Version 1.0. General Certificate of Education (A-level) January Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

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1 Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm

2 Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions, by a panl of subjct tachrs. This mark schm includs any amndmnts mad at th standardisation vnts which all aminrs participat in and is th schm which was usd by thm in this amination. Th standardisation procss nsurs that th mark schm covrs th studnts rsponss to qustions and that vry aminr undrstands and applis it in th sam corrct way. As prparation for standardisation ach aminr analyss a numbr of studnts scripts: altrnativ answrs not alrady covrd by th mark schm ar discussd and lgislatd for. If, aftr th standardisation procss, aminrs ncountr unusual answrs which hav not bn raisd thy ar rquird to rfr ths to th Principal Eaminr. It must b strssd that a mark schm is a working documnt, in many cass furthr dvlopd and pandd on th basis of studnts ractions to a particular papr. Assumptions about futur mark schms on th basis of on yar s documnt should b avoidd; whilst th guiding principls of assssmnt rmain constant, dtails will chang, dpnding on th contnt of a particular amination papr. Furthr copis of this Mark Schm ar availabl from: aqa.org.uk Copyright 0 AQA and its licnsors. All rights rsrvd. Copyright AQA rtains th copyright on all its publications. Howvr, rgistrd schools/collgs for AQA ar prmittd to copy matrial from this booklt for thir own intrnal us, with th following important cption: AQA cannot giv prmission to schools/collgs to photocopy any matrial that is acknowldgd to a third party vn for intrnal us within th cntr. St and publishd by th Assssmnt and Qualifications Allianc. Th Assssmnt and Qualifications Allianc (AQA) is a company limitd by guarant rgistrd in England and Wals (company numbr 67) and a rgistrd charity (rgistrd charity numbr 07). Rgistrd addrss: AQA, Dvas Strt, Manchstr M5 6EX.

3 Ky to mark schm abbrviations M mark is for mthod m or dm mark is dpndnt on on or mor M marks and is for mthod A mark is dpndnt on M or m marks and is for accuracy B mark is indpndnt of M or m marks and is for mthod and accuracy E mark is for planation or ft or F follow through from prvious incorrct rsult CAO corrct answr only CSO corrct solution only AWFW anything which falls within AWRT anything which rounds to ACF any corrct form AG answr givn SC spcial cas OE or quivalnt A, or (or 0) accuracy marks EE dduct marks for ach rror NMS no mthod shown PI possibly implid SCA substantially corrct approach c candidat sf significant figur(s) dp dcimal plac(s) No Mthod Shown Whr th qustion spcifically rquirs a particular mthod to b usd, w must usually s vidnc of us of this mthod for any marks to b awardd. Whr th answr can b rasonably obtaind without showing working and it is vry unlikly that th corrct answr can b obtaind by using an incorrct mthod, w must award full marks. Howvr, th obvious pnalty to candidats showing no working is that incorrct answrs, howvr clos, arn no marks. Whr a qustion asks th candidat to stat or writ down a rsult, no mthod nd b shown for full marks. Whr th prmittd calculator has functions which rasonably allow th solution of th qustion dirctly, th corrct answr without working arns full marks, unlss it is givn to lss than th dgr of accuracy accptd in th mark schm, whn it gains no marks. Othrwis w rquir vidnc of a corrct mthod for any marks to b awardd.

4 MPC Q Solution Marks Total Commnts (a) y B B all 7 valus corrct (and no tra) (PI by 7 corrct y valus) 5 or mor corrct y valus, act,... or valuatd (in tabl or in formula) A [ corrct substitution of thir 7 y-valus into ] M Simpson s rul 9 or 5.5 or 7 A CAO 6 (b)(i) f( ) + 8 or g( ) 8 f (. ) 0. or g(. ) 0. f (. ) 0.7 or g(. ) 0.7 M attmpt at valuating f(.) and f(.) AWRT ± 0. and ± 0.7 altrnativ mthod condon f (.) < 0, f (.) > 0 if f is. 5., dfind. 6., M chang of sign. < α <. A at. LHS < RHS (f() must b dfind and all working corrct) at. LHS > RHS. < α <. A (ii) ( ). B ( ). B ths valus only Total 8

5 MPC (cont) Q Solution Marks Total Commnts (a) f () f (6) M sight of and f ( ) A allow f() rplacd by f, y (b)(i) y 6 6 y ( y ) 6 or bttr M M rvrs, y on corrct stp Eithr ordr f 6 ( ) + OE A condon y (ii) 6 + 6, or bttr + on corrct stp from thir (b)(i), M or f() ( ) A not: scors / fg( ) 6 B (c)(i) ( ) (ii) 6 6 or bttr M on corrct stp from thir (c)(i) 6 OE A g ( + 8) ( 8) 0, or ± ONLY A Total

6 MPC (cont) Q Solution Marks Total Commnts (a) dy 6 d B do not ISW (b) 6+ d ( ) () ln 6 M + 6 () A ln ( 6 + ) 6 ln ( 6 + ) 6 m ln 9 ln 6 6 AF 9 ln or ln 6 6 A 5 Total 6 ( + ) k ln 6, k is a constant k 6 corrct substitution in F() F(). condon poor us or lack of brackts. kln 9 kln only follow through on thir k or if using th substitution u 6+ du k M u ln u A 6 thn, ithr chang limits to and 9 m thn AF Aas schm or changing back to, thn m AF A as schm (a) sc θ... B corrct us of sc θ + tan θ quadratic prssion in scθ with all sc θ + sc θ 0 ( 0) M trms on on sid ( θ )( θ ) sc + 5 sc 0 m attmpt at factors of thir quadratic, ( scθ ± 5)( scθ ± ), or corrct us of quadratic formula sc θ 5, A cos θ, 5 B 60, 00,0.5, 58.5 (AWRT) B 6 corrct, ignor answrs outsid intrval all corrct, no tras in intrval (b) 0 60, 0 5,58 5,00 M 0 any of thir (60), all thir answrs from (a), BUT must hav 70, 5,68 5,0 AF scord B 7 5, 7 9,67.,77 5 (AWRT) A CAO, ignor answrs outsid intrval Total 9

7 MPC (cont) Q Solution Marks Total Commnts 5(a) strtch I SF II MA I + (II or III) in y-dirction III ithr ordr translat E 0 B accpt in positiv -dirction (b) M A mod graph, in connctd sctions, both in th first quadrant, touching -ais curv touchs -ais at + (or.7 or bttr), and lablld (ignor scal) + A corrct curvatur, including at thir +, appro. asymptot at (c)(i) ln ( ) ( ) ( ) ln ln or bttr M must s quations, condon omission of brackts ( ) do not ISW A accpt valus of AWRT 5., 5., 5. + or ( ) + do not ISW A accpt valus of AWRT.08,.09 if M0 thn with or without working scors SC ( ) (ii) B accpt valus of AWRT 5., 5., 5. < + B accpt valus of AWRT.7,.08,.09 if B not arnd, thn SCfor any of +, < < +, < + Total

8 MPC (cont) Q Solution Marks Total Commnts 6(a) ( ) d sinθ 0 cosθ dθ sin θ M ± sinθ k ± cosθ quotint rul sin θ whr k 0 or must s th 0 ithr in th quotint or in A g d u 0 dθ cosθ cosθ or sin θ sinθsinθ or quivalnt coscθ cotθ A CSO, AG must s on of th prvious prssions (b) coscθ d coscθ cotθ dθ B OE, g d coscθ cotθdθ Rplacing cosc θ by cot θ, or bttr B at any stag of solution ( ) coscθcotθ ( θ ) cosc θ cosc dθ M all in trms ofθ, and including thir attmpt at d, but condon omission of dθ A coscθ cotθ ( dθ ) cosc θ cotθ ( dθ ) coscθ cosθ A, θ 0.5 AWRT B, θ AWRT fully corrct and must includ dθ (at som stag in solution) A OE g sinθ( dθ) corrct chang of limits ± cos ± OE or ( ) θ ( ) m c's F( 0.5 ) F( 0.79 ) 0.59 A 9 Total substitution into ± cos θ only or

9 MPC (cont) Q Solution Marks Total Commnts 7(a) M p, q constants dy p + q d A p and q E or 0 impossibl OE (may b sn latr) ( a + b ) 0 m or ( a + b ) 0 0, 8 A 0, y 0 A 8, y 6 B 7 condon 8 y 8 tc ignor furthr numrical valuation (b)(i) d d v u d du v k d M whr k is a constant k A (d ), or bttr AF corrct substitution of thir trms u m dv n d du m v n d m both diffrntiation and intgration must b corrct d 8 ( ) ( 0 ) Al + [ ] [ ] m (dp on M only) corrct substitution and attmpt at subtraction in a + b + c (may b in stags) 8 0 A 7 or 8 0 ignor furthr numrical valuation (ii) ( ) ( 0 ) ( ) v π 9 d M condon omission of brackts, limits 9π 8 0 AF 9π (thir act b(i)) Total 6 TOTAL 75

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