A-level Mathematics Mark scheme Pure Core 3 …

1 A-level Mathematics MPC3 pure core 3 mark scheme 6360 June 2016 Version: Final 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.

2 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.

3 Further copies of this mark scheme are available from Copyright 2016 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. mark scheme A-level Mathematics MPC3 JUNE 2016 3 of 14 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 A2,1 2 or 1 (or 0)

4 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) 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 .

5 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. mark scheme A-level Mathematics MPC3 JUNE 2016 4 of 14 Q1 Solution Mark Total Comment (a) 32d(41) cos 2(41) sin 2dym xx n xxx 2 and4 3[ 12]mn isw M1 A1 ,0mn 2 (b) 2222d(34)4(23)6d(34)yxxxxxx oe 222(34)xx M1 A1 Or 22221(23)( 1)(34) 6(34) 4xxxxx 2 (c) 22d1their ( )d2334ybixxx PI 2222d(34)2d(23) (34)yxxxxx isw 222(23)(34)xxx M1 A1 their b(i) must be in the correct form 22(34)kxx Or (using rules of logs) 2222ln(23)

6 Ln(34)dM1d23340,04,6A1yxxyaxbxxxxabab 2 Total 6 Notes: Allow recovery from poor use of brackets in each part. (a) If expanded, 232d() sin 2(1)2 cos 2M1d192,96,12,64,48,12A1yaxbx cxdxexfxxxabcdef (b) For M1, 2222d(34)4(23)6d(34)yxxxxxx or 22221(23)( 1)(34) 6(34) 4xxxxx For A1, accept p= 2 mark scheme A-level Mathematics MPC3 JUNE 2016 5 of 14 Q2 Solution Mark Total Comment a f ( )5xxx PI f(2) = 1 f(3) = 22 Change of sign(or different signs) M1 A1 2 (or reverse)

7 Both values correct Must have both statement and interval in words or symbols OR comparing 2 sides: at 2, 225 ; at 3, 335 (M1) (A1) b ln 5(5lnln 5)lnln 5ln 5lnexxxxxxxxxx M1 A1 A1 Taking logs and using rule of logs Must see this line AG, all correct (including middle line) 3 c B1 B1 Ignore any further values 2 di x y B1 B1 All 7 correct x values (and no extras used) PI by correct y values At least 5 correct y in exact form or decimal values, rounded or truncated to 3dp or better (in table or formula) (PI by correct answer)

8 [ 4( ) 2( )] M1 A1 Correct use of Simpson s rule using 1/3 and oe and their 7 y values (of which 5 are correct to 2dp), either listed or totalled. CAO 4 dii 6 - their (di) = M1 A1F PI by correct answer 2 SC1 for Total 13 Notes: a condone 23x , allow x , root for , but not it di with no working scores 0/4 dii scores 2/2, with NMS scores 0/2 mark scheme A-level Mathematics MPC3 JUNE 2016 6 of 14 Q3 Solution Mark Total Comment 256[ 0][]2, 3 PIxxx 256 0[] 6, 1 PIxxx 6312xxx B1 B1 B1 B1 B1 B1 can be earned for any correct 2 solutions or 63xx And no extras seen 5 Total 5 Notes.

9 Correct inequalities implies correct critical values if not seen explicitly A candidate may use a quartic to find the critical values, but marks are only earned for correct solutions as above, eg solutions of 1, 2 scores B1 If strict inequalities are used throughout then penalise 1 mark but if some correct answers and some strict inequalities then mark as scheme . mark scheme A-level Mathematics MPC3 JUNE 2016 7 of 14 Q4 Solution Mark Total Comment a Stretch I [Parallel to] x[-axis] II (or line y = 0) [SF] III then Translation 0k OR Translation 0k 50 then Stretch I [Parallel to] x[-axis] II [SF] III M1 A1 M1 A1 (M1) (A1) (M1) (A1) 4 I and II or III I + II + III or (2nd)

10 Stretch [parallel to] y[-axis] SF 5e (for the 2 stretch method, if the y direction stretch is first, marks can only be earned if there is a second stretch in x direction. The stretches can be in either order) I and II or III I + II + III b 25d2edxyx Grad normal = 1their gradient (equation normal) 1e(2)2y ex oe (At A 0y ) 222ex oe (At B 0x ) 21e1e+eey oe 222(e1) 2(1+e )(Area ) 223(e1)e B1 B1F B1 M1 A1 A1 Condone expression in terms of x Must be exact values Attempt to find at least one intercept from their normal, subst x = 0 or y = 0 in any straight line equation Both x and y values correct 6 Total 10 Notes.