Unit H Mark scheme

Transcription

1 AQA Qualifications GCSE Mathematics Unit H Mark scheme 43602H June 2015 Version 1.0 Final Mark Scheme

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 crect way. As preparation f standardisation each associate analyses a number of students scripts: alternative answers not already covered by the mark scheme are discussed and legislated f. 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 wking 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.g.uk Copyright 2015 AQA and its licenss. All rights reserved. AQA retains the copyright on all its publications. However, registered schools/colleges f AQA are permitted to copy material from this booklet f their own internal use, with the following imptant exception: AQA cannot give permission to schools/colleges to photocopy any material that is acknowledged to a third party even f internal use within the centre.

3 Glossary f Mark Schemes GCSE examinations are marked in such a way as to award positive achievement wherever possible. Thus, f GCSE Mathematics papers, marks are awarded under various categies. If a student uses a method which is not explicitly covered by the mark scheme the same principles of marking should be applied. Credit should be given to any valid methods. Examiners should seek advice from their seni examiner if in any doubt. M A B Q ft SC M dep B dep Method marks are awarded f a crect method which could lead to a crect answer. Accuracy marks are awarded when following on from a crect method. It is not necessary to always see the method. This can be implied. Marks awarded independent of method. Marks awarded f Quality of Written Communication Follow through marks. Marks awarded f crect wking following a mistake in an earlier step. Special case. Marks awarded within the scheme f a common misinterpretation which has some mathematical wth. A method mark dependent on a previous method mark being awarded. A mark that can only be awarded if a previous independent mark has been awarded. Or equivalent. Accept answers that are equivalent. 1 eg, accept 0.5 as well as 2 [a, b] Accept values between a and b inclusive Accept answers which begin 3.14 eg 3.14, 3.142, Use of brackets It is not necessary to see the bracketed wk to award the marks. 3 of 18

4 Examiners should consistently apply the following principles Diagrams Diagrams that have wking on them should be treated like nmal responses. If a diagram has been written on but the crect response is within the answer space, the wk within the answer space should be marked. Wking on diagrams that contradicts wk within the answer space is not to be considered as choice but as wking, and is not, therefe, penalised. Responses which appear to come from increct methods Whenever there is doubt as to whether a candidate has used an increct method to obtain an answer, as a general principle, the benefit of doubt must be given to the candidate. In cases where there is no doubt that the answer has come from increct wking then the candidate should be penalised. Questions which ask candidates to show wking Instructions on marking will be given but usually marks are not awarded to candidates who show no wking. Questions which do not ask candidates to show wking As a general principle, a crect response is awarded full marks. Misread miscopy Candidates often copy values from a question increctly. If the examiner thinks that the candidate has made a genuine misread, then only the accuracy marks (A B marks), up to a maximum of 2 marks are penalised. The method marks can still be awarded. Further wk Once the crect answer has been seen, further wking may be igned unless it gs on to contradict the crect answer. Choice When a choice of answers and/ methods is given, mark each attempt. If both methods are valid then M marks can be awarded but any increct answer method would result in marks being lost. Wk not replaced Erased crossed out wk that is still legible should be marked. Wk replaced Erased crossed out wk that has been replaced is not awarded marks. Premature approximation Rounding off too early can lead to inaccuracy in the final answer. This should be penalised by 1 mark unless instructed otherwise. 4 of 18

5 Alternative method (0) 0.2(0) 36 their eg crect method to find 4 3 of their 36 their their dep dep on 2 nd 1.35 A1 Alternative method (0) eg their 5.4(0) their their dep dep on 2 nd 1.35 A1 135 A0 crossed out and 135p A1 Do not allow further wk to add on subtract from their 27 f third method mark eg = 27 followed by = 63 and 63 5 Allow rounding, truncation exact decimal f their 27 in third method mark eg = 35, = 26.25, 26 x 5 (= 130) M0A0 A0 5 of 18

6 and 200 and A1 SC without wking 1900 from x = 81 and y = 19 Condone x = 19 and y = 81 B2 100 (a square number) crectly evaluated 100 (a prime number) crectly evaluated A list of square numbers up to and including 81 with one err omission and a list of prime numbers up to and including 19 with one err omission A crectly evaluated trial of a square number plus a prime number. eg = 102 B2 x = 9 2 and y = 19 B2 x = 9 and y = 19 with 9 2 = in wking x = 9 and y = 19 without wking B2 49 and 51 implies 100 (a square number) crectly evaluated 91 and 9 implies 100 (a square number) crectly evaluated 6 of 18

7 5x 3x 2x 3x + 5x 2x x = 13 2x = 13 A A1ft 2 ft rearrangement with one err if awarded Igne further wk after crect fraction 2 ( ) ( ) 40 conditional on one prime fact in a crect product equal to 200 one prime fact shown in a crect section on a fact tree starting from 200 Any der allow on prime fact tree repeated division using 2 5 crectly condone 100 ( ) 2 ( ) 1 etc f this mark 5(a) 2 ( ) 2 ( ) 2 ( ) 5 ( ) 5 A Q1ft Any der allow on prime fact tree repeated division Strand (i) crect index notation Any der ft crect product of prime numbers in index fm from their wking A1Q0 (200 =) 2 ( ) 2 ( ) 5 ( ) 5 and is minimum Q1ft = ( ) 10 ( ) 10 as a product shown on a crect section of fact tree 20 ( ) 5 ( ) 2 as a product shown on a crect section of fact tree 20 ( ) 5 ( ) 4 as a product shown on a crect section of fact tree M0 7 of 18

8 5(b) 4 and 60 and 12 and 20 B2 one crect one crect and one increct two crect and one increct Any indication Alternative method their their 2400 their 400 ( 100) dep dep 20(%) A1 6 Alternative method their their % = % = and crectly finds multiplier using build up division to find percentage equivalent to total their 400 dep Crect build up to find percentage equivalent to total their (their ) their (2000 their 2400) implies M3 20(%) A1 8 of 18

9 Alternative method their 2400 ( 100) 120(%) their their 1.2(0) 1(.00) 100 their 120 1(.00) their 1.2(0) 0.2 dep dep 20(%) A1 6 (cont) 20% on answer line and no wking A (= 2400) from 5 years sces minimum 60 x 40 = 1800 and 200 sces minimum x 40 = 1800 and 200 and 2000 A x 40 = 1800 and 2000 A (= 1.2) ds not sce second method mark on ALT3 their < n 8 9, 10, 11, 12, 13, 14, 15, , 5, 5.5, 6, 6.5, 7, 7.5, 8 4, 5, 6, 7, 8 5, 6, 7 10, 12, 14, 16 Accept 4 < n and n 8 List of numbers in any der 5, 6, 7, 8 A1 Any der Embedded answer fully crect 2 5 = 10, 2 6 = 12, 2 7 = 14, 2 8 = 16 A0 4, 5, 6, 7 M0A0 9 of 18

10 (8 1 =) 8 (8 0 =) 1 9 A1 SC (a) with answer 9 1 A with answer 8 1 M0A0 8 on answer line without wking M0A with answer 8 M0A0 8 1 = 8 and 8 0 = 0 with answer 8 M0A0 8(b) x 7 y 5 B2 two terms crect 8(c) 8x 7 y 5 15x 6 y 5 15x 7 y 5 8x 7 y 5 15x 7 y 6 15x 12 y 6 15x 7 + y 5 8x 7 + y 5 B0 B0 B0 9(a) y = 3x + 2 9(b) ( PQ =) ( 9, 14) x = 9 ( RS =) : 5 A1 Accept if seen on LHS of ratio (PQ) as denominat in a gradient calculation f PR 10 of 18

11 and and 2499 Must be seen as a linked pair 549 A1 SC2 550 x 2 (+) 9x (+) 5x (+) 45 Allow one err Any der x x + 45 A1 Any der 11(a) Terms may be seen in a multiplication grid Do not igne attempts to factise after crect answer seen x(x + 14) + 45 A0 x x + 40 with no wking seen is one err x x + 45 with no wking seen is two errs x 2 + 5x + 45 with no wking seen A0 M0A0 M0A0 11(b) 5x(x 2y) B2 5(x 2 2xy ) x(5 x 10y) Condone missing final bracket 5x(x 2y 5x (x 2y) Condone missing final bracket 5(x 2 2xy B2 12 (3a b)(3a + b) B2 (3a b)(3a b) (3a + b)(3a + b) (3a b) 2 (3a + b) 2 (9a + b)(a b) (9a b)(a + b) (3a b) (3a + b) 11 of 18

12 13(a) x + y < 7 13(b) 2y x + 4 Alternative method 1 Method to show 4 divided by 9 with answer 0.44(...) method to show 1 divided by 9 = 0.11(...) and ( ) Alternative method 2 ( x = 0.44 x = 10x = x = 9x = 4 4 x = ) 4.4 Q1 Q1 Strand (ii) full calculation explanation seen Strand (ii) full calculation explanation seen 14(a) Alternative method = = = = 9 Q1 Strand (ii) full calculation explanation seen Minimum of two 4 digits seen 10x = 4.4 9x = 4 Q1 x = 9 4 x = x = 4.4 9x = 4 Q0 x = of 18

13 Alternative method A1 Alternative method 2 14(b) 10x = 9. 4 and 100x = 100x 10x = 100x 10x = 85 90x = x x = x = ( x = ) A1 10x = 9.44 and 100x = 94.4 is minimum requirement to sce May be recovered by a fully crect answer to sce A1 Igne further wking from crect fraction 15(a) 63 5(y + 1) 5y + 5 (4 + 1)(y + 1) 4y y (b) Condone (4 + 1) (y + 1) Condone 5 (y + 1) 5 y + 5 Condone missing final bracket 5 (y + 1 Do not igne further increct wk 13 of 18

14 15(c) (x + 1)(y + 1) x(y + 1) + y + 1 y(x + 1) + x + 1 xy + x + y + 1 Condone (x + 1) (y + 1) Condone x (y + 1) + y + 1 Do not igne further increct wk 15(d) (2x + 1)(y + 1) 2x(y + 1) + y + 1 y(2x + 1) + 2x + 1 2xy + 2x + y + 1 Condone (2x + 1) (y + 1) Condone 2x (y + 1) + y + 1 Do not igne further increct wk 14 of 18

15 x (2y 3 ) 2x y 3x 2x y 3x = 5y + 4 dep 2xy 5y = 3x + 4 y(2x 5 ) = 3x + 4 5y 2xy = 3x 4 dep 3x + 4 2x 5 3x 4 5 2x is M3 y(5 2x ) = 3x 4 16 y = 3x + 4 2x 5 y = 3x 4 5 2x A1 If there is choice mark the wking linked to the answer line 2x y 3x = 5y + 4 is as minimum 2xy 5y = 3x + 4 y ( 2x 5 ) = 3x + 4 is as minimum Condone x (2y 3 ) a = 4 (3x 1)(4x + b ) 3ax 2 + 3bx ax b 3b a = 19 12x 2 + 3bx 4x b 3bx 4x = 19x This mark implies M2 17 3b 4 = 19 3b = 15 b = 5 (3x 1)(4x 5) a = 4 and b = 5 and c = 5 A1 3ax 2 + 3bx 1ax b 3ax 2 + 3bx ax 1b Condone 3x 2 a and 3xb and xa 18(a) of 18

16 (b) A1 24 ds not sce alone without further wking M of 18

17 Alternative method 1 x 2 6x 20 = 4 x x 2 5x 24 ( = 0 ) ft one err in collection of terms with all terms crectly collected on one side ( x 8 )( x + 3 ) ( = 0 ) ( x + a )( x + b ) ( = 0 ) where ab = ± their 24 a + b = ± their 5 ft their quadratic quadratic fmula (allow one err) x = 8 and y = 4 x = 3 and y = 7 A1 x = 8 and y = 4 and x = 3 and y = 7 A1 SC2 f both ( 8, 4 ) and ( 3, 7 ) by trial and improvement SC1 f either ( 8, 4 ) ( 3, 7 ) by trial and improvement Alternative method 2 19 y = ( 4 y ) 2 6 ( 4 y ) 20 y = 16 8y + y y 20 y = y 2 2y 28 allow one err in rearrangement of y = 4 x y 2 3y 28 ( = 0 ) ft one err in expansion and collection of terms with all terms crectly collected on one side ( y 7 ) ( y + 4 ) ( = 0 ) ( y + a ) ( y + b ) ( = 0 ) where ab = ± their 28 a + b = ± their 3 ft their quadratic quadratic fmula (allow one err) y = 4 and x = 8 y = 7 and x = 3 A1 y = 4 and x = 8 and y = 7 and x = 3 A1 SC2 f both ( 8, 4 ) and ( 3, 7 ) by trial and improvement SC1 f either ( 8, 4 ) ( 3, 7 ) by trial and improvement 17 of 18

18 19 (cont) Substituting x = y 4 into quadratic is two errs in rearrangement of y = 4 x Substituting x = y 4 into quadratic followed by collection of terms with all terms crectly collected on one side y 2 15y + 20 ( = 0 ) (allow one err) Substituting x = y 4 into quadratic followed by y 2 15y + 20 ( = 0 ) followed by attempt to factise quadratic where ab = ± their 20 a + b = ± their 15 M0 M0 M0 18 of 18