PMT. Version : klm. General Certificate of Education. Mathematics MPC2 Pure Core 2. Mark Scheme examination - June series

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1 Version : klm Generl Certificte of Eduction Mthemtics 660 MPC Pure Core Mrk Scheme 009 exmintion - June series

2 Mrk schemes re prepred by the Principl Exminer nd considered, together with the relevnt questions, by pnel of subject techers. This mrk scheme includes ny mendments mde t the stndrdistion meeting ttended by ll exminers nd is the scheme which ws used by them in this exmintion. The stndrdistion meeting ensures tht the mrk scheme covers the cndidtes responses to questions nd tht every exminer understnds nd pplies it in the sme correct wy. As preprtion for the stndrdistion meeting ech exminer nlyses number of cndidtes scripts: lterntive nswers not lredy covered by the mrk scheme re discussed t the meeting nd legislted for. If, fter this meeting, exminers encounter unusul nswers which hve not been discussed t the meeting they re required to refer these to the Principl Exminer. It must be stressed tht mrk scheme is working document, in mny cses further developed nd expnded on the bsis of cndidtes rections to prticulr pper. Assumptions bout future mrk schemes on the bsis of one yer s document should be voided; whilst the guiding principles of ssessment remin constnt, detils will chnge, depending on the content of prticulr exmintion pper. Further copies of this Mrk Scheme re vilble to downlod from the AQA Website: Copyright 009 AQA nd its licensors. All rights reserved. COPYRIGHT AQA retins the copyright on ll its publictions. However, registered centres for AQA re permitted to copy mteril from this booklet for their own internl use, with the following importnt exception: AQA cnnot give permission to centres to photocopy ny mteril tht is cknowledged to third prty even for internl use within the centre. Set nd published by the Assessment nd Qulifictions Allince. The Assessment nd Qulifictions Allince (AQA) is compny limited by gurntee registered in Englnd nd Wles (compny number 6447) nd registered chrity (registered chrity number 074). Registered ddress: AQA, Devs Street, Mnchester M5 6EX Dr Michel Cresswell Director Generl

3 MPC - AQA GCE Mrk Scheme 009 June series Key to mrk scheme nd bbrevitions used in mrking M m or dm A B E mrk is for method mrk is dependent on one or more M mrks nd is for method mrk is dependent on M or m mrks nd is for ccurcy mrk is independent of M or m mrks nd is for method nd ccurcy mrk is for explntion or ft or F follow through from previous incorrect result MC mis-copy CAO correct nswer only MR mis-red CSO correct solution only RA required ccurcy AWFW nything which flls within FW further work AWRT nything which rounds to ISW ignore subsequent work ACF ny correct form FIW from incorrect work AG nswer given BOD given benefit of doubt SC specil cse WR work replced by cndidte OE or equivlent FB formule book A, or (or 0) ccurcy mrks NOS not on scheme x EE deduct x mrks for ech error G grph NMS no method shown c cndidte PI possibly implied sf significnt figure(s) SCA substntilly correct pproch dp deciml plce(s) No Method Shown Where the question specificlly requires prticulr method to be used, we must usully see evidence of use of this method for ny mrks to be wrded. However, there re situtions in some units where prt mrks would be pproprite, prticulrly when similr techniques re involved. Your Principl Exminer will lert you to these nd detils will be provided on the mrk scheme. Where the nswer cn be resonbly obtined without showing working nd it is very unlikely tht the correct nswer cn be obtined by using n incorrect method, we must wrd full mrks. However, the obvious penlty to cndidtes showing no working is tht incorrect nswers, however close, ern no mrks. Where question sks the cndidte to stte or write down result, no method need be shown for full mrks. Where the permitted clcultor hs functions which resonbly llow the solution of the question directly, the correct nswer without working erns full mrks, unless it is given to less thn the degree of ccurcy ccepted in the mrk scheme, when it gins no mrks. Otherwise we require evidence of correct method for ny mrks to be wrded.

4 MPC - AQA GCE Mrk Scheme 009 June series MPC Q Solution Mrks Totl Comments () 5 = cosθ M Use of the cosine rule must be correct (PI by the correct line below) cos θ = ( = = ) 7 8 m Rerrngement θ = 8.. = 8.º (to nerest 0.º) A CSO (Must see either exct vlue for cosθ or t lest 4sf vlue for either cosθ or θ before the printed nswer 8.º) AG (b) Are = 7 8sinθ OE eg Are = 0(0 5)(0 8)(0 7) M (= 00 ) = 7. {cm } to sf A Condone 7. to 7. inclusive Totl 5 () (n =) 4 B Accept x 4 (b) (c) + x = B,,0 x x + dx = x 6x x + c M x A,,0 Apply ISW fter B stge (B if correct but unsimplified seen) At lest one power of x correctly obtined in the integrtion of n expnsion A terms correct nd + c (AF two terms in x correct ft on expnsion provided integrting x to negtive power) (d) + dx = x 6 x x x = ( 6 ) = 6 7 M A CSO; Totl 8 Deling correctly with limits; F() F() (must hve ttempted integrtion to get F) OE provided vlue is exct, eg ISW dec vlue fter exct vlue seen NMS scores 0/ 80 40, 9 7 ;

5 MPC - AQA GCE Mrk Scheme 009 June series MPC (cont) Q Solution Mrks Totl Comments () 4 = 6k + M Condone with 0.75 (OE) subst for k k = 6 = 0.75 A AG; OE frction; if verifiction must explicitly stte the conclusion (b) u = 0 B u 4 = 4.5 BF ft on 0.75 cnd s u + (c)(i) L = 0.75L + M Replcing u n+ nd u n by L (ii) L = k = 0.75 m L = 48 A Totl 7 4() h = B PI g(x) = x + I h/{ } { } = g(0) + g(6) + [g() + g(4)] M PI, but previous M must be scored SC: (c)(i) incorrect nd then in (c)(ii) L = 0.75L + leding to L = 48 scores B OE summing of res of the trpezi.. Cn wrd even if MR expression for g(x) but must be using from 0 to 6 { } = ( + 65) ( ) A OE Accept dp evidence for surds (I ) = 7.86 (to 4sf) A 4 Must be 7.86 (b) f(x) = ( x ) + = 8x + M kx +, k or 0 or f(x) = g(x) A Either form cceptble Totl 6

6 MPC - AQA GCE Mrk Scheme 009 June series MPC (cont) Q Solution Mrks Totl Comments 5() dy 45 5 x x dx M One power correctly obtined A,,0 A for ech term on the RHS coeffs simplified (b) 45 5 x x = 0 M cnd s () = 0 5 ( 9 ) 0 x x = Must be solving eqn of form x m +bx n = 0, m m nd n non-zero, with t lest one of m nd n non-integer nd reching stge from which the non-zero vlue of x cn be stted PI. Must del with powers of x p correctly nd ny squring of kx terms or expressions must be correct. At M, x = 9 A y M = 6 A 4 M must be scored, else 0/4 (c) At P(, 4), d y = 45 5 = 0 M Attempt to find y () dx Tngent t P: y 4 = m(x ) m m = cnd s vlue of y () y 4 = 0x 0; y = 0x 6 A CSO; AG (d) Tngent t M: y = 6 BF ft y = cnd s y M At R, 6 = 0x 6; x = 8.4 M Solving cnd s numericl y M = 0x 6 to find vlue for x Distnce RM = xm xr = = 0.6 AF ft on coordintes of M Totl 6 {Are of sector =} r θ M r θ seen or used for the re; PI.75 r θ r =... or r =... r = 7.5 A PI eg by correct rc length {Arc =} rθ M rθ seen or used for the rc length. = 9 AF ft on. cnd s r provided the two M s scored; if not explicit, PI by ft on. cnd s r for perimeter {Perimeter =} 4 {cm} A 6 CAO Totl 6

7 MPC - AQA GCE Mrk Scheme 009 June series MPC (cont) Q Solution Mrks Totl Comments 7()(i) r = 75; r 4 = 8 B For either OE or PI by next line 75r = 8 M Elimintion of OE r = 8 75 = 7 5 = 0.6 r = 0.6 A CSO AG Full vlid completion SC: Cler explicit verifiction, with sttement mx B out of. (If considers uniqueness then is possible) (ii) 0.6 = 75 M OE; PI = 65 A (b) (c) r = M used with vlue of r < 0.6 r S = 65 = 56.5 AF ft on cnd s vlue for. ie un = 6 u = 0.6 u n 5 un M u (= 5) nd u 4 = 0.6 u (= 5) M Vlid method to either find u nd u4 or n ( r ) use of { Sn = } for either n = 5 or r n = 6 5 u = (= 44) A n un = =.5 A 4 6 Alterntive for (c): Recognise tht the sum to infinity with first term u 6 is required Vlid method to find u 6 (= 0.6 u 5 ) (M) (M) un = (A) =.5 (A) Totl

8 MPC - AQA GCE Mrk Scheme 009 June series MPC (cont) Q Solution Mrks Totl Comments 8() sinθ cosθ = 4 cosθ cosθ tnθ = 4 M sinθ tnθ = stted or used cosθ tnθ = 5 A AG; CSO (b)(i) cos x sin x = ( sin x) sin x = M Use of cos x + sin x = sin x sin x = sin x + sin x = 0 A AG; CSO (ii) (sin x + )(sin x ) = 0 M Fctoristion or use of formul; PI by both correct vlues for sin x sin x =, sin x = 0.5 A Need both (sin x = ) so x = 70º B (sin x = 0.5) so x = 0º A 0º s the only cute ngle x = 80 0 = 50º BF 5 ft for nd ngle from c s sinx = non-integer Totl 9 Ignore vlues outside intervl 0º 60º but extrs inside intervl lose the corresp. B, A or BF mrk. If using rds, ccepting either equivlent exct vls (in terms of pi) or dp vlues insted of degrees, penlise mx of mrk from ny of the finl three mrks (BABF) wrded NMS: 70º (B); 0º, 50º (B) [mx /5]

9 MPC - AQA GCE Mrk Scheme 009 June series MPC (cont) Q Solution Mrks Totl Comments 9()(i) 5 = 5 5 = 5 5 M OE eg 5 = 5 or.5 5 seen 5 p = 5 p =.5 A Correct vlue of p must be explicitly stted Alterntive for ()(i): p log 5 = log5 (M) plog5 = log5 p= (A) OE eg p log5 = log.8 or eg p = log5 5 Correct vlue of p must be explicitly stted (ii) x p 5 = 5 = 5 x = 0.5p = 0.75 BF Must be 0.5 c s vlue of p SC: x = 0.75 with c s ns ()(i) BF.5 5 scores (b) = 0.05 (x )log = log0.05 M Tke logs of both sides nd use rd lw of logs. PI eg by x = log 0.05 seen log 0.05 log 0 x = + m Correct rerrngement to x =. PI 0 = 0.864(65 ) = to 4dp A Condone > 4dp. Must see logs clerly used in solution, so NMS scores 0/ (c) ( ) log x = log + log log = log ( ) M A vlid lw of logs used = log (6 ) M Another vlid lw of logs used = log 6 log B log = quoted or used x x or log = = OE k k 6 x = log 6 6 x = A 4 CSO Must be x = or x = 6 Totl 0 TOTAL 75