Version 1.0: hij. General Certificate of Education. Mathematics MFP2 Further Pure 2. Mark Scheme examination - June series

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1 Version.0: 0609 hij General Certificate of Education Mathematics 6360 MFP Further Pure Mark Scheme 009 examination - June series

2 Mark schemes are prepared by the Principal Examiner and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation meeting attended by all examiners and is the scheme which was used by them in this examination. The standardisation meeting ensures that the mark scheme covers the candidates responses to questions and that every examiner understands and applies it in the same correct way. As preparation for the standardisation meeting each examiner analyses a number of candidates scripts: alternative answers not already covered by the mark scheme are discussed at the meeting and legislated for. If, after this meeting, examiners encounter unusual answers which have not been discussed at the meeting they are required to refer these to the Principal Examiner. It must be stressed that a mark scheme is a working document, in many cases further developed and expanded on the basis of candidates 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 examination paper. Further copies of this Mark Scheme are available to download from the AQA Website: Copyright 009 AQA and its licensors. All rights reserved. COPYRIGHT AQA retains the copyright on all its publications. However, registered centres for AQA are permitted to copy material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to centres 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 ) and a registered charity (registered charity number ). Registered address: AQA, Devas Street, Manchester M5 6EX Dr Michael Cresswell Director General

3 MFP - AQA GCE Mark Scheme 009 June series Key to mark scheme and abbreviations used in marking M m or dm A B E mark is for method mark is dependent on one or more M marks and is for method mark is dependent on M or m marks and is for accuracy mark is independent of M or m marks and is for method and accuracy mark is for explanation or ft or F follow through from previous incorrect result MC mis-copy CAO correct answer only MR mis-read CSO correct solution only RA required accuracy AWFW anything which falls within FW further work AWRT anything which rounds to ISW ignore subsequent work ACF any correct form FIW from incorrect work AG answer given BOD given benefit of doubt SC special case WR work replaced by candidate OE or equivalent FB formulae book A, or (or 0) accuracy marks NOS not on scheme x EE deduct x marks for each error G graph NMS no method shown c candidate PI possibly implied sf significant figure(s) SCA substantially correct approach 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. However, there are situations in some units where part marks would be appropriate, particularly when similar techniques are involved. Your Principal Examiner will alert you to these and details will be provided on the mark scheme. 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. 3

4 MFP - AQA GCE Mark Scheme 009 June series MFP (a) 4πi 4 = 6e M Allow M if π π = 6 cos + isin 3 3 A 4πi 4 = e πi OE could be ae 3 π π a cos isin = i; a = 8 AF 3 ft errors in (b) For other roots, r = B for realising roots are of form e iθ π kπ M for strictly correct θ θ = + 4 MA π i.e must be their + kπ 3 4 (a) 7πi 5πi πi Roots are e, e, e A =, B = A,, 0 F 5 Total 8 B, BF (b) Method of differences clearly shown M Sum = n + A n = n + A 3 AG 4 or ft error in π or r π kπ + i 4 accept e k =,, For either A or B For the other (c) n < 0.00 or > n+ ( n ) M Condone use of equals sign < 0.004n or n > 0.998n n > or 0.004n> A OE n = 50 AF 3 ft if say used If method of trial and improvement used, award full marks for a completely correct solution showing working Total 8 4

5 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) 3(a) + 3i B (b)(i) αβ = 3 B (ii) αβ+ βγ + γα = 5 M MA0 for -5 (no ft) γ ( α+ β) = AF γ = 3 AF 3 ft error in αβ (iii) p = α = 7 M AF M for a correct method for either p or q q = αβγ = 39 AF 3 ft from previous errors p and q must be real for sign errors in p and q allow M but A0 Alternative for (b)(ii) and (iii): (ii) Attempt at ( 3i)( 3i) + (M) 4+ 3 (A) cubic is ( )( ) γ = 3 (A) (3) (iii) Multiply out or pick out coefficients (M) p = 7, q = 39 (A, A) (3) Total 8 4(a) Sketch, approximately correct shape B Asymptotes at y =± B B0 if curve touches asymptotes lines of answer booklet could be used for asymptotes be strict with sketch sinh x (b) Use of u = cosh x M x x x e e e = or x x x e + e e + A ( ) x x x x u e + e = e e M M for multiplying up x ( + u) = ( u) x e e A for factoriing out e s or M for A attempt at + u and u in terms of e x x + u e = u m + u x = ln u A 6 AG 5

6 MFP - AQA GCE Mark Scheme 009 June series 4(c)(i) Use of tanh x = sech x M Printed answer A 3tanh x tanh x = 0 M Attempt to factorise tanh x E Accept tanh x written down but not ignored or just crossed out tanh x = 3 A M x = ln AF 5 ft Total 5 k cosθ + i sinθ = (ii) ( )( ) 5(a) ( ) ( cos kθ i sin kθ)( cosθ i sinθ) + + M Multiply out A Any form = cos( k + ) θ + isin ( k + ) θ A Clearly shown True for n = shown B P k P k + and P true E 5 provided previous 4 marks earned ( ) ( ) ( ) (b) cos i sin = n cos n i sin n n θ n θ θ + θ = MA n + = cosnθ n A 3 AG n or = cos( nθ) + isin ( nθ) SC ( cosθ + i sinθ) n quoted as cos nθ i sin nθ earns MA only (c) + = cosθ = M π θ = 4 A 0 0π + = cos 0 4 M = 0 AF 4 Total M0 for merely writing 0 + = cos0θ 0 6

7 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) 6(a) Centre i or (, ) B M Radius 5 ft incorrect centre if used AF + + i = 5or i = 5 AF 4 ft + + i = 0 earns M0B (b) ( ) C correct centre, correct radius ft errors in (a) but fit circles need to intersect and enclose ( 0, 0) C correct centre, correct radius B Touching x-axis BF 3 error in plotting centre (c) OO = 3 5 MA Correct length identified Length is allow if circles misplaced but OO is still 3 5 m M AF 5 ft if r is taken as 0 Total 7

8 MFP - AQA GCE Mark Scheme 009 June series MFP (cont) 7(a)(i) d s d y s = + = + dx dx MA 4 = + s A 3 AG ds (ii) = d x 4 + s M s Allow M for s = + dx then A for d y dx For separation of variables; allow without integral sign s sinh = x + C A Allow if C is missing C = 0 s = sinh x A A 4 AG if C not mentioned allow 3 4 SC incomplete proof of (a)(ii), differentiating x ds s = sinh to arrive at = 4 + s dx allow MA only ( 4 ) (iii) dy sinh x dx M y = cosh x+ C A Allow if C is missing C = 0 A 3 Must be shown to be ero and CAO (b) y x = 4 + sinh M Use of = 4 + s A AG Total TOTAL 75 cosh = + sinh 8

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