Version,0 General Certificate of Education (A-level) June 0 Mathematics MPC (Specification 660) Pure Core Mark Scheme
Mark schemes are prepared by the Principal Eaminer and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation events which all eaminers participate in and is the scheme which was used by them in this eamination. The standardisation process ensures that the mark scheme covers the students responses to questions and that every eaminer understands and applies it in the same correct way. As preparation for standardisation each eaminer analyses a number of students scripts: alternative answers not already covered by the mark scheme are discussed and legislated for. If, after the standardisation process, eaminers encounter unusual answers which have not been raised they are required to refer these to the Principal Eaminer. It must be stressed that a mark scheme is a working document, in many cases further developed and epanded on the basis of students reactions to a particular paper. Assumptions about future mark schemes on the basis of one year s document should be avoided; whilst the guiding principles of assessment remain constant, details will change, depending on the content of a particular eamination paper. Further copies of this Mark Scheme are available from: aqa.org.uk Copyright 0 AQA and its licensors. All rights reserved. Copyright AQA retains the copyright on all its publications. However, registered schools/colleges for AQA are permitted 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 acknowledged to a third party even for internal use within the centre. Set and published by the Assessment and Qualifications Alliance. The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales (company number 67) and a registered charity (registered charity number 07). Registered address: AQA, Devas Street, Manchester 5 6EX.
Key to mark scheme abbreviations M mark is for method m or dm mark is dependent on one or more M marks and is for method A mark is dependent on M or m marks and is for accuracy B mark is independent of M or m marks and is for method and 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 rounds to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE deduct marks for each error NMS no method shown PI possibly implied SCA substantially correct approach c candidate sf significant figure(s) dp decimal place(s) No Method Shown Where the question specifically requires a particular method to be used, we must usually see evidence of use of this method for any marks to be awarded. Where the answer can be reasonably obtained without showing working and it is very unlikely that the correct answer can be obtained by using an incorrect method, we must award full marks. However, the obvious penalty to candidates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the candidate to state or write down a result, no method need be shown for full marks. Where the permitted calculator has functions which reasonably allow the solution of the question directly, the correct answer without working earns full marks, unless it is given to less than the degree of accuracy accepted in the mark scheme, when it gains no marks. Otherwise we require evidence of a correct method for any marks to be awarded.
MPC - AQA GCE Mark Scheme 0 June series y 0.5.96 0.7.878 0.9 0.950. 0.77 B All correct values (and no etras used) + y decimal values rounded or truncated to dp or better (in table or in formula) (PI by correct answer) 0. y m ( = 0. 7. ) =. A CAO Total Correct substitution of their y values (of which are correct), either listed or totalled
MPC - AQA GCE Mark Scheme 0 June series (a) f ln Or reverse f 0.5.5 must haveboth valuescorrect f.5 0. Allow f0.5 0 and f.5 0only if f() defined Change of sign 0.5 5 A f() must be defined and all working correct, including both statement and interval (either may be written in words or symbols) (b) ln or e Must be seen e B AG; no errors seen OR comparing sides: ln 0.5.8 0.5 0.7 () ln.5.6.5. At 0.5, LHS < RHS; at.5, LHS > RHS 0.5.5 (A) (c).9 B. B If B0B0 scored but either value seen correct to or dp, score SC (d) Vertical line from to curve (condone omission from -ais to y = ) and then horizontal to y = A nd vertical and horizontal lines, and, (not the values) must be labelled on -ais Total 7
MPC - AQA GCE Mark Scheme 0 June series dy (a) ln d p q ln where p and q are integers A p, q (b)(i) dy d e e lne e Substituting e for in their d y, but must d have scored in (a) y elne e B y e e e A OE but must have evaluated ln e (twice) for this mark (must be in eact form, but condone numerical evaluation after correct equation) (ii) e e e or e e OE Correctly substituting y = 0 into a correct tangent equation in (b)(i) e A CSO; ignore subsequent decimal evaluation Total 7 (a) u e 6 d dv 6 e (d ) du v ke (d ) 6 A All terms in this form, k 6 k,or 6 6 6 6 Correct substitution of their terms into e e d 6 AF 6 parts formula 6 6 e e c OE A No ISW for incorrect simplification 6 6 (b) 6 π e d 0 V B Must include π, limits and d 6 6 π e e 6 6 6 5 6 π e 6 6 Correct substitution of 0 and into their answer in (a), must be of the form 6 6 ae be, where a > 0, b > 0 and F() F(0) seen A CAO; ISW Total 7
MPC - AQA GCE Mark Scheme 0 June series 5(a) f 0 A Condone y 0 f 0, f 0, 0, y 0, range 0 (b)(i) 0 fg 5 0 5 OE B No ISW (ii) 0 5 5 0 5 5 Correctly squaring their fg() and correctly isolating their term A No ISW (c)(i) y 5 Swap and y either order Correctly squaring (ii) 5 f ( ) A 9 or if 9 or seen Candidate must have scored full marks in (c)(i) (ie no follow through) = and = rejected A Must see both Total 0
MPC - AQA GCE Mark Scheme 0 June series 6 u du d B or du d 7 d ( ) 7 Either epression all in terms of u ku ( ) ku ( ) du du or including replacing d, but condone u u ( u ) omission of du du m u u ln u u A k au bu du, where k, a, b are constants Must have seen du on an earlier line where every term is a term in u ln u ln u 0 ln ln ln m Dependent on previous A Correct change of limits, correct substitution and F() F() or correct replacement of u, correct substitution and F() F(0) A 6 OE in eact form Total 6
MPC - AQA GCE Mark Scheme 0 June series 7(a) Modulus graph, sections touching -ais at,, A Correct, A Correct with maimum at lower than maimum at and correct cusps at =, = and = The maimums need to be at = and (appro) (b) Symmetrical about y-ais, from original curve for 0 < < and > A Correct graph including cusp at = 0 (c) Translate 0 Stretch (I) either order sf (II) / / y-ais (III) E B A I and (either II or III) I + II + III (d) B y 5 B Each value may be stated or shown as coordinates Total
MPC - AQA GCE Mark Scheme 0 June series 8(a) LHS cos cos cos cos Combining fractions A Correctly simplified cos m Use of sin cos sin cosec cosec 6 A AG; no errors seen OR cos cos cos cos cos (A) sin (m) cosec 6 (A) (b) cosec y ( ) 6 or better (PI by further or sin y ( ) working) 6 or better (y =) 0.5, (.889,) (.9,) (6.0,) ( 0.5) B Sight of any of these correct to dp or better (y =) 0.5,.89,.9 (or better) A Must see these answers, with or without either/both of 0.5 or 6.0 Ignore answers outside interval 0.5 to 6.0 but etras in this interval scores A0 () = 0.,.7, (.00), 0.7 B B 5 Total 9 correct (must be dp) All correct (must be dp) and no etras in interval (ignore answers outside interval)
MPC - AQA GCE Mark Scheme 0 June series d cosy cos y sin y sin y 9(a) dy cos y Condone incorrect signs, poor notation, omission of d dy or using d y d cos y sin cos y y A RHS correct with terms squared, including correct notation Must see this line cos y d d or tan sec y y A y CSO Must see one of these AG; all correct including correct use of d dy throughout sec y (b) OE A Correct use of sec y tan y and in terms of AG; must see sec y, ( ) epanded and no errors seen (c) d dy dy d sec y or Must be seen dy d B AG and no errors seen
MPC - AQA GCE Mark Scheme 0 June series 9 cont (d)(i) y tan ln dy d Must be correct dy 0 d bc ( 0) m Epression in this form (generous), where b and c 0 0 A Must see correct equation = 0, A Both answers must be seen The two A marks are independent (ii) y p q where p and q are constants A p =, q = including correct brackets y (iii), y Must have scored full marks in (d)(i) and (ii) At, y 0 min Must see y 0 or in words When =, y = 0 hence on -ais A Both statements fully correct Total TOTAL 75