Version hij. General Certificate of Education. Mathematics MPC3 Pure Core 3. Mark Scheme examination - January series

Transcription

1 Version.0 00 hij General Certificate of Education Mathematics 6360 MPC3 Pure Ce 3 Mark Scheme 00 eamination - January series

2 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 crect way. As preparation f 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 f. 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 wking 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 licenss. All rights reserved. COPYRIGHT AQA retains the copyright on all its publications. However, registered centres f AQA are permitted to copy material from this booklet f their own internal use, with the following imptant eception: AQA cannot give permission to centres to photocopy any material that is acknowledged to a third party even f 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 3673) and a registered charity (registered charity number 07333). Registered address: AQA, Devas Street, Manchester M5 6EX Dr Michael Cresswell Direct General

3 MPC3 - AQA GCE Mark Scheme 00 January series Key to mark scheme and abbreviations used in marking M m dm A B E mark is f method mark is dependent on one me M marks and is f method mark is dependent on M m marks and is f accuracy mark is independent of M m marks and is f method and accuracy mark is f eplanation ft F follow through from previous increct result MC mis-copy CAO crect answer only MR mis-read CSO crect solution only RA required accuracy AWFW anything which falls within FW further wk AWRT anything which rounds to ISW igne subsequent wk ACF any crect fm FIW from increct wk AG answer given BOD given benefit of doubt SC special case WR wk replaced by candidate OE equivalent FB fmulae book A, ( 0) accuracy marks NOS not on scheme EE deduct marks f each err G graph NMS no method shown c candidate PI possibly implied sf significant figure(s) SCA substantially crect 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 f 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 wking and it is very unlikely that the crect answer can be obtained by using an increct method, we must award full marks. However, the obvious penalty to candidates showing no wking is that increct answers, however close, earn no marks. Where a question asks the candidate to state write down a result, no method need be shown f full marks. Where the permitted calculat has functions which reasonably allow the solution of the question directly, the crect answer without wking 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 crect method f any marks to be awarded. 3

4 MPC3 - AQA GCE Mark Scheme 00 January series MPC3 y e e y = Ae a+ b ± Be + = + (a) ( ) + M = e ( ) where A and B are non-zero constants A All crect e - 6 e +0e = e 5 3 A 3 AG; all crect with no errs, y= e + e e nd line (OE) must be seen Condone increct der on final line y = e + e +. e + e + 8e = e 6e + 0e ( ) (M) (A) = e 5 3 (A) A e + Be + Ce + De + Ee All crect AG; all crect with no errs, 3 rd line (OE) must be seen (b) ( )( ) + 5 ( = 0) M OE Attempt at factisation ( ± ± 5)( ± ± ) fmula with at most one err 5 =, A Both crect and no errs =, y= e m F e b =, y= e AF SC = only sces MA0 y= a attempted Either crect, follow through only from increct sign f A 5 CSO solutions only Total 8 Note: withhold final mark f etra solutions Note: approimate values only f y can sce m only

5 MPC3 - AQA GCE Mark Scheme 00 January series (a)(i) A B crect shape passing through igin and stopping at A and B B A π, π B, B B 3 SC A(, 90) and B (, 90) sces B (ii) line intersecting their curve (positive gradient, positive y intercept) M Crect statement A one solution only, stated indicated on sketch - must be in the first quadrant (ie curve intersects line once) (b) () () LHS 0.5 = 0.5 RHS 0.5 =. LHS =.6 RHS =.3 At 0.5 LHS < RHS, At LHS > RHS 0.5 < α < f = sin f( 0.5) = 0.6 AWRT f() = 0.3 M Change of sign 0.5 < α < (A) f = sin + f ( 0.5) = 0. Attempt (M) f () = 0. Change of sign 0.5 < α < (A) A CSO Must have sced B f graph in (a)(i) 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 =.3 (M) Change of sign 0.5 < α < (A) 5

6 MPC3 - AQA GCE Mark Scheme 00 January series (c)(i) = 0.90 M Sight of AWRT 0.90 AWRT = 0.9 A These values only (ii) M Staircase, (vertical line) from to curve, hizontal to line, vertical to curve O 3 A, 3 appro crect position on -ais Total 6

7 MPC3 - AQA GCE Mark Scheme 00 January series 3(a) sin =, 3 sight of ± 0.3, ± 0.π ± 9.7 ( better) M = 0.3,.8( 0) AWRT A Penalise if increct answers in range; igne answers outside range (b) cosec = cosec M Crect use of cot = cos ec cosec cosec 0 + = A ( cosec )( cosec 3)( 0) + = m Attempt at Facts Gives cosec when epanded Fmula one err condoned cosec =, 3 A Either Line sin =, 3 sin = = 3.39, 6.03 AWRT 3 crect their two answers from (a) BF and 3.39, ,.8( 0 ) AWRT B 6 crect and no etras in range igne answers outside range SC 9.7, 60.53, 9.8, 35.5 B Alternative cos sin = sin cos = sin sin sin = sin sin 0 sin sin (M) = (A) ( )( ) Crect use of trig ratios and multiplying by sin 0 = sin + 3sin (m) Attempt at facts as above sin =, 3 (A) (BF) (B) Total 8 As above 7

8 MPC3 - AQA GCE Mark Scheme 00 January series (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 crect answer = 6 B Second crect 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 crect answer Second crect answer and no etras and no further increct statement eg 6 < < < > 6 SC 6, sces B values crect PI 3+ y values crect to sf better eact values.98, 3.8 / 9,. / 5,.77 f y ( better) = 3 y =. A (Note:. with evidence of mid-dinate rule with four strips sces /) (b)(i) y = ln ( + 5) y e 5 y = + OE = e 5 B AG Must see middle line, and no errs (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 crect notation must see dy ISW if evaluated seen, condone ln +... B + 3 A 3 CSO mark final answer (no ISW) Total 8

9 MPC3 - AQA GCE Mark Scheme 00 January series f > 3 M > 3, 3 f 3 6(a) > (b)(i) y = e 3 y + 3= e ( y ) A Allow y > 3 ln + 3 = M swap and y ( f ) ln( 3) M = + A 3 Alternative e -3 ln + 3 (M) (M) ln( + 3) y = (A) attempt to isolate: ln ( y± A) = B reverse OE with no further increct wking Condone y =.. (ii) + 3= M (c)(i) f putting their p( ) = from kln ( p ) in their part (b)(i) = A CSO SC: B = - with no wking, if full marks gained in part (b)(i) (gf = ) 3e ( 3 ) + substituting f into g either OE B (=) ISW 3e 5 (ii) 3e 5 = = OE M Crect removal of their fraction 3e 5 e = = ln m Crect use of logs leading to k = ln a b = ln OE A 3 CSO No ISW ecept f numerical evaluation Total 9

10 MPC3 - AQA GCE Mark Scheme 00 January series 7(a) dy cos. cos sin. sin ± Acos ± Bsin = M d cos cos cos + sin = better cos A Both terms crect = + tan CSO A 3 All crect dy cos. cos sin. sin = d cos (M) ± Acos ± Bsin cos (b) cos cos sin sin = + cos cos cos cos (A) better = ( + tan ) CSO (A) All crect d y tan d M A tan f() sec m f() = B sec = 3tan sec AF ft 8 their p ( y ) = 3 tan + tan m = 3y + A 5 CSO Alternative Solutions sin tan cos from part (a) Previous two method marks must have been earned y = + = + y = 3 3 (M) Acos ± Bsin cos sin cos + s in cos sin where A and B are cos cos constants 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) = 3 tan sec (m) ktan sec ( y ) = 3y + (A) CSO dy sec d = d y sec.sectan d = (M) A sec f() (m) f() = B sec tan = 3 sec tan (AF) ft 8 their p from part (a) = 3 ( + tan ) tan Previous two method marks must have (m) been earned = 3 y + y (A) CSO 0

11 MPC3 - AQA GCE Mark Scheme 00 January series 7(b) 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) = 3 u( + u ) (m) 8(a) sin ( ) = 3 y( + y ) (A) Total 8 d u= dv d = sin ( ) M sin f, attempted d d du = v= cos( ) d A All crect condone omission of brackets ( = ) cos( ) m crect substitution of their terms into parts cos ( ) ) = cos( ) + cos( ) (d ) A All crect 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 All in terms of u All crect PI from later wking B u + + ln u 8 A 6 ( + ) = + ln ( ) + c 8 CSO condone missing + c only Total TOTAL 75 ISW