Version 1.0. General Certificate of Education (A-level) June 2012 MPC1. Mathematics. (Specification 6360) Pure Core 1. Mark Scheme

Transcription

1 Version 1.0 General Certificate of Education (A-level) June 01 Mathematics (Specification 660) Pure Core 1 Mark Scheme

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 events which all examiners 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 examiner understands and applies it in the same correct way. As preparation for standardisation each examiner 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, examiners encounter unusual answers which have not been raised 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 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 01 AQA and its licensors. All rights reserved. Copyright 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. 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 107). Registered address: AQA, Devas Street, Manchester 5 6EX.

3 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,1 or 1 (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.

4 - AQA GCE Mark Scheme 01 June series m1 correct 8 7 (Numerator ) (Denominator = 1 9 ) B1 must be seen as denominator (a)(i) CSO; accept 16 9 Total 7 y x grad AB = y 'their grad' x c y xk y or x with correct points condone slip in rearranging if gradient is correct; condone 1. or better and attempt to use x, y 5 or y5 'theirgrad AB' x or x y = k and attempt to find k using x = and y = 5 y5 x 7 or y x x y = 7 correct equation in any form but must simplify to + integer coefficients in required form eg 8x + 6y = 5 (b) x y 7 and x y must use correct pair of equations and 8x9x1 1 etc attempt to eliminate x or y (generous) x 5, 5 (c) 7 y or D k k sub x k, y k into x y 7 k 8 6k 9 7 k Total 10 and attempt to multiply out with all k terms on one side (condone one slip)

5 - AQA GCE Mark Scheme 01 June series p 1 ( 1) ( 1) 5( 1) 6 p 1 attempted not long division (a)(i) p x 1is a factor CSO; correctly shown = 0 plus statement long division as far as constant term Quad factor in this form: x bx c or comparing coefficients, or b 1 or c 6 by inspection x x x x x x 6 correct quadratic factor p 1 must see correct product (b) p0 6;p1 8 p0 p1 both p0 andp(1)attempted and at least one value correct AG both values correct plus correct statement involving p(0) and p(1) (c) y 1 x cubic with one max and one min with, 1, marked correct with minimum to right of y-axis AND going beyond and Total 10

6 - AQA GCE Mark Scheme 01 June series (a)(i) x x + xyxyxy xy correct expression for surface area 6x 8xy (x xy xy) etc x xy 16 AG be convinced V x y OE correct volume in terms of x and y 16 x x 16 x or x x 9x 1 x OE CSO AG be convinced that all working is correct (b) dv 7 1 x dx one of these terms correct all correct with 9 evaluated (no + c etc) (c)(i) dv 7 dv x 1 attempt to sub x into ' their' dx dx dv dx 9 or or 1 0 etc 6 dv 0 stationary value CSO; shown = 0 plus statement dx d V 7x dv OE B1 FT for their a bx dx dx d V d V or sub of x into their when x, 0 maximum E1 dx dx maximum d V E0 if numerical error seen FT "minimum" if their 0 dx Total 10 5

7 - AQA GCE Mark Scheme 01 June series 5(a)(i) x or p = 1.5 stated 11 x Mark their final line as their answer x x B1 1 correct or FT their x = p (b)(i) x x 5 x 5 x x x0 x y 9 eliminating x or y and collecting like terms (condone one slip) or y 5 y5 5 y y y x x 5x c one of these terms correct another term correct all correct (need not have + c) (iii) Area trapezium x 5 y Area of shaded region must have earned in part(b) their x F 0 correctly sub d F B condone 17. but not 16 etc or 10 6 etc 1 B B B1 FT their numerical values of xb, y Area = 1 1 (= 8) CSO; Total 1, accept 10.7 or better B 6

8 - AQA GCE Mark Scheme 01 June series x 5 y 8 B1 6(a) (b)(i) = 5 B1 condone or 5 AC = 9+16 = 5 hence AC = 5 ; ( also radius = 5) A lies on circle B1 1 CSO radius = AC A lies on circle (must have concluding statement and circle equation correct if using equation) (must have concluding statement & RHS of circle equation correct or r =5 stated if Pythagoras is used) grad AC B1 Gradient of tangent is y 1 ' their tangent grad' x B1 FT their 1/ grad AC or y = their tangent grad x + c & attempt to find c using x =, y = 1 1 y 1 x or y x etc correct equation in any form x y 0 CSO; must have integer coefficients with 5 all terms on one side of equation accept 0 8 y 6 x 8 etc (c)(i) CM (7 5) (1 8) or CM CM 0 CM 5 0 PM PC CM 5 0 Pythagoras used correctly PM 5 Area PCQ CSO Total 1 d 5 5 eg 7

9 - AQA GCE Mark Scheme 01 June series 7(a)(i) d Increasing y 0 dx either 0 x 6 x 16 0 correct interpretation of y increasing 6 x 0x16 0 must see at least one of these steps before or () (10 x x 8) 0 final answer for x 10x8 0 CSO AG no errors in working x x CVs are and correct factors or correct use of quadratic equation formula as far as 10 6 condone 8 6 and 1 6 line + + sketch or sign diagram x or x accept x AND x Mark their final line as their answer but not x OR x nor x, x here but not in final 8

10 - AQA GCE Mark Scheme 01 June series 7(b)(i) x ; dy dy 0 16 sub x into and simplify terms dx d x dy 0 dx tangent at P is parallel to the x-axis must be all correct working plus statement x dy ; must attempt to sub x = into d dx d ( = ) 10 Gradient of normal = 1 10 Normal: y 1 'theirgrad' x 1 " their 10" m1 normal attempted with correct coordinates used and gradient obtained from their d y dx value 1 any correct form, eg 10 y x 1 but y1 x 10 must simplify to + (Equation of tangent at P is ) y B1 x 7 CSO; R, y x Total 15 TOTAL 75 9