Version : klm. General Certificate of Education. Mathematics MPC1 Pure Core 1. Mark Scheme examination - June series

Transcription

1 Version : klm General Certificate of Education Mathematics 660 MPC Pure Core Mark Scheme 009 examination - June series

2 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: Copyright 009 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 6447) and a registered charity (registered charity number 074). Registered address: AQA, Devas Street, Manchester M 6EX Dr Michael Cresswell Director General

3 MPC - AQA GCE Mark Scheme 009 June 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.

4 MPC - AQA GCE Mark Scheme 009 June series MPC (a)(i) y = x+ M Attempt at y = f(x) Or correct expression for gradient using two correct points Or answer = or x gets M But answer of x gets M0 (Gradient of AB =) A Correct answer scores marks. Condone error in rearranging formula if answer for gradient is correct. (ii) mm = M Used or stated Gradient of perpendicular = A ft their answer from (a)(i) or correct y = ( x ) OE A x y= 7; or CSO 7 y = x+ c, c= etc (b) Eliminating x or y but must use M An equation in x only or y only x+ y = & x+ y = 8 x = 7 A 7, scores marks y = A Answer only of ( ) (a) Total M Numerator = m Condone one error or omission Denominator = 9 7 (= ) B Must be seen as the denominator (b) ( ) or ( ) (Answer = ) A 4 = 0 = 8 B Either correct their ( ) ( ) M Condone missing brackets and x ( x = 0 8 ) x = B, M ( x = ) A ± scores A0 Total 7 Answer only of scores B0, M0 Answer only of scores marks 4

5 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) (a) M One of these powers correct 4 x 40x A One of these terms correct A All correct (no + c etc) = = + M Substitute x = into their d y = 6+ ( 40 ) = 0 P is stationarypoint A CSO Shown = 0 plus statement eg st pt, as required, grad = 0 etc Or their 0 n x = k (M) x = 8 x= (A) CSO x = 0 need not be considered (b) x ( ) 4 ( 40 ) (c)(i) d y d y x = 0 ( ) + 40 M Subst x = into their second derivative (= ) = 0 A CSO 0x 40 x = + B Correct ft their d d (ii) Maximum (value) E Accept minimum if their ci ( ) answer> 0 their c(i) answer must be < 0 and correctly interpreted Other valid methods acceptable provided maximum is the conclusion Parts (i) and (ii) may be combined by candidate but 0 must be seen to award A in part (c)(i) (d) ( x ) When = y = B When x =, 40 = + M Sub x = into their d y y = (their 4)x + k OE m ft their d y Tangent has equation y 4 ( x ) = A 4 CSO OE y = 4 x+ c, c= Total

6 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) 4(a)(i) p ( ) = M p() attempted (Remainder) = 0 A Or long division up to remainder (M) Quotient= x + x+ 8 and remainder = 0 clearly stated or indicated (A) (ii) p ( ) = M p( ) attempted : NOT long division p ( ) = 0 x + is factor A Shown = 0 plus statement Minimum statement required factor May make statement first such as x+ is a factor if p( ) = 0 (iii) b = B c = B No working required for B + B Try to mark first using B marks or long division/comparing coefficients (M) Award M if B0 earned and a clear method is used p( x) = ( x+ )( x x+ ) Must write final answer in this form if (A) long division has been used to get A (iv) b 4 ac= ( ) 4 M Discriminant correct from their quadratic M0 if b =, c = 6 used (using cubic eqn) b 4ac= 8 (or < 0) CSO All values must be correct plus A no (other) real roots statement Or ( x ) + (M) Completion of square for their quadratic ( x ) + > 0 therefore no real roots Or ( x ) = has no real roots (A) Shown to be positive plus statement regarding no real roots (b)(i) ( y B = ) 6 B Condone (0, 6) (ii) 4 x x + 6x 4 0 ( ) = 0 4 M A A One term correct Another term correct All correct (ignore + c or limits) m F( ) attempted = 0 A CSO Clearly from F(0) F( ) (iii) Area of Δ= 6 M Condone and ft their y B value 0 Or ( + 6)d and attempt to integrate = 6 A Must be positive allow 6 converted to +6 Shaded region area = 0 6 = 4 A CSO 0 must come from correct working Total 7 6

7 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) C, B (a)(i) ( ) (ii) Radius = (or 69 ) B ± 69 or ± as final answer scores B0 ( ) + or + 44 (b)(i) = 69 circle passes through O B Correct arithmetic plus statement y Eg O lies on circle, as required etc (ii) Sketch O 0 x Freehand circle through origin and cutting B positive x-axis with centre in 4 th quadrant Condone value 0 missing or incorrect + ( p + ) = M Or doubling their y C -coordinate ( p + ) = ± p = 4 A Condone use of y instead of p SC B for correct value of p stated or marked on diagram 69 (c)(i) (ii) +7 grad AC = M correct expression, but ft their C +7 = A Condone grad tangent = B their grad AC y+ 7 = ( x+ 7) M ft their must be tangent and not AC x y+ 49 = 0 A OE with integer coefficients with all terms on one side of the equation Total 7

8 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) 6(a)(i) ( x 4) or p = 4 B ISW for p = 4 if ( 4) + or q = B x seen (ii) (Minimum value is) B Correct or FT their q (NOT coords) (iii) (Minimum occurs when x = ) 4 B Correct or FT their p may use calculus Condone (p, ** ) for this B mark (b)(i) ( ) x = x 0x+ B (ii) ( x ) + ( 7 x 4) = ( x ) + ( x) M Condone one slip in one bracket May be seen under sign = x 0x x+ x A From a fully correct expression AB = x 6x + 4 ( x x ) = A AG CSO (iii) Minimum AB = their (a)(ii) M Or use of their x = 4 in expression Or use of their B(4, ) and A(, 4) in distance formula Minimum AB = A Total 0 M0 if calculus used Answer only of their (a)(ii) scores M, A0 8

9 MPC - AQA GCE Mark Scheme 009 June series MPC (cont) k x + = x+ 7(a) ( ) kx x + k = 0 B AG OE all terms on one side and = 0 (b)(i) Discriminant = ( ) 4 ( ) k k Condone one slip (including x is one slip) M Condone or 4 as first term = 4 k + 8k A condone recovery from missing brackets Two distinct real roots b 4ac > 0 4 k + 8k > 0 (ii) ( k )( k ) B their discriminant in terms of k > 0 Not simply the statement b 4ac > 0 k 8 k 4 < 0 Change from > 0 to < 0 and divide by 4 k k < 0 A 4 AG CSO + M Correct factors or correct use of formula Critical values and A May score M, A for correct critical values seen as part of incorrect final answer with or without working Use of sign diagram or sketch M If previous A earned, sign diagram or sketch must be correct for M + + Otherwise, M may be earned for an attempt at the sketch or sign diagram using their critical values. < k < or > k > A 4 condone < k AND k < for full marks but not OR or, instead of AND Total 9 TOTAL 7 Full marks for correct final answer with or without working loses final A mark Answer only of < k < or k < ; k < etc scores M,A,M0 since the correct critical values are evident Answer only of < k < etc where critical values are not both correct gets M0,M0 9