Version 1.0. General Certificate of Education (A-level) June Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

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

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 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 6447) and a registered charity (registered charity number 074). Registered address: AQA, Devas Street, Manchester 5 6EX.

3 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.

4 MPC- AQA GCE Mark Scheme 0 June series (a) or ( ) or No ISW in part, mark their final line as their answer. Or (a) y 4 tan dy d 4 4 tan sec Total 5 Or Or or for 4 tan sec non-zero constant. 4 A k for k = may have (sec ) or cos OE where A is a all correct ISW if attempt to simplify is incorrect. dy ( ) ( ) d ( ) ( ) Use of the quotient rule ( ) ( ) ( ) Simplification not required dy d 4 or 0.75 OE Obtained from correct d y d Total 6

5 MPC- AQA GCE Mark Scheme 0 June series (a) f() 0.(8) f( ) e Both values correct f(4) 0.0(8) Change of sign 4 ( e n n ) Must have both statement and interval in words or symbols..880 Do not accept Do not accept.97 (c) y Staircase to curve from including at least two stairs between curve and line y. and marked on the -ais. Do not accept marking on the Curve or on the line. Total 6

6 MPC- AQA GCE Mark Scheme 0 June series 4 8sec sec tan 8sec sec sec sec 8sec 0 (sec )(sec ) 0 Or sec 8 ( 8) 4()( ) () sec, (or 0.) sec cos cos or 0.., 5.05 m 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 64 equation. sec 6 6 Both correct. PI sec is impossible One correct. Must have earned for correct quadratic, but independent of the second. Both correct and no etras in 0 π. CAO Total 7

7 MPC- AQA GCE Mark Scheme 0 June series 5(a) i 0.4( ).( ) ( ).8( ).6( ) All 5 -values correct, PI by 5 correct y- y values. i At least 4 correct y-values rounded or truncated to at least 4 s.f. or in surd form 7 (0.4), 7 (.),etc. or 7.064, 8.78, etc. or sight of yi ( ) Correct use of mid-ordinate rule using 0.8 with candidate s 5 y-values. Dependent on first CAO (must be eactly this) and no error seen Could be gained without answering part (a) Diagram showing curve through the midpoint of the top of rectangle. May have one or more rectangles. Smaller OE E Total 6 Dependent on

8 MPC- AQA GCE Mark Scheme 0 June series 6(a) y π Correct sketch of cos. -, π and, 0 Stated y π Correct sketch of π cos Must touch negative -ais. -, 0 and, π Stated Total 4

9 MPC- AQA GCE Mark Scheme 0 June series 7(a) y Reflection in the -ais. O 4 y 4 Intersection with the -ais and y-ais marked and 4. Accept (, 0) and (0, 4) instead of marking on the aes. Reflection of 0< <6 part in the y-ais giving two connected sections 6 O 6 Correct curve beyond 6, correct curvature and correct cusp at =0 (generous) 6 and 4 marked correctly Accept (6, 0), (0, 4) and (-6,0) instead of marking on the aes. (c) Reflection in the y-ais (followed by) ) stretch ) parallel to the -ais ) by factor 4 and either or, and OR Vice versa Total 9

10 MPC- AQA GCE Mark Scheme 0 June series 8(a)(i) f( ) ln( ) e y Either order: y e for antilog for replacing f() or y with (ii) (iii) f ( ) (e ) OE f ( ) - y Correct epression in Do not condone f ( ), y, range, f Correct shape crossing y-ais and above -ais marked on the y-ais O (i) ln() gf( ) e 4 Correct composition ln() e 4 m PI by correct epression ( ) 4 (ii) fg( ) ln((e 4) ) ln(e ) ln 5 e 5 OE OE correct composition Correct antilog of correct equation e 8 ln8 ln8 Total OE eact solution, e.g. ln 8 or ln or ln

11 MPC- AQA GCE Mark Scheme 0 June series 9 V π dy 6 ( y8) ( 8) y 6 (6) V (π) ( y8) (d y) (0) 6 (6) V (π) ( y8) y 6 V (π) (6 8) (6) 6 ( 8) 6 60 V π (0) 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 48

12 MPC- AQA GCE Mark Scheme 0 June series 0(a)(i) dv u ln d ln d & d attempted d du All correct v d ln d ln (d ) m Correct substitution of their terms into parts ln C 4 All correct (constant needed) (ii) dv u (ln ) d d u ln v d 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 4 All correct (constant needed) including correct use of brackets. Do not penalise missing constant if already penalised in part (i) ISW du or d d u du (4) () d u (d u) u u () () 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( 4 ) ln( ) F F 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 4 ln ln 7 Total 5 TOTAL 75 OE ISW

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