Version.0: 007 abc General Certificate of Education Mathematics 0 MPC Pure Core Mark Scheme 007 examination - January series
Mark schemes are prepared by the Principal Examiner 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 examiners and is the scheme which was used by them in this examination. The standardisation meeting ensures that the mark scheme covers the candidates responses to questions and that every examiner understands and applies it in the same correct way. As preparation for the standardisation meeting each examiner 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, examiners encounter unusual answers which have not been discussed at the meeting they are required to refer these to the Principal Examiner. It must be stressed that a mark scheme is a working document, in many cases further developed and expanded 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 examination paper. Further copies of this Mark Scheme are available to download from the AQA Website: www.aqa.org.uk 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 exception: 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 7) and a registered charity (registered charity number 07). Registered address: AQA, Devas Street, Manchester 5 EX Dr Michael Cresswell Director General
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 explanation 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 x EE deduct x 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 Examiner 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
MPC - AQA GCE Mark Scheme 007 January series MPC (a)(i) dx dy, 8t B, B CAO (ii) (b) (c) (a) dy dy 8t t dx dx AF BF, mt, mn BF x y y y+ x x A x t x y f 7 + Chain rule in correct form ft on sign coefficient errors (not power of t) ft on d y if f(t) dx Use candidate s (x, y) and m N Any correct form; ISW; CAO A Substitute for t Simplification not required but CAO Or equivalent methods / forms: y y x x, t, x y Total A Substitute ± in f(x) (b) g 0 d + 0 d A AG (convincingly obtained) SC Written explanation with not seen/clear E,,0 g 0 (c) a, b B, B Inspection expected By division: complete method A CAO Multiply out and compare coefficients: evidence of use A both a and b correct Total
MPC - AQA GCE Mark Scheme 007 January series MPC (cont) (a) cos x sin x B (b)(i) sin x cosx sin x( sin x) Candidate s sin x + sin x A AG cos x or sin x (ii) sin x+ sin x 0 Soluble quadratic form ( sin x )( sin x+ ) 0 Attempt to solve (allow one error in formula, allow sign errors) sin x x 0 x 50 sin and two solutions ( 0 < x < 0 ) A A0 if radians Allow misread for sin x+ sin x 0 () Soluble quadratic form ± 7 sin x () Use of formula (allow one error) x.,.7 (A) Max / (c) x sinx ( cos x) ( + c) A solve integral, must have terms for sin x from (a) 9 (a)(i) x 5 By division: + x x B, B B for, B for or B x By partial fractions: multiply by x and using values of x, A both correct x x x x c AF + d x x and attempt at integrals ft on A and B; condone omission of brackets around x (ii) + d + ln( )( + ) Alternative: By substitution u x x 5 u+ dx du x u x + ln x (A) Correct, in x () Integral in terms of u ( ) ( ) (b)(i) x 5 P(x 5) + Q(x+ 5) Clear evidence of use of cover-up rule M x 5 x 5 m 0 0 Q 0 0P Q P A (ii) + d x+ 5 x5 x ln ( x + 5) + ln ( x 5 )( + c ) Attempt at ln integral aln x+ 5 + bln x 5 ( ( ) ( )) AF ft on P and Q; must have brackets Total 0 5
MPC - AQA GCE Mark Scheme 007 January series MPC (cont) 5(a) ( ) ( ) + x + x+ x A + x + kx (b)(i) ( ) 8 x 8 + B 8 ( + kx) + x x 8 9 8 + x x ( ) ( ) Replacing x with kx in answer to (a) A For numerical expression which would evaluate to answer given Alternative: 5 B all powers of 8 correct: 8 8 8 powers of x (condone x ) + x 9x 5 9 8 8 A see some arithmetic processing must see 9s in last term x Using x in given answer 57 + 599 9 A Any correct numerical expression 599 88 88 88 Total 7 (ii) : ( ) 8++
MPC - AQA GCE Mark Scheme 007 January series MPC (cont) 5 (a)(i) BA 0 A Attempt ± BA (OA OB or OB OA) (ii) BC ( ( ) ( ) ) BA + + () 5 BA BC B BF A Allow CB ; or BC or CB May not see explicitly Calculate modulus of BA or BC ; for finding modulus of one of vectors they have used Attempt at BA BC with numerical answer; or AB CB for 0, or correct if done with multiples of vectors 0 5 cos ABC A 5 AG (convincingly obtained) 5 5 7 Cosine rule: attempt to find sides A lengths of sides cosine rule AF correct A rearrange to get 5 cos ABC 7 (ft on length of sides) 7
MPC - AQA GCE Mark Scheme 007 January series MPC (cont) (cont) 8 A λ verified in three equations (b)(i) + λ ( λ ) 8 + λ for λ + λ A for λ shown for all three equations 8 λ λ A SC: λ written and nothing else: SC (ii) same direction or same gradient or parallel (c) OD OC + BA E B PI; OD correct vector expression which may involve AD 7(a) (b) 9 0 + D is ( 9, 0, 0) A for substituting into vector expression 0 for OD NMS / Total tan x + tan x tan x tan( x+ x) A B x used tan x tan x tan x A tan x( tan x) tanx Substitute from (a) tanx tan x tan x tan x ( )( + ) Simplification tan x ( tan x ) ( tan x)( ( + tan x) ) tan x + tan x ( tanx) A AG (convincingly obtained) Total ( tan x ) ( tan x) Any equivalent method 8
MPC - AQA GCE Mark Scheme 007 January series MPC (cont) 8(a)(i) dy sint y Attempt to separate and integrate ln y y cost+ C A,A A for ln y; A for cost; condone missing C cost Ae A cost+ C A present; or y e (ii) y t A A Substitute y 50, t π to find constant A Can have 50 e + C if substituted in above 50 e C e cos 50e t y e A AG (convincingly obtained) cosπ 50, π : 50 e e Alternative: Alternative: Must have a constant in answer to (a)(i) Substitute y 50, t π into ln y cos t + c cos t y Ae or cost+ c y e or ln y cost+ c ln y cos t + ln 50 A y + c () ln cost (AG) A 50 + c 50 Ae 50 e ln y cost+ ln 50 (A) cosπ cosπ+ c 50 Ae 50 e ln 50 cos π y 50e y e ln cost e 50 cos t cost 50 y (A) (b)(i) cos t : y 50e e 7.07... 7cm A Degrees.8 SC 7 or 7.0 for A dy π sin 0 0 B Condone x for t d y dy ycos t + sin t (ii) t ( t ) t π d y dy y cos π + sin π 50 max A For attempt at product rule including d y d y term; must have A Accept y, with explanation that y is never negative 9
MPC - AQA GCE Mark Scheme 007 January series MPC (cont) 8(b)(ii) Alternative: (cont) ( + cos t) 50 cost y 50e e e dy 50 e cost sin t 0 at t π e (B) y e e Substitute π 50 max d 50 cos t 50 cost e cost+ e sin t () (A) t (A) Total TOTAL 75 Attempt at product rule Correct 0