A-LEVEL Mathematics. Pure Core 4 MPC4 Mark scheme June Version 1.1: Final

Transcription

1 A-LEVEL Mathematics Pure Core MPC Mark scheme 0 June 05 Version.: Final

2 Mark schemes are prepared by the Lead Assessment Writer 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 associates participate in and is the scheme which was used by them in this examination. The standardisation process ensures that the mark scheme covers the students responses to questions and that every associate understands and applies it in the same correct way. As preparation for standardisation each associate 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, associates encounter unusual answers which have not been raised they are required to refer these to the Lead Assessment Writer. It must be stressed that a mark scheme is a working document, in many cases further developed and expanded 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 examination paper. Further copies of this Mark Scheme are available from aqa.org.uk Copyright 05 AQA and its licensors. All rights reserved. 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 exception: 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.

3 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 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 explanation 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 x EE deduct x 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. Page of

4 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Q Solution Mark Total Comment (a) 9x A x B5 x M Correct equation and attempt to find a value for A or B. A A B A NMS or cover up rule; A or B correct SC A and B correct SC. (b) 5x x pln 5 x qln x M ln 5 Aft ft on their A x ln x Aft ft on their B ln ln5 ln5 ln m 0 ln 5 ln 5 A Condone missing brackets OE Either term in a correct form Substitute limits correctly in their integral; F() F(0) ACF. ln 0 PI 8 ln 5 A Total 9 CSO Condone equivalent fractions or recurring decimal Q Solution Mark Total Comment (a) R 9 B Allow 5. or better 5 Their 9 9 cos, 9 sin 5 or tan M Note cos or sin 5 is M0.9 A Must be exactly this (b)(i) R x R x cos or cos or x or x 0 or x M x 5.09 A Candidate s R and Must be exactly this (ii) cosx R M Candidate s R and ; PI x and A Rounded or truncated to at least dp; Ignore extra solutions x0.57 and x. A Total 8 Condone x 0.58 ; x. must be correct NMS is 0/ A0 if extra values in interval 0 x Page of

5 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Q Solution Mark Total Comment (a) f d M to d A NMS is 0/ (b)(i) x is a factor B OE x g( x) x x bx M Attempt to evaluate f and equated Attempt to find quadratic factor or a second linear factor using Factor Theorem g( x) x x 8x g( x) x x x A OE if x is used OE; must be a product NMS : SC if product is correct SC if one or two factors are correct (ii) x g( x) x B d k x x x (Derivative is) negative, or 0 hence decreasing M A E Attempt to differentiate simplified h Correct derivative Explanation and conclusion required Derivative must be correct Total 9 (b)(ii) Special case h x B x x is an increasing function, so x is a decreasing function E h Award only if x x is correct Page 5 of

6 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Q Solution Mark Total Comment (a) 5x kx M k any non-zero numerical expression 5 x x A Simplified to this (b) (i) 8 8 x 8 x B ACF for 8 x 8 Expand correctly using their 5 x 8 x x M Condone poor use of or missing brackets x x 5 5 A Accept x x (ii) x M x used in their expansion from (b)(i) 0. (dp) A Note in th decimal place Total 7 Page of

7 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Q5 Solution Mark Total Comment (a) dy sin t cos t dt dt B dy cost sint M At t gradient mt A Both correct Correct use of chain rule with their derivatives of form asin t, bcost (b) Gradient of normal mn Bft ft gradient of tangent; For m N, allow their m T with a change of ycos mn xsin M sign or the reciprocal at sin,cos or, y x A Must be in this y mx c form Alternative for M Use y mx cto find c with their sin cos c gradient m at sin,cos or, N m N mt (c) cos q sin q B Seen or used in this form Use parametric equations and candidate s sin qsin q M cos q in the form ksin q 8sin q sin q 0 OE A sin q sin q x 8 A Collect like terms; must be a quadratic equation Must come from a correct quadratic equation with the previous marks awarded A 5 Previous marks must have been awarded Total Page 7 of

8 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Mark scheme Alternative Q5 Solution Mark Total Comment (a) dy x y y or y dy B Find a correct Cartesian equation and differentiate implicitly correctly dy M Use y sin or y in their d y d sin ; PI x At t gradient mt A CSO (b) Gradient of normal Bft ft gradient of tangent, For m N, allow their m T with a change of ycos mn xsin M sign or the reciprocal at sin,cos or, y x A CSO Alternative for M Use y mx cto find c with candidate s sin cos c gradient m at sin,cos or, N m N mt (c) x y B PI by xx y y M y y 0 8sin qsin q 0 A sin q sin q A x A 8 Use their Cartesian equation and normal to eliminate x Collect like terms; must be a quadratic equation Must come from a correct quadratic equation with the previous marks awarded 5 Previous marks must have been awarded Total Page 8 of

9 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Q Solution Mark Total Comment (a) AB Or BA B AB () ( ) () M Correctly ft on their AB 5 cos BAC Correct use of formula with consistent m vectors; ACF angle BAC 0 A or / ; NMS 0 scores 0/ (b) 5 BC 0 AB BC 0 B M 5 0 A C is at 5,, A BC ACF Correct scalar product with their AB, their BC, equate to 0 and solve for Accept as a column vector NMS (5,,) scores 0/ (c) is at,0,0 E B Accept as a column vector 5 7 OD OC AB B 7 OE Correct vector expression with their and M their OD E is at,, 8 A Accept as a column vector Total (b) Alternative by Pythagoras 5 BC 0 5 B M BC ACF AC AB BC Correct Pythagoras expression, attempt to expand and solve for 8 0 A C is at 5,, A Accept as a column vector Page 9 of

10 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 (b) Alternative by cos 0 AB 5 B AC 5 M Square and simplify or A C is at 5,, A Accept as a column vector (c) Alt (i) Alt (ii) Alternatives 5 OE OB AC 8 E is at,0,0 B 5 M OE OB BE B` E is at,, 8 A 5 OD OB AC 8 D is at 7,, B 7 OE OD AC 8 M E is at,, 8 A 5 OE OB AC 8 E is at,0,0 B Correct vector expression with their BE All correct Correct vector expression with their OD and their AC Page 0 of

11 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Q7 Solution Mark Total Comment (a) ln k e ln ln ln 8 8 B ln Clear use of and e 8 8 Accept ln 8 (b) y y d pye d y x x B qe M ye e x x dy A 0 B Both required and no other terms dy dy m dy or.75 8 A Substitute x x ln or e and y into their expression 8 Total 7 Page of

12 MARK SCHEME A-LEVEL MATHEMATICS MPC JUNE 05 Q8 Solution Mark Total Comment (a)(i) d x dt 5x 5 B a t 5x or b t M OE 5 x 5 A OE ( ) t 5 C A OE m Use x 0, t 0 C 5x t A OE t x 5 5 A 7 Correct separation and notation seen on a single line somewhere in their solution ACF eg a 5 x or b t 5 x 5 5t 0,0 to find a constant 5t x 0 t 5 (b)(i) dr dt r dr dt k r B M r A Seen; allow R for r seen ; allow R for r Any constant k including c but not including variable t Must use R or r consistently (ii) d r.5 k or.5 c dt (metres) r M r A Use d r.5 with r dt to find a value for the constant Total Page of