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

Transcription

1 Version General Certificate of Education (A-level) January Mathematics MPC (Specification 66) 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 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 7). Registered address: AQA, Devas Street, Manchester M5 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 ) 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 January series MPC (a) f f 6 must have both values f M correct allow f and f 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 BB scored and =.67 Total 5

5 MPC - AQA GCE Mark Scheme January series (a) y y. y. B all 5 -values PI by 5 correct y-values y.7 y. at least y-values eact or rounded or B 8 truncated to at least sf...7. M correct use of Simpson s rule using and and correctly with candidate s 5 y-values =. A CAO (must be eactly this value) (b) d ln 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 /5 Total 9

6 MPC - AQA GCE Mark Scheme 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 A 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)

7 MPC - AQA GCE Mark Scheme January series (a) M reflection in the -ais for the negative f and remainder as given on sketch A correct curvatures, correct cusp at = condone straight lines for < and > marked on -ais (b) Either. Stretch M and either or. -ais. by factor.5 A, and (followed by) translation E.5 B or translation (E) (B) (followed by). Stretch (M) and either or. -ais. by factor.5 (A), and Total 6

8 MPC - AQA GCE Mark Scheme 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 (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 /, 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, AF AF eact values PI by correct answers 6, A CAO, rejecting the positive Total 5 5

9 MPC - AQA GCE Mark Scheme 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 sin A Shown, with no errors cosec A AG (No errors, omissions or poor notations seen) (b) cosec cosec cosec cosec B must have = 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. and.768 or.767 A PI 6,5, 5, B B 6 B for any three values correct AWRT B for all four values correct AWRT and no etras in the interval 8 8 (c) 6 M where is a written value from candidate s (b) in degrees PI by their answer, 5 A CSO Ignore solutions outside interval 9 Total 6

10 MPC - AQA GCE Mark Scheme January series 7(a) (b) y cos dy cos sin d gradient of the tangent π π π Acos B sin π A must have π an equation of the tangent is π yπ M A m anything reducible to Acos Bsin where A and B are non-zero integers OE, all correct substituting π into candidate s derived function using correct d y d A 5 OE, dependent on previous A π cosd 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 π π 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 7

11 MPC - AQA GCE Mark Scheme January series 8(a) e d ke or e e k M where k is a rational number ln ln e d e or e e ln A correct integration condone missing limits e e eliminating ln e 8 A AG, be convinced ln () e e 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 u this could be gained by changing u to tan after the integration and using π B u and π sec tan d 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 A 8 CAO Total TOTAL 75 8