Y434/01 Mark Scheme June 2022 3 2. Subject-specific Marking Instructions for AS Level Mathematics B (MEI) a Annotations must be used during your marking. For a response awarded zero (or full) marks a single appropriate annotation (cross, tick, M0 or ^) is sufficient, but not required. For responses that are not awarded either 0 or full marks, you must make it clear how you have arrived at the mark you have awarded and all responses must have enough annotation for a reviewer to decide if the mark awarded is correct without having to mark it independently. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. Award NR (No Response) - if there is nothing written at all in the answer space and no attempt elsewhere in the script - OR if there is a comment which does not in any way relate to the question (e.g. ‘can’t do’, ‘don’t know’) - OR if there is a mark (e.g. a dash, a question mark, a picture) which isn’t an attempt at the question. Note: Award 0 marks only for an attempt that earns no credit (including copying out the question). If a candidate uses the answer space for one question to answer another, for example using the space for 8(b) to answer 8(a), then give benefit of doubt unless it is ambiguous for which part it is intended. b An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not always be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner. If you are in any doubt whatsoever you should contact your Team Leader. Y434/01 Mark Scheme June 2022 4 c The following types of marks are available. M A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. A method mark may usually be implied by a correct answer unless the question includes the DR statement, the command words “Determine” or “Show that”, or some other indication that the method must be given explicitly. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. B Mark for a correct result or statement independent of Method marks. E A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument. d When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation ‘dep*’ is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given. e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only – differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. If this is not the case, please escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be ‘follow through’. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question. Y434/01 Mark Scheme June 2022 5 f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question. (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km, when this would be assumed to be the unspecified unit.) We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so.  When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value.  When a value is not given in the paper accept any answer that agrees with the correct value to 2 s.f. unless a different level of accuracy has been asked for in the question, or the mark scheme specifies an acceptable range. NB for Specification A the rubric specifies 3 s.f. as standard, so this statement reads “3 s.f” Follow through should be used so that only one mark in any question is lost for each distinct accuracy error. Candidates using a value of 9.80, 9.81 or 10 for g should usually be penalised for any final accuracy marks which do not agree to the value found with 9.8 which is given in the rubric. g Rules for replaced work and multiple attempts:  If one attempt is clearly indicated as the one to mark, or only one is left uncrossed out, then mark that attempt and ignore the others.  If more than one attempt is left not crossed out, then mark the last attempt unless it only repeats part of the first attempt or is substantially less complete.  if a candidate crosses out all of their attempts, the assessor should attempt to mark the crossed out answer(s) as above and award marks appropriately. h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate’s data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A or B mark in the question. Marks designated as cao may be awarded as long as there are no other errors. If a candidate corrects the misread in a later part, do not continue to follow through. E marks are lost unless, by chance, the given results are established by equivalent working. Note that a miscopy of the candidate’s own working is not a misread but an accuracy error. i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers provided that there is nothing in the wording of the question specifying that analytical methods are required such as the bold “In this question you must show detailed reasoning”, or the command words “Show” and “Determine. Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader. j If in any case the scheme operates with considerable unfairness consult your Team Leader. Y434/01 Mark Scheme June 2022 6 Question Answer Marks AO Guidance 1 (a) sinh2 = 3.62686(041)… and cosh2 = 3.76219(5691).. so both values are from rounding B1 1.1 Both values shown, need to see3.62686…and 3.76219… [1] 1 (b) 0.99974523−1 1 ‒0.000255 or 0.000255 M1 A1 1.1 1.1 allow 1−0.99974523 1 NB ± 0.00025477 to 2 or more sf unsupported implies M1 B2 for ± 0.000255 unsupported [2] 1 (c) not the same, since the order of operations is different oe B1 2.4 must refer to order of operations being different [1] 2 (a) 1.072858−1 3.25−3 oe 0.29143 cao M1 A1 [2] 1.1a 1.1 NB 0.072858 0.25 implies M1 2 (b) 1.072858−0.920799 3.25−2.75 oe 0.30412 cao M1 A1 [2] 1.1a 1.1 NB 0.152059 0.5 implies M1 Y434/01 Mark Scheme June 2022 7 Question Answer Marks AO Guidance 2 (c) central difference method is a 2nd order method whereas forward difference method is 1st order oe or central difference method uses values either side of 3 oe whereas forward difference method steps in the positive x- direction. E1 [1] 2.4 must refer to both methods must refer to both methods 2 (d) 0.024 × 0.182 soi ±0.004368 or ±0.00437 or ± 0.0044 M1 A1 [2] 3.1a 1.1 may be embedded mark the final answer 3 (a) Secant (method) B1 [1] 2.2a 3 (b) −0.7576546×−0.000174−(−0.7540834)×(−0.0020574) (−0.000174−(−0.0020574)) oe ‒0.7537535 ‒0.7537502 M1 A1 A1 [3] 1.1 1.1 1.1 allow sign errors; may be implied by equivalent cell formulae or by correct answers to 3 or more dp allow ‒0.7537501 from working with values rounded to 7 dp 3 (c) ‒0.75375 cao since the last two approximations agree to this precision or ‒0.753750 is possible since convergence is faster than 1st order oe B1 [1] 2.2b must see value and justification Y434/01 Mark Scheme June 2022 8 Question Answer Marks AO Guidance 3 (d) the spreadsheet uses the numbers to a higher precision than is displayed; the calculator uses the displayed values oe B1 [1] 2.4 B0 if answer spoiled by eg spreadsheet stores exact values 3 (e) ‒1.4629×10‒9 or ‒ 0.000 000 001 462 9 B1 [1] 1.1 4 (a) with 𝑥0 = 1 the value of 𝑥1 is further away from α oe with 𝑥0 = 2 the value of 𝑥1 is closer to α so better to use 𝑥0 = 2 B1 B1 [2] 2.2a 2.2a (initially) diverges starting at 1 allow converges more slowly with 𝑥0 = 1 B1 for one correct statement and B1 for 2nd correct statement and better to start at 2 4 (b) can use either (but better to start at x = 2 because it’s nearer) because the magnitude of the gradient of (𝑦=) ln(𝑥2 + 𝑥+ 1.1) is less than 1 at the point of intersection with y = x oe B1 B1 [2] 2.4 2.2a allow gradient of the curve for gradient of (𝑦=) ln(𝑥2 + 𝑥+ 1.1) Y434/01 Mark Scheme June 2022 9 Question Answer Marks AO Guidance 4 (c) (i) difference between 𝑥𝑛+1 and 𝑥𝑛 oe B1 [1] 2.2a 4 (c) (ii) ratio of differences B1 [1] 2.2a 4 (d) ratio of differences decreasing so convergence faster than first order which suggests she used Newton-Raphson method B1 B1 [2] 2.2a 2.2b or ratio of differences not constant so convergence not first order which means she didn’t use fixed point iteration – she probably used Newton-Raphson B0B0 for reasoning based on values in columns M or N 5 (a) 𝑀1+𝑇1 2 used =(G4+H4)/2 oe M1 A1 [2] 1.1a 1.1 Y434/01 Mark Scheme June 2022 10 Question Answer Marks AO Guidance 5 (b) 𝑆2 = 2𝑀1+𝑇1 3 𝐨𝐫 𝑆4 = 2𝑀2+𝑇2 3 or 4𝑇2−𝑇1 3 or 𝑇2 = 𝑀1+𝑇1 2 used soi Mn Tn S2n 0.2436699 0.1479020 0.2117473 0.2306967 0.1957860 0.2190598 M1 A1 A1 A1 [4] 3.1a 1.1 1.1 1.1 use of appropriate formula, may be implied by one correct answer if all correct values given to greater precision allow M1A1A1A0 if table incorrect or incomplete, allow up to M1A1A1 for work seen in the space below 5 (c) 16×𝑡ℎ𝑒𝑖𝑟 0.2190598−𝑡ℎ𝑒𝑖𝑟 0.2117472 15 oe 0.2195473 – 0.21954731 0.220 is possible due to increased accuracy from Richardson’s extrapolation M1 A1 A1 [3] 3.1a 1.1 1.1 or 0.2190598 + 0.2190598−0.2117473 16 ; may be implied by 0.2195168… allow 0.22 is secure by comparison with S4 Y434/01 Mark Scheme June 2022 11 Question Answer Marks AO Guidance 6 (a) 𝑥+ 𝑑× 𝑟 1−𝑟 or 𝑥+ 𝑑× 𝑟 𝑥= −0.596806, 𝑑= 0.003598, 𝑟= −0.95889 𝑜𝑟−0.9589 𝑜𝑟− 0.959 𝑜𝑟−0.96 awrt −0.598567 to awrt −0.598568 −0.5986 or −0.599 is possible due to increased accuracy from extrapolation oe M1 A1 A1 A1 [4] 3.1a 2.1 1.1 2.2b at least two values from information in table; allow sign errors allow−0.60 is certain by comparison with 𝑥108 oe the final mark is only available if all other marks earned if M0 allow SC1 for −0.60 is certain by comparison of 𝑥108 and 𝑥107 and confirmation by correct change of sign test 6 (b) r xr difference ratio 0 0 1 ‒0.4291502 ‒0.429 2 ‒0.5817139 ‒0.153 0.356 3 ‒0.5983772 ‒0.0167 0.109 4 ‒0.5985695 ‒0.000192 0.0115 5 ‒0.5985697 ‒1.99×10-7 0.00103 6 ‒0.5985697 ‒1.82×10‒10 0.000914 M1 A1 A1 A1 [4] 1.1 1.1 1.1 1.1 two iterates seen to 3 or more dp iterates differences ratios allow M1A1A1A0 for correct values to greater precision 6 (c) ‒0.5985697 cao B1 [1] 2.2a Y434/01 Mark Scheme June 2022 12 Question Answer Marks AO Guidance 6 (d) ratio of differences is not constant B1 [1] 2.4 6 (e) to adapt an iterative scheme which does not converge into one which will converge oe B1 [1] 1.2 7 (a) t M ΔM Δ²M Δ³M 0 88.3 ‒8.25 10 80.05 6.9 ‒1.35 ‒5.4 20 78.7 1.5 0.15 30 78.85 B1 1.1 [1] 7 (b) 𝑀= 88.3 − 8.25𝑡 10 + 6.9𝑡(𝑡−10) 2!×102 − 5.4𝑡(𝑡−10)(𝑡−20) 3!×10³ 𝑀= −0.0009𝑡3 + 0.0615𝑡2 −1.35𝑡+ 88.3 M1 A1 A1 3.3 1.1 1.1 soi; allow bracket and/or sign errors; allow use of different variable two correct coefficients in cubic three correct coefficients FT their ‒5.4 and their 6.9 FT first A1 only A1 1.1 all correct A0 if different variable; must see “M =” at some stage NB 𝑏= 123 2000 [4] Y434/01 Mark Scheme June 2022 13 Question Answer Marks AO Guidance 7 (c) evaluation of M at t = 40 and t = 50 75.1 and 62.05 obtained respectively, so a poor fit for these data oe M1 A1FT 3.4 3.5a FT their cubic, dependent on award of M1 in (b) may be stated separately [2] 7 (d) t W ΔW Δ²W Δ³W Δ4W 0 88.3 ‒8.25 10 80.05 6.9 ‒1.35 ‒5.4 20 78.7 1.5 3.6 0.15 ‒1.8 30 78.85 ‒0.3 3.6 ‒0.15 1.8 40 78.7 1.5 1.35 50 80.05 B1 B1 1.1 2.4 difference table correct 4th differences same (so quartic may be good model) [2] 7 (e) −0.0009𝑡3 + 0.0615𝑡2 −1.35𝑡+ 88.3 + 3.6𝑡(𝑡−10)(𝑡−20)(𝑡−30) 4!×104 𝑊= 0.000015𝑡4 −0.0018𝑡3 + 0.078𝑡2 − 1.44𝑡+ 88.3 M1 M1 A1 [3] 3.5c 3.3 1.1 for addition of their cubic to extra term quartic term; FT their 3.6 condone use of other variable NB may see 3 200000 , − 9 5000 , 39 500 , − 36 25 Y434/01 Mark Scheme June 2022 14 Question Answer Marks AO Guidance 7 (f) the model predicts that Sam’s weight will continue to increase oe allow eg 𝑡→∞, 𝑀→∞ so not appropriate (in long run) M1 A1 [2] 3.5a 3.5b must be in context; dependent on award of 2nd M1 in (e) do not allow eg increases exponentially Need to get in touch? 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