Version.0 00 General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 00 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 alrea 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 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 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 67) and a registered charity (registered charity number 07). Registered address: AQA, Devas Street, Manchester 5 6EX Dr Michael Cresswell Director General
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 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.
MPC - AQA GCE Mark Scheme 00 January series MPC (a) p ( ) ( ) must attempt p NOT long division (b) (a)(i) grad 7 9 0 x is factor x x bx c x x obtained A A shown 0 plus statement Full long division, comparing coefficients or by inspection either b or c or A for either x or x clearly found using factor theorem x x x A CSO; must be seen as a product of factors NMS full marks for correct product SC B for x x Total 5 7 AB (must simplify /) A y x or ( x )( x )( ) or ( x )( x )( x ) NMS correct expression, possibly implied (ii) grad BC 7 9 Condone one slip NOT Pythagoras or cosine rule etc grad AB grad BC ABC 90 or AB & BC perpendicular A convincingly proved plus statement SC B for /(their grad AB) or statement that mm for perpendicular lines if M0 scored (b)(i) M 0, 6 B B + B each coordinate correct (ii) AB 7 either expression correct, simplified or BC 7 9 unsimplified (iii) AB or BC or 0 found as a length AB BC AB BC or AB 0 and BC 0 A A Must see either AB =.., or BC =..., 7 6 grad BM ft their M coordinates 0 or /(grad AC) attempted A correct gradient of line of symmetry BM has equation y x 6 A CSO, any correct form Total
MPC - AQA GCE Mark Scheme 00 January series MPC (cont) (a)(i) one term correct t t A another term correct 8 A all correct (no + c etc) unsimplified (ii) (b) d y t 8 ft one term correct A correct unsimplified (penalise inclusion of +c once only in question) t ; 8 Substitute t into their d t 0 stationary value A CSO; shown = 0 plus statement t d y ; 6 Sub t y has MINIMUM value A CSO d y into their d t (c)(i) t ; Substitute t A OE; CSO into their d t NMS full marks if d y correct (ii) 0 (depth is) INCREASING E allow decreasing if states that their d y 0 Reason must be given not just the word increasing or decreasing Total (a) 50 5 ; 8 ; 8 At least two of these correct 5 A Answer A CSO or 8 or 8 or 5 9 any correct expression all in terms of or with denominator of 8, or simplifying numerator eg 00 8 (b) 7 7 5 7 5 7 5 numerator = 7 7 0 7 5 m expanding numerator ( condone one error or omission) denominator = B (seen as denominator) Answer 7 A Total 7 OE 5
MPC - AQA GCE Mark Scheme 00 January series MPC (cont) 5(a) x 8x 5 B Terms in x must be collected, PI their x ( k ) ft x A x p for their quadratic ISW for stating p if correct expression seen (b)(i) y shape in any quadrant (generous) 7 O x A correct with min at (, ) stated or and marked on axes condone within first quadrant only B crosses y-axis at (0, 7) stated or 7 marked on y-axis (ii) y = k y constant y A Condone y 0x (c) Translation (not shift, move etc) E and no other transformation One component correct with vector or ft either their p or q CSO; condone across, up; or two A separate vectors etc Total 6
MPC - AQA GCE Mark Scheme 00 January series MPC (cont) 6(a)(i) terms correct x 9 6x dx A all correct (no + c etc) when x, dx 8 9 m gradient = 5 A CSO (ii) grad of normal 5 y 6 their x 5 or y their x c and c evaluated 5 using x = and y = 6 B ft their answer from (a)(i) ft grad of their normal using correct coordinates BUT must not be tangent condone omission of brackets x 5y 8 0 A CSO; condone all on one side in different order (b)(i) one term correct 9 x x x A another term correct A all correct (ignore +c or limits) 8 8 m F attempted A 5 CSO; withhold A if changed to + here (ii) Area 6 6 B condone 6 Shaded region area 6 difference of = 8 A CSO Total 5 7
MPC - AQA GCE Mark Scheme 00 January series MPC (cont) 7(a)(i) x = or y 6 or ( x ) ( y 6) C, 6 A correct (ii) r 6 5 (RHS = ) their ( ) + their (6) 5 r 5 A Not 5 or 5 (b) explaining why yc r ; 6 5 E Comparison of y C and r, eg 6 + 5 = or indicated on diagram full convincing argument, but must have correct y C and r E Eg highest point is at y = scores E E: showing no real solutions when y 0 +E stating centre or any point below x- axis (c)(i) PC 5 k 6 ft their C coords 9 k k 6 and attempt to multiply out PC k k 5 A AG CSO (must see PC at least once) (ii) PC r PC k 5 k 0 0 B AG Condone k k k 5 5 k 0 0 (iii) k k 0 Correct factors or correct use of formula k, k 0 are critical values A Use of sketch or sign diagram: May score, A for correct critical values seen as part of incorrect final answer with or without working. 0 + + + 0 If previous A earned, sign diagram or sketch must be correct for, otherwise may be earned for an attempt at the sketch or sign diagram using their critical values. k, k 0 A k, k 0 loses final A mark Condone k OR k 0 for full marks but not AND instead of OR Take final line as their answer Total TOTAL 75 Answer only of k, k 0 etc scores, A, M0 since the critical values are evident. Answer only of k, k 0 etc scores M0, M0 since the critical values are not both correct. 8