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 MPC (a)(i) 5 6 AB Multiply by denominator and use two values of. 0 A B Alternative: equate coefficients 6 A 5 AB A B () () Set up and solve simultaneous equations for values of A and B. (ii) dln +ln C Bft Bft their A ln their B ln ( ) and no other terms; condone B ln (b)(i) 5 - p q r 5 6 6 5-5 Division as far as p q with p 0, q 0, PI PI by PI by q seen seen and must state p=, q =, r = 5 eplicitly or write out full correct RHS epression Alternative : 5 p pqqr p 0 () pq5 qr p q r 5 () () Clear attempt to equate coefficients, PI by p = Alternative : 5 pq r 5 r r 5 p, q () () () used to find a value for r
MPC - AQA GCE Mark Scheme 0 June series MPC (b)(ii) 5 p q r kln C 5 ln C ft ft on p and q CSO Total (a) R 0 B Accept. or better. Can be earned in (b) tan 7.6 or better OE; M0 if tan seen 7.56505... (b) sin R or 9. 7.6. or their R and/or their ; PI or better Condone. 7.6 9. m must see 9 and 7 or better PI by 9 or better as answer Condone etra solutions 9 Total 7 Condone 90.8 or better CSO Withhold final if more than two answers given within interval
MPC - AQA GCE Mark Scheme 0 June series MPC (a) k (b)(i) 8 8 B OE Condone missing brackets and use of instead of 6 56 CSO 0.5 0.065 0.07(875) Alternative using formula from FB 5 6 56 () (A) Condone one error and missing brackets CSO Must be fully correct (b)(ii) or and B Condone Must be and; not or not, (comma) (c) 6 56 7 6 56 product of their epansions CSO 0.5.065 0.86(8...) Total 8
MPC - AQA GCE Mark Scheme 0 June series MPC (a)(i) 5 000.0 60 B Condone missing sign;60 only. (ii) 000 000 00 ln n ln.0 n B Condone = or < used throughout Take logs, any base, of their initial epression correctly n N.9... Condone (b) n.5 000 500 00 00 n B Condone use of T for n Condone = or < used throughout ln000 nln.0 ln500 nln.05 ln.5 n.0 ln.05 n7.6... T 8 Total 8 Take logs, any base, of their initial epression correctly Correct epression for n or T Condone 7
MPC - AQA GCE Mark Scheme 0 June series MPC 5 (a)(i) dy d dy d d d 6cos sin 6 sin sin 6sin cosec m condone coefficient errors Use cos sin a 6 b (a)(ii) dy 6 6 d gradient normal Bft Bft substituted into their d y 6 d and evaluated ft d y, provided non-zero d (b) y 6sincos 6 cos cos 6 9 y Correct epansion of sin and use cos to eliminate Correct elimination of 9 p OE and shown Alternative using verification y 9sin 6sin cos cos sin 9 p OE () () () Total 9 must be squared 9 or y
MPC - AQA GCE Mark Scheme 0 June series MPC 6 9 6y y 8 0 B =0 PI dy 6y6 d B or d(6 y ) d y 6y 6 seen separately d d dy 8y B dy d 6 8y 6y 8 d dy Use 0 d y or y CSO y 9 6 Substitute y ainto equation m and solve for a value of or y. Condone missing brackets. 7 OE,, ft 8 Total 8 Both values of or y required. ft on their y = CSO Correct corresponding simplified values of and y. Withhold if additional answers given
MPC - AQA GCE Mark Scheme 0 June series MPC 7(a) 8 Use the first two equations to set up and 5 attempt to solve simultaneous equations for or. Must not assume q =., q 5 q 5 Use rd equation to show q AG. P is at 6,, B Condone as a column vector (b) 0 5 0 perpendicular B or 0 seen and conclusion (condone = 90) (c)(i) A is at (,, ) 6 AP 0 Candidate s AP CAO NMS AP 0 A0 (ii) 5 8 6 PB 5 55 Clear attempt to find of BP or PB in terms PB 55 m Find distance BP in terms of 5 90 5 0 5 9 8 5 0 m Attempt to set up three-term quadratic in and equate to their AP 5 0 m Solve quadratic equation to obtain two values of 5 and 6 5 B is at,, and,, 6 Both values correct. Both sets of coordinates required. Condone column vectors. SC one value of correct and corresponding coordinates of B correct scores A0.
MPC - AQA GCE Mark Scheme 0 June series MPC Alternative 5 8 6 AB 5 55 () Clear attempt to find of AB or BA in terms AB 6 55 (m) Find distance AB in terms of 5 90 65 0 5 9 8 5 0 (m) Attempt to set up three-term quadratic in and equate to their their AP As before Alternative PB k 5 k 5 0 k OB OP their value of k 5 6 5 B is at,, and,, () (m) (m) () (m) () m for LHS m for equating to their 0 May score m0m Total
MPC - AQA GCE Mark Scheme 0 June series 8(a) dh dt B derivative h Use of h or h ; is a constant or epression in dh k h All correct; must be h dt (b)(i) d d t 5 t 5 Substitute u d u u du d u u u 5 u u 5 C, t 0 u, t 0 5 C 0 C 5 5 5 t Alternative (Parts) As before dv d u, du vk d d 5 C 5, t 0 C 0 5 C 5 5 t 5 (ii) t. minutes B B 8 (BB) () () () () () () Correct separation and notation; condone missing integral signs. h and/or t. Suitable substitution and attempt to write integral in terms of u including d replaced by du or du. need not be seen Integration correct including Use, t 0 to find a value for constant C from equation in and t. C 0.666... C 0.67 or better ISW t Attempt to use parts Condone missing d Use, t 0 to find a value for constant C from equation in and t C 0.666... C 0.67 or better ISW t B. or better (.7 ) Total TOTAL 75