Conone eg π/ for π/ if either one point correct or two correct solutions of cosθ 0.5 ALT π MN sin Perp. from L to MN π cos π Area OMLN sin Area LMN Area LMN (c) r + r cosθ r + B r cosθ state or use r r ± ( ) 9( + y ) ( ) r + y use 9y ( ) 9 5 CSO ACF for f() eg 9y 5 8+ Total 5

52 MFP - AQA GCE Mark Scheme 009 January series MFP (cont) 6(a)(i) e Clear use of in epansion of e ACF (ii) {f()} e (+ ) 5 ( ) 0 ( + ) + ( ) + {f() } m ft First three terms as + ( ) + k OE Dep on both prev MS Conone one sign or numerical slip in mult CSO AG A0 if binominal series not use (b)(i) y y ln( + sin ) cos + sin y (+ sin )( sin ) cos (cos ) (sin + ) ( + sin ) ( + sin ) Chain rule Quotient rule OE with u an v non constant ACF (ii) y(0) 0, y (0), y (0) McL Thm.: { ln( sin ) +.. CSO AG (c) lim f ( ) 0 ln( + sin ) lim lim m Total 6 Using epansions Division by stage before taking limit. CSO 6

53 MFP - AQA GCE Mark Scheme 009 January series 7 MFP (cont) 7(a) t e t { } B OE y t y t t y t e Chain rule OE eg y y t t y t t t y y t t e e ) ( ) ( t t OE + e e t y t y t t t Prouct rule OE. + e e e t y t y t t t. + t y t y t y t y y 7 OE CSO AG Completion. Be convince (b) 0 y y t y t y t y t y t y CSO AG Completion. Be convince (c) 0 5 t y t y (*) Aul eqn m 5m 0 PI m(m 5) 0 m 0 an 5 CF: 5 ( ) e t C y A B + ft wrong values of m provie arb. constants in CF. conone for t here PI: ( ) P y t B GS of (*) {y} A + B e 5t t Bft 5 ft on c s CF + PI, provie PI is non-zero an CF has two arbitrary constants () y A + B 5 ln y () 5B ft Must involve ifferentiating a ln ft slip Using bounary conitions to fin A & B B ; A ; { y + 5 ln } ;ft 5 ft a slip. Total 9 TOTAL 75

54 Version.0: 0609 hij General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 009 eamination - June series

55 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: Copyright 009 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 5 6EX Dr Michael Cresswell Director General

56 MFP - AQA GCE Mark Scheme 009 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.

57 MFP - AQA GCE Mark Scheme 009 June series MFP (a) y(.) y() (0..).6 Conone > p if correct (b) y(.) y() + (0.)[f(., y(.))]. + (0.)[ ( ) ] F ft on caniate s answer to (a) (89..).70 CAO Must be.70 Total 6 tan IF is e Awar even if negative sign missing ln(cos ) ( + c) e OE Conone missing c (k) cos F ft earlier sign error y cos ytan cos sin cos ( ycos ) sin cos LHS as (y IF) PI ycos sin cos F ft on c s IF provie no ep or logs ycos sin m Double angle or substitution OE for integrating sin cos y cos cos ( + c) ACF + c m Bounary conition use to fin c 5 c 5 y cos cos + 9 ACF eg ycos + sin Apply ISW after ACF Total 9

58 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) (a) Centre of circle is M(, ) B PI A(6, 8) B (b)(i) k OA 0 ya tanα A B B SC r 0 an tan 8 θ 6 B only (b)(ii) + y 6 8y B If polar form before epansion awar the B for correct epansions of both ( rcos m) an( rsin n) ( mn, ) (,) or ( mn, ) (,) θ θ where r 6r cos θ 8r sin θ 0 st for use of any one of + y r, r cos θ, y r sin θ n for use of these to convert the form + y + a+ by 0 correctly to the form + cosθ+ sinθ 0 r ar br {r 0, origin} Circle: r 6cosθ + 8sinθ NMS Mark as or 0 ALTn Circle has eqn r OA cos(α θ ) (M) r OAcosα cosθ +OAsinα sinθ (m) OE Circle: r 6cosθ + 8sinθ () Total 8 5

59 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) ln ln( + ){ + c} + B OE lim I a a + a lim ln ln(+ ) a a [ ] lim a a ln ln a a + 5 m ln a ln ( a+ ) ln a + an previous score lim ln ln a 5 a + m ln ln a a an + + a previous m score 5 ln ln ln 5 5 CSO Total 5 5(a) ksin + kcos + 5ksin 8sin + cos Differentiation an subst. into DE k (b) Aul eqn m + m m ± Formula or completing sq. PI m ± i CF: { yc} e ( Asin + Bcos ) F ft provie m is not real GS {y} e ( Asin + Bcos ) + k sin BF ft on CF + PI; must have arb consts When 0, y B BF y e ( Asin+ Bcos ) + e (Acos Bsin ) + k cos Prouct rule When 0, y B +A+k PI A y e sin + cos + sin 8 CSO Total 6

60 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) 6(a)(i) f( ) 9+ tan ( ) so f () ( 9 + tan ) sec f () ( 9 + tan ) sec + ( 9 + tan ) (sec tan ) (a)(ii) f(0) B f (0) ( 9) ; 6 f (0) (9) 08 f() f(0)+ f (0)+ f (0) ( ) Chain rule Prouct rule, OE ACF 9+ tan + CSO AG 6 6 Both attempte an at least one correct ft on c s f () an f () (b)... f( ) 6 6 sin ( )...! lim f( ) 0 sin 8 m Total 0 Using series epns. Diviing numerator an enominator by to get constant term in each 7

61 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) 7(a) θ π Area + 6e θ r θ π π π e 6e θ θ + + θ 0 B Correct epansion of + 6e B Correct limits θ π θ π e θ π 8π e θ π π 0 m Correct integration of at least two of the three terms, p e θ π, q e θ π π (6 6e 9e ) 5 ACF (b) B Going the correct way roun the pole 0 B Increasing in istance from the pole (c) En-points (, 0) an (e, π) B,,0 Correct en-points B for each pair or for an graph in correct positions e θ π θ π + 6 e Elimination of r or θ [r + 6 r ] e shown on θ θ π π e e θ θ π e π e 0 m m θ e π Forming quaratic in or in e or in r. [r r 6 0] θ e π > 0 so θ π e E Rejection of negative solution PI [r ] Polar coorinates of P are (, π ln ) 5 OE θ π Total 8

62 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) 8(a)(i) t t B PI or for t y t y t OE Chain rule y... or y... t t y y t y y so AG t (a)(ii) y y t t y t t t (f ( t)) (f ( t)) OE t eg (g( )) (g( )) t t + y y t t Prouct rule OE t y y y + t t y y y + AG Completion (b) y y + (+ ) y 0 y y y ( + ) + ( ) y 0 y y + y 0 t t Use of either (a)(i) or (a)(ii) AG Completion (c) y y + y 0 (*) t t Aul. Eqn. m +m 0 (m +)(m ) 0 PI m an PI GS of (*) {y} Ae t + Be t Ae y Ae + Be Total TOTAL 75 + Be scores M0 here 9

63 Version.0: 00 klm General Certificate of Eucation Mathematics 660 MFP Further Pure Mark Scheme 00 eamination - January series

64 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: 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 5 6EX Dr Michael Cresswell Director General

65 MFP - AQA GCE Mark Scheme 00 January 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.

66 MFP - AQA GCE Mark Scheme 00 January series MFP (a) y + 0. [ ln( + ) ] + 0.ln8.68(...) y(.).68 (to p) Conone greater accuracy (b) k 0. ln ( ) BF PI ft from (a), p or better k 0. f (.,.68(...)) 0.. ln 8.88(..) F PI; ft on 0.. ln[6. + answer(a)] [ ( ) y(.) + [ (..) ] m.69( ).69 to p 5 CAO Must be.69 Total 8 (a) y + Chain rule y ( + ) 9( +) for quotient (PI) or chain rule use (b) Clear attempt to use Maclaurin s theorem ln (+) ln + y (0) + y (0) +.. with numerical values for y (0) an y (0) 9 First three terms: ln + F ft on c s answers to (a) provie y (0) an y (0) are 0. Accept.8(6..) for ln (c) 9 ln ( ) ln BF ft in c s answer to (b) () + ln ln( + ) ln( ) 9 9 ln + ln + + CSO AG Total 8

67 MFP - AQA GCE Mark Scheme 00 January series MFP (cont) (a) y u y u u u CSO AG Substitution into LHS of + u + u DE an completion (b) IF is ep ( ) e ln ; ; ep ( k ), for k ±, ± an integration attempte ( u ) LHS as ifferential of u IF u + A u + A 5 Must have an arbitrary constant (c) y + A an with integration attempte y A + A y + B F ft only if IF is A0A0 Total 9 (a) sin () +.5 +! B (b) cos () +.! B lim cos sin 0 5 lim lim.5 + ( o( ))... m Total 5 Using epansions Division by stage to reach relevant form of quotient before taking limit. CSO OE 5

68 MFP - AQA GCE Mark Scheme 00 January series MFP (cont) 5(a) y Prouct Rule use y PI pe pe pe y pe pe + pe pe + pe +pe 6pe + pe e. Sub. into DE pe e p F ft one slip in ifferentiation 5(b) Au. eqn. m + m + 0 m, B CF is Ae +Be ft on real values of m only GS y Ae +Be e. BF Their CF + their PI must have arb consts When 0, y A + B BF Must be using GS; ft on wrong nonzero values for p an m y Ae Be e +e When 0, BF Must be using GS; ft on wrong nonzero values for p an m y 0 A B 0 BF Must be using GS; ft on wrong nonzero values for p an m an slips in fining y () Solving simultaneously, eqns each in two m arbitrary constants A 6, B ; y 6e e e. 8 CSO Total 6

69 MFP - AQA GCE Mark Scheme 00 January series MFP (cont) 6(a) The interval of integration is infinite E OE (b)(i) y y y ln ln y )( y ) ( y y y ln y y y ln y y CSO AG (ii) ln y y y y y y y ln y... ky ln y ± f ( y) y involving the original ln y y + c Conone absence of + c with f(y) not y ln y lim y ln y y 0 a 0 y ln y y a lim a 0 a 0 a ln a lim since a ln a 0 a 0 5 CSO Must see clear inication that can has correctly consiere lim a k ln a 0 a 0 ft on minus c s value as answer to (b)(ii) (iii) So ln BF Total 9 7 Au. eqn. m + 0 m ± i B CF is Acos + Bsin OE. If m is real give M0 F ft on incorrect comple value for m PI: Try a + b + csin a csin+a +b+csin 8 + 9sin Awar even if etra terms, provie the relevant coefficients are shown to be zero. a, b, Dep on relevant M mark c Dep on relevant M mark (y ) Acos + Bsin + + sin BF 8 Their CF + their PI. Must be eactly two arbitrary constants Total 8 7

70 MFP - AQA GCE Mark Scheme 00 January series sinθ sinθ Elimination of r or θ { r [ (/r)]} 8(a) ( ) sin θ sinθ + 0 { r r + 0 } (sinθ ) 0 sin θ 0.5 m Vali metho to solve quaratic eqn. PI { (r ) 0 r } π 5π A, for any two of the three. θ, θ, r 6 6 π [ P, 5π Q, 6 6 ] π SC: Verification of P, scores ma 6 5 5π of B & a further B if Q, 6 state 8(b) Area triangle OPQ rq sin POQ Any vali metho to correct (ft eg on r Q ) epression with just one remaining unknown Angle POQ 5π π π m 6 6 π Area triangle OPQ sin Unshae area boune by line OP an π arc OP [( sin θ )] θ π 6 8 ( sinθ + sin θ) θ Vali metho to fin remaining unknown either relevant angle or relevant sie Use of r θ for relevant area(s) (conone missing/wrong limits) B Correct epn of ( sinθ ) cosθ Attempt to write sin θ in terms of cosθ 8 sin θ + θ θ sin θ 8 θ + cosθ + (+ c) F Correct integration ft wrong coeffs π π ( θ) θ 6 8 sin θ sinθ 8 + cosθ π π 6 π π π π π π 8 { + cos sin } 6 6 m F F OE for relevant area(s) 6 π { π 7 } F ft one slip; accept terms in π an left unsimplifie Shae area Area of triangle OPQ π OE π [( sin θ )] θ 6 Shae area 6 π π CSO Accept m 5, n 8 Total 6 TOTAL 75 8

71 Version.0 klm General Certificate of Eucation June 00 Mathematics MFP Further Pure Mark Scheme

72 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: 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 5 6EX

73 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.

74 MFP - AQA GCE Mark Scheme 00 June series (a) y(.) y() + 0.[ + + sin] (7..).8 to p Conone > p (b) y(.) y() + (0.){f[., y(.)]} (a). + (0.){.++sin[.8(7..)]} F Ft on can s answer to (a).09 to p CAO Must be.09 Note: If using egrees ma mark is /6 ie A0;FA0 Total 6 k sin + k sin sin Substituting into the ifferential equation k Accept correct PI (b) (Au. eqn m + 0) m ± i B PI CF: Acos + B sin F M0 if m is real OE Ft on incorrect comple values for m For the F 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 ke k e for non-zero k e e {+c} F Conone absence of +c (c) I e a lim a e lim { ae a e a } a 5 e F(a) F() with an inication of limit a a lim a e 0 For statement with limit/ limiting a process shown I 5 e CSO Total 7

75 MFP - AQA GCE Mark Scheme 00 June series MFP (cont) IF is ep ( ) an with integration attempte e ln PI y ( ) + LHS. Use of c s IF. PI y ( ) 5 0 A m ( ) 5 k + Conone missing A 5 () + A 5 0 m Use of bounary conitions in attempt to fin constant after intgr. Dep on two M marks, not ep on m A ; (*) 5 9 ACF. The 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 5

76 MFP - AQA GCE Mark Scheme 00 June series MFP (cont) 5(a) ( ) ( ) Clear attempt to replace by in cos + epansion of cos conone! missing brackets for the M mark 8 + (b)(i) y Chain rule ( e ) e y ( e )( e ) ( e )( e ) Quotient rule OE ( e ) ACF e ( e ) y ( e ) ( e ) ( e ) ( e )( e ) m All necessary rules attempte (ep on previous M marks) ( e ) 6 ACF (ii) y(0) 0; y (0) ; y (0) ; y (0) 6 At least three attempte Ln( e ) y(0)+y (0)+ y (0)+ y (0) 6. CSO AG (The previous 7 marks must have been aware an no ouble errors seen) (c) ln( e )... cos 8 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 CSO 8 Total Division by stage before taking the limit 6

77 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 + y ( y) ACF (ii) ( ) + ( y + ) F Centre (, ); raius F (b)(i) Area ( + sinθ ) θ Use of r θ. π (6 8sin sin ) + θ + θ θ B 0 B π 0 Correct epn of [+sinθ ] Correct limits (8 + sinθ + 0.5( cos θ)) θ Attempt to writesin θ in terms of cos θ π 8θ cosθ + θ sin θ 8 0 F Correct integration ft wrong coefficients 6.5π 6 CSO (ii) For the curves to intersect, the eqn (cosθ sinθ ) + sinθ must have a solution. Equating rs an simplifying to a suitable form cosθ sinθ R cos( θ + α ), OE. Forming a relevant eqn from which vali eplanation can be state irectly where R + an cos α R 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 ) 6.5π π.5π F Ft on (a)(ii) an (b)(i) Total 9 7

78 MFP - AQA GCE Mark Scheme 00 June series MFP (cont) 7(a)(i) y y t t OE Chain rule t y y so y y t t t CSO A.G. (a)(ii) (b) (c) t y t t ( f ( t) ) ( f ( t) ) t t t O.E. eg ( g( ) ) ( g( ) ) t t y + y y t t t t t m y y t + t t CSO A.G. y y t t y t y t + t y t ( 8t + )t 5 + t y t y t 5 + y y t 6t t t t y t Divie by t gives y y + y t t t CSO A.G. Prouct rule O.E. use ep on previous being aware at some stage Subst. using (a)(i), (a)(ii) into given DE to eliminate all y y Solving + y t (*) t t Aul. Eqn. m m + 0 (m )(m ) 0 PI m an CF Ae t + Be t Conone for t here; ft c s real values for m For PI try y pt + q OE p + pt + q t p, q 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 5 (8 + ) + y is e e 7 y A + B + Total TOTAL 75 8

79 Version.0 General Certificate of Eucation (A-level) January 0 Mathematics MFP (Specification 660) Further Pure Mark Scheme

80 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 events which all eaminers participate in an is the scheme which was use by them in this eamination. The stanarisation process 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 stanarisation each eaminer analyses a number of caniates scripts: alternative answers not alreay covere by the mark scheme are iscusse an legislate for. If, after the stanarisation process, eaminers encounter unusual answers which have not been raise 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 from: aqa.org.uk Copyright 0 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 5 6EX.

81 Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE euct marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) 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. 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.

82 Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP k ( ) 0. + (0.5) k 0. f (.,.5) k 0. (.+.5) PI accept p or better y(.) y() + [ + ] k k m Dep on previous two Ms an numerical values for k s y(.).5 5 Must be.5 Total 5 (a) pcos qsin + 5 psin + 5qcos cos Differentiation an subst. into DE p+ 5q ; 5 p q 0 m Equating coeffs. 5 p ; q OE Nee both (b) Au. eqn. m PI. Or solving y ()+5y0 as far as y ( ycf 5 ) Ae OE 5 5 c s CF + c s PI with eactly one ( ygs ) Ae + sin+ cos BF arbitrary constant OE Total 6 (a) r + rcosθ r + B rcosθ state or use r + y ( ) r + y use y 5 Must be in the form y f() but accept ACF for f(). (b) Equation of line: rcosθ Use of rcosθ OE y y ± ; [Pts, ± ] Distance between pts (0.75, ) an (0.75, ) is Altn: 5 ( elimination of either r or θ) At pts of intersection, r an cosθ OE () 5 (For A conone slight prem appro.) Distance PQ r sinθ () Or use of cosine rule or Pythag. 5 5 () Must be from eact values. Total 9

83 Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) IF is e Awar even if negative sign missing ln( ) ( + c) ln( ) ( + ) e OE Conone missing c e c (k) F Ft earlier sign error y y e ( y) e LHS as /(y IF) PI y e (e ) e e Integration by parts in correct irn y e e (+c) ACF When, y e so m Bounary conition use e e e + c to fin c after integration. 5 e c 5 y e e e 9 Must be in the form y f() Total 9 5

84 Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) 5(a) (+ )(+ ) (+ )(+ ) B Accept C 5 (b) 0 ( )( ) [ ln( ) ln( ) ] + + (+c) OE I lim a a 0 (+ )(+ ) replace by a an lim a (OE) [ a a ] a lim lim a ln( + ) ln( + ) (ln 5 ln 5) ln a + a + 6 ln ln 9 lim a + ln a + a m,m 6 CSO Total 7 Limiting process shown. Depenent on the previous 6

85 Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) sin θ cos θ θ Use of r θ 6 Area ( ) π cos θ sin θ θ 0 B ( ) ( ) B r cosθsin θ or better Correct limits π 6sin θ cos θ θ 0 sin k sin cos ( ) θ θ θ (k>0) π 8sin θ (-sin θ) sinθ 0 m Substitution or another vali metho to π 8sin θ 8sin θ F integrate sin 7 CSO AG θcos θ Correct integration of p sin θ cos θ Alternatives for the last four marks π cosθ 0 cos θ cos θ θ () Area ( ) ( cos θ cos θ) θ (cos θsinθ sin θcos θ) 5 Area ( 0) + ( ) [ 0 (0)] (m) (F) () cosθsin θ λcosθ + μ cos θcosθ ( λ, μ 0) Integration by parts twice or use of cos θ cosθ ( cos5θ + cosθ) Correct integration of p cos θcosθ [eg p sin 5θ + sin θ 0 6 ] CSO AG 6 { + } Total 7 7

86 Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) 7(a)(i) B Accept coeffs unsimplifie, even! cos + sin + 6 for 6. (ii) 9 ln(+ ) () + () + 9 B Accept coeffs unsimplifie (b)(i) tan y e, y tan sec e Chain rule ACF eg ysec y tan tan sec tan e sec e + m Prouct rule OE ACF tan sec e ( tan + sec ) y (tan tan + + ) y y ( + tan ) 5 AG Completion; CSO any vali metho. (ii) y y y ( + tan ) sec + ( + tan ) y When 0, CSO (iii) y(0) ; y (0) ; y (0) ; y (0) ; y() y(0) + y (0) + y (0) +! y (0) (c) tan e tan lim e (cos + sin ) 0 ln( + ) lim lim lim CSO AG m Total Using series epns. Diviing numerator an enominator by to get constant terms. OE following a slip. 8

87 Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) 8(a) y t y Chain rule t e t y y t y y t CSO AG (b) y t y ; y t t y y OE t t y t y y y + t t m Prouct rule (ep on previous M) y y y + t OE y y + y ln becomes y y y + y ln t y y t + y lne (using (a) t t y y + y t t t m 5 CSO AG (c) Aul eqn m m + 0 PI (m ) 0, m PI CF: ( y ) ( )e t C At+ B Ft wrong value of m provie equal roots an arb. constants in CF. Conone for t here PI Try ( yp ) at+ b If etras, coeffs. must be shown to be 0. a+ at+ b t a b Correct PI. Conone for t here GS {y} (At +B)e t +0.5(t + ) BF 6 Ft on c s CF + PI, provie PI is non-zero an CF has two arbitrary constants an RHS is fn of t only () y (Aln + B) +0.5(ln + ) y.5 when B F Ft one earlier slip y () (A ln + B ) + A m Prouct rule y () 0.5 A F Ft one earlier slip y ( ln ) + (ln + ) 5 ACF Total 8 TOTAL 75 9

88 Version.0 General Certificate of Eucation (A-level) June 0 Mathematics MFP (Specification 660) Further Pure Final Mark Scheme

89 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 events which all eaminers participate in an is the scheme which was use by them in this eamination. The stanarisation process 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 stanarisation each eaminer analyses a number of caniates scripts: alternative answers not alreay covere by the mark scheme are iscusse an legislate for. If, after the stanarisation process, eaminers encounter unusual answers which have not been raise 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 from: aqa.org.uk Copyright 0 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 5 6EX.

90 Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE euct marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) 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. 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.

91 MFP k + + PI. May be seen within given formula 0. [ ln ( )] 0.586(9 ) ( *) Accept sf roune or truncate or better as evience of the line k 0. f (., + *...) 0. [. + ln (+.586 )] 0. [.+ln(++c s k )]. PI May be seen within given formula. 0.66(8.) p or better. PI by later work k k [ ] y(.) y()+ [ + ] m Dep on previous two Ms but ft on c s numerical values (or numerical epressions) for k s following evaluation of these. (.5877 ).585 to p 5 CAO Must be.585 Total 5 SC For those scoring M0 who have k 0.56(78..), an final answer.5 (ie p) for y(.) awar a total of marks [B]

92 MFP (cont) (a) PI: y PI p+ qe y qe qe Prouct rule use ' PI y qe + qe '' PI qe + qe + qe qe p qe 9e Subst. into DE q 9 an p m Equating coefficients q 9 so q ; p so p ; B 5 y e ] [ PI (b) Au. eqn. m + m 0 (m )(m+) 0 Factorising or using quaratic formula OE PI by correct two values of m seen/use y Ae + Be CF y Ae + Be + e GS BF ( ) y c s CF + c s PI, provie GS arbitrary constants (c) 0, y A + B BF Only ft if eponentials in GS y Ae Be + e 6e As, ( e 0 an) e 0 E As, y 0 B When A 0, 0 +B B 6 y 6e + e B y 6e + e OE Total

93 MFP (cont) (a) ln ln... k ln ± f ( ), with f() not involving the original ln ln (+ c) 9 Conone absence of + c (b) Integran is not efine at 0 E OE (c) e ln 0 { lim e ln a 0 a } lim a a ln a a 0 e e ln e 9 But lim a 0 a ln a 0 OE 9 E F(e) lim a 0 [F(a)] Accept a general form eg lim k ln 0 e So e ln 0 9 CSO Total 7 y (cot ) y sin IF is ep ( cot ) an with integration attempte ln(sin) (+c) e (k) sin OE Conone missing +c IF sin scores y sin (cos y ) sin sin [ ysin ] sin sin LHS as ifferential of y IF PI 0 y sin sin sin F Ft on c s IF provie no ep. or logs y sin sin cos B sin sincos use y sin sin (sin ) m ep on both Ms y sin sin (+c) π π sin sin + c 6 6 m c 6 sin Total 0 Use of relevant substitution to stage s s or further or by inspection to ksin ACF ep on both Ms Bounary conition use in attempt to fin value of c after integration CSO no errors seen accept equivalent forms

94 MFP (cont) 5(a) y sec Chain rule + tan ACF for y () y (+ tan )(sec tan ) sec (sec ) (+ tan ) Quotient rule OE in which both u an v are not const. or applie to a correct form of y ACF for y () (b) McC. Thm: y(0) + y (0) + y (0) (y(0) 0); y (0) ; y (0) Attempt to evaluate at least y (0) an y (0). PI ( tan ) ln + Dep on previous 5 marks (c) ln( )... ( tan ) ln + ln( ) So lim 0 ( tan ) ln + ln( ) B m Ignore higher power terms Epansions use Diviing num. an en. by to get constant term in each an non-const term in at least num. or en. F ft c s answer to (b) provie answer (b) is in the form ± p± q... an B aware Total 0

95 MFP (cont) 6(a) y u y Differentiating subst wrt, two terms correct u DE becomes u ( + )( + ) ( u+ ) u ( + ) + + u 6 Substitute into LHS of DE as far as no ys DE becomes u ( + ) u CSO AG (b) Separate variables OE PI u u + ln u ln + + ln A ; ln u; ln( + ) ( ) F u A( + ) OE RHS Applying law of logs to correctly combine two log terms or better y A ( ) + + m u f() to y f( ) ± ± y A B m 8 Total Solution with two arbitrary constants an both previous M an m score OE RHS

96 MFP (cont) 7(a) r sinθ r rsinθ + y y A, OE () either for r +y or for rsinθ y SC If M0 give B for r +y or for rsinθ y use (b)(i) sinθ tanθ Equating rs sinθcosθ sinθ sinθ( cosθ ) 0 Both solutions have to be consiere if m not in factorise form π sinθ 0 θ 0 ; cosθ θ Alternative: sin θ sinθ θ 0, π θ 0 r 0 ie pole O (0,0) B Inep. Can just verify using both eqns +statement. π π θ r P, CSO (ii) π π At A, θ, r sin π Substitute θ into the equations of π π At B, θ, r tan both curves. Since >, A is further away (from the pole than B.) E CSO (iii) P Area boune by line OP an curve C π sin θ θ 0 Use of r θ ; ignore limits here ( cos θ) θ m Attempt to write sin θ in terms of cosθ only θ sin θ Ignore limits here π 0 π PI Area boune by line OP an curve C O C C π tan θ θ 0 Use of r θ ; ignore limits here (sec θ ) θ m Using tan θ ± sec θ ± PI [ θ θ] Ignore limits here π π 0 6 PI π π Can awar earlier eg if we see Require area 6 π π sin θ θ tan 0 θ θ 0 π a, b 0 CSO Total 9 TOTAL 75

97 Version.0 General Certificate of Eucation (A-level) January 0 Mathematics MFP (Specification 660) Further Pure Final Mark Scheme

98 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 events which all eaminers participate in an is the scheme which was use by them in this eamination. The stanarisation process ensures that the mark scheme covers the stuents responses to questions an that every eaminer unerstans an applies it in the same correct way. As preparation for stanarisation each eaminer analyses a number of stuents scripts: alternative answers not alreay covere by the mark scheme are iscusse an legislate for. If, after the stanarisation process, eaminers encounter unusual answers which have not been raise 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 stuents 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 from: aqa.org.uk Copyright 0 AQA an its licensors. All rights reserve. Copyright AQA retains the copyright on all its publications. However, registere schools/colleges 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 schools/colleges 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 5 6EX.

99 Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE euct marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) 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. 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.

100 (a) y(.) y() (b) y(.) y() + (0.){f[., y(.)]}.0. + (0.).0 +. F ft on c s answer to (a) to p CAO Must be.06 Total ( O ) Attempt to use binomial theorem OE The notation O( n ) can be replace by a term of the form k n + + O( + + ) + O( ) + m Division by stage before taking the limit + lim 0 + CSO NMS 0/ Total m + m PI m ± i Complementary function is (y ) e ( Acos + B sin ) F OE Ft on incorrect comple value of m Particular integral: try y ke k + k + 0k 6 k (GS y ) e ( Acos + B sin ) + e BF c s CF+ c s non-zero PI but must have arb consts 0, y 5 5 A + so A BF ft c s k ie A 5 k, k 0 y e ( Asin + Bcos Acos Bsin ) + e Attempt to ifferentiate c s GS (ie CF + PI) B A + ( B ) y e (cos + sin ) + e 0 CSO Total 0

101 (a) IF is ep ( ) an with integration attempte e ln PI [ y ] ln y LHS; PI ln (ln ) y ln + A 9 { y ln + A } 7 9 Attempt integration by parts in correct irection to integrate p ln (b) Now, as 0, k ln 0 E Must be state eplicitly for a value of k > 0 As 0, y 0 A 0 B Const of int 0 must be convincing RHS y ln 9 When, y BF 9 Total 0 ft on one slip but must have mae a realistic attempt to fin A

102 5(a) The interval of integration is infinite E OE (b) u e + u (e e ) u/ or better ( ) + e ln u ( )e e + u u ln e + {+ c} + c ( ) OE Conone missing c. Accept later substitution back if eplicit (c) I ( ) + e a a + e ( ) lim lim a e a ln( + )} {ln ( e ) a + Uses part (b) an F(a) F(/) ln{ lim Now lim e a } ln( + ) ( e a + ) a a ( a ) a e I ln ln( e 0 E State eplicitly (coul be in general form) + ) CSO ACF Total 8

103 6(a) y ln cos y () ( sin ) cos Chain rule y () sec m λ sec OE y () 8sec (sec tan ) K sec tan OE {y () 6tan (sec )} y () 6[sec (sec ) + tan (sec (sec tan ))] 6 Prouct rule OE ACF (b) y(0) 0, y (0) 0, y (0), y (0) 0, y (0) ln cos ( ) ( )! BF ft c s erivatives CSO throughout parts (a) an (b) AG (c) ln (sec ) ln (cos ) PI 8 + Total

104 7(a) u y u y Prouct rule OE y + OE y y y + ( + ) u OE y y + ( + ) + y( + ) 8 y y y ( + ) + 6 ( + y) + 9y 8 u u u 8 CSO AG Be convince (b) u u u 8 CF: Au eqn m + 6m PI ( m + ) 0 so m PI CF: (u ) e (A + B) F PI: Try (u ) p + q PI. Must be more than just state 0 + 6p + 9(p + q) 8 9p 8, 6p + 9q 0 m p ; q 9 Both e c s CF + c s PI but must have constants, u ( A + B) + BF also must be in the form u f() e y ( A + B) + y {e ( A + B) + } 8 Total

105 8(a) Area ( cos ) + θ θ Use of π (9 cos cos + θ + θ ) π 0 0 θ B B (.5 + 6cos θ + ( + cos θ)) θ π π r θ or r 0 θ Correct epn of [ + cosθ ] Correct limits Attempt to write cos θ in terms of cos θ.5θ + 6sinθ + θ + sin θ 0 F Correct integration ft wrong coefficients π 6 CSO (b)(i) + y r 8r cosθ r 8cosθ Use of any two of r cosθ, y r sinθ, + y r At intersection, 8 cosθ + cosθ cos θ 6 Equating rs or equating cosθ s with a further step to solve eqn. (OE eg r + r r r ) π Points, an 5π, OE π AB sin Vali metho to fin AB, ft c s r an θ values 6 OE sur (ii) Let Mcentre of circle then AMB π B Accept equiv eg AMO π Length of arc AOB of circle π Use of arc ( AMB in ras) Perimeter of segment AOB 8π + Total 5 Alternative to (b)(i): Writing r + cosθ in cartesian form () Fining cartesian coorinates of points A an B ie (, ± ) () Fining length AB () TOTAL 75

106 Version.0 General Certificate of Eucation (A-level) June 0 Mathematics MFP (Specification 660) Further Pure Mark Scheme

107 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 events which all eaminers participate in an is the scheme which was use by them in this eamination. The stanarisation process ensures that the mark scheme covers the stuents responses to questions an that every eaminer unerstans an applies it in the same correct way. As preparation for stanarisation each eaminer analyses a number of stuents scripts: alternative answers not alreay covere by the mark scheme are iscusse an legislate for. If, after the stanarisation process, eaminers encounter unusual answers which have not been raise 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 stuents 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 from: aqa.org.uk Copyright 0 AQA an its licensors. All rights reserve. Copyright AQA retains the copyright on all its publications. However, registere schools/colleges 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 schools/colleges 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 5 6EX.

108 Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE euct marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) 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. 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.

109 MFP : June 0 k 0.5 ( + 9 ) (.5) k 0.5 f (.5, 9 +.5) k 0.5 ( ) k.(07 ) y (.5) y () + [k + k ] [.5 +. (07 )] (07 ) y (.5) (to p) m 5 PI. May see within given formula Either k 0.5 f (.5, 0.5) state/use or k 0.5 ( c 's k ) PI. May see within given formula k.(07 ) p or better PI by later work Dep on previous two Ms an y () 9 an numerical values for k s CAO Must be 0.9 Total 5 (a) () () 5 sin +...! 5! B Accept ACF even if unsimplifie (b) lim sin 0 ln( + k) 5 lim ( +...) 5 0 k (k) lim k k lim O( ) 0 k O( ) 6 k k B m Using series epansions. Epansion of ln ( + k) k ( ) Diviing numerator an0 enominator by to get constant term in each. Must be at least a total of terms ivie by OE eact value. Dep on numerator being of form /(OE) + λ... (λ 0) an enominator being of form k + µ.. (µ 0) before limit taken Total 5

110 Use of r (θ ) Area ( + tan θ ) (θ ) 0 π ( + tanθ ) θ 0 [θ + ln secθ ] π π π 0 + ln sec π π π ln ln ln B B Correct limits. If any contraiction use the limits at the substitution stage k ( + tanθ ) (θ ) k (θ + ln secθ ) ACF ft on c s k CSO AG Total (a) PI IF is e + e ln( +) ( + c ) ln( +) ( +c ) e (A)(+) F Either O.E. Conone missing + c Ft on earlier e λ ln(+), conone missing A ( + ) y + ( + ) y ( + ) 7 [( + ) y] ( + ) 7 LHS as / (y c s IF) PI an also RHS of form p ( + ) q ( + ) y ( + ) 7 ( + ) y ( + ) 8 (+ c) (GS): y ( + ) 6 + c( + ) BF 7 Correct integration of p ( + ) q to p( + ) q + (+ c) ft for q> only (q + ) Must be in the form y f (), where f () is ACF (b) y ( + ) 6 + c( + ) When 0, y 0 y y ( + ) 5 c( + ) B Using bounary conition 0, y 0 an c s GS in (a) towars obtaining a value for c Either y or correct epression for y/ in terms of only c so y ( + ) 6 + ( + ) CSO Total 0

111 5(a) e e e k e k e () for k ± e + { e e } e e e ( +c ) m e λe λe () for λ ± in n application of integration by parts Conone absence of + c (b) lim a I e e 0 0 lim { a e a ae a e a } [ ] a lim a k e a 0, (k>0) a e 0 a E F(a) F(0) with an inication of limit a an F() containing at least one n e, n > 0 term For general statement or specific statement for either k or k CSO Total 6(a) y ln( + sin ), y (cos ) + sin Chain rule OE ACF eg e y cos (b) y ( + sin )( sin ) cos (cos ) ( + sin ) y sin y e ( + sin ) + sin e y Quotient rule OE, with u an v non constant ACF CSO AG Completion must be convincing (c) y e y y y y y e y + e y y e y y (e y ) B y ACF for Prouct rule OE an chain rule y OE in terms of e y an only () y(0) 0; y (0) ; y (0) ; y() y(0)+y (0)+ y (0)+ y (0)+ y (iv) (0)!! y (0) ; y (iv) (0) BF Ft only for y (0); other two values must be correct Maclaurin s theorem applie with numerical values for y (0), y (0), y (0) an y (iv) (0). M0 if missing an epression for any one of the st, r or th erivatives ln( + sin ) + 6 A0 if FIW Total

112 7(a) y t y t e t y y y y t t y y y y ; t t t t y t y OE Relevant chain rule eg t OE eg y y e t t OE. Vali st stage to ifferentiate y () oe with respect to t or to ifferentiate y (t) oe with respect to. y y y + t t y y y + t y y + 6 y + 0 sin(ln ) becomes y y y + 6 y + 0 sin(ln ) t y y y + 0 sin(ln e t ) t t y 5 y + 6 y + 0 sin t t t m m 7 Prouct rule (ep on previous M) y y y OE eg e t e t + e t t {Note: e t coul be replace by /} Substitution to reach a one-step away stage for LHS. Dep on previous M M m CSO AG (b) Aul eqn m 5m (m )(m ) 0, m, CF: ( y ) Ae t + Be t C P.Int. Try ( y P ) a + b sin t + c cos t (y (t)) b cos t c sin t (y (t)) b sin t c cos t Substitute into DE gives a 0.5 5c + 5b 0 an 5c 5b 0 b c GS ( y ) Ae t + Be t + sin t + cos t + F F B BF PI Ft wrong values of m provie real roots, an arb. constants in CF. Conone for t here Conone a missing here ft can be consistent sign error(s) Substitution an comparing coefficients at least once OE Ft on c s CF + PI, provie PI is non-zero an CF has two arbitrary constants an RHS is fn of t only (c) y A + B + sin (ln ) + cos (ln ) B CAO Total 9

113 8(a) y 8 r cos θ r sin θ 8 r sin θ 8 m Use of sin θ sin θ cos θ r 6 6 cosec θ sin θ AG Completion (b)(i) (At N, r is a minimum sin θ ) N, π BB B for each correct coorinate. (ii) At pts of intersection, ( ) 6 cosec θ sin θ θ π 5π, 6 6 π 5π,, PI by cosec θ an a correct eact or SF value for θ or θ PI OE eact values Both require, written in correct orer (iii) POQ 5π π π or PON π ( QON) 6 BF Ft on c s θ P, θ Q, θ N as appropriate OE N N PN ( ) + (r ) ( ) r cos POQ PT sin Fining the lengths of two unequal sies or POQ of ΔPNQ or ΔPNT or ΔQNT, where T is or PT the point at which ON prouce meets PQ. Any vali equivalent methos eg fining tan OPN or fining sin ONP. or NT cos POQ r N PN (8 6 6 ) [.96(7855 )] NQ or PT [.8(87 )] or PQ or NT 6 [0.898(979 )] tan α PT [.66 ] OE NT 6 or α π π tan or PN ( cos α ) cos α 6 m Two correct unequal lengths of sies of ΔPNQ or ΔPNT or ΔQNT PI OE eg tan OPN / ( ) or sin ONP /( ) Vali metho to reach an eqn in α (or in α ) only; ep on prev M but not on prev A. Alternative choosing eg obtuse ONP then α π.87(85...) α ; α to sf 5 Total TOTAL 75.5 Conone >sf.

114 Version General Certificate of Eucation (A-level) January 0 Mathematics MFP (Specification 660) Further Pure Final Mark Scheme

115 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 events which all eaminers participate in an is the scheme which was use by them in this eamination. The stanarisation process ensures that the mark scheme covers the stuents responses to questions an that every eaminer unerstans an applies it in the same correct way. As preparation for stanarisation each eaminer analyses a number of stuents scripts: alternative answers not alreay covere by the mark scheme are iscusse an legislate for. If, after the stanarisation process, eaminers encounter unusual answers which have not been raise 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 stuents 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 from: aqa.org.uk Copyright 0 AQA an its licensors. All rights reserve. Copyright AQA retains the copyright on all its publications. However, registere schools/colleges 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 schools/colleges 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 5 6EX.

116 Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE euct marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) 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. 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.

117 MFP - AQA GCE Mark Scheme 0 January series (a) y(.) y() to p Conone >p (b) y(.) y() + (0.){f [., y(.)]}. 5 + (0.) F Ft on can s answer to (a) ( 5 + (0.) ) 6.89 to p CAO Must be 6.89 Total 6 Ignore higher powers beyon throughout this question (a) e B (b) 5 / / ±+k or +k±7.5 OE +7.5 OE (simplifie PI) e / ( )( +7.5 ) Prouct of c s two epansions with an attempt to multiply out to fin term term(s): Total 5

118 MFP - AQA GCE Mark Scheme 0 January series PI: k e y PI y ' PI y '' PI ke k e ke k e k e m Prouct rule use in fining both erivatives ke ke k e ke k e + 6e m Subst. into DE k e k 6 ; k ; y PI e CSO (GS: y ) e (A+B) + e BF 5 e (A+B) + k e, ft c s k. Total 5 (a) Integran is not efine at 0 E OE (b) 5 5 ln ln k ln f, with f() not involving the original ln 5 5 ln (+ c) 5 5 ln { 0 5 lim a 0 ln } a lim 5 5 a a [ ln a ] a Limit 0 replace by a limiting process an F() F(a) OE But lim a 0 5 a ln a 0 E lim Accept k ln 0 for 0 any k>0 So ln Dep on M an A marks all score Total 7

119 MFP - AQA GCE Mark Scheme 0 January series 5 y sec y tan tan (a) sec IF is ep ( ) tan an with integration attempte e ln(tan) tan AG Be convince (b) y tan (sec ) y tan y tan tan LHS as ifferential of y IF PI y tan tan y tan (sec ) m Using tan ± sec ± PI or other vali methos to integrate tan y tan tan (+c) Correct integration of tan ; conone absence of +c. tan tan c m Bounary conition use in attempt to fin value of c c so y tan tan + y +( ) cot 6 ACF Total 8

120 MFP - AQA GCE Mark Scheme 0 January series ln e cos ln e + ln cos + ln cos B 6(a)(i) y y ( sin ) cos y tan CSO AG Chain rule for erivative of ln cos (ii) y y sec ; sec (sec tan ) B; for /{ [f()] } f()f () y sec (sec tan ) tan sec ACF (b) Maclaurin s Thm: ( iv) y(0)+ y (0)+ y (0)+ y (0) + y (0)!!! y(0) ln 0; y (0) ; y (0) ; (iv) y (0) 0; y (0) ln e 0 cos 0...!!! F Mac. Thm with attempt to evaluate at least two erivatives at 0 At least of 5 terms correctly obtaine. Ft one miscopy in (a) CSO AG Be convince (c) {ln(+p)} p p B accept (p) for p ; ignore higher powers; ()(i) {ln e cos ln( p)} O( ) p p O( ) Law of logs an epansions use; For lim 0 e cos ln to eist, p p convincingly foun p (ii) lim p p Divie throughout by before ( ) O( ) m taking limit. (m can be aware 0 before or after the above) p Value of limit. Must be convincingly obtaine Total

121 MFP - AQA GCE Mark Scheme 0 January series 7(a) y y t Solving 6 0y e (*) t t Aul. Eqn. m 6m (m ) + 0 PI Completing sq or using quaratic formula to fin m. m ± i CF (y CF ) e t (A cos t + B sin t) OE Conone for t here; ft c s non-real values for m. (b) For PI try (y PI ) ke t Conone for t here k k 0k k GS of (*) is (y GS ) e t t CF +PI with arb. constants an (A cos t + B sin t) + e BF 6 both CF an PI functions of t only y t y t OE Chain rule y y t OE y y t y y () + t t t t y y ()() + t t t ( f ( t) ) ( f ( t) ) OE t eg ( g( ) ) ( g( ) ) t t y y y t 5 CSO A.G. t t m Prouct rule OE use ep on previous being aware at some stage (c) t t y t t y t y y { 6 + 0y} t t t t 0 so ivie by t gives y ( t )t t + 0t y t e t e t y y t 6 0y e t (*) CSO A.G. t Subst. using (b) into given DE to eliminate all () y e ( Acos B sin ) e B OE Must inclue y Total

122 MFP - AQA GCE Mark Scheme 0 January series 8(a)(i) 9 r sin ( cos ) ; (ii) ON ( )/8 Polar eqn of PN is r cos ON r sec 8 AG Be convince (iii) Area Δ ONP 0.5 r N r P sin (/) OE With correct or ft from (a)(i) (ii), values for r P an r N. (b)(i) n sin cos u n u n sin n (+c) Be convince PI (ii) Area of shae region boune by line OP an arc OP sin cos B Use of r Correct limits ( cos ) + sin cos cos sin cos B sin cos sin cos cos sin 8 + (sin sin ) cos 5 sin sin sin 8 5 m Correct integration of 0.5( cos) Writing n integran in a suitable form to be able to use (b)(i) OE PI Last two terms OE CSO Total 6 TOTAL 75

123 Version.0 General Certificate of Eucation (A-level) June 0 Mathematics MFP (Specification 660) Further Pure Final Mark Scheme

124 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 events which all eaminers participate in an is the scheme which was use by them in this eamination. The stanarisation process ensures that the mark scheme covers the stuents responses to questions an that every eaminer unerstans an applies it in the same correct way. As preparation for stanarisation each eaminer analyses a number of stuents scripts: alternative answers not alreay covere by the mark scheme are iscusse an legislate for. If, after the stanarisation process, eaminers encounter unusual answers which have not been raise 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 stuents 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 from: aqa.org.uk Copyright 0 AQA an its licensors. All rights reserve. Copyright AQA retains the copyright on all its publications. However, registere schools/colleges 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 schools/colleges 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 5 6EX.

125 Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE euct marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) 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. 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.

126 MFP- AQA GCE Mark Scheme 0 June series k 0. ( ) ( 0. ) PI. May be seen within given formula. k 0.6(0 ) ( *) Accept p or better as evience of the line. 0. f (., *...) 0. (..6...) (. c's k) (. c' s k) PI May be seen within given formula.. 0.(96.) p or better. PI by later work [ ] y(.) y()+ k k m Dep on previous two Ms but ft on c s numerical values for k an k following evaluation of these. (.95 ). to p 5 CAO Must be. SC Any consistent use of a MR/MC of printe f(,y) epression in applying IEF, mark as SC for a correct ft final p value otherwise SC0. Total 5 8 y y + 6 y ( 00) B OE If polar form before epn of brackets awar the B for correct epansions of both (rcos m) an (rsin n) where (m,n) ( 8, 6) or (m,n) (6, 8) r + 6r cos r sin 0 st for replacement using any one of {[ + y r, r cos, y r sin ](*)} {r0, origin} Circle: r sin 6cos Total n for use of (*) to convert the form +y +a+by0 correctly to the form r +arcos +brsin 0 or better

127 MFP- AQA GCE Mark Scheme 0 June series (a) y y y 8e P. Integral : ypi y ' PI b ce y '' PI a b c e ce Prouct rule use at least once giving 6ce 9ce 6ce 9ce b ce a b ce 8e 6ce terms in the form ± pe ± qe Substitution into LHS of DE b ; b a 0; c 8 m Dep on n M only Equating coeffs to obtain at least two of these correct eqns; PI by correct values for at least two constants b ; c ; a A,,0 5 Dep on m all aware if any two correct; A if all three correct but o not awar the n A mark if terms in e were incorrect in the line [ y PI e ] (b) Au. eqn. m m 0 (m+)(m ) 0 Factorising or using quaratic formula OE PI by correct two values of m seen/use ( ycf ) Ae Be ( ygs ) Ae Be e c s CF + c s PI with arbitrary constants, BF non-zero values for a,b an c an no trig or ln terms in c s CF (c) 0, y A + B BF Only ft if previous BF has been aware Ae y Be e 6e As, ( e 0 an) e 0 E Must treat e separately y (As, so) B 0 5 When B 0, A A B B0, where B is the coefficient of e. y 5 e e Total

128 MFP- AQA GCE Mark Scheme 0 June series ln( +) ln(+) {+c} B B OE (I ) lim a a 0 lim replace by a (OE) an a seen or taken at any stage lim a ln ln() a 0 Remaining marks are ep on getting pln( +)+qln(+) after integration, where p an q are non-zero constants lim a lim a lna ln a ln ln a ln (a ) ln ln 9 Dealing with the 0 limit correctly an using lnp lnqln(p/q) at least once at any stage either before or after using F( ) F(0). OE lim a ln a 9 a a ln ln 9 Writing F(a) OE in a suitable form when consiering a. OE I 0 ln 9 ln ln CSO Total 6

129 MFP- AQA GCE Mark Scheme 0 June series 5(a) ln ln ln B ACF (b)(i) y y 9 ln An IF is ep { [ /( ln )] () }. an with integration attempte e ln(ln) ln AG Must see e ln(ln) before ln (ii) y ln y 9 ln y ln 9 ln LHS as ifferential of y ln PI y ln 9 ln y ln ln [ ] ln m k ln or better () p ln p () y ln ln (+c) ACF Conone missing +c When e, y e, e e e + c c e m Dep on previous m. Bounary conition use in attempt to fin value of c after integration is complete y ln ln + e ( e ) y ln 6 ACF Total 9

130 MFP- AQA GCE Mark Scheme 0 June series 6(a) y sin so y +sin y y y y cos y y cos y (a) Altn y / sin (cos ) () Chain rule y y cos () () (b) y y y + sin When 0, y, y y, + 0 F Correct ifferentiation of y y Ft on RHS of line as ksin y y + y y y + y cos m Correct LHS When 0, y y CSO (b) Altn y ( sin ) / / y 8 (cos ) ( sin ).5 sin (cos ) sin ( sin ) ( cos sin ) sin (cos )( sin ) sin cos.5 () () (m) () Sign an numerical coeffs errors only. ACF Sign an numerical coeffs errors only. ACF When 0, y () (5) CSO (c) McC. Thm: y(0) + y (0) + y (0) +! y (0) Maclaurin s theorem use with c s numerical values for y(0), y (0), y (0) an y (0), all foun with at least three being non-zero. sin CSO Previous 6 marks must have been score Total 9

131 MFP- AQA GCE Mark Scheme 0 June series 7(a) sin y y sin cos y sin cos y u sin y u sin + u cos y u u u sin + cos + cos u sin Both erivatives attempte an prouct rule use at least twice. Both correct u sin u + cos sin usin u sin cos u sin cos +usin sin cos m Substitution into original DE (b) u sin + usin [ sin cos +] sin cos u Nee to see clear use of sin + usin [ sin + sin ] sin cos the trig ientity (Divie throughout by sin u,) u sin cos u u sin 5 AG Completion, be convince u For u sin, au eqn, m +0 m ± i PI CF: (u ) Asin + B cos OE For PI try (u) psin p sin p sin sin Conone etra terms provie their coefficients are shown to be zero p Correct Particular integral GS for u Asin + B cos GS: y Asin + B sin cos sin BF ug(), where g() c s (CF+PI) with two arb. constants, PI 0 an all real. Can be implie by net line. sin sin 6 yf() with ACF for f() Total

132 MFP- AQA GCE Mark Scheme 0 June series 8(a) At intersections of r an r sin sin Elimation of r 7 sin, π, π Any one correct solution of sin 6 6 (P) 7π, 6, (Q) π 7π,, 6 6 an π, 6 (b)(i) Angle between OA an initial line π BF If not correct, ft on P π 6 π π When, r sin ; 6 6 π, 6 B (ii) OA, OQ π Angle AOQ π Q P If not correct, ft on π Q P. BF OE eg Cartesian coors of A an Q both attempte an at least one correct ft. AQ ()() cos AOQ () Vali metho to fin AQ (or AQ ). Ft on c s r A for OA AQ ACF but must be eact sur form. (iii) Since ( ) so 90 E Justifying why (angle OQA) 90º OE angle OQA90º AQ is a tangent (c) Area of minor sector OPQ of circle E () Q P π Area of minor region OPQ of curve π 6 (sin sin 9) 7π 6 Must have convincingly shown that OQA 90º () Q P PI by combine Use of ( cos sin 9) sin cos 9 F π 6 77π 6 π { } { π Area of shae region π 7 π π 6 7π OE term later. π/ r or use of r B r sin +sin +9 P Use of cos ±±sin with k r ( ) Ft wrong non zero coefficients, ie for correct integration of a + bcos + csin OE eg } () eg π 77π 6 6 π π 6 6 Q P 9 CSO π Total 9 TOTAL 75 6 Q ( sin) P. (m, n )

133 A-LEVEL MATHEMATICS Further Pure MFP Mark scheme 660 June 0 Version/Stage: v.0 Final

134 Mark schemes are prepare by the Lea Assessment Writer an consiere, together with the relevant questions, by a panel of subject teachers. This mark scheme inclues any amenments mae at the stanarisation events which all associates participate in an is the scheme which was use by them in this eamination. The stanarisation process ensures that the mark scheme covers the stuents responses to questions an that every associate unerstans an applies it in the same correct way. As preparation for stanarisation each associate analyses a number of stuents scripts: alternative answers not alreay covere by the mark scheme are iscusse an legislate for. If, after the stanarisation process, associates encounter unusual answers which have not been raise they are require to refer these to the Lea Assessment Writer. It must be stresse that a mark scheme is a working ocument, in many cases further evelope an epane on the basis of stuents 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 from aqa.org.uk Copyright 0 AQA an its licensors. All rights reserve. AQA retains the copyright on all its publications. However, registere schools/colleges 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 schools/colleges to photocopy any material that is acknowlege to a thir party even for internal use within the centre.

135 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE euct marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) 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. 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. of 0

136 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE Q Solution Mark Total Comment DO NOT ALLOW ANY MISREADS IN THIS QUESTION ln( 6 + ) PI. May be seen within given formula k 0. (0.8) ln k 0. f (6., + k) ln( ) ln( c' s k) 0. ln.8 0. ln( + c' s k) PI. May be seen within given formula k (59 ) or better. PI by later work + [ (59...) ] m + [ c' s k + c' s k ] but epenent on previous two Ms score. PI by.78 or.779. ( ).78 (to p) 5 CAO Must be.78 Total 5 y(6.) y(6)+ [ k + k ] Q Solution Mark Total Comment (a) y a + bsin + c cos y b cos c sin B Correct epression for y bcos csin+(a+bsin+ccos) ( 0 0cos) Differentiation an substitution into LHS of DE a 0; b c 0; b + c 0 m Equating coefficients OE to form equations at least two correct. PI by net line a 5, b, c (b) Au. eqn. m + 0 PI Or solving y ()+y0 as far as yae ± OE ( y CF ) Ae OE ( y GS ) Ae + 5 sin cos BF c s CF + c s PI with eactly one arbitrary constant When 0, y A y e + 5 sin cos y e + 5 sin cos ACF Total 8 of 0

137 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE Q Solution Mark Total Comment r r cosθ use r + r + 6 r ( + ) + y ) ( ) 6 ( + y y r use Must be in form y f() but accept ACF ( + 7)( ) for f() eg y 6 Accept y Total ( + ) 6 an apply ISW if incorrect simplification after seeing this form. 6 Q Solution Mark Total Comment Au eqn m m 0 ( m )( m + ) 0 Correctly factorising or using quaratic formula OE for relevant Au eqn. PI by correct two values of m seen/use. ( y CF ) Ae + Be Try ( y PI ) ae ( y' PI ) ae ae Prouct rule OE use to ifferentiate e ( y' ' PI ) ae + ae in at least one erivative, giving terms in the form ±e ±e ae + ae ( ae ae ) ae (e ) m Subst. into LHS of DE a A0 if terms in e were incorrect in m a line BF ( ygs ) c s CF + c s PI, must have eactly ( y GS ) Ae + Be e two arbitrary constants As, e 0 (an e 0) E As, e 0 OE. Must be treating e term separately y 0 so B0 B B 0, where B is the coefficient of e ( y' ( ) Ae 0.5e + 0.5e ) ( y '(0) A 0.5 A.5) 5 y 5 e e B 0 y e e OE Total 0 5 of 0

138 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE Q Solution Mark Total Comment 5(a). sin 8 sin 8 () 8 8 k sin 8 k sin 8 (), with k,, 8, 8, /8 or /8 sin 8 sin 8 () 8 8 sin 8 + cos8 (+c) 8 6 (b) sin lim 0 + O( ) sin Ignore higher powers of. PI by answer. + O( ) CSO Must see correct intermeiate step [ ] (c) cot an / are not efine at 0 E Only nee to use one of the two terms. Conone Integran not efine at lower limit OE () ( ( cot + cos8 ) ) ln sin ln + sin 8 + cos8 BF Ft c s answer to part (a) 8 6 ie ln sin ln + c s answer to part (a) π lim (...) (...) 0 a 0 π lim Limit 0 replace by a (OE) an a 0 a seen or taken at any stage with no remaining lim relating to π/. π sin 8 cos8 π / (...) ln 0 lim sin a ln(π/) a 0 ln a lim sin a a 0 ln a π ln 6 6 lim sin a a 0 ln a F(π/) F(0), with ln[(sin)/] a term in F(), an at least all non ln terms evaluate π π ln ln ln OE single term in eact form, eg ln. π (a) Total 0 Eample: u, v cos 8 ; u, v sin 8 an. uv 8 v u all seen an substitution into uv v u with no more than one miscopy, awar the 6 of 0

139 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE Q Solution Mark Total Comment 6(a) + IF is e PI With or without the negative sign ln( ) ( ) e + +c ln( + ) ( +c) e Either O.E. Conone missing +c (A)( +) F λ ln( ) Ft on earlier e +, conone missing A ( u + ) ( + ) u (b) [ ( + ) u] LHS as /(u c s IF) PI ( + ) u (+C) Conone missing +C here. + (GS): u ( + C)( ) 6 Must be in the form u f(), where f() is ACF y u so u y y + u y y ± ± p, p 0 y y ( + ) + 8 u y + [ u y + u + ( ) ] ( ) ( ) u y + 8 m Substitution into LHS of DE an correct ft simplification as far as no y s present. Given DE becomes: u ( + ) u ( + ) u u ( ) + + CSO AG (c) From (a), u ( + C )( + ) y ( + C)( + ) y c's f( ) answer to part (a) So state or use y C C (b) C y ln + + C + D OE Total u u ± ± pu u y Altn: y, p 0 () () ( ) ( ) 7 of 0

140 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE Q Solution Mark Total Comment 7(a)(i) y sin + cos Chain rule OE (sign errors only) y ln (cos +sin ), cos + sin ACF eg e y y () cos sin (cos + sin ) ( sin + cos ) y (cos + sin ) m Quotient rule (sign errors only) OE eg e y [y ] +e y y ±cos ± sin (a)(ii) (cos + sin (cos + sin ) ) + cos sin y CSO AG Completion must be convincing + sin y ( sin ) y + cos B ACF for (b)(i) y(0) 0; y (0) ; y (0) ; y (0) BF Ft only for y (0) an y (0) y() y(0)+y (0)+ y (0)+ y (0)! Maclaurin s theorem applie with numerical vals. for y (0), y (0) an y (0). M0 if can is missing an epression OE for the st or r erivatives y() + 6 CSO AG Dep on all previous 7 marks + aware with no errors seen. (b)(ii) (c) B ln( cos sin ) cos ln ln cos ( ) e ln(cos) ln[ ( cos + sin )( cos sin )] B ln ( cos + sin ) + ln( cos sin ) B cos ln e + + CSO Must have use Hence. Total (a)(i) For guiance, working towars AG may inclue y [y ] 8 of 0

141 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE Q Solution Mark Total Comment 8(a) π (Area) π ( tan θ ) sec θ (θ ) Use of (b) (i) π (or) ( tan ) sec ( ) 0 θ Let utan θ so (Area) ( u ) u θ θ B r (θ ) or use of ( ) r θ 0 Correct limits () Vali metho to integrate tan n θ sec θ, (0) n or, coul be by inspection. 5 Correct integration of k ( tan θ ) sec θ u u (Area) u + OE; ignore limits at this stage CSO AG + ( 0) ( tan θ ) secθ sec θ Elimination of r or θ. [ r (r) r ] m tan θ ( + tan θ ) Using + tan θ sec θ OE to reach a correct equation in one unknown. tan π θ ; θ ± ; r 6 π π Coorinates,, 6 6 (b) (ii) π 7 7 sinα () sin π α OE 6 BF OE eg AP or eg sin α. 7 8 π π sinα sin cosα + cos sinα 6 6 B Or cosα 7 8 tanα OE Vali metho to reach an eact numerical epression for tan α. tanα (k ) Altn for the two B marks π π (BF) OE Any two correct ft. PI eg NP/ ON cos ; AN sin ; 6 6 (N is foot of perp from A or B to OP) OP tan OPA (B) tan OPA OE or tan PAN OE [Then ()() as above] (b)(iii) Since tan α is negative, α is obtuse so point A lies insie the circle. (If A was on EF Ft c s sign of k. the circle α woul be a right angle.) Total TOTAL 75 Altn (a) Converts to Cartesian eqn. y ( ) (); sets up a correct integral with correct limits for the area using the (b)(ii) alt sym of the curve (B); vali metho to integrate ( ) (); 8/5 obtaine convincingly () + π Altn epressions for : tan tan π α + OPA ; tanα tan + PAN 6 π OE 9 of 0

142 MARK SCHEME A-LEVEL MATHEMATICS MFP JUNE 0 of 0

143 A-LEVEL Mathematics Further Pure MFP Mark scheme 660 June 05 Version/Stage: Final Mark Scheme V