Unit 2_higher mark_scheme_march_2011 unit 2
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The document is a mark scheme for GCSE Mathematics (Modular) - 5MB2H Paper: 01 from March 2011. It provides guidance to examiners on marking principles, such as awarding full marks for correct answers and following through marks. It also includes notes on specific types of questions and acceptable ways to show working and arrive at solutions. The mark scheme then provides detailed mark allocations and guidance for 14 sample questions testing various mathematical concepts.
Unit 2_higher mark_scheme_march_2011 unit 2
- 1. Mark Scheme (Results) March 2011 GCSE GCSE Mathematics (Modular) – 5MB2H Paper: 01 Edexcel Limited. Registered in England and Wales No. 4496750 Registered Office: One90 High Holborn, London WC1V 7BH
- 2. Edexcel is one of the leading examining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers. Through a network of UK and overseas offices, Edexcel’s centres receive the support they need to help them deliver their education and training programmes to learners. For further information, please call our GCE line on 0844 576 0025, our GCSE team on 0844 576 0027, or visit our website at www.edexcel.com If you have any subject specific questions about the content of this Mark Scheme that require the help of a subject specialist, you may find our Ask The Expert email service helpful. Ask The Expert can be accessed online at the following link: http://www.edexcel.com/Aboutus/contact-us/ March 2011 Publications Code UG026940 All the material in this publication is copyright © Edexcel Ltd 2011
- 3. NOTES ON MARKING PRINCIPLES 1 All candidates must receive the same treatment. Examiners must mark the first candidate in exactly the same way as they mark the last. 2 Mark schemes should be applied positively. Candidates must be rewarded for what they have shown they can do rather than penalised for omissions. 3 All the marks on the mark scheme are designed to be awarded. Examiners should always award full marks if deserved, i.e if the answer matches the mark scheme. Examiners should also be prepared to award zero marks if the candidate’s response is not worthy of credit according to the mark scheme. 4 Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and exemplification may be limited. 5 Crossed out work should be marked UNLESS the candidate has replaced it with an alternative response. 6 Mark schemes will indicate within the table where, and which strands of QWC, are being assessed. The strands are as follows: i) ensure that text is legible and that spelling, punctuation and grammar are accurate so that meaning is clear Comprehension and meaning is clear by using correct notation and labeling conventions. ii) select and use a form and style of writing appropriate to purpose and to complex subject matter Reasoning, explanation or argument is correct and appropriately structured to convey mathematical reasoning. iii) organise information clearly and coherently, using specialist vocabulary when appropriate. The mathematical methods and processes used are coherently and clearly organised and the appropriate mathematical vocabulary used. 1
- 4. 7 With working If there is a wrong answer indicated on the answer line always check the working in the body of the script (and on any diagrams), and award any marks appropriate from the mark scheme. If working is crossed out and still legible, then it should be given any appropriate marks, as long as it has not been replaced by alternative work. If it is clear from the working that the “correct” answer has been obtained from incorrect working, award 0 marks. Send the response to review, and discuss each of these situations with your Team Leader. If there is no answer on the answer line then check the working for an obvious answer. Any case of suspected misread loses A (and B) marks on that part, but can gain the M marks. Discuss each of these situations with your Team Leader. If there is a choice of methods shown, then no marks should be awarded, unless the answer on the answer line makes clear the method that has been used. 8 Follow through marks Follow through marks which involve a single stage calculation can be awarded without working since you can check the answer yourself, but if ambiguous do not award. Follow through marks which involve more than one stage of calculation can only be awarded on sight of the relevant working, even if it appears obvious that there is only one way you could get the answer given. 9 Ignoring subsequent work It is appropriate to ignore subsequent work when the additional work does not change the answer in a way that is inappropriate for the question: e.g. incorrect canceling of a fraction that would otherwise be correct It is not appropriate to ignore subsequent work when the additional work essentially makes the answer incorrect e.g. algebra. Transcription errors occur when candidates present a correct answer in working, and write it incorrectly on the answer line; mark the correct answer. 10 Probability Probability answers must be given a fractions, percentages or decimals. If a candidate gives a decimal equivalent to a probability, this should be written to at least 2 decimal places (unless tenths). Incorrect notation should lose the accuracy marks, but be awarded any implied method marks. If a probability answer is given on the answer line using both incorrect and correct notation, award the marks. If a probability fraction is given then cancelled incorrectly, ignore the incorrectly cancelled answer. 2
- 5. 11 Linear equations Full marks can be gained if the solution alone is given on the answer line, or otherwise unambiguously indicated in working (without contradiction elsewhere). Where the correct solution only is shown substituted, but not identified as the solution, the accuracy mark is lost but any method marks can be awarded. 12 Parts of questions Unless allowed by the mark scheme, the marks allocated to one part of the question CANNOT be awarded in another. 13 Range of answers Unless otherwise stated, when an answer is given as a range (e.g 3.5 – 4.2) then this is inclusive of the end points (e.g 3.5, 4.2) and includes all numbers within the range (e.g 4, 4.1) Guidance on the use of codes within this mark scheme M1 – method mark A1 – accuracy mark B1 – Working mark C1 – communication mark QWC – quality of written communication oe – or equivalent cao – correct answer only ft – follow through sc – special case dep – dependent (on a previous mark or conclusion) indep – independent isw – ignore subsequent working 3
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- 7. 5MB2H_01 Question Working Answer Mark Notes 1 (a) 21 1 B1 cao (b) 4n + 1 2 M1 for 4n + k (k ≠ 1) A1 oe NB n = 4n+1 gets M1 only 2 300 ÷ (2 + 3 + 5) × 5 150 2 M1 for 300 ÷ ( 2 + 3 + 5 ) or × 5 = 300 ÷ 10 × 5 or 30 seen = 30 × 5 or 60 : 90 : 150 A1 cao 5
- 8. 5MB2H_01 Question Working Answer Mark Notes 3 y = 2x + 3 Line 3 (Table of values) M1 for at least 2 correct attempts to find points x –3 –2 –1 0 1 by substituting values of x y –3 –1 1 3 5 M1 ft for plotting at least 2 of their points (any points plotted from their table must be correct) A1 for correct line between –3 and 1 (No table of values) M2 for at least 2 correct points (and no incorrect points) plotted OR line segment of 2x + 3 drawn (ignore any additional incorrect segments) (M1 for at least 3 correct points with no more than 2 incorrect points) A1 for correct line between –3 and 1 (Use of y = mx + c) M2 for at least 2 correct points (and no incorrect points) plotted OR line segment of 2x + 3 drawn (ignore any additional incorrect segments) (M1 for line drawn with gradient of 2 OR line drawn with a y intercept of 3 and a positive gradient A1 for correct line between –3 and 1 6
- 9. 5MB2H_01 Question Working Answer Mark Notes 4 LCM (40, 24) = 120 Rolls 5 M1 attempts multiples of either 40 or 24 (packs) 3 (at least 3 but condone errors if intention is clear) Rolls 120 ÷ 40 = Sausages 120 ÷ 24 = Sausages M1 attempts multiples of both 40 and 24 (trays) 5 (at least 3 but condone errors if intention is clear) OR M1 (dep on M1) for a division of 40 or 24 Hot dogs 120 or counts up ‘multiples’ Rolls 40 is (implied if one answer is correct or answers reversed) 2×2×2 (×5) Sausages 24 is A1 rolls (packs) 3, sausages (trays) 5 2×2×2 (×3) OR any multiple of 3,5 A1 hot dogs 120 or ft on both of their packs or ft ‘common multiple’ OR M1 expansion of either number into factors M1 demonstrates one of the expansions that includes 8 oe M1 demonstrates a 2nd expansion that includes 8 oe A1 cao for rolls (packs) 3, sausages (trays) 5 A1 hot dogs 120 7
- 10. 5MB2H_01 Question Working Answer Mark Notes 5 Triangular face: 360 4 M1 for ½ × 5 × 12 ( = 30) oe ½ × 5 × 12 = 30 cm2 M1 for 2 + of (13×10) and (12×10) and (5×10) oe Rectangular faces: (13×10), (12×10), A1 cao (5×10) NB: No marks awarded for calculating volume Area: 30 + 30 + 130 B1 (indep) units stated (cm2) + 120 + 50 = 6 (i) 180° – 160° = 20 3 B1 cao (ii) Exterior angles sum 18 M1 for 360 ÷ “20” to 360° A1 cao So 360 ÷ ‘20’ = 7 ½ × 6(10 + 8) – ½ × 36 3 M1 for ½ × 6(10 + 8) or ½ × 3(7 + 5) oe 3(7 + 5) M1(dep) for ½ × 6(10 + 8) – ½ × 3(7 + 5) oe = 54 – 18 A1 cao 8
- 11. 5MB2H_01 Question Working Answer Mark Notes 8 Park Palace: Park Palace 6 M1 for identifying correct week for holiday £3492 (eg use of 854 for DG, eg circle correct row) 810 + 80 + 80 = £970 per adult 1/5 of 970 = 194 970 – 194 = £776 per child M1 for using 7 nights for at least one hotel 970 + 970 + 776 + 776 = £3492 M2 for complete correct method for reduction Dubai Grand: of 1/5 and 15% for at least 5 nights (M1 for correct method to get 1/5 or 15% 854 + 53 + 53 = £960 per adult or 4/5 or 85% of a total for at least 5 nights) 10% + 5% of 960 = 96 + 48 = 144 A1 for one correct total (3492 or 3552) 960 – 144 = £816 per child 960 + 960 + 816 + 816 = £3552 A1 for 34 92 and 3552, with Park Palace (or 3492) indicated as the best choice. 9 8 km per 30 seconds 600 3 M1 convert to km/h by × 2 × 60 or 960 seen = 16 km per minute or use of speed = distance ÷ time = 16 × 60 = 960 km per hour M1 convert distance to miles by × 5 ÷ 8 oe or sight of 5 miles 960 km/hr × 5 ÷ 8 = 600 miles A1 cao per hour 9
- 12. 5MB2H_01 Question Working Answer Mark Notes 10 (2, 5, 6) to (–1, – 4, 2) (– 4, –13, – 2) 2 M1 for a complete correct method for at least is (– 3, – 9, – 4) 1 coordinate (–1 –3, –4 – 9, 2 – 4) (could be implied by 2 out of 3 coordinates correct) 2+x A1 cao or = –1, 2 5+ y 6+z = – 4, =2 2 2 11 (a) 3x + 6 2 M1 for attempted expansion of the bracket eg 3× x and 3 × 2 seen or 3x + k or kx + 6 A1 for 3x + 6 (b) 6xy(2x2 – 3y) 2 M1 or 6xy (two terms involving x and/or y) or correct partial factorisation by taking out two from 6 (or 3 or 2) or x or y A1 cao (c) 2x2 + 8x – 3x –12 2x2 + 5x – 12 2 M1 for 3 out of 4 correct terms with correct signs, or all 4 terms ignoring signs A1 cao (d) 10 x7 y5 2 B2 for 10 x7 y5 (B1 for product of two of 5×2 oe, x4+3, y3+2 ignore × signs ) 10
- 13. 5MB2H_01 Question Working Answer Mark Notes 12 (i) 1 3 B1 cao (ii) 1 B1 for 1 or 0.2 5 5 (iii) 3 B1 cao (accept ± 3) 13 (a) y = 5x + c 1 B1 for y = 5x + c oe c ≠ 6 (b) 1 1 1 3 3 1 1 gradient = − =− y =− x +4 M1 recognition that gradient = − = − oe m 5 5 5 m 5 1 y = − + c x = –2, y = 5 M1 substitution of x = –2, y = 5 in y = mx + c 5 1 1 where m = – , or –5 5 5 2 5= +c 5 1 3 A1 y = − x + 4 oe 2 3 5 5 c=5– =4 5 5 1 3 y =− x +4 5 5 11
- 14. 5MB2H_01 Question Working Answer Mark Notes 14 x − 2 x − 15 ( x − 5) ( x + 3) 2 x −5 3 M1 attempt to factorise numerator = (at least one bracket correct) or (x ± 5)(x ± 3) x 2 − 4 x − 21 ( x − 7)( x + 3) x −7 M1 attempt to factorise denominator (at least one bracket correct) or (x ± 7)(x ± 3) A1 oe 15 Unknown length = x + 3 – x – x = 10x2 + 24x– 4 B1 for x + 3 – x – x oe or 3 – x seen 3–x 18 or x – 1 + 2x + x – 1 oe or 4x – 2 seen Cross-sectional area = (x + 3)(x –1) + (x + 3)(x –1) + M1 for correct expression for 1 area from (3 – x)(2x) cross-section or for 1 volume of cuboid(s) = x2 +2x – 3 + x2 + 2x – 3 + 6x – 2x2 = 4x – 6 + 6x (brackets not needed) = 10x – 6 Volume =(10x – 6)(x + 3) M1 for correct method for total cross-sectional area = 10x2 + 24x – 18 OR at least 2 volumes added OR volume of surrounding cuboid – at least 1 vol OR (brackets needed) Unknown length = x + 3 – x – x = 3–x A1 for 10x2 + 24x – 18 oe Volume = (x + 3)(x + 3)(x –1) + (x + 3)(x + 3)(x –1) + (2x)(3 – x)(x + 3) = (10x – 6)(x + 3) = 10x2 + 24x –18 12
- 15. 5MB2H_01 Question Working Answer Mark Notes 15 OR cont.. Unknown length = (2x – 2) + 2x = 4x – 2 Surrounding area = (4x – 2)(x + 3) = 4x2 + 10x – 6 So A = 4x2 + 10x – 6 – 4x2 = 10x – 6 So V = (10x – 6)(x + 3) = 10x2 + 24x – 18 OR Unknown length = (2x – 2) + 2x = 4x – 2 Surrounding volume = (4x – 2)(x + 3) (x + 3) V = (4x – 2)(x + 3) (x + 3) – 2x(2x) (x + 3) 13
- 16. 5MB2H_01 Question Working Answer Mark Notes 16 DE = AE, and AE = EB Proof 4 B1 for DE = AE or AE = EB (tangents from an external point are (can be implied by triangle AED is isosceles equal in length) so DE = EB or triangle AEB is isosceles or indication on the diagram) AE = EC (given) OR tangents from an external point are equal Therefore AE = DE = EB = EC in length So DB = AC B1 for AE = DE = EB = EC If the diagonals are equal and bisect each other then the quadrilateral is a rectangle. B1 for DB = AC, (dep on B2) OR consideration of 4 isosceles triangles in ABCD OR C1 fully correct proof. If AE = DE = EB = EC Proof should be clearly laid out with technical then there are four isosceles language correct and fully correct reasons triangles ADE, AEB, BEC, DEC in which the angles DAB, ABC, BCD, CDA are all the same. Since ABCD is a quadrilateral this makes all four angles 90°, and ABCD must therefore be a rectangle. 14
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