Version. General Certificate of Education (A-level) January Mathematics MPC4. (Specification 6360) Pure Core 4. Final.

Transcription

1 Version General Cerificae of Educaion (A-level) January 0 Mahemaics MPC (Secificaion 660) Pure Core Final Mark Scheme

2 Mark schemes are reared by he Princial Examiner and considered, ogeher wih he relevan uesions, by a anel of subjec eachers. This mark scheme includes any amendmens made a he sandardisaion evens which all examiners ariciae in and is he scheme which was used by hem in his examinaion. The sandardisaion rocess ensures ha he mark scheme covers he sudens resonses o uesions and ha every examiner undersands and alies i in he same correc way. As rearaion for sandardisaion each examiner analyses a number of sudens scris: alernaive answers no already covered by he mark scheme are discussed and legislaed for. If, afer he sandardisaion rocess, examiners encouner unusual answers which have no been raised hey are reuired o refer hese o he Princial Examiner. I mus be sressed ha a mark scheme is a working documen, in many cases furher develoed and exanded on he basis of sudens reacions o a aricular aer. Assumions abou fuure mark schemes on he basis of one year s documen should be avoided; whils he guiding rinciles of assessmen remain consan, deails will change, deending on he conen of a aricular examinaion aer. Furher coies of his Mark Scheme are available from: aa.org.uk Coyrigh 0 AQA and is licensors. All righs reserved. Coyrigh AQA reains he coyrigh on all is ublicaions. However, regisered schools/colleges for AQA are ermied o coy maerial from his bookle for heir own inernal use, wih he following imoran exceion: AQA canno give ermission o schools/colleges o hoocoy any maerial ha is acknowledged o a hird ary even for inernal use wihin he cenre. Se and ublished by he Assessmen and Qualificaions Alliance. The Assessmen and Qualificaions Alliance (AQA) is a comany limied by guaranee regisered in England and Wales (comany number 67) and a regisered chariy (regisered chariy number 07). Regisered address: AQA, Devas Sree, Mancheser 5 6EX.

3 Key o mark scheme abbreviaions M mark is for mehod m or dm mark is deenden on one or more M marks and is for mehod A mark is deenden on M or m marks and is for accuracy B mark is indeenden of M or m marks and is for mehod and accuracy E mark is for exlanaion or f or F follow hrough from revious incorrec resul CAO correc answer only CSO correc soluion only AWFW anyhing which falls wihin AWRT anyhing which rounds o ACF any correc form AG answer given SC secial case OE or euivalen A, or (or 0) accuracy marks x EE deduc x marks for each error NMS no mehod shown PI ossibly imlied SCA subsanially correc aroach c candidae sf significan figure(s) d decimal lace(s) No Mehod Shown Where he uesion secifically reuires a aricular mehod o be used, we mus usually see evidence of use of his mehod for any marks o be awarded. Where he answer can be reasonably obained wihou showing working and i is very unlikely ha he correc answer can be obained by using an incorrec mehod, we mus award full marks. However, he obvious enaly o candidaes showing no working is ha incorrec answers, however close, earn no marks. Where a uesion asks he candidae o sae or wrie down a resul, no mehod need be shown for full marks. Where he ermied calculaor has funcions which reasonably allow he soluion of he uesion direcly, he correc answer wihou working earns full marks, unless i is given o less han he degree of accuracy acceed in he mark scheme, when i gains no marks. Oherwise we reuire evidence of a correc mehod for any marks o be awarded.

4 MPC Q Soluion Marks Toal Commens (a) f 7 Evaluae f (b) (i) g =0 + d 0 d gx x +x x7 g x x +x x x x x x x x g x x x B B division., no long Or f d 0 All ses seen wih conclusion AG Allow verificaion wih 0seen, and conclusion ; herefore facor a (iii) x x x xxx xxx x x xx x m Clear aem o facorise denominaor; facors needed. A leas one correc facor cancelled (b)(iii) x g x x x x Alernaive gx x 6x x x x x x x x x x x x x Toal 7 CSO ar (a)(iii) NMS is 0/ uadraic + x x x

5 Q Soluion Marks Toal Commens 7x xbx (a) Use wo values of x o find A x x m and B. Or solve A B AB 7 Or cover u rule A B (b) (i) x x x x x 9x Condone missing brackes x x x x x kx x x 9 7x xx x x x 9x 9 9 x x is ouside he range of validiy, because 0.. B B 7 Toal Condone missing brackes Aem o use PFs o combine exansions, or exand 7x x x and simlify o OE Acce abx cx 0.

6 Q Soluion Marks Toal Commens (a)(i) R an.7 minimum value when x cos x.7 B Bf Acce.6 or beer OE Acce.6or beer; f R NMS 0/ Calculus used 0/ (b)(i) LHS sin x co x co x 0 co x 0 co x0 or 0 x 90, 5, 5 x m m Exress co x in erms of and ; ACF Facor ou and sin All correc Boh euaions correc Condone missing 70 All correc x (b) (i) Alernaives RHS co x sin x sin x co x Toal m Exress co x in erms of and, cos x ACF sin mulily ou and simlify. All correc. x and co x 0 sin sin cos 0 x x x Rearrange o exression 0 and facor ou co x ; Exress co x,cos x and in erms of and, ACF sin sin cos 0 x x x 0 m sin x used Simlified, wih all correc

7 (b) Alernaive co x 0 0 sin x 0 or 0 x 90, 5, 5 m Boh euaions

8 Q Soluion Marks Toal Commens dy (a)(i) xy 0 d x Correc differeniaion dy x dx y dy, subsiued ino correc a, dx derivaive or x y saed AG angen a, y x B ACF angen a, y x B ACF add y 0 Solve angen euaions for y. conclusion y 0 inersec on Ox Conclusion reuired (b) x y x y Aem o suare x and y and subrac. All correc AG Allow Toal

9 (a)(i) Alernaive dy y x x x dx (a)(i) Alernaive dy dx d d dy x dx y dy a, dx x angen a, y x y x angen a, y When y 0 x and x x is on boh lines, so inersec on x axis B B Aem arameric derivaives and use chain rule., subsiued ino correc derivaive. ACF ACF Subsiue y 0ino boh candidae s angens and solve for x. Conclusion (b) x y x y x yx y x y Aem o eliminae

10 Q Soluion Marks Toal Commens 5(a) xx dx x By insecion or subsiuion x C (b) y e dy x x dx B Correc searaion and noaion Condone missing inegral signs e y B Euae o resul from (a) wih x C consan. C m Use, 0o find consan. CAO C 6 y ln x m Solve for y, aking logs correcly. y ln x 7 CSO Toal 9

11 Q Soluion Marks Toal Commens 6 (a)(i) 5 Mus see OC OA in correc AC OC OA 5 5 comonens. B n 5 (b) (c)(i) BC cosACB cos ACB cos ACB vecor euaion r B BC or CB correc Correc form of formula using consisen vecors; condone use of or a wrong angle and a missing mulile of 5 Correc scalar roduc and moduli. AG Mus see, or rearrangemen 5 5 cos ACB or 7 5 a d OE Euae vecor euaions for AC and BD. OE Se u euaions and solve for ; mus find a value for 6 AB CD 6 6 Clear aem o find he vecors of he sides. AD BC 6 6 All sides are of same lengh, 7 ; hence rhombus. m Toal 5 All vecors correc Find he lenghs of he sides, or sae hey all 9 if all correc. Each side 7 and conclusion. Or adjacdn sides 7 and oosie sides are arallel.

12 (c) Alernaive AC BD 55 0 AC and BD are erendicular inersecion is a midoin of AC and BD Calculae scalar roduc of AC and BD 0 from correc AC and BD and conclusion Find value of and aem o use in argumen abou oin of inersecion Diagonals bisec each oher a righ angles; hence rhombus, wih all sides eual o 7 Fully correc conclusion. Mus show diagonals bisec

13 Q Soluion Marks Toal Commens 7 (a)(i) 0 N 50 N 5 B B Mus be 5 (no 5.5..) (iii) (b) (i) (b) 500 9e 9e 00 e 6 ln e e d N N N N N 500 N d 000 d N N N 500 N 0 N 50 T 9e T e 9 T ln Alernaive, by insecion Max of N500 N m m m Toal TOTAL 75 occurs a N 50 B Correc algebra seen Or e 6 or 6ln 6 Clear aem a chain rule or uoien rule. 500 Use e 9 N o. eliminae e Correc algebra o AG Differeniae and aem o find N a max value d d Condone for d T 7 or beer CSO Acce 7,, 7.5, 7.6

14 (b)(i) Alernaives Alernaive imlici differeniaion 500 N e 9N d 500 e 9N N d 000 9N N N N N 500 N d 000 use e o ge d m Correc exressions for e and aem o use imlici differeniaion Fully correc Aem o eliminae e using correc exression Alernaive exlici differeniaion 500 N ln 9N 500 N d 9N 9N 500 N 9N 9N 9 9N 500 N 500 9N 500 N N 500 N d 000 m Correc exression for and aem a differeniaion wih use of chain rule and roduc for ln derivaive. Clear fracions wihin fracions All correc Or ln 500 N ln 9N d N 9N 500 N N N 500 N N500 N = N N N N 500 d 500 m Correc exression for and ln derivaives, condone sign errors Common denominaor o combine fracions All correc Alernaive solve differenial euaion

15 N d 500 N 000 d 500 N 500 N ln N ln 500 N C N 50 C ln 9 9N 9N ln e 500 N 500 N N9 e 500e 500e 500 N 9e 9 e m Searae variables, and aem o form arial fracions and inegrae o ln erms k C Use 50,0o find C and obain e f N Maniulae correcly o original given euaion.