PhysicsAndMathsTutor.com PMT

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1 PhysicsAndMathsTutor.com PMT

2 Version.0: 006 General Certificate of Education abc Mathematics 660 MPC Pure Core Mark Scheme 006 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 already 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. Copyright 006 AQA and its licensors. All rights reserved.

3 MPC AQA GCE Mark Scheme, 006 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.

4 AQA GCE Mark Scheme, 006 January series MPC MPC (a) dy sec d M for sec SC/sec A Alternative Use of product/quotient rule (M) Good attempt cos + sin cos (A) Correct B (b) ( + ) ( + ) + ( + ) ( + ) dy 6 6 = = d = ( + ) M use of quotient rule A A AG (no errors) Alternative Alternative: = ( )( ) ( ) ( + ) + d (MA) (A) Total 5 MA ( ) y = + dy = ( + ) A d = ( + ) AG y () () ( ) ( ) ( ) B B correct SC B for all correct epressions but all correct wrongly evaluated y() + y() + A= 0.5 ( y (.5) + y (.5) ) + ( y ()) M use of Simpson s rule = 0.7 A Total

5 MPC AQA GCE Mark Scheme, 006 January series MPC (cont) (a)(i) dy f' = = + d B (ii) + d + = ln + + c ( ) ( ) M A For kln ( + ) By substitution kln u M correct A (b)(i) u = + du = d B + d= u du u M Must be in terms of u only incl. du = u u du A AG (ii) 9 d = du B Or changing u's to 's at end 0 u u u u = 5 5 M A 5 = ( 9) ( 9) 5 5 = ! Sight of any of these lines = 9.86 = 9.9 A AG Total 0

6 AQA GCE Mark Scheme, 006 January series MPC MPC (cont) cos ec = 5 cot (a) ( ) cot 5 5cot + = M use of cosec = + cot cot 5cot 0 + = A AG cot cot + = 0 M or + 5t t = 0 Or in tan ( t)( + t) = 0 (b) ( )( ) cot =, tan =, (c) =.,.0 = 0.,.8 5(a) a = 8 AWRT A AG B Any correct In degrees: B0 B Any correct B B correct B Total 7 B e 9= 0 M e = 9 = ln9 = ln A AG Condone verification (b) ( ) e 9 = e 8e + 8 B AG (c) V π y ( d) = B = ( π) e 8e + 8 d M e = ( π) 9e + 8 ln 0 ln8 e ln 9 = ( π) 9e + 8ln 9 M A m ST or nd term correct All correct Attempt at limits with ln (d) = π[ 8ln 5] A 6 M Modulus graph AF All correct a Total 5

7 MPC AQA GCE Mark Scheme, 006 January series MPC (cont) 6(a) f ( 0.5) = M f() = Change of sign root A (b) + = 0 = B = AG (c)(i) = 0.5 M = AWRT A = 0.66 A (ii) M A For cobweb, to curve For A All correct 7(a) π, π, OE in decimals Total 9 B B Or for and (b) M Translation in + ve direction M Correct shape A Correct Graph Through (,0) touching y ais Total 5 6

8 AQA GCE Mark Scheme, 006 January series MPC MPC (cont) 8(a) ( Range of f ) 0 B (b)(i) fg( ) = ( + ) B OE Maybe in part (ii) (ii) ( + ) = ( + ) = Or + M ( ) = ( ) + = ± 5 =, M ( )( ) A A = 0 (c)(i) Not one to one E OE (ii) = y + M y y + = y = M A Total 0 Attempt to isolate 7

9 MPC AQA GCE Mark Scheme, 006 January series MPC (cont) 9(a) y = ln dy = ln d M A A Use of product or quotient each term ln = A Convincing argument AG = (b) ln d u = ln dv= M du = v= A Attempt at integration by parts ln d = + = ln c + ( ) A A (c)(i) At A, d y 0 d = ln = 0 ln = M Attempt at ln = k = e A R = ln + (ii) ( ) 5 = ( ln 5 + ) + (ln + ) 5 = ( ln5 ) 5 R = Their b M ( ) 5 A OE A convincing argument; AG Total TOTAL 75 8

10 Version.0: 0606 abc General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 006 eamination - June 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 already 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. Copyright 006 AQA and its licensors. All rights reserved.

11 MPC AQA GCE Mark Scheme, 006 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 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.

12 AQA GCE Mark Scheme, 006 June series MPC MPC (a) f( ) = f(.) =+ 0.6 M both attempted change of sign < α <. A (b) 7= 0 = + 7 = + 7 B AG (c) = M = A AWRT.08 = AWRT.09 =.09 A Total 6 (a) ( ) 0 y = d y 0 ( d ) 9 M A M for ( ) 9 = 0 ( ) 9 (b) ( ) 8 + d u = + du = d B OE u du = 9 8 d u u u 0 9 u u = 0 9 a where a = constant 8 = u M all in terms of u. Condone omission of du 0 9 ( + ) ( + ) ( c ) B 0 9 u u p + q 0 9 = + A OE; CAO 0 6 Total 6 SC: correct answer, no working/parts in (B)

13 MPC AQA GCE Mark Scheme, 006 June series MPC (cont) (a) sec = 5 cos = 0. M =.7,.9 AWRT AA (b) tan = sec + 9 sec = sec + 9 M for using sec = + tan OE sec sec 0 = 0 A AG (c) ( sec 5)( sec + ) = 0 M or use of formula (attempt) sec = 5, A cos = 0., 0.5 =.7,.9 BF any correct or ft their answers in (a).09,.9 A all correct, no etras Total 9 (a)(i) (ii) B M y = branches mod graph > 0 for y = 0 A for, (b)(i) =, = B = M = A OE one value only Alternative: = ( ) M =, AA (ii) < < M A Total 8, ( ft ) identified as etremes CAO

14 AQA GCE Mark Scheme, 006 June series MPC MPC (cont) 5(a) y = e 0e + (i) dy e 0e d = + B B e remaining terms correct, no etras (ii) d d y e 0e = BF ft slip (b)(i) e 0e + = 0 e 5e + 6 = 0 B AG (be convinced) (ii) z 5z+ 6= 0 M use of z = e oe z =, z =, e = M finding e = their, = ln z =, e = = ln A all correct AG SC: verification ln (B) ln (B) (iii) = ln : ln ln y = e 0e + ln M or 0 + ln = 0 + ln = 6 + ln A = ln : ln ln y = e 0e + ln = ln = + ln A either substitution of their = ln e = ( e ) = or their = ln ( ) (iv) = ln : d y d ln ln e 0e = M use of; in either of their e, their = 6 0 = maimum A CSO = ln : d y ln ln e 0e d = 6 0 = 6 minimum A CSO Total d y d = into 5

15 MPC AQA GCE Mark Scheme, 006 June series MPC (cont) 6(a) ln = ( ln.5 + ln.5+ ln.5+ ln.5) M use of.5,.5, ; or correct values A AWFW to. =.08 A CAO (b)(i) y = ln dy M use of product rule (only differentiating, = + ln d terms with + sign) = ln + A ln + d = ln M OE; attempt at parts with u = ln ln d = ln ( + c ) A (ii) ( ) 5 (iii) ln d = [ ln ] 7(a) (b) 5 ( 5ln5 5) ( ln ) = M correct substitution of limits into their (ii) provided ln is involved 5ln5 A ISW Total 9 sin z = cos dz cos cos sin sin = d cos ( ) M A ± cos ± sin use of quotient rule cos = cos = sec A AG (be convinced) M correct shape including asymptotic behaviour and symmetrical about = 0 and y > 0 A use of V = k sec d M = k tan A (c) ( ) ( )[ ] 0 =.89 A CAO Total 8 6

16 AQA GCE Mark Scheme, 006 June series MPC MPC (cont) 8(a) ( ) f = e Range: f ( ) > (or y > or f > ) M for only (b) y = e A eactly correct y = e M y y e = + y + e = + y = ln M attempt to isolate A all correct with no error AG (be convinced) (c) ( ) f = + OE M A for differentiation of ln; for k their ( ± ) = 0 f ( ) = A A CSO all correct Alternative f ( ) = ln( + ) ln f ( ) = ( ) = f 0 ( + ) MA A A Total 9 CSO 7

17 MPC AQA GCE Mark Scheme, 006 June series MPC (cont) 9(a) π = y= ( or.57,sin ) B ignore 90! (b)(i) y = sin sin y = and sin y= B AG (be convinced) (ii) (c) d cos y dy = B dy k = MA M for d cosy cos y sin y = and sin + cos = M use of to get cos y or cos y cos y = dy = d Total 7 TOTAL 75 A AG; condone omission of proof of sign 8

18 Version.0: 007 abc General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 007 eamination - January series

19 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 already 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: Copyright 007 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

20 MPC - AQA GCE Mark Scheme 007 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. Jan 07

21 MPC - AQA GCE Mark Scheme 007 January series MPC =.5,.5,.5,.5 M Method A values y = y = AWRT y = A correct y s y = A = ( y+ y + y + y) =.08 A Total Stretch (I) SF M For I + (II or III) (II) Parallel to ais (III) A All correct Translate E Allow translation 0 B Correct vector or description ( ) Total (a) f ( ) MA M for f <, Condone y, f, range (b)(i) y = + M Attempt to obtain as a function of y or + = y y as a function of M y at any stage = y A Any correct form y/g ( ) = = (ii) ( g ( ) ) B (c)(i) M h( ) = + A = = + ( )( ) (ii) ( ),, R + B Condone omit is real Allow Total 9

22 MPC - AQA GCE Mark Scheme 007 January series MPC (cont) (a) sin d u= dv sin d = M For differentiating one term and integrating other du v cos d = = = cos cos ( d ) m For correctly substituting their terms into parts formula A ( ) = cos+ sin + c A CSO (b) u= + 5 du = d = u u ( d ) M A d ( u) ku condone omission of du but M0 if d k = OE u = A Ft ku d u = ( + 5) (+c) A CSO SC ( + 5) with no working B 6 (c) y = 9 = y + 9 V = π dy B Must have π and, condone omission of dy, but B0 if d = π ( y+ 9)dy y = (π) + 9y or ( ) ( ) y+9 π = (π)[ 0 9 ] m = 0 π M "their "dy integrated π not Limits and substituted in necessary correct order including sign A CSO Total 5

23 MPC - AQA GCE Mark Scheme 007 January series MPC (cont) cosec + 5 cosec =0 M 5(a)(i) ( ) cosec + 5cosec 0 = 0 cosec 5 cosec = 0 + A AG (ii) ( cosec )( cosec + ) = 0 M Attempt to solve A Condone answers with no method shown cosec = or sin = or A AG θ 0. = 0.7,., 0.5,.89 B correct values, may be implied later AWRT (.8,8., 65.5,.5) θ = 0.8,.5, 0.5,.79 AWRT B correct answers B + correct answers and no etra within range Total 8 (b) ( ) 6(a)(i) ( ) 0 (ii) y= + + dy 9 0 ( ) ( 8 ) d = M A y= tan M Product rule d d (b)(i) = y + ln y d = 6y + dy y (ii) At (, ) y sec tan = + A B For f() ( ) 9 where f( ) d 6 7 dy = + = M dy d = A May be implied 7 ( ) A OE y = ( ) 7 Total 8 k and is linear 6

24 MPC - AQA GCE Mark Scheme 007 January series MPC (cont) 7(a) B (b) M A A Shape inverted V in all four quadrants Symmetrical about y ais Coordinates (c) = M Attempt to solve = = A + = = A And no others (d) M Either correct < < A Other solution and no etras SC B Total 9 8(a) A (,π) B B π 0, (b) cos = 0 B f( 0.) = 0.7 allow 0., 0. M Or comparing sides f ( 0. ) = 0. allow 0. Change of sign root A (c) = 0. M = = 0.57 A = 0.78 = 0.8 = 0. A Total 7 7

25 MPC - AQA GCE Mark Scheme 007 January series MPC (cont) 9(a)(i) ( ) e d = e ( + c) (ii) ln ln e = 0 ln 0 = ln e 0 e 0 ( ) ( ) M B B e Substitute both ln and 0 correctly into an integrated epression Convincing = ln + = ln A AG (b)(i) = 0 y = = B (ii) At B, y = 0 e = 0 M Or reverse argument e = = ln A AG (c) dy e d B ln = ln, Gradient = e M = ln into k e = 8 Gradient normal = = ln 8 e A OE Equation y = ln 8 8 A OE (d) When = 0 M Attempt to integrate their line and y = ln 8 substitute = 0, ln Area = ( ln ) condone ve sign A ( y ) ln 6 = 0.0 Total area = ln + ( ln) =.0 6 A CSO AWRT Total Total 75 8

26 Version.0: 0607 abc General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 007 eamination - June series

27 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 already 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: Copyright 007 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

28 MPC - AQA GCE Mark Scheme 007 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 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. June 07

29 MPC - AQA GCE Mark Scheme 007 June series MPC (a) y = ln penalise + c once on (a) or (a) dy = d B (b) = ( + ) y ln dy = ( + ) + ln d (c) = ( + ) y ln dy = + + ln d dy = : = + = d Grad normal = M A M M A product rule substitute = into their d y d use of mm = CSO y = ( ) A OE Total 7 (a) ( ) or in epanded form B allow ( ) (b) ( ) M V= (π) d ( ) = π ( ) M m (π) y d ( ) k( ) π or in epanded form correct substitution of limits into k( ) = π 8 = 80π A CAO (c) Translate E 0 B OE Stretch (I) SF (II) M for I and (II or III) // y ais (III) A for I and II and III Total 9

30 MPC - AQA GCE Mark Scheme 007 June series MPC (cont) (a) cosec = M 0 scores M implied sin = = 0, 50 A and no etras in range (b)(i) B (ii) M A all positive, U shapes minima consistent > 0, not intersecting with each other or y-ais (c) = 0, 50, 0, 0 BF correct values from their (a), which must be θ,80 θ B all correct and no etras in range Total 7 5

31 MPC - AQA GCE Mark Scheme 007 June series MPC (cont) (a) y 0 B values PI.5.98() () B ( +) y values correct (5) 9 A = ( ) M Simpson s rule = 5.6 A CAO (b)(i) ( ) = ( ) ( ) f f 0.5 =.77 change of sign root f.5 = MA (ii) = + ln = ln + M correct use of logs ( ) ( ) ( + ) ln = ln + ln = A correct with no mistakes; AG ln (iii) = 0.5 ( =. ) M =.9 =. A CAO (iv) M A staircase, correct and labelled on -ais Total 6

32 MPC - AQA GCE Mark Scheme 007 June series MPC (cont) 5(a) f( ) 0 allow y 0 M > 0 or f 0 or 0 A (b)(i) B (ii) = M squaring their (b)(i) in an equation = OE A = A CSO (c) y = 6(a) y y = M attempt to isolate; condone slip = y M reverse y = + A 5 e d Total 9 5 = d = e M integrate one term, differentiate one term u v du = 5 v= e 5 A 5 5 = e e d 5 5 A 5 5 = e e ( + c) 5 5 A (b)(i) (ii) u = du = d M = du + u A correct with no errors; AG 9 d = du m + u ( u) = ln + kln + = ln ln A ISW OE ( = ln) Total 9 M for ( ) correct limits used in correct epression, ignoring k 7

33 MPC - AQA GCE Mark Scheme 007 June series MPC (cont) e 7(a)(i) y = ( ) dy ( ) e e d = + M A product rule d y (ii) ( ) e e e e d = M A product rule from their d y d dy (b)(i) 0 d e ( + ) = 0 M e f( ) = 0 from d y d e ( + )( ) = 0 m attempt at factorising or use of formula =, A first correct solution A second correct solution, and no others SC No working shown: = B, = B (ii) = y = e ma ( 0.) M Condone slip = y = e min (0.9) A Total 0 8(a) tan (+ c) B cos = sin sin cos f ( ) = ± sin ± cos M quotient rule sin A sin = sin A use of sin + cos = = cosec A AG CSO (b) f ( ) Special cases cot f ( ) = cosec cot 0 f' ( ) = M = cosec A (ma /) Or f ( ) = tan tan 0 sec f' ( ) = tan M A sec = tan = = cosec sin A (ma /) 8

34 MPC - AQA GCE Mark Scheme 007 June series MPC (cont) (c) LHS = tan + cot + tan cot M epanding = tan + + cot + M correct use of trig identities = sec + cosec A CSO = RHS tan + cot d = sec + cosec d M use of identity (d) ( ) = [ tan cot ] 0.5 M A = =. A AWRT Total TOTAL 75 ± tan ± cot OE 9

35 Version.0: 008 abc General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 008 eamination - January series

36 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 already 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: 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

37 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.

38 MPC - AQA GCE Mark Scheme 008 January series MPC (a)(i) y = ( 5+ ) 0 dy 9 0( 5 ) ( 5) d = + OE M A 9 chain rule 0( ) f ( ) with no further incorrect working (ii) y= cos dy sin cos d = + M A product rule ± sin ± cos CSO (b) y = ( ) ( ) dy = 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 (b) cot + cosec = + cot = cot + M Correct use of cosec = + cot ( ) ( cot cot+ = 0) cot cot = 0 A AG; correct with no slips from line with no fractions cot + cot = 0 M Attempt to solve cot =, A tan =, 0.5 = 0.6,.6,.0, 5.8 B correct Allow.6(0) B correct (with no etras in range) AWRT SC Degrees 6.57, B for correct 6.57, Total 8 (c) ( )( )( )

39 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 (b) = ( + ) = + 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 ( ) (b)(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

40 MPC - AQA GCE Mark Scheme 008 January series MPC (cont) 5(a)(i) y = 8+ (ii) dy = 8 d 6 d 8+ 6 = ln 8 + = [ ln 7 ln ] m B = ln 9 = ln A (b) ( ) 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 (b) y ( y ) M B Correct values correct y values 0. y = 5.99 B correct h used correctly =.59 A Total 6 6

41 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 (b) 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 = 0 = =± A (ii), BF correct statements, BF correct statements SC c(ii) mark penalty for strict inequalities Total 7

42 MPC - AQA GCE Mark Scheme 008 January series MPC (cont) 8(a) e = = ln M = ln A OE ISW (b) 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 dy = e + 6 = 0 d M k e + 6 = 0 dy = 0 ( e ) = 0 d = ln A OE y = + 6 ln M Correct substitute of their valid = ln A OE ISW d y = = d > 0 e M minimum A () Other methods need justification d y Allow error in or -value, but not d both (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 (b) ( 0) A All correct = π e 9e + = π 5 9e e =. B 5 AWRT Total 7 TOTAL 75 8

43 Version.0: 0608 abc General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 008 eamination - June series

44 MPC - AQA GCE Mark Scheme 008 June 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 already 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: 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

45 MPC - AQA GCE Mark Scheme 008 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 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.

46 MPC - AQA GCE Mark Scheme 008 June series MPC (a) d y 5 ( ) d M k( + ) = 5 + A with no further errors (w.n.f.e) ( ) (b) dy = d + M k + A w.n.f.e (c) (a) dy d = + ( + ) + ln( + ) 5( +) 5 ( ) 5ln( ) ( ) 5ln ( ) = = M A A Total 7 product rule uv + u v (from (a) and (b)) either term correct CSO with no further errors = cos M PI =., 5.05 (0.9π,.6π) A,A AWRT ( for each error in range) SC 70.5, 89.7 B (b) sec = sec + M use of sec = + tan sec sec = 0 A AG; CSO (c) sec sec = 0 sec sec + = 0 M attempt to solve ( )( ) cos = or o.e A =., 5.05, Bf ( answers in range from (a)) AWRT. (π) B all correct and no etras in range SC 70.5, 89.7, 80 B Total 9 (Etra +c penalised once throughout paper)

47 MPC - AQA GCE Mark Scheme 008 June series MPC (cont) dy (a) sin cos d = + M product rule ksin ± cos A no further incorrect working (b)(i) αsin α + cosα = 0 M replacing = α and writing equation equal to zero (at any line) αsinα= cosα either αtanα= αtan α = 0 A AG; CSO (ii) f(0.)= 0. awrt o.e. M (0.9 s unsubstantiated scores M0) f(0.5)= -0.6 Change of sign 0. < α < 0.5 A (iii) tan = tan = either = tan = tan B AG; CSO (iv) = 0. = M = 5.7 = = 0. A (c) y = cos u = du= sin M dv= cos v= m sin sin = (d ) sin cos = + (0.5) (0) A differentiate one term must be ksin integrate one term correct substitution of their values into parts formula using u = sin cos cos 0 correctly substituting values from = + m previous method marks = A 5 AWRT Total 5

48 MPC - AQA GCE Mark Scheme 008 June series MPC (cont) f 0 B allow f 0, y 0, 0 (a) ( ) (b)(i) y = y M swap and y ( y ) = y = y = + M attempt to isolate + y= = g ( ) o.e. A w.n.f.e ( ) (ii) g ( ) B (c) = 9 B =± M 5 =, o.e. A Total 8 square root and invert (condone missing ± ) alternative: attempt to solve a quadratic that comes from + 9= o.e. 9 Alternative (b)(i) divide into y + + divide into y y + y M 6

49 MPC - AQA GCE Mark Scheme 008 June series MPC (cont) 5(a)(i) B B shape coordinates (ii) B B shape coordinates (b)(i) Translation M OR I stretch M I + (II or III) II SF 0 A III y-ais A Stretch I M I + ( II or III ) (I + II + III) Translation M SF II y-ais III A A I + II + III B 0 Translation B both All correct and no mistakes on order etc A 6 All correct A Alternative: y = ln( + ) = ln( + ) (B) Translation (M) (A) Stretch I (M) I + ( II or III ) SF II y-ais III (A) I + II + III All correct and no mistakes on order etc (A) (6) 7

50 MPC - AQA GCE Mark Scheme 008 June series MPC (cont) y = ln + 5(b)(ii) ( ) = 0 y= B y = 0 ln( + ) = ( + ) = M isolate ln ( ) ln + = e A + = e k = e o.e. A CSO isw Total 6(a) ( ) y = e + d e e d y ( ) = + M ( ) e + A A = ln : dy ( ) e ln8 e ln8 d = + M = 8 = A 5 e + = or (allow ) w.n.f.e ( + ) correct substitution into their d y d (must use ln8 or ln ) (b) y (5) (5) B correct values ().75.8() B or correct y values s.f. or better = 0.5 y P.I M =.7 A sc.7 with no working (c) v= π y d = ( π) ( e + ) (d ) M = ( π) e + () (0) 6 0 = ( π) e + e = π e + π ( 6 5) e = + A ke m A CSO Total + correct substitution into f ( e ) 8

51 MPC - AQA GCE Mark Scheme 008 June series MPC (cont) 7(a) sinθ y = cosθ dy cosθcos θ sin θ( sinθ) M ± cos θ ± sin θ = dθ cos θ A cos θ = o.e. ( + tan θ ) cos θ = sec θ A AG; CSO (b) (c) = sinθ OR LHS = sinθ = sin θ sin θ sinθ = cos θ = cosθ M use of cos θ + = sinθ tanθ = cosθ = = tanθ AG A AG; CSO ( ) d = sinθ d = cosθ dθ o.e. = = cos θ (d θ) ( ) sin θ cosθ ( ) cos θ (d θ ) M m A = sec θ (d θ) A = tanθ d cosθ dθ =± all in terms of θ = ( + c) A 5 CSO including d θ 's Total 0 TOTAL 75 Alternative tanθ 7(a) y = dy sec θ 0 = dθ = sec θ M A A 9

52 Version : klm General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 009 eamination - January series

53 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 already 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: 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 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

54 MPC - AQA GCE Mark Scheme 009 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.

55 MPC - AQA GCE Mark Scheme 009 January series MPC y 0.5 B values and no etra values 0.66(0) (0) B 7 0.7() + correct y values or etc = ( ) + ( + ) + ( ) M Correct application of Simpson s rule for their values ( odd) =.6 A CSO must be sf Total V = π y d ( ) ( ) ( ) 5 = π d M ( ) ( ) 6 = π A 6 limits not required 6 = ( π ) 6 6 = 0.5π A 6 allow equivalent fraction π etc 6 (AWRT 0.5 or 0.5π m, A0) Total m correct substitution into ( π ) k( ) 6

56 MPC - AQA GCE Mark Scheme 009 January series MPC (cont) f = + 5 (a) ( ) ( ) f() = f 0.5 =.75 M Condone f (0.5) rounding to. Change of sign 0.5 < α < A Both statements needed (b) + 5 = 0 5 = Must be seen = ( ) B AG 5 (c) = 0.5 (d) ( ) ( ) = = M For or = ( sf ) 0 = A M A From 0.5 vertical to curve then horizontal to line CAO Total 7 5

57 MPC - AQA GCE Mark Scheme 009 January series MPC (cont) (a) sec = cos = = 8, B correct (Condone answers rounding to) B correct and no etras in interval (b) tan = 0 5sec 5(a) ( ) ( ) sec = 0 5sec M Use of trig identity correctly sec 5sec ( 0) ( sec )( sec+ )( = 0) + = A sec =, either of these cos =, m A Attempt to solve or factorise slip using formula = 8,, 0, 56 B AWRT correct condone 05 or 55 B 6 All correct and no etras in interval Alternative: sin 5 = 0 (M) cos cos sin = 0 cos 5cos cos = 0 cos 5cos (A) cos 5 cos = 0 then rest of scheme as above Total 8 f, f, y B, f ( ) <, B y <, f < (b) f ( ) is not one to one E Allow many to one or numerical eample (c)(i) ( ) (ii) fg = = 6 = ( ) = 6 B M =± M =, A Total 7 Correct handling of fourth root Must have ± Correct handling of reciprocal 6

58 MPC - AQA GCE Mark Scheme 009 January series MPC (cont) 6(a) y = e ( ) dy d ( ) = e M Product rule; allow slip d d e 6 8 ( ) + e A y e ( 8 ) = + M Factorising e ( a ) ( ) A or = 0 e 0 ( )( + ) = 0 m Solving term quadratic Dependent on both M marks =, A 6 And no etras eg = 0 (b)(i) d y d e. ( ) e = + ( ) e e ( ) + + Or d y d ( ) ( ) = e e M A M A Product rule from their d y in form d e 8 e (quadratic) ( ) (ii) d y 8 Their s in their = : y " = e ( 0) > 0 MIN M d only of form e (quadratic) = : y " = e ( 0) < 0 MAX A CSO Both correct Allow values either side of y or y Total 0 7(a) e = e = M = ln A (b)(i) e + 0e = 9 0 y + = 9 or e + 0 = 9e y y y+ = B AG (ii) ( y )( y 5) = 0 y =,5 B = ln, ln 5 M A Total 6 ln (their + ve y s) 7

59 MPC - AQA GCE Mark Scheme 009 January series MPC (cont), π, 80 8(a) P ( ) B Condone ( ) Q (, 0) B (b) Translate E 0 B or equivalent in words Stretch SF y-ais M Stretch + one other correct A all correct (c) B B Correct shape in st quadrant π and marked correctly y = M y cos = y = cos + A (d)(i) cos ( ) (ii) y sin M A k sin (...) d dy correct d At y =, = sin dy A Condone AWRT 0. Total 8

60 MPC - AQA GCE Mark Scheme 009 January series MPC (cont) 9(a) y = dy ( ). ( ) = Must use quotient rule M d Condone one slip = ( ) ( ) A k = (b)(i) y = ln ( ) d y. = + ln ( ) M d m A ( ) f + g ( ) f() may be constant k + ln ( ) (ii) = y = 0 B dy d = M Sub = into their d y d y = any correct form A CSO Must have full marks in (b)(i) ( ) (c)(i) u = du = d M d = u + d u A u = ( ) + (d u) m u = ( ln ) u+ u Or d = + d = + du etc u = ( ) + ln( ) ( + c) A CSO Condone missing du (ii) ( ) ln d dv u = ln ( ) = M In correct direction d du = v= d = ln ( ) d A m = ln( ) ( ) ln( ) + A ( + c) ln ( ) their (c)(i) Total 6 TOTAL 75 9

61 Version : klm General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 009 eamination - June series

62 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 already 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: 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 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

63 MPC - 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 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.

64 MPC - AQA GCE Mark Scheme 009 June series MPC (a)(i) cos f ( ) = + OE π π f( 0) = ; f = M = 0 LHS=, = LHS= 0 Change of sign π 0 < α < A Either side of, 0< α < π (ii) cos = + cos= + or, cos = + cos = Either line = cos B AG; or cos = All correct with no errors (iii) = 0 = 0.5 = 0.78 M A Attempt at iteration allow = 0.5, = 0.8, 0. ( ) CAO (b)(i) dy ( )( ) = d ( + ) + sin cos M Attempt at quotient rule: ± + sin± cos ( ) ( + ) A Either term correct A All correct ISW (ii) = 0 dy = m Correctly subst. 0 d = into their d y d Gradient of normal = A CSO Total 0

65 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) f 0 For 0, f > 0 (a) ( ) M ( ) (b)(i) y = + 5 A Correct; allow y 0, f 0 = y+ 5 M y f = y+ 5 M Attempt to isolate, squaring first 5 ( ) = A condone ( y = ) (ii) 0 BF ft their (a), but must be (c)(i) h( ) = fg( ) = (ii) + 5 = = 9 + = + either + = or 6+ = = or equiv 8 (a) tan = 0. B M A A CSO Total 0 one correct step from (c)(i), squaring M Sight of ± 0. or 8. =.8, 5.96 (b) ( ) tan + = 5tan+ 5 A A tan 5tan = 0 B AG (c) ( )( ) a correct answer AWRT for any etra in range, ignore etra answers not in range. [SC 6.57,.57 AWRT MA (ma /)] tan + tan = 0 M Attempt at factorisation/formula tan =, A =.,.5,.8, 5.96 AWRT B correct [SC =.,.5 + their two answers from (a)] B correct, no etras in range [SC 6.57,.57, 6.,. AWRT B (ma /)] Total 8 5

66 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) (a) y 50 M Modulus graph, section, condone shape inside + outside ± 50 ( 50) O ( ) 50 A A Cusps + curvature outside ± 50 Value of y and shape inside ( ± 50) (b) 50 = 50 = = 6 50 = = 6 M Either =± 6, ± 8 A correct, from correct working A All correct, from correct working (c) 6 < < 6 B > 8, < 8 B (d) Reflect in -ais 0 Translate 50 Reflect in y = 5 scores / 5(a) ln = 5 (b) M,A E, B Total 5 5 ln = = e B 5 ln + = ln ln ( ) ln 5 0 ( ln 5)( ln ) 0 + = M Reflect in y = a or 0 Translate 50 a 0 Translate or 50 Reflect in ais 0 or Translate a 50 Reflect in y = a Forming quadratic equation in ln, condone poor notation = m Attempt at factorisation/formula 5 ln =, condone ln = 5 A 5 = e, e A,A 5 Total 6 [SC for substituting = e 5 or equivalent into equation and verifying B ( 5 )] 6

67 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) 6(a) V = π dy B PI (b) 6(c)(i) (ii) ( π) ( ) V = 00 y dy ( ) π y = 00 y ( ) ( 0) ( 0) M A ( ) k 00 y dy may be recovered their d y, epanded Allow ( ) π 000 = m For F(0) F(0) 500π = A 5 OE CSO SC: if rotated about -ais 5 V = π 00 M = π A ma /5 y (0) or better (0) B M A Correct + correct y to sf All y correct A = y = 9.6 A ( 9.6 scores ) dy 00 8 ( ) ( ) = M Chain rule ( ) f ( ) d ; allow f() = k f ( ) = ( 8 ) = dy = = ( 00 6) d A = or equivalent A CSO y 8 = ( ) M y 8= their ( ) ( y 6 = + 9) y+ = 5 A dy d dy or y = their + cand subst. (,8) to d find c AG; all correct with no slips, full marks in part (i) 7

68 MPC - AQA GCE Mark Scheme 009 June series MPC 6(d) 5 = 0 y = or equivalent B y= 0 5 = B OE 5 5 Area of Δ = M for ( y) ( ) or sin ab C Area = Area Δ (b) m PI Δ > (b) Required area =.5 AWRT A 5 Condone. AWRT (d) Alternative 5 5 d Area Δ = ( ) ( ) 0 (B) (B) = (M) For integration and 5 f( ) f(0) = 65 7(a) ( ) t lnt dt Total 9 u = ln t dv = t dt M Differentiate + integrate, correct direction du t = v = t dt t A All correct t t = t ln t t ( dt) t t t ( ) = t ln t dt A Condone missing brackets t t = t ln t + t + c ( ) A CAO 8

69 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) 7(a) Alternative t lnt u= lnt v = t ( ) (M) ( ) ( t ) ( t ) = ln t dt t t ( ) t t t + ln t d t t ( ) t ln t t + dt (A) t ( ) t t ln t t+ ln t t = ln t tln t + ln t t + t lnt ( ) (A) t (A) u = v= t t = ln + + t t t t c () (b) t = + dt dt = d (RHS) M ( LHS) d = = t, m OE dt = ( t ) lnt A AG (c) [ ] [ t] = M Limit becoming 0 t t = t ln t + t 9 9 = ln m Correctly sub., into their (a) = ln A CSO or ( + ) ( + ) = ( + ) ln( + ) + ( + ) (M) Condone slip = ln (m) Correctly sub. 0, = ln (A) () CSO Total 0 TOTAL 75 9

70 MPC - AQA GCE Mark Scheme 009 June series 0

71 Version.0 00 hij General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 00 eamination - January series

72 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 already 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: Copyright 00 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

73 MPC - AQA GCE Mark Scheme 00 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.

74 MPC - AQA GCE Mark Scheme 00 January series MPC y e e y = Ae a+ b ± Be + = + ( ) (a) ( ) ( ) + M ( ) ( ) = e or ( ) where A and B are non-zero constants A All correct e - 6 e +0e = e 5 A AG; all correct with no errors, or y= e + e e nd line (OE) must be seen Condone incorrect order on final line y = e + e +. e (M) + e + 8e (A) = e 6e + 0e = e ( 5 ) (A) A e + Be + Ce + De + Ee All correct AG; all correct with no errors, rd line (OE) must be seen (b) ( )( ) + 5 ( = 0) M OE Attempt at factorisation ( ± ± 5)( ± ± ) or formula with at most one error 5 =, A Both correct and no errors SC = only scores MA0 =, y= e m For e b 5 0 =, y= e AF y= a attempted Either correct, follow through only from incorrect sign for A 5 CSO solutions only Total 8 Note: withhold final mark for etra solutions Note: approimate values only for y can score m only

75 MPC - AQA GCE Mark Scheme 00 January series MPC (cont) (a)(i) A B correct shape passing through origin and stopping at A and B B A π, π B, B B SC A(, 90) and B (, 90) scores B (ii) line intersecting their curve (positive gradient, positive y intercept) M Correct statement A one solution only, stated or indicated on sketch - must be in the first quadrant (ie curve intersects line once) Must have scored B for graph in (a)(i) (b) ( ) ( ) () () LHS 0.5 = 0.5 RHS 0.5 =. LHS =.6 RHS =. At 0.5 LHS < RHS, At LHS > RHS 0.5 < α < or ( ) ( ) f = sin f( 0.5) = 0.6 AWRT f() = 0. M Change of sign 0.5 < α < (A) or f ( ) sin = + f ( 0.5) = 0. Attempt (M) f () = 0. Change of sign 0.5 < α < (A) or ( ) A CSO f( ) must be defined (M) Allow f( 0.5) < 0 f( ) > 0 f( ) must be defined f = sin f( ) must be defined ( ) () f 0.5 =. attempt f =. (M) Change of sign 0.5 < α < (A) 5

76 MPC - AQA GCE Mark Scheme 00 January series MPC (cont) (c)(i) = 0.90 M Sight of AWRT 0.90 or AWRT 0.9 = 0.9 A These values only (ii) M Staircase, (vertical line) from to curve, horizontal to line, vertical to curve O A, appro correct position on -ais Total 6

77 MPC - AQA GCE Mark Scheme 00 January series MPC (cont) (a) sin =, or sight of ± 0., ± 0.π or ± 9.7 (or better) M = 0.,.8( 0) AWRT A Penalise if incorrect answers in range; ignore answers outside range (b) cosec = cosec M Correct use of cot = cos ec cosec + cosec ( = 0) A ( cosec + )( cosec )( = 0) Attempt at Factors m Gives cosec or when epanded Formula one error condoned cosec =, A sin =, Either Line sin = =.9, 6.0 AWRT correct or their two answers from (a) BF and.9, ,.8( 0 ) AWRT B 6 correct and no etras in range ignore answers outside range SC 9.7, 60.5, 9.8, 5.5 B Alternative cos sin = sin cos = sin sin Correct use of trig ratios and multiplying (M) by sin sin = sin sin 0 = sin sin (A) 0 = ( sin+ )( sin ) (m) Attempt at factors as above sin =, (A) (BF) As above (B) Total 8 7

78 MPC - AQA GCE Mark Scheme 00 January series MPC (cont) (a) y Modulus graph V shape in st quad going M into nd quad, touching -ais. Must cross y-ais Condone not ruled A and 8 labelled (b) = B One correct answer = 6 B Second correct answer and no etras Condone answers shown on the graph, if clearly indicated (c) > 6 < 5(a) y B B Total 6 B M A One correct answer Second correct answer and no etras and no further incorrect statement eg 6 < < or < > 6 SC 6, scores B values correct PI + y values correct to sf or better or eact values.98,.8 / 9,. / 5,.77 for y ( or better) = y =. A (Note:. with evidence of mid-ordinate rule with four strips scores /) (b)(i) y = ln ( + 5) y e 5 y = + OE = e 5 B AG Must see middle line, and no errors (ii) ( π) ( e y 5) ( dy) (c) 8 M ( ) 0 ( π) e y 5y ( 5) = A 0 5 ( π) ( e 50) ( e 5) = m F ( 0 ) F( 5 ) 0 5 V = π e e 5 A ( y = )ln M Condone omission of brackets around f (y) throughout CSO including correct notation must see dy ISW if evaluated seen, condone ln +... B + A CSO mark final answer (no ISW) Total 8

79 MPC - AQA GCE Mark Scheme 00 January series MPC (cont) f > M >, f 6(a) ( ) > or ( ) (b)(i) y = e y + = e ( y ) A Allow y > ln + = M swap and y ( f ( ) ) ln( ) M = + A Alternative e - ln + (M) (M) ln( + ) y = (A) attempt to isolate: ln ( y± A) = B or reverse OE with no further incorrect working Condone y =.. (ii) + = M (c)(i) ( ) for putting their p( ) = from kln ( p ( ) ) in their part (b)(i) = A CSO SC: B = - with no working, if full marks gained in part (b)(i) (gf = ) e ( ) + substituting f into g either OE B (=) ISW e 5 (ii) = e 5 = e 5 OE M Correct removal of their fraction e = = ln m Correct use of logs leading to k = ln a b = ln OE A CSO No ISW ecept for numerical evaluation Total 9

80 MPC - AQA GCE Mark Scheme 00 January series MPC (cont) 7(a) dy cos. cos sin. sin ± Acos ± Bsin = M d cos cos cos + sin = or better cos A Both terms correct = + tan CSO A All correct ( ) or dy cos. cos sin. sin = d cos (M) ± Acos ± Bsin cos (b) cos cos sin sin = + cos cos cos cos (A) or better = + tan CSO (A) All correct d y d ( ) tan = M A tan sec m f() = f() B sec = tan sec AF ft 8 their p ( ) ( y ) = tan + tan m from part (a) Previous two method marks must have been earned = y + A 5 CSO Alternative Solutions sin y = + tan = + cos y = (M) Acos ± Bsin cos sin cos + s in cos sin where A and B are cos cos constants or trig functions. (m) Where A is msin and B is ncos 8 sin cos cos + sin = cos (AF) ft 8 their p from part (a) = tan sec (m) ktan sec = y ( + y ) (A) CSO or dy sec d d y sec.sectan d (M) A sec f() (m) f() = B sec tan = sec tan (AF) ft 8 their p from part (a) = ( + tan ) tan Previous two method marks must have (m) been earned = y + y (A) CSO ( ) 0

81 MPC (cont) MPC - AQA GCE Mark Scheme 00 January series 7(b) or dy = ( + tan ) d u = tan dy = + u d d y du = (8) u d d (M) du tan u d (m) d y 8 u( u ) d (A) = u( + u ) (m) 8(a) sin ( ) = y( + y ) (A) Total 8 d u= dv d = sin ( ) M sin f ( ), ( ) attempted d d du = v= cos( ) d A All correct condone omission of brackets ( = ) cos( ) m correct substitution of their terms into parts cos ( ) ) = cos( ) + cos( ) (d ) A All correct condone omission of brackets CSO condone missing + c and d = cos( ) + sin ( ) + c A 5 Condone missing brackets around if recovered in final line ISW (b) u= 'du= d ' M OE ( u + ) m A d d u = u u + u+ d = u 8 u = u+ + du 8 u u = + u+ lnu 8 ( ) = + ( ) + ln ( ) + c 8 A B All in terms of u All correct PI from later working ( ) u + or + ln u 8 ( ) A 6 + or = + ln ( ) + c 8 CSO condone missing + c only Total TOTAL 75 ISW

82 Version.0 klm General Certificate of Education June 00 Mathematics MPC Pure Core Mark Scheme

83 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 already 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: Copyright 00 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

84 MPC - AQA GCE Mark Scheme 00 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 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.

85 MPC - AQA GCE Mark Scheme 00 June series MPC (a) f ( ) = 0 + (or reverse) f() = 6 f( ) = 7 M Attempt to evaluate f() and f() Change of sign < α < A All working must be correct plus statement OR LHS () = RHS () = 9 LHS ( ) = 9 RHS ( ) = At LHS < RHS, < α < At LHS > RHS (M) (A) Must be these values (b)(i) = 0 = 0 This line must be seen = 0 B AG (ii) ( = ) =.9 M Sight of AWRT.9 or AWRT. =. A Both values correct Total 5

86 MPC - AQA GCE Mark Scheme 00 June series MPC Condone marked at A, A = etc (a)(i) ( y = ) B but not cos0, sec 0 (ii) y M Modulus graph y > 0 O A + sections roughly as shown, condone sections touching, variable minimum heights A Correct graph with correct behaviour at asymptotes but need not show broken lines; and roughly same minima (b) cos = or cos seen M π or sight of ± 60 or ±, ±.05 (AWRT) = 60, 00 A Condone etra values outside 0 < < 60, but no etras in interval (c) sec( 0 ) =, sec( 0 ) = cos ( 0 ) = or cos ( 0 ) = 0 = 60, 00 or 0 = 0, 0 (ignore values outside 0 < < 60 ) M A Either of these, PI by further working Both correct values from one equation or correct values and no wrong values from both equations, but must have 0 = PI by = 70,0,50,0 = 5,65,5,55 B correct (and not more than etra value in 0 < < 80 ) B All correct (and no etras in interval) Total 0 5

87 MPC - AQA GCE Mark Scheme 00 June series MPC (cont) (a)(i) y= ln (5 ) M k 5 dy 5 5 = A No ISW, eg = d 5 5 (ii) y= sin dy = cos d M k cos A (b)(i) f ( ) ln0.5 or f( ) M ln A (ii) ( gf ( ) ) sin ln ( 5 ) or gf ( ) sin ln 5 = ( ) ( ) = B Condone sin ln 5 ( ) or sin ( ln ( 5 ) ) but not sin (ln 5 ) or sin ln 5 (iii) gf ( ) = 0 ( ) ln( 5 ) 0 sin ln 5 = 0 = M Correct first step from their (b)(ii) = m Their ( ) k ( ( ) ) 5 = A 5 f = from ln f = 0 Withhold if clear error seen other than omission of brackets (iv) = sin y sin ( g ( ) ) = y ( or sin y = ) M Correct equation involving = sin A Total sin 6

88 MPC - AQA GCE Mark Scheme 00 June series MPC (cont) (a) y = B values correct PI = 0.5 = = = = 0. 9 = B M At least 5 y values that would be correct to sf or better, or eact values. May be seen within working. Clear attempt to use their y values within Simpson s rule = 0.5[ ] = A Answer must be with no etra sf (Note with no evidence of Simpson s rule scores 0/) (b) 0 d + ln = ( + ) M kln ( ) = ln ( + ) ln A m + condone missing brackets Correct. A may be recovered for missing brackets if implied later F() ( F(0)) = ln A ln must not be left in final answer Alternative u = l + du= d du = u (M) = [ ln u] (A) = ln ln (m) du du correct and integral of form k d Correct substitution of correct u limits or conversion back to and F() ( F(0)) u = ln (A) ln must not be left in final answer Total 8 7

89 MPC - AQA GCE Mark Scheme 00 June series MPC (cont) 5(a) 0cosec = 6 cot 0( + cot ) = 6 cot 0cot cot = B AG Must see evidence of correct identity and no errors. (b) Attempt at factors, giving when epanded. ± ± 0cot 6 M Use of formula: condone one error ( 5cot )( cot ) ( 0) + = A Correct factors cot =, 5 5 tan =, A,A st A must be earned Condone AWRT 0.67 ISW if values attempted Alternative 0cot + cot 6 = 0 cos cos = 0 sin sin 0cos + cos sin 6sin = 0 5cos sin cos+ sin = 0 (M) ( )( ) ( ) ( 5cos = sin cos= sin ) 5 = tan = tan (A) (A), (A) Alternative 0 + tan 6 tan = 0 ( 5 tan)( + tan) ( = 0) (M) (A) 5 (A), tan =, (A) Total 5 Attempt at factors, gives ± 0cos ± 6sin when eplained As above st A must be earned Condone AWRT 0.67 ISW if values attempted Attempt at factors gives ± 0 ± 6tan st A must be earned Condone AWRT 0.67 ISW if values attempted 8

90 MPC - AQA GCE Mark Scheme 00 June series MPC (cont) 6(a) ln y = (when) y= 0 = or (, 0) Both coordinates must be stated, not B simply shown on diagram (b) dy = d ln ln = or ln ± ± ln M Quotient/product rule A OE must simplify At B, ln = 0 m Putting their d y = 0 or numerator = 0 d = e A CSO condone = e or e y = e A 5 CSO must simplify ln e (c) Gradient at = e Gradient of normal lne = M (e ) Substituting = (condone slip) but must have scored M in (b) = or e 6 6 e A PI e 6 = A CSO simplified to this Total 9 dy e into their d 9

91 MPC - AQA GCE Mark Scheme 00 June series MPC (cont) 7(a)(i) cos d u= dv d = cos M cos, ( ) d attempted d du sin = v = d A All correct sin sin Correct substitution of their terms into = d parts formula sin cos = + ( + c) A OE with fractions unsimplified 6 m (ii) dv sin d u = = sin d d M du cos sin, ( ) attempted = v= d d cos cos = d A cos = + cos d cos = + [ ] sin cos Clear attempt to replace integral using + 6 m their answer from part (a)(i) cos sin cos = + + ( + c) A OE with fractions unsimplified 8 ( 0.) (b) V ( π) ( 6) sin ( d) = M ( 0) cos sin cos Must see evidence of their (a)(ii) result or = ( π 6) starting again obtaining terms of the form ± A cos ± Bsin ± Ccos = m AND F(0.) F(0) attempted π[.0959 ] = 0.99 AWRT A Accept AWRT π Total 0

92 MPC - AQA GCE Mark Scheme 00 June series MPC (cont) 8(a) y = e e Stretch (I) scale factor (II) M I + (II or III) in -direction (III) A I + II + III Translation E Allow translate 0 B OE unit down etc (b) = 0 y = 6 or (0, 6) B Both coordinates must be stated, not simply 6 marked on diagram (c)(i) e = e + e e = + e e e = + e or ( ) (e ) e 0 M Multiplying both sides by = A AG With no errors seen (ii) ( e )( e + ) M ( e ± )( e ± ) = ln or ln Reject e A = OE A eg e > 0, e e, impossible etc e d (I) (d) ( + ) e = + ln 0 ln e = + ln + 0 = + ln++ = + ln ( e ) d (II) e = ln 0 ln e = ln 0 = ln = ln A = + ln ln M m A B I or II attempted and e correctly or e F[ their ln from (c)(ii)] F[0] Both I and II correctly integrated Attempt to find difference of their I their II integrated = ln or ln 8 or ln OE A 5 CSO must be eact

93 MPC - AQA GCE Mark Scheme 00 June series MPC (cont) 8(d) Alternative A = e + d e d ( ) ( ) (B) Condone functions reversed ( ln ) ( e e ) d = + ( 0) ln e e = + 0 ln ln = e e + ln (M) (A) (m) e or e correctly integrated Correct substitution of their ln from (c)(ii) into their integrated epression = ln or ln 8 or ln OE (A) CSO must be eact Total 5 TOTAL 75

94 Version.0 General Certificate of Education (A-level) January 0 Mathematics MPC (Specification 660) Pure Core Mark Scheme

95 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 candidates 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 candidates 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 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 from: aqa.org.uk Copyright 0 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.

96 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 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.

97 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (a) dy = k( ) 5 d M Where k is an integer or function of 6 ( ) 5 = (ISW) A But note d y k ( ) 5 p d = + M0 Or u = y = u ( ) ( 6 ) dy d 6u 5 u = and d ( ) 5 du = M = 6 A Note dy = 6 ( ) 5 + c scores M A0 d (penalise + c in differential once only in paper) (b)(i) dy ln d =± ± M Product rule attempted and differential of ln = + ln (ISW) A Must have replaced ln e by (ii) ( = e) y = e PI B Condone y =.7 (AWRT) dy lne ( = ) d = + M Correct substitution into their d y d But must have scored M in (b)(i) ( ) y e= e or y = e OE, ISW A Must have replaced ln e by Total 7

98 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) (a) ( ) ( ) ( ) f = ln + 5 Or reverse ( ) ( ) ( ) ( ) f.5 = 0.9 M M f.6 = 0. But must see f( ) = 5 ( ) ln( + ) before A may be earned Condone f(.5 ) < 0 Only if f ( ) defined M f(.6 ) > 0 Or =.5 y=. ( < 5) M =.6 y= 5. ( > 5) < < OE A Either side of 5,.5 < α <.6 OE A f.5 = 0.9 f.6 = 0. Attempt at evaluating both f (.5) and f (.6) Change of sign,.5 α.6 (b) ( ) ln( + ) = 5 5 = ln ( + ) 5 = + ln =± ( + ) + 5 ln ( + ) M AG A Either of these lines correct Condone poor use of brackets for M only Must have both middle lines and no errors seen (c) ( = ) 5 =.578 CAO B =.568 CAO B Total 6 Sight of AWRT.58 or.57 scores B B0 Or ±.578 or ±.568 scores B B0 =.578, =.568 scores BB0 5

99 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) (a)(i) Where k is an integer d ksec ( y ) dy M Condone omission of d dy But dy = ksec ( y+ ) d scores M A0 = sec y + ISW A Alternative methods ( ) y= ( tan ) d k( ) dy = + M ( ( y )) = + tan + A Or sin = cos ( y + ) ( y + ) kcos ( y ) ksin ( y ) ( y+ ) d ± + ± + = dy cos = ( y + ) cos M A (ii) d y = sec + M Substitution of y = into their d d dy or = sec 0 dy d BUT must have scored M in (a)(i) dy = d CSO A Condone 0. or better Or dy = d sec (y+ ) = As above sec 0 = (b) M A Total 6 Appro correct shape with no turning points, through (0,0) and only curve π Asymptotic at both ± and both values shown Condone ± 90 (degrees) Condone y= tan also drawn but clearly identified, otherwise M0 6

100 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) (a) f ( ) M, < f( ) < < f <, < y< f <, < f A Allow y, f (b)(i) y = cos y = cos y cos = M Or cos = y = cos Either order y= cos M Swap and y f ( ) = cos A (ii) If incorrect in (b)(i) BUT answer = cos M in form p cos ( q ) (condone p, q =) Then q = cos M or = f() M p = cos ISW A = cos A (c)(i) ( ) gf = cos B (ii) π π Modulus graph in st quadrant, starting from a +ve y-intercept, at least M continuous parts, first descending, then second increasing IGNORE CURVE OUTSIDE RANGE A Correct curvature, curves reaching -ais, condone multiple curves (no turning points at ais) A Approimately symmetrical graph with, π, π indicated (must have scored previous marks) Condone y = cos also drawn but clearly identified, otherwise M0 (d) STRETCH + direction M Either in -direction or y-direction s.f., parallel to y-ais A Either order s.f., parallel to -ais A Total 7

101 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) 5(a)(i) d + = kln + M Where k is a rational number ( ) = ln ( + ) + c A Or if substitution u= +, du= d du = = kln u u M = ln ( + ) + c A (b) u= dv= sin M d sin ( d) = kcos, ( ) = d where k is a constant du = v= cos A All correct Correct substitution of their terms into = cos cos ( d) m parts formula (watch signs carefully) = cos + sin + c A CAO Total 6 8

102 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) 6(a) y B Using correct -values, PI 0.05 cos.5 = cos.5 = cos.75 = cos.05 = 0.86 M At least correct y-values, (condone unsimplified correct epressions), Or correct values rounded to s.f. or truncated to s.f. 0. Σ y m Used and must be working in radians = 0. CAO A Must be s.f. (b) du d = M du= d OE u ± = u kdu m = ± All in terms of u, with k = or Condone omission of du u u ( du) 9 m p ( ) u ± u d u 5 u u = = A m (must have scored first marks) OE Must have earned all previous method marks and then correct substitution, into their integral, of, for u or 0, for and subtracting 6 = ISW A 6 Or equivalent fraction 5 Total 0 9

103 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) 7(a) cos = 0. M Or tan =± ( ) =.77,.5 AWRT A One correct value A Second correct value and no etra values in interval 0 to 6.8 Ignore answers outside interval (b) LHS cosec cosec cosec + cosec = + cosec cosec ( ) ( ) ( )( ) M SC =.8,.5 with or without working M A A0 SC (using degrees) 0.5, 8.5 M A A0 0.5, 8.5 M A0 A0 SC No working shown correct answers / correct answer / Correctly combining fractions but condone poor use, or omission, of brackets cosec cosec cosec cosec = A Allow recovery from incorrect brackets cosec cosec ( + cot ) Correct use of relevant trig identity = or m cot cot eg cosec = + cot sec = 50 All correct with no errors seen sec = 5 AG A INCLUDING correct brackets on st line Or cosec cosec = 50 + cosec cosec cosec cosec cosec + cosec = 50 + cosec cosec ( ) ( ) ( )( ) cosec cosec cosec cosec = 50 cosec 8cosec = 50 sin ( ) (M) (A) = cos = (m) 5 5 sec = 5 AG (A) Correctly eliminating fractions but condone poor use, or omission, of brackets Allow recovery from incorrect brackets Correct use of relevant trig identity eg sin = cos All correct with no errors seen INCLUDING correct brackets on st line 0

104 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) 7(c) sec =± 5 M Or cos =± 0. Or tan =± =.77,.5,.7,.9 (AWRT) A correct A correct and no other answers in interval Ignore answers outside interval SC.8,.5,.,.9 With or without working M A SC their answers from (a) +.7,.9 (AWRT) / SC For this part, if in degrees ma mark is M A0 Total 0 SC No working shown correct answers / correct answers / 0,, correct answers 0/

105 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) 8(a) e = = ln M ln = ln ISW A OE, eg ln, ln, (b)(i) ( y = ) B Condone ( 0,) but not (,0 ) (ii) y = 0 e e = 0 e = 0 M a ± e ± b =0 e = or e = A = ln ISW A OE, eg ln, ln, ln and no etra solutions Or e = e ln = (M) = ln (A) OE = ln (A) OE (iii) ( ) y = 8e + e B e = 8e e = 0 or e = 0 or e = or e = or ln = ln8 ln M Equating d y = 0 and getting d ± dy ae ± b =0 from pe qe d = + ln ln8 and no etra solutions = ISW A OE, eg ( )

106 Mark Scheme General Certificate of Education (A-level) Mathematics Pure Core January 0 MPC (cont) 8(b)(iv) Must be completely correct including d ln seen on this line or net line V = π ( e e ) d B 0 Limits, brackets and π PI from later working 8 6 = ( π) 6e + e 8e ( d) B Correct epansion, PI from later working e π e e 8 ( ) 8 = + 6 = + 8 ( ) ln ( 0) ln 8ln 6ln ( π ) e e e B B B M 6 e OE 8 e OE 8 8 e 6 OE may be two separate terms 6 Correct substitution of = ln and 0 into their integrated epression 6 8 must be of form ae + be + ce ( ) and subtracting. e e + e 8 PI 57 = π A 7 OE eact fraction eg 5856 π Total 6 TOTAL 75

107 Version.0 General Certificate of Education (A-level) June 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme

108 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 candidates 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 candidates 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 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 from: aqa.org.uk Copyright 0 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.

109 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.

110 MPC June 0 (a) (b) 6 or,0 6 dy = d B condone 0.67 AWRT M k where k =, 6 or 6 A k = (c) y ln 6 =.798 ln =.89 ln8 =.890 ln =.78 5 ln 0 =.0 6 ln 6 = ln =.777 M A 5+ y-values correct, either eact or correct to SF (rounded or truncated) or better all 7 y-values correct (and only these 7 values), either eact or correct to SF (rounded or truncated) or better A = ( ) ( ) + ( ) M correct use of Simpson s rule on their 7 y-values, condone missing square brackets = 8. A CAO this value only Total 7

111 MPC (cont) (a)(i) y = e dy = e + e d M A A ISW ke + le where k and l are s or s k = Independent of each other l = ( = e ( + ) ) (ii) = dy d e = M tangent: y e e ( ) correct substitution of = into their d y d but must have earned M in part (i) = OE A CSO (no ISW), must have scored first marks common correct answer: y= e e (b) sin y = + cos dy ( + cos )6cos sin ( sin ) = d ( + cos ) M ± p( + cos ) cos± qsin (sin ) ( + cos ) where p and q are rational numbers condone poor use/omission of brackets PI by further working = 6cos + 6cos + 6sin ( + cos ) A this line must be seen in this form (ie in terms of cos and sin ), but allow sin replaced by cos condone denominator correctly epanded 6cos + 6 = ( + cos ) 6 = + cos A CSO m Total 9 correct use of ksin + kcos = k or ksin = k cos ( )

112 MPC (cont) note: if degrees used then no marks in (a) and (c) f = cos e or reverse f( 0.) = 0. sight of ±0. (AWRT) AND 0. M f ( 0.5 ) = 0. (AWRT) change of sign 0. < α < 0.5 A CSO, note f () must be defined, condone 0. α 0.5 alternative method (M) e 0. =.5, cos ( 0. ) =.8 e 0.5 =.65, cos ( 0.5 ) =.57 (a) ( ) ( ) at 0. e < cos ( ) at 0.5 e > cos ( ) cos = e (b) ( ) = cos(e ) (A) = ( cos(e ) + ) = + cos(e ) B 0. < α < 0.5 AG must see middle line, and no errors seen, but condone cos e (c) = 0. = 0.59 B CAO = 0.8 B CAO Total 5

113 MPC (cont) sin ± 0.5 = ±.5 PI by sight of 9.5 etc M condone ±. θ = 9.5, 5.5 (AWRT) A no etras in interval, ignore answers outside interval (a)(i) ( ) (ii) cot (+ 0) = 7 cosec(+ 0) condone replacing + 0 by Y (cosec (+ 0) ) = 7 cosec(+ 0) M correct use of cosec Y = + cot Y cosec (+ 0) + 7cosec(+ 0) ( = 0) A must be in this form (cosec(+ 0) ± )(cosec(+ 0) ± ) ( = 0) m attempt at factorisation cosec( + 0) = or A must be this line using f ( + 0) + 0 = 9.5, 5.5 = 8., 57.8 (AWRT) B one correct answer, allow 8., ignore etra solutions B 6 CAO both answers correct and no etras in interval, ignore answers outside interval (b) stretch (I) scale factor (II) parallel to -ais (III) M I and either II or III A I + II + III translate E 5 B condone 5 to left or 5 in 0 (direction) alternative method translate (E) 0 0 (B) stretch (M) as above scale factor parallel to -ais (A) as above Total

114 MPC (cont) 5(a) ( ) f not E OE (b) y = + = y + y + = M M swap and y a correct net line ( ) g = OE A [ ] y = either order (c) g ( ) 0.5 B sight of 0.5 OE (d) = + + ( + ) = ( + ) B sight of + or ( + ) or + = + + one correct step, must be one of these four M or = lines + or + = = 0 A CSO Total 8 6(a) ln = ln = = e B ISW. Condone e (b) 0 ln + = 9 ln (ln ) + 0 = 9ln M correctly multiplying by ln. (ln ) 9ln + 0( = 0) A (ln ± )(ln ± 5) ( = 0) m use of formula, or completing the square must be correct ln =, 5 A 5 = e, e A 5 condone e Total 6

115 MPC (cont) 7(a)(i) M A modulus graph, approimate V shape, touching negative -ais and crossing y- ais, marked, graph symmetrical, straight lines (ii) (b)(i) + = ( + = ) ( ) modulus graph in sections, touching M -ais and crossing positive y-ais correct curvature A their >, their < independent correct curve A and = ±, y = marked 0 = A M either A or B seen, all terms on one side =, A,A + = ( ) ( ) + + = 0 B =, A,A 5 =,, SC NMS or partial method correct value /5 correct values /5 independent of correct values 5/5 method mark more than distinct values ma /5 (ii) >, < M,A > their largest, < their smallest; CAO Total

116 MPC (cont) cos ( + tan ) u = + tan 8 du = sec d du u OE d M = m u = = u = ( + tan ) + ( c) du condone = asec where a is a d constant k (d u u ), where k is a constant d u u correct integral of their epression but must have scored M m A correct, or ( ) AF A 5 CSO, no ISW Total 5

117 MPC (cont) 9 (a) ln d dv u = ln ( d ) M correct direction and sight of du l, = v = ( d) = ln ( d) A = ln ( + c) A (b) y = (ln ) dy = ln d M k ln A k = where k =,or (c) y = ln e ( ) ( ) V = π ln d B dv u = = ( ) ( ln ) ( d) du = ln v = d M = ( ln ) ln ( d) m all correct, incl brackets, π, limits and d (but d may be seen BEFORE this line) du k = ln where d correct direction with ( ) k =,or and sight of correct substitution of their terms into the parts formula = ( ln ) ln ( d) A integral needs to be simplified to ln = ( ln ) ( ln ) OE e V = (π) ( ln ) ( ln ) e correct substitution of and e into their = (π) e 0 + m epressions of the form p(ln ) + qln + rwhere p, q and r are non-zero rational numbers, and an intention to subtract Do not condone F() F(e) π = e OE A 6 e π etc Total TOTAL 75

118 General Certificate of Education (A-level) January 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme

119 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 M5 6EX.

120 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.

121 MPC (a) y B B all 7 values correct (and no etra) (PI by 7 correct y values) 5 or more correct y values, eact,... or evaluated (in table or in formula) A = [ correct substitution of their 7 y-values into ] M Simpson s rule = 9 or 5.5 or 7 A CAO 6 (b)(i) f( ) = + 8 or g( ) = 8 f (. ) = 0. or g(. ) = 0. f (. ) = 0.7 or g(. ) = 0.7 M attempt at evaluating f(.) and f(.) AWRT ± 0. and ± 0.7 alternative method condone f (.) < 0, f (.) > 0 if f is. = 5., 8. = 5.6 defined. = 6., 8. = 5. M change of sign. < α <. A at. LHS < RHS (f() must be defined and all working correct) at. LHS > RHS. < α <. A (ii) ( = ). B ( = ). B these values only Total 8

122 MPC (cont) (a) f () = f (6) = M sight of and f ( ) A allow f() replaced by f, y (b)(i) y = 6 = 6 y ( y ) = 6 or better M M reverse, y one correct step Either order f 6 ( ) = + OE A condone y = (ii) 6 + = 6, or better + = one correct step from their (b)(i) =, M or = f() ( = ) A note: scores / fg( ) = 6 B (c)(i) ( ) (ii) 6 = = 6 or better M one correct step from their (c)(i) = = 6 OE A eg ( + 8) ( 8) = 0, or = ± = ONLY A Total

123 MPC (cont) (a) dy = 6 d B do not ISW (b) 6+ d ( ) () ln 6 M = + 6 () A = ln ( 6 + ) 6 ln ( 6 + ) 6 m = ln 9 ln 6 6 AF 9 = ln or = ln 6 6 A 5 Total 6 ( + ) k ln 6, k is a constant k = 6 correct substitution in F() F(). condone poor use or lack of brackets. kln 9 kln only follow through on their k or if using the substitution u = 6+ du = k M u = ln u A 6 then, either change limits to and 9 m then AF Aas scheme or changing back to, then m AF A as scheme (a) sec θ =... B correct use of sec θ = + tan θ quadratic epression in secθ with all sec θ + sec θ 0 ( = 0) M terms on one side ( θ )( θ ) sec + 5 sec = 0 m attempt at factors of their quadratic, ( secθ ± 5)( secθ ± ), or correct use of quadratic formula sec θ = 5, A cos θ =, 5 B 60, 00,0.5, 58.5 (AWRT) B 6 correct, ignore answers outside interval all correct, no etras in interval (b) 0 = 60, 0 5,58 5,00 M 0 = any of their (60), all their answers from (a), BUT must have = 70, 5,68 5,0 AF scored B = 7 5, 7 9,67.,77 5 (AWRT) A CAO, ignore answers outside interval Total 9

124 MPC (cont) 5(a) stretch I SF II MA I + (II or III) in y-direction III either order translate E e 0 B accept e in positive -direction (b) M A mod graph, in connected sections, both in the first quadrant, touching -ais curve touches -ais at + e (or.7 or better), and labelled (ignore scale) e +e A correct curvature, including at their + e, appro. asymptote at = e (c)(i) ln ( e ) = ( ) ( ) ln e = ln e = or better M must see equations, condone omission of brackets ( =) e do not ISW A accept values of AWRT 5., 5., 5. ( = ) e + e or ( ) = e + e do not ISW A accept values of AWRT.08,.09 if M0 then = e with or without working scores SC (ii) e B accept values of AWRT 5., 5., 5. e < e + e B accept values of AWRT.7,.08,.09 if B not earned, then SCfor any of e e+, e< < e +, e < e+ e e e Total

125 MPC (cont) 6(a) ( ) d sinθ 0 cosθ = dθ sin θ M ± sinθ k ± cosθ quotient rule sin θ where k = 0 or must see the 0 either in the quotient or in A eg d u 0 dθ = cosθ cosθ or = sin θ sinθsinθ or equivalent = cosecθ cotθ A CSO, AG must see one of the previous epressions (b) = cosecθ d cosecθ cotθ dθ B OE, eg d = cosecθ cotθdθ Replacing cosec θ by cot θ, or better B at any stage of solution ( ) = cosecθcotθ ( θ ) cosec θ cosec dθ M all in terms ofθ, and including their attempt at d, but condone omission of dθ A cosecθ cotθ ( dθ ) cosec θ cotθ ( dθ ) cosecθ = cosθ A =, θ = 0.5 AWRT B =, θ = AWRT fully correct and must include dθ (at some stage in solution) = A OE eg sinθ( dθ) correct change of limits ± cos = ± OE or ( ) θ ( ) m c's F( 0.5 ) F( 0.79 ) = 0.59 A 9 Total substitution into ± cos θ only or

126 MPC (cont) 7(a) M p, q constants dy = p e + qe d A p = and q = e + = 0 e 0 E or e = 0 impossible OE (may be seen later) e ( a + b ) = 0 m or e ( a + b ) = 0 = 0, 8 A = 0, y = 0 A = 8, y = 6e B 7 condone 8 y = 8 e etc ignore further numerical evaluation (b)(i) d e d v u = = e d du v ke d M where k is a constant k = A e e (d ), or better AF correct substitution of their terms u = m dv = ne d du m v ne d = = m both differentiation and integration must be correct e = + 8 e + e d e e 8e = ( ) ( 0 ) Al = + [ ] [ ] e m (dep on M only) correct substitution and attempt at subtraction in a e + be + ce (may be in stages) = 8 0 A 7 e or 8 0e ignore further numerical evaluation (ii) ( ) ( 0 ) ( ) v= π 9 e d M condone omission of brackets, limits = 9π 8 0 e AF 9π (their eact b(i)) Total 6 TOTAL 75

127 Version,0 General Certificate of Education (A-level) June 0 Mathematics MPC (Specification 660) Pure Core Mark Scheme

128 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 M5 6EX.

129 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.

130 MPC - AQA GCE Mark Scheme 0 June series y B M 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 ( = ) =. A CAO Total Correct substitution of their y values (of which are correct), either listed or totalled

131 MPC - AQA GCE Mark Scheme 0 June series (a) f ln Or reverse f must haveboth valuescorrect f.5 0. M Allow f0.5 0 and f.5 0only if f() defined Change of sign 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 (M) ln At 0.5, LHS < RHS; at.5, LHS > RHS (A) (c).9 B. B If B0B0 scored but either value seen correct to or dp, score SC (d) M 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

132 MPC - AQA GCE Mark Scheme 0 June series dy (a) ln d M p q ln where p and q are integers A p, q (b)(i) dy d e e lne e M Substituting e for in their d y, but must d have scored M 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 M 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 M A All terms in this form, k 6 k,or 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 π e 6 6 Correct substitution of 0 and into their M 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

133 MPC - AQA GCE Mark Scheme 0 June series 5(a) f 0 M A Condone y 0 f 0, f 0, 0, y 0, range 0 (b)(i) 0 fg OE B No ISW (ii) M Correctly squaring their fg() and correctly isolating their term A No ISW (c)(i) y 5 M M Swap and y either order Correctly squaring (ii) 5 f ( ) A 9 or if 9 or seen M Candidate must have scored full marks in (c)(i) (ie no follow through) = and = rejected A Must see both Total 0

134 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 M 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

135 MPC - AQA GCE Mark Scheme 0 June series 7(a) M 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) M 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 M 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

136 MPC - AQA GCE Mark Scheme 0 June series 8(a) LHS cos cos cos cos M 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 M 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 (M) = 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)

137 MPC - AQA GCE Mark Scheme 0 June series d cosy cos y sin y sin y 9(a) dy cos y M 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 M (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

138 MPC - AQA GCE Mark Scheme 0 June series 9 cont y tan ln (d)(i) dy d dy 0 d M Must be correct 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) M y p q where p and q are constants y A p =, q = including correct brackets (iii), y M 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

139 Version General Certificate of Education (A-level) January 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme

140 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 M5 6EX.

141 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.

142 MPC - AQA GCE Mark Scheme 0 January series MPC (a) f f 6 must have both values f 0 M correct allow f 0 and f 0 only if f is defined and no errors seen change of sign A must have both statement and interval which may be written in words/symbols (b) 6 or or B AG (c) 6.66 must see one of these lines and no errors at least sf needed B.5 PI by correct.6 B SC if B0B0 scored and =.67 Total 5

143 MPC - AQA GCE Mark Scheme 0 January series (a) y0 0 y 0. y 0. B all 5 -values PI by 5 correct y-values y 0.7 y 0. at least y-values eact or rounded or B 8 truncated to at least sf M correct use of Simpson s rule using and and correctly with candidate s 5 y-values =.0 A CAO (must be eactly this value) (b) d ln 0 M A for k ln all correct; limits not needed ln8 ln AF For k (ln8 ln) ln 9 AF combining candidate s logarithms correctly (must be seen) ln A 5 CAO (must be eactly this) NMS scores 0/5 Total 9

144 MPC - AQA GCE Mark Scheme 0 January series dy B B for one term correct (a) e d B B all correct (b)(i) cos cos sin sin cos du d cos cos cos sin sin cos cos sin cos cos cos M A Acso cos clear attempt at quotient/product rule condone poor use of brackets any correct form seen AG be convinced correct use of brackets and correct notation used throughout (eg A0 if cos etc seen) (ii) dy cos d sin cos sin OE M correct use of chain rule cosec A AG, must see Total 7 and no errors seen; sin condone incorrect use of brackets only if penalised in part (b)(i)

145 MPC - AQA GCE Mark Scheme 0 January series (a) M reflection in the -ais for the negative f and remainder as given on sketch 0 A correct curvatures, correct cusp at = condone straight lines for < 0 and > marked on -ais (b) Either. Stretch M and either or. -ais. by factor 0.5 A, and (followed by) translation E B or translation (E) 0 (B) (followed by). Stretch (M) and either or. -ais. by factor 0.5 (A), and Total 6

146 MPC - AQA GCE Mark Scheme 0 January series 5(a) M f,f,range f A (b)(i) (ii) BF correct or FT from (a) y y M either order M for correctly changing the subject or reversing f M operations; M for replacing y with f A (dependent on both M marks) correct sign 0 (c)(i) M Or e or OE A CAO, NMS OE scores / (ii) g has NO inverse because two values of map to one value (of y) or it is many-one or it is not oneone or it is two-one B must indicate no inverse with valid reason; do not accept contradictory reasons (iii) ln M ln 5 A NMS scores 0/, condone k = 5 after correct epression seen (iv) ln (or or e or e seen) M k etc, for candidate s positive integer, k 6, or candidate s k or k 6, 6, 0 6, AF AF eact values PI by correct answers A CAO, rejecting the positive Total 5 5

147 MPC - AQA GCE Mark Scheme 0 January series 6(a) sec sec sec sec sec sec tan used M M for correct use of sec tan at least once or cosec cot sec tan or tan tan cos tan or cot A Shown, with no errors sin cosec A AG (No errors, omissions or poor notations seen) (b) cosec cosec cosec cosec 0 B must have = 0 correct solution of the quadratic, or by completing the square cosec or (... and...) M cosec PI by values for sin sin BF BF for cosec seen or implied sin 0. and or A PI 6,5, 50, 0 B B 6 B for any three values correct AWRT B for all four values correct AWRT and no etras in the interval (c) 60 M where is a written value from candidate s (b) in degrees PI by their answer, 5 A CSO Ignore solutions outside interval 0 90 Total 6

148 MPC - AQA GCE Mark Scheme 0 January series 7(a) (b) y cos dy cos sin d gradient of the tangent π π π Acos B sin M A m anything reducible to Acos Bsin where A and B are non-zero integers OE, all correct π A must have π an equation of the tangent is π yπ substituting π into candidate s derived function using correct d y d A 5 OE, dependent on previous A π cosd 0 u A dv cos d M all terms in this form seen or used du A v Bsin d B or A = and B =, etc π π sin sind m 0 0 π π 0 0 sin cos AF π A 5 OE, eact value correct substitution of candidate s terms into integration by parts formula condone missing limits candidate's second integration completed correctly FT on one error including coefficient condone missing limits Total 0 7

149 MPC - AQA GCE Mark Scheme 0 January series 8(a) e d ke or e e k M where k is a rational number ln ln e d e or 0 0 e e ln 0 A correct integration condone missing limits e e e e eliminating ln e 8 A AG, be convinced ln (0) A correct (no decimals) (b) u tan du sec d M PI below, condone du sec d Replacing d by du in integral A or du sec u sec u B PI below 0 u 0 this could be gained by changing u to tan after the integration and using 0 π B u and π sec tan d 0 du all in terms of u including replacing d u ud u or u u M u all correct, condone omission of du 5 u u du A must be in this form 7 u u 7 A accept correct unsimplified form 0 A 8 CAO Total TOTAL 75 8

150 Version.0 General Certificate of Education (A-level) June 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme

151 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 M5 6EX.

152 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.

153 MPC- AQA GCE Mark Scheme 0 June series (a) B M or ( ) or A (b) No ISW in part(b), mark their final line as their answer. B Or (a) y tan dy tan sec d B Total 5 M A Or Or or for B B tan sec non-zero constant. A k A for k = may have (sec ) or cos OE where A is a A A all correct ISW if attempt to simplify is incorrect. (b) dy ( ) ( ) d ( ) ( ) M A Use of the quotient rule ( ) ( ) ( ) Simplification not required dy d or 0.75 OE A Obtained from correct d y d Total 6

154 MPC- AQA GCE Mark Scheme 0 June series (a) f() 0.(8) f( ) e M Both values correct f() 0.0(8) (b) Change of sign ( e n n ) A Must have both statement and interval in words or symbols..880 B Do not accept B Do not accept.97 (c) y M Staircase to curve from including at least two stairs between curve and line y. A and marked on the -ais. Do not accept marking on the Curve or on the line. Total 6

155 MPC- AQA GCE Mark Scheme 0 June series 8sec sec tan 8sec sec sec sec 8sec 0 (sec )(sec ) 0 Or sec 8 ( 8) ()( ) () sec, (or 0.) sec cos cos or 0.., 5.05 M A m A B A A 7 Using tan sec and NOT replacing sec with tan. Correct factors or correct use of quadratic equation formula or completing the square for their 8 6 equation. sec 6 6 Both correct. PI sec is impossible One correct. Must have earned A for correct quadratic, but independent of the second A. Both correct and no etras in 0 π. CAO Total 7

156 MPC- AQA GCE Mark Scheme 0 June series 5(a) i 0.( ).( ) ( ).8( ).6( ) B All 5 -values correct, PI by 5 correct y- y values. i B At least correct y-values rounded or truncated to at least s.f. or in surd form 7 (0.), 7 (.),etc. or 7.06, 8.78, etc. or sight of yi ( ) 5.6 M A Correct use of mid-ordinate rule using 0.8 with candidate s 5 y-values. Dependent on first B CAO (must be eactly this) and no error seen (b) Could be gained without answering part (a) B Diagram showing curve through the midpoint of the top of rectangle. May have one or more rectangles. Smaller OE E Total 6 Dependent on B

157 MPC- AQA GCE Mark Scheme 0 June series 6(a) y π B Correct sketch of cos. -, π and, 0 B Stated (b) y π B Correct sketch of π cos Must touch negative -ais. -, 0 and, π B Stated Total

158 MPC- AQA GCE Mark Scheme 0 June series 7(a) y M Reflection in the -ais. (b) O y A M Intersection with the -ais and y-ais marked and. Accept (, 0) and (0, ) instead of marking on the aes. Reflection of 0< <6 part in the y-ais giving two connected sections 6 O 6 A Correct curve beyond 6, correct curvature and correct cusp at =0 (generous) A 6 and marked correctly Accept (6, 0), (0, ) and (-6,0) instead of marking on the aes. (c) Reflection in the y-ais (followed by) ) stretch ) parallel to the -ais ) by factor M A M A and either or, and OR Vice versa Total 9

159 MPC- AQA GCE Mark Scheme 0 June series 8(a)(i) f( ) ln( ) e y Either order: M y e M for antilog M M for replacing f() or y with (ii) (iii) f ( ) (e ) OE f ( ) - y A B M A Correct epression in Do not condone f ( ), y, range, f Correct shape crossing y-ais and above -ais marked on the y-ais O (b)(i) ln() gf( ) e M Correct composition ln() e m PI by correct epression ( ) A (ii) fg( ) ln((e ) ) ln(e ) ln 5 e 5 OE M A OE correct composition Correct antilog of correct equation e 8 ln8 ln8 A Total OE eact solution, e.g. ln 8 or ln or ln

160 MPC- AQA GCE Mark Scheme 0 June series 9 V π dy 6 ( y8) ( 8) y 6 (6) V (π) ( y8) (d y) 6 (0) 6 V (π) ( y8) y (6) V (π) (6 8) (6) 6 ( 8) 6 60 V π (0) B M A A A 5 Total 5 OE Accept 'their' in terms of y Condone missing limits and wherever bracketed OE, for correct integration of correct integrand OE, correct use of correct limits in correct epression, PI by correct answer. OE eact value, 560 eg π 5 or π 5. or π 8

161 MPC- AQA GCE Mark Scheme 0 June series 0(a)(i) dv u ln d ln d M & d attempted d du A All correct v d ln d ln (d ) m Correct substitution of their terms into parts ln C A All correct (constant needed) (ii) dv u (ln ) d d u ln v d M A d(ln ) & d d All correct attempted (ln ) d (ln ) ln (d ) m OE correct substitution of their terms into parts (ln ) ( ln ) C OE A All correct (constant needed) including correct use of brackets. Do not penalise missing constant if already penalised in part (i) ISW (b) du or d d u du () () d u (d u) u u () () B M u All in terms of u including attempt at replacing d (not simply writing du), condone missing limits and du () (d u) u () ln( u ) () () ln(+) ln(+) or ln( ) ln( ) A A AF AF Integrand correct unsimplified k FT their (d u) u correct use of correct limits k u or kln on ln( ) 9 ln or ln or ln ln A 7 Total 5 TOTAL 75 OE ISW

162 A-LEVEL MATHEMATICS Pure Core MPC Mark scheme 660 June 0 Version/Stage:.0 Final

163 Mark schemes are prepared by the Lead Assessment Writer 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 associates 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 associate understands and applies it in the same correct way. As preparation for standardisation each associate 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, associates encounter unusual answers which have not been raised they are required to refer these to the Lead Assessment Writer. 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. 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.

164 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 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. of

165 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment y ( 0) = 0 π y = 0.666(57068) π y =.5(7) π y =.085(088) y ( ) π = 0 B B All 5 -values*, PI by 5 correct y-values All 5 y-values eact** or correct to at least SF (rounded or truncated) π { [ ] + [.5] } M Correct use of Simpson s rule π using (or ) and and correctly with their 5 y- values from any -values =.9 A CAO (must be eactly this value) Total * Accept decimals 0.78(598...),.5( ),.(569...),.(59...) ** π π y = π sin, etc. The minimum evidence for M is the correct non-zero values of y in any form and sight of.90(97...), but condone omission of the two zeros. If a candidate s calculator setting is in degrees, they may earn the first B for 0, π, etc, and then B0, but M is available. NMS: An answer of.9 without anything else gains 0/. of

166 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment (a) dy ( e ) k = or d ( e ) ( e ) M M for or k( e ), k ( e ) A OE, all correct (b) dy dy = = Substituting e for in their PI d ( e e) e M d Gradient of the normal e e e m Must have also earned M in part (a) = = (c)(i) y = ln( e e ) ( = ) B e e e y = ( e ) or y = + A ( ) = ( ) ( ) = ( ) f ln e or g ln e f () =.98 or g() =.98 f () =. or g() =. M OE, but must have simplified the gradient and replaced ln(e e) with Must have both values correct rounded or truncated to sf. Allow f() > 0 and f() < 0 only if f() is defined. OR evaluating both sides of ln( e ) = : ln ( e ) =.98 = ( M) ln ( e ) =.78 =.98 > and.78 < < α < (A) (ii) (iii) Change of sign < α < A =.980 ( ) B Not.98 =.798 ( ) B. y y = M All working must be correct together with correct statement If B0, B0 scored but both values given correct to sf or more than dp, then SC. Vertical line from to curve( condone omission from - ais to y = ) and then horizontal line from the curve to y = * ( ) y = ln e A Second vertical and horizontal lines * and, ( or the values) must be labelled on - ais * * c(iii) O Total * On diagram, the solid lines may be dotted and the dotted lines need not be shown. ** Condone correct values (unrounded or dp) marked on the -ais instead of and. 5 of

167 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment (a)(i) k + or ku M Attempted use of the chain rule ( ) 5 ( ) + A OE, all correct (ii) (b) e B Differentiating e correctly 5 dy e ( ) = + + e ( their part (a)(i)) d M OE 5 dy 5 e = ( + ) + e ( + ) d dy ( When = 0 ) = d A Substituting = 0 and CSO ( + ) ( ) ± + ± M M for ( + ) ( + ) A All correct ( ) ( ) ( ) ( ) = 0 or = 0 m Forming a quadratic equation with all terms on one side a + b + c ( = 0) b 0, c 0 + = 0 A OE correct factors or using the ± 5 formula as far as = or completing the square as far as ( )( ) ( ) =, = A 5 Total 0 = ± 5 6 CAO, simplified answers 6 of

168 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment (a) Reflection in the -ais for the positive M f() and the remainder as given in the 0 sketch. A Correct < < with minimum at < 0 lower than minimum at > 0 and correct cusps at =, 0,. (b) A M Correct branches for > and <, including the curvature of both branches and and marked * Symmetrical about the y-ais using only the original curve for >0 0 A and labelled on the -ais and correct cusp at =0 (c)(i)* ( I) Stretch s.f. // -ais III ( II ) ( ) M A (I) and either (II) or (III) (I) and (II) and (III) (followed by) Translation E Not shift, move, etc. 0 B Or in words Not (, 0) (ii) (, ) B B B for each coordinate (a) *(c)(i) Or =, y = Total * The two A marks are independent. Condone straight lines for the branches for > and <- but not curves which are concave upward. Alternative: Translation E 0 B (followed by) Stretch (I) s.f. (II) // ais (III) M : (I) and either (II) or (III) A: (I), (II) and (III) 7 of

169 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment 5(a) M f ( ) >, or ******, (b) f( ) y = ( ) ( ) = ± y+ A M = ± y+ A condone = + y+ y = ± + B interchanging and y at any stage ( ) f ( ) = + + A negative clearly rejected. must have ± earlier. (c)(i) ( ( ) ) gf = or 6 B (ii) ' their ' = 6 M and attempt to solve term quadratic ' their ' = 6 M = 7 = A = 5 = = 5, = 7 E and attempt to solve term quadratic all four solutions seen and correct values and clearly rejected (a) ( ) Total f >, f,,, range, y score M only y >, etc scores M0 (two errors) (b) Alternative y = 6+ 5 ( ) y = 0 ( y) 6 ± 6 5 = correctly solving M 6 ± 6 + y = A B for swapping and y and A for having rejected minus sign 8 of

170 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment 6(a) sin d (b) du u = = d dv = sin v = cos d ( sin d = ) cos+ cos d du u = = d dv = cos v = sin d ( sin d = ) ( ) cos+ sin sin (d ) M A AF m A = cos+ sin + cos+ C A π V = π sin d 0 ( ) B 6 du = k, k = or and v = pcos d p =±, ±, ± 0.5 All correct Correct substitution of their terms into parts formula Correct follow through unsimplified from their first integral above Correct OE, must have constant of integration Fully correct including d and limits π π = ( π) cos π + sin π + cos π π = π 8 π M Attempt at F F(0) FT their epression from part (a) A Total 9 OE in eact form with sin π evaluated cos π and 9 of

171 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment 7 du d = or du = d du Condone d = or du = d and substituting for d and in terms of u M for M ( u) du u A OE correct unsimplified integral in terms of u only with du seen on this line or later = (d u ) u A PI by the net line u = ln u () () = ln ln ln + ln or ln AF m A 6 Total 6 FT on their b a+ du u Correct use of correct limits in u for epression of form au +blnu or in terms of OE eact value 0 of

172 FINAL MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE Q Solution Mark Total Comment 8(a) ( sin ) + cos ( ) sin cos + = cos sin cos sin sin + sin + cos = cos sin ( ) ( ) M Combining fractions correctly sin + = cos sin m Using sin + cos = ( ) ( ) ( ) sin sin = or cos sin cos sin A Must have factorised denominator = cos (b) = sec tan = sec A AG, both epressions seen sec = sec Using tan = sec, OE sec sec ( = 0) B Or cos + cos ( = 0) ( sec )( sec ) ( 0) + = M Correctly factorising their epression or substituting into formula sec = or A sec = = 7, 89 B B sec = = 80 B 6 Or cos = or no etras inside the interval 0 < 60, EE (c) θ 0 = 70.5, 80, 89.5 M For RHS accept any -value from part (b) PI θ = 50, 05, 60 A Allow 5, 05, 60 Total TOTAL 75 (b) = 70 and 90 scores B0 B0 AWRT = 7 and 89 both not given to the nearest degree earns SC. (c) Condone correct answers not given to the nearest degree if already penalised in part (b), AWRT θ = 50 or 5, 05, 60 of

173 A-LEVEL Mathematics Pure Core MPC Mark scheme 660 June 05 Version/Stage:.0 Final