PMT. Version 1.0: abc. General Certificate of Education. Mathematics MPC4 Pure Core 4. Mark Scheme examination - January series

Transcription

1 Version.: 8 abc General Cerificae of Educaion Mahemaics 66 MPC Pure Core Mark Scheme 8 eaminaion - January series

2 Mark schemes are prepared by he Principal Eaminer and considered, ogeher wih he relevan quesions, by a panel of subjec eachers. This mark scheme includes any amendmens made a he sandardisaion meeing aended by all eaminers and is he scheme which was used by hem in his eaminaion. The sandardisaion meeing ensures ha he mark scheme covers he candidaes responses o quesions and ha every eaminer undersands and applies i in he same correc way. As preparaion for he sandardisaion meeing each eaminer analyses a number of candidaes scrips: alernaive answers no alrea covered by he mark scheme are discussed a he meeing and legislaed for. If, afer his meeing, eaminers encouner unusual answers which have no been discussed a he meeing hey are required o refer hese o he Principal Eaminer. I mus be sressed ha a mark scheme is a working documen, in many cases furher developed and epanded on he basis of candidaes reacions o a paricular paper. Assumpions abou fuure mark schemes on he basis of one year s documen should be avoided; whils he guiding principles of assessmen remain consan, deails will change, depending on he conen of a paricular eaminaion paper. Furher copies of his Mark Scheme are available o download from he AQA Websie: Copyrigh 8 AQA and is licensors. All righs reserved. COPYRIGHT AQA reains he copyrigh on all is publicaions. However, regisered cenres for AQA are permied o copy maerial from his bookle for heir own inernal use, wih he following imporan ecepion: AQA canno give permission o cenres o phoocopy any maerial ha is acknowledged o a hird pary even for inernal use wihin he cenre. Se and published by he Assessmen and Qualificaions Alliance. The Assessmen and Qualificaions Alliance (AQA) is a company limied by guaranee regisered in England and Wales (company number 67) and a regisered chariy (regisered chariy number 7). Regisered address: AQA, Devas Sree, Mancheser 6EX Dr Michael Cresswell Direcor General

3 MPC - AQA GCE Mark Scheme 8 January series Key o mark scheme and abbreviaions used in marking M m or dm A B E mark is for mehod mark is dependen on one or more M marks and is for mehod mark is dependen on M or m marks and is for accuracy mark is independen of M or m marks and is for mehod and accuracy mark is for eplanaion or f or F follow hrough from previous incorrec resul MC mis-copy CAO correc answer only MR mis-read CSO correc soluion only RA required accuracy AWFW anyhing which falls wihin FW furher work AWRT anyhing which rounds o ISW ignore subsequen work ACF any correc form FIW from incorrec work AG answer given BOD given benefi of doub SC special case WR work replaced by candidae OE or equivalen FB formulae book A, or (or ) accuracy marks NOS no on scheme EE deduc marks for each error G graph NMS no mehod shown C candidae PI possibly implied Sf significan figure(s) SCA subsanially correc approach Dp decimal place(s) No Mehod Shown Where he quesion specifically requires a paricular mehod o be used, we mus usually see evidence of use of his mehod for any marks o be awarded. However, here are siuaions in some unis where par marks would be appropriae, paricularly when similar echniques are involved. Your Principal Eaminer will aler you o hese and deails will be provided on he mark scheme. 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 penaly o candidaes showing no working is ha incorrec answers, however close, earn no marks. Where a quesion asks he candidae o sae or wrie down a resul, no mehod need be shown for full marks. Where he permied calculaor has funcions which reasonably allow he soluion of he quesion direcly, he correc answer wihou working earns full marks, unless i is given o less han he degree of accuracy acceped in he mark scheme, when i gains no marks. Oherwise we require evidence of a correc mehod for any marks o be awarded.

4 MPC - AQA GCE Mark Scheme 8 January series MPC (a) k( + + ) OE A + B 6A 6B + k A or eg pu, k + k 9 (b) ln( ) + ln( + ) 9 (( ) ( )) ln ln ln ln ln AF aln( ± ) f on k AF accep ln Toal f only for sign error in inegral: ln ( ) 8

5 MPC - AQA GCE Mark Scheme 8 January series (a)(i) f + 8( ) + 8 use of ± subsiued in f () arihmeic seen and conclusion facor A minimum seen: (ii) f( ) ( )( 8) + BB or p, q 8 (iii) ( + ) ( )( + )( ) ( )( ) A CAO numeraor correc; aemp o facorise denominaor (algebraic fracion no required) A + + B+ C (b) ( )( ) A B any equivalen mehod using PFs (see alernaive mehod) A+ C A+ B equae coefficiens or use values of o find B and C C B A boh B and C correc ALTERNATIVE METHOD + + () complee division + A (B) B (A) C (A) ALTERNATIVE METHOD A + D + E + + ( )( ) A( + )( ) + D( ) + E( + ) 9 8 8E E () 8D D, A+ 9 ( )( ) + ( )( ) A (B) D E ( + ) ( ) ( ) + 9( + ) ( + )( ) 6 ( + )( ) () ( + )( ) (A) Toal find D and E recombine o required form CAO

6 MPC - AQA GCE Mark Scheme 8 January series MPC (con) (a) ( ) k 8 A replaced by condone missing (b) A correc evaluaion brackes, bu no incorrecly placed brackes eg ( ) alernaively, sar again and find correc epression manipulaion o k ( answer o (b)) and (c) + k + 8 evaluaed a+ b+ c A a, b, c fracions or decimals only Or use ( a+ ) n formula (condone one error for ) Toal 6 (a)(i) A B (ii) k 6 A ( ) 6 k.7977 A AG; or log k 6. or 6 or ln 6 e e or e or or.7977(6) seen 888 (iii) P k 78 A CAO neares (b) equae prices.7977 as a single inde, or correc log.879 epression a his sage log.7 log.9979 m epression for A SC Answer only/trial and error 6 seen ( ou of ) 99 ( ou of ) Toal 9 6

7 MPC - AQA GCE Mark Scheme 8 January series (a)(i) + y ( ) ( ) y A (ii) d d + A A and d d aemped in boh derivaives use chain rule; epressions can be in erms of or evaluaed CAO or any equivalen fracion (no decimals) 9 y + ( ) BF f on, y and gradien 7 if y m + c used, c mus be found correcly and he equaion mus be rewrien (b) eiher correc epression y + y or boh of y and + y + y ( y) eliminae ( y)( + y) or ( y)( + y) ( ) A k alone, no marks Toal 6 + y y aemp implici differeniaion A produc A chain B consan A CSO ALTERNATIVE METHOD y+ solve for epression in y and y () differeniae wih respec o y y (AA) y, d () subsiue y (A) CSO Toal 7

8 MPC - AQA GCE Mark Scheme 8 January series MPC (con) 7(a)(i) R B R an α 8, α. 6 (ii) sin ( +.).7 BF For α; f incorrec R +.. AF one correc answer ; f α and R.6 or.7,.,9.6 or 9.7 A oher correc answers ignore eras. or., 7.6 or 7.7, A four soluions CAO (wih decimal place discrepancies). or.,.6 or.7 Answers only: / (b)(i) sin sin cos cos cos sin sin sin cos cos sin ( ) sin cos cos sin sin an B B A ideniies for sin and cos in any correc form use of candidae s double angle formulae AG, CSO (ii) an an an ± A (see * below), B,, A if answers given wihou working, B ma if an seen and followed by an correc answers wihou working ou of Toal * Commens for 7(b)(ii) If hence ignored, so working in sines and cosines, mus simplify as far as: cos sin or cos or cos or cos ± or sin for sin ± for A 8

9 MPC - AQA GCE Mark Scheme 8 January series MPC (con) 8 y cos aemp o separae and inegrae py qsin seen implies separaion sin ( ) y C sin π (, ) sin π + C use ( π, ) o find consan C y sin+ 6 A CSO (in any correc form) Toal 9(a)(i) AB A for ± ( OA OB) (ii) ( ) λ r + BF f on AB ; OE (b)(i) μ + + μ μ μ μ ALTERNATIVE METHOD μ, which is saisfied by μ 6 μ found and verified or saemen μ saisfies all componens A μ alone B (ii) PQ OQOP wih OQ in parameric + λ form in erms of λ (can be inferred laer) PQ + λ 8λ 6+ λ λ A or λ 7λ + λ 8 λ PQ wih PQ in erms of λ λ (can be inferred laer) ( + λ) + ( )( λ) m linear epression in λ equaed o λ. AF f on sign/arihmeic error in PQ or equaion Q A 6 CAO is (,,.) Toal TOTAL 7 9