GCSE Mathematics. Mark Scheme for November Unit J560/06: Higher Tier Paper 6. General Certificate of Secondary Education


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1 GCSE Mathematics Unit J560/06: Higher Tier Paper 6 General Certificate of Secondary Education Mark Scheme f November 07 Oxfd Cambridge RSA Examinations
2 OCR (Oxfd Cambridge RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of cidates of all ages abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills, Key Skills, Entry Level qualifications, NVQs vocational qualifications in areas such as IT, business, languages, teaching/training, administration secretarial skills. It is also responsible f developing new specifications to meet national requirements the needs of students teachers. OCR is a notfprofit ganisation; any surplus made is invested back into the establishment to help towards the development of qualifications suppt, which keep pace with the changing needs of today s society. This mark scheme is published as an aid to teachers students, to indicate the requirements of the examination. It shows the basis on which marks were awarded by examiners. It does not indicate the details of the discussions which took place at an examiners meeting befe marking commenced. All examiners are instructed that alternative crect answers unexpected approaches in cidates scripts must be given marks that fairly reflect the relevant knowledge skills demonstrated. Mark schemes should be read in conjunction with the published question papers the rept on the examination. OCR will not enter into any discussion crespondence in connection with this mark scheme. OCR 07
3 . Annotations used in the detailed Mark Scheme. Annotation Meaning Crect Increct BOD Benefit of doubt FT Follow through ISW Igne subsequent wking (after crect answer obtained), provided method has been completed M0 Method mark awarded 0 M Method mark awarded M Method mark awarded A Accuracy mark awarded B Independent mark awarded B Independent mark awarded MR Misread SC Special case Omission sign These should be used whenever appropriate during your marking. The M, A, B etc annotations must be used on your stardisation scripts f responses that are not awarded either 0 full marks. It is vital that you annotate these scripts to show how the marks have been awarded. It is not maty to use annotations f any other marking, though you may wish to use them in some circumstances. SubjectSpecific Marking Instructions. M marks are f using a crect method are not lost f purely numerical errs. A marks are f an accurate answer depend on preceding M (method) marks. Therefe M0 A cannot be awarded. B marks are independent of M (method) marks are f a crect final answer, a partially crect answer, a crect intermediate stage. SC marks are f special cases that are wthy of some credit. 3
4 3. Unless the answer marks columns of the mark scheme specify M A marks etc, the mark scheme is bed, then if the crect answer is clearly given is not from wrong wking full marks should be awarded. Do not award the marks if the answer was obtained from an increct method, ie increct wking is seen the crect answer clearly follows from it. 4. Where follow through (FT) is indicated in the mark scheme, marks can be awarded where the cidate s wk follows crectly from a previous answer whether not it was crect. Figures expressions that are being followed through are sometimes encompassed by single quotation marks after the wd their f clarity, eg FT 80 (their ), FT 300 (their ). Answers to part questions which are being followed through are indicated by eg FT 3 their (a). F questions with FT available you must ensure that you refer back to the relevant previous answer. You may find it easier to mark these questions cidate by cidate rather than question by question. 5. Where dependent (dep) marks are indicated in the mark scheme, you must check that the cidate has met all the criteria specified f the mark to be awarded. 6. The following abbreviations are commonly found in GCSE Mathematics mark schemes.  cao means crect answer only.  figs 37, f example, means any answer with only these digits. You should igne leading trailing zeros any decimal point eg 37000,.37,.370, would be acceptable but would not.  isw means igne subsequent wking (after crect answer obtained).  nfww means not from wrong wking.  oe means equivalent.  rot means rounded truncated.  seen means that you should award the mark if that number/expression is seen anywhere in the answer space, including the answer line, even if it is not in the method leading to the final answer.  soi means seen implied. 7. Make no deductions f wrong wk after an acceptable answer unless the mark scheme says otherwise, indicated f example by the instruction mark final answer. 4
5 8. As a general principle, if two me methods are offered, mark only the method that leads to the answer on the answer line. If two ( me) answers are offered, mark the poer (poest). 9. When the data of a question is consistently misread in such a way as not to alter the nature difficulty of the question, please follow the cidate s wk allow follow through f A B marks. Deduct mark from any A B marks earned recd this by using the MR annotation. M marks are not deducted f misreads. 0. Unless the question asks f an answer to a specific degree of accuracy, always mark at the greatest number of significant figures even if this is rounded truncated on the answer line. F example, an answer in the mark scheme is 5.75, which is seen in the wking. The cidate then rounds truncates this to 5.8, 5 6 on the answer line. Allow full marks f the If the crect answer is seen in the body the answer given in the answer space is a clear transcription err allow full marks unless the mark scheme says mark final answer cao. Place the annotation next to the crect answer. If the answer space is blank but the crect answer is seen in the body allow full marks. Place the annotation next to the crect answer. If the crect answer is seen in the wking but a completely different answer is seen in the answer space, then accuracy marks f the answer are lost. Method marks would still be awarded. Use the M0, M, M annotations as appropriate place the annotation next to the wrong answer.. Ranges of answers given in the mark scheme are always inclusive. 3. F methods not provided f in the mark scheme give as far as possible equivalent marks f equivalent wk. If in doubt, consult your Team Leader. 4. Anything in the mark scheme which is in square brackets [ ] is not required f the mark to be earned, but if present it must be crect. 5
6 (a) 00 B f soi Eg. answer with (5 0) + seen (b) ( s ut) a oe t M f s ut = ½ at² 30 November 3 B f identified as LCM answer 30 M f listing at least four multiples of 3 at least three multiples of 7 3, 6, 9,, 5, 8,, 7, 4,, M f listing/identifying at least four dates f running at least three dates f cycling 3 (a) B f.5, 4.5, 7.5, 0.5, 3.5 MFT f soi 9,,,, M f their , 5, 8,, 6, 3, 30, At least 4 midpoints crect FT midpoints either end of range values consistently used Allow one numerical err Four crect products 33.5 imply B M (b) Exact times f each customer are not recded oe Do not accept, Because the midpoint is used comments on the method used. 6
7 4 (a) 3 Alternative = = = of 8 = = 7 (b) Integer from 3000 to 3334 M f oe Accept integer from 3000 to 4000 after M (c) The growing conditions on the farm may be different to the garden oe Sample too small oe If M0 then SC f figs 333 seen Mere reference to facts that affect growth is insufficient 5 (a) M f oe Eg If 0 sced SC f figs 4 in answer (b) 4.[3 ] B f oe as answer M f their If 0 sced SC f figs 43[...] in answer Eg. 43.[36] 0 7 their converted from info in (a) 7
8 (c) (i) 60 3 B f 6447 to May be in stages. M f oe figs 6447 in NB: 5 5 = 80 (5 5) answer (ii) Each machine makes the same amount of sweets There are no breakdowns oe Machines running at same rate oe All machines run f the same time oe 6 (a) (i) of Bag A s counters [are red] 5 The ratio of red to yellow in Bag B is :3 Accept : 4 = 5 Accept 4 = : 3 Equivalents may be percentages decimals Eg. Bag A: 0% red, Bag B: 5% red. (ii) Crect answer is any integer multiple of this. Red Yellow Bag A 4 6 Bag B B f (Bag A) yellow = 4 red A total = B total B f (Bag B) yellow = 3 red If 0 sced SC f crect figures but transposed hizontally
9 (b) 0 nfww 3 B f two ratios equivalent to 3:4 6:8, 9:, :6, 5:0, M f their 5:0 reduced to (53):0 Alternative approach their 5:0 any ratio but not 3:4 using equivalent fractions: B f two fractions equivalent to 7 3 Eg M f their reduced to their any fraction but not ( ) 6 50 M3 f 00 B f 8.[0] seen B f 09.[0] seen M f their 09.[0] 50 [ 00] 9
10 M4 f oe soi by 3.67 to 3.68 M3 f 6 oe.5 oe soi M f 6 oe.5 oe soi oe soi M f 6 oe. 5 oe soi If 0 sced SC f 6 oe.5 oe soi by 4.. by.6 to.7 SC f 6 oe.5 oe soi by 4.. by by 6.67 to 6.68 by by to Method marks may be awarded f multiples of π seen in crect wking. Eg
11 9 6 4 M f 5b = 80 oe Eg. b + 80 = 6b B f [b = ] 36 M f their their 36 6 provided their answer would be between soi by final answer to in figs wds M f attempt to find gradient using figures from the graph e.g. (7.4.6) (05 95) Could be in billions Eg. ( ) (05 95) F M, condone increct conversion used consistently f both population figures. people/year (a) [ 9 4 ] B f 9 5 at least one fraction with denominat 8 f second card
12 (b) 5 3 oe FT their probabilities from (a) MFT f oe MFT f oe soi by oe 7 (a) 5 (b) (k =) 5 (r = ) nfww 5 B f 06 = 4k + r B f 03 = 06k + r M f 65k = 85 A f k = 5 r = If no partial method shown, allow full marks f final answer crect If 0 sced, allow SC f final crect answers interchanged Condone attempt to reduce to one variable by sub. elim. With max of one err 3 (a) M f possibly soi by figs 4355,
13 (b) Yes (Trev is crect) because Eg 0 87³ 00³ = 334.[. ] ³ 00³ = 9.9 to 0 3 M f 0 87³ 00³ ³ 00³ B f 87³ soi 4 (a) (34 36) (5 ) = 99 M f either soi by 4 5 3
14 (b) Eg. If M = n L = (n )(n + ) = n² T = (n 0)(n + 0) = n² 00 L T = (n² ) (n² 00) = 99 5 B f defining relative positions algebraically Eg. n, n +, n 0, n + 0 Or equivalent algebraic representation of relative positions. Condone po notation f B marks eg B f n n + n 0 n + 0 B f at least two relative positions defined algebraically AND M f [L =] (n )(n + ) = n² [T =] (n 0)(n + 0) = n² 00 M f [L =] (n )(n + ) = n [T =] (n 0)(n + 0) = n 00 L T = (their (n )(n + ) (their (n 0)(n + 0)) If 0 sced, allow SC f one further numerical example F M marks, follow through allowed f wking with their relative positions described algebraically as linear expressions: ie. L = Multiplication of their left right expressions T = Multiplication of their top bottom expressions M could be awarded by expressing their L their T, even if increctly exped M may be embedded [.0 ] 3 M f 5 5 oe B f [.4 ] [.5 ] 5 4
15 6 (a) (i) b a (b) (c) (ii) b a b a FT from (a)(i) EF EB BF = b a + b M f their part (a)(ii) + 4 leading to 3b a as given AG b a B f AG b a AG EF oe so are parallel. b oe M f b + (their part (a)(i)) oe (a)(ii) must be in terms of a b Allow vects found in reverse throughout eg. GA instead of AG Condone AG EF are multiples of each other Full marks dependent on both AG EF in crect simplified fms 7 (a) x² + y² = 00 oe (b) 8² + (6)² = 00, so it s on the circle oe M f 8² + ([]6)² seen f substituting x = 8 y = 6 into their part (a) Alternative using Pythagas their part (a) must be an equation in both x y. 5
16 (c) 3y 4x + 50 = 0 oe 5 4 Equivalents include: B f [tangent gradient = ] oe y = x Condone decimals with at least M f oe decimal places rot: 8 6 Eg. y =.33x 6.67 AND M f y + 6 = their 3 4 (x 8) oe M f y = their 3 4 x + c Equivalent f M includes y = their 3 4 x + c then attempt to find c by substituting in y = 6 x = 8 8 (a) y y x 8 oe 3 B f [ gradient =]  soi M f suitable method to find equation of line y 8 ( their ) x 5 y ( their ) x 8 eg. If both inequalities are wrong way round, condone once (max penalty mark) Or M f y = their x + c with a point from the line substituted in to find c F M allow use of an inequality symbol in place of = (b) y = 6 shown as a solid line crect region shaded B f line drawn at y = 6 Accept dashed line f B B f crect squares shaded but no line 6
17 (c) 8 5 oe M f soi by 8 5 M f 4h their 8 3 oe h is top of triangle A f [h =].5 AND M f [k =] 4 their.5 oe Alternative method M f 4 8 ' t' M f their 4 8 ' t' = 3 oe A f [t =] 3.5 t is top of trapezium Must be a trapezium AND M f [k =] 4 (6 their 3.5) oe 7
18 OCR (Oxfd Cambridge RSA Examinations) Hills Road Cambridge CB EU OCR Customer Contact Centre Education Learning Telephone: Facsimile: F staff training purposes as part of our quality assurance programme your call may be recded monited Oxfd Cambridge RSA Examinations is a Company Limited by Guarantee Registered in Engl Registered Office; Hills Road, Cambridge, CB EU Registered Company Number: OCR is an exempt Charity OCR (Oxfd Cambridge RSA Examinations) Head office Telephone: Facsimile: OCR 07
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