Version.0: 008 abc General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 008 eamination - January series
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 meeting attended by all eaminers and is the scheme which was used by them in this eamination. The standardisation meeting ensures that the mark scheme covers the candidates responses to questions and that every eaminer understands and applies it in the same correct way. As preparation for the standardisation meeting each eaminer analyses a number of candidates scripts: alternative answers not alrea covered by the mark scheme are discussed at the meeting and legislated for. If, after this meeting, eaminers encounter unusual answers which have not been discussed at the meeting 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 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 eamination paper. Further copies of this Mark Scheme are available to download from the AQA Website: www.aqa.org.uk Copyright 008 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 eception: 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 67) and a registered charity (registered charity number 07). Registered address: AQA, Devas Street, Manchester M5 6EX Dr Michael Cresswell Director General
MPC - AQA GCE Mark Scheme 008 January 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 eplanation 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 EE deduct 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 Eaminer 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.
MPC - AQA GCE Mark Scheme 008 January series MPC (a)(i) y = ( 5+ ) 0 9 0( 5 ) ( 5) d = + OE M A 9 chain rule 0( ) f ( ) with no further incorrect working (ii) y= cos sin cos d = + M A product rule ± sin ± cos CSO y = ( ) = d ( ) 6 = ( ) ( ) = ( ) M A A CSO ± vu ' ± uv ' quotient rule ( ) condone missing brackets Total 7 (a) cot = tan = 0.5 M = 0.6,.6 A AWRT; no others within range cot + cosec = + cot = cot + M Correct use of cosec = + cot ( ) ( cot cot+ = 0) cot 0 cot = A AG; correct with no slips from line with no fractions cot + cot = 0 M Attempt to solve (c) ( )( )( ) cot =, A tan =, 0.5 = 0.6,.6,.0, 5.8 B B Total 8 correct Allow.6(0) correct (with no etras in range) AWRT SC Degrees 6.57, 06.57 B for correct 6.57, 96.57
MPC - AQA GCE Mark Scheme 008 January series MPC (cont) (a) + ( + ) = 0 f ( 0.) = 0. f ( 0.) = 0.0 M Change of sign 0.< < 0. A AWRT; allow + ve, ve = ( + ) = + M Attempt to isolate = A AG (c) = 0. ( 0.) ( ) = AWRT M = 0.9 AWRT A = 0.9 A Total 7 real values B No in answer, unless f () (a) all ( ) (i) fg ( ) = B ISW (ii) = 6 = M = M Invert = A (c)(i) y = = y M Swap and y ( y ) = y = M attempt to isolate + y = = g ( ) or + A ( ) (ii) ( real values ) ( ) g B Total 9 5
MPC - AQA GCE Mark Scheme 008 January series MPC (cont) 5(a)(i) y = 8+ (ii) = 8 d 6 d 8+ 6 = ln 8 + = [ ln 7 ln ] m B = ln 9 = ln A ( ) d MA M for k ln ( + ) u = du = d B OE 8 ; allow k ln u Correct substitution into 8+ or, 7 into k ln u k ln ( ) 6(a) = u + u ( du) 9 u 5 u u ( c) AF Must be terms with correct indices = + + u 9 5 only ft for = 5 = ( ) + ( ) +c A CSO OE 5 7 Total 9 M terms in with rational indices M A Correct shape Verte y 0.5 6.69 0.5.0 0.5.96 0.5.99 ( y ) M B Correct values correct y values 0. y = 5.99 B correct h used correctly =.59 A Total 6 6
MPC - AQA GCE Mark Scheme 008 January series MPC (cont) 7(a) Stretch (I) Scale factor (II) parallel to -ais (III) M A I + (II or III) All correct (Or scale factor parallel to y-ais) Translation M 0 5 OE A Alternatives 0 translate 5, stretch sf y-ais 0 translate, stretch sf -ais 5 M Mark translation first. Mark stretch as above, but relative to their translation. Modulus graph symmetrical about y-ais A left of 5 and right of 5 A (0, 5), cusps drawn and no straight lines between cusps (c)(i) 5= = 9 =± OE B 5= M 6 0 + 9 = 0 = =± A (ii), BF correct statements, BF correct statements SC c(ii) mark penalty for strict inequalities Total 7
MPC - AQA GCE Mark Scheme 008 January series MPC (cont) 8(a) e = = ln M = ln A OE ISW e d (c)(i) (ii) dv u = = e d du = v = e M differentiating and integrating d m correct subs of their values into parts = e + e ( d) formula A = e e + c A No further incorrect working y= e + 6 = e + 6 = 0 d M k e + 6 = 0 = 0 ( e ) = 0 d = ln A OE y = + 6 ln M Correct substitute of their valid = ln A OE ISW d y = e = M Other methods need justification d > 0 d y Allow error in or -value, but not d both minimum A () (iii) ( V) = π y d ( π) ( e + = 6) ( d) M Either 0 ( 0) () ( ) ( ) = π e + e + 6 d B Correct epansion ( 0) () A correct terms; 6, correct or = ( π) e 6e e + their ( 0) A All correct = π e 9e + = π 5 9e e =. B 5 AWRT Total 7 TOTAL 75 8