Y414/01 Mark Scheme June 2023 7 12. Subject Specific Marking Instructions 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. 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 Y414/01 Mark Scheme June 2023 8 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. Y414/01 Mark Scheme June 2023 9 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. 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. Y414/01 Mark Scheme June 2023 10 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” or “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. Y414/01 Mark Scheme June 2023 11 Question Answer Mark AO Guidance 1 (a) (i) 1.30258 × 10‒9 or 0.000 000 001 302 58 B1 1.1 [1] 1 (a) (ii) Because the values in L22 and M22 are stored to a higher precision than they are displayed B1 1.1 Accept ‘greater accuracy’ but do not allow e.g. ‘the spreadsheet stores values to a different degree of accuracy than it displays’ And they are not equal B1 1.1 Accept “they are both not equal to 1” [2] 1 (b) (i) 1.051… soi B1 1.1 ‒0.000 271 (096 376) to 3 or more sf B1 1.1 Condone 0.000 271 [2] 1 (b) (ii) It will be different because (although the numbers and operations are the same) the order of operations is different B1 1.2 Must be a written explanation [1] 2 (𝑥−5)(𝑥−8) (3−5)(3−8) × 9.26 + (𝑥−3)(𝑥−8) (5−3)(5−8) × 19.3 + (𝑥−3)(𝑥−5) (8−3)(8−5) × 37.96 M1 A1 1.1 1.1 Allow sign errors and one substitution error All substitutions correct, may be implied by later work 𝑦= 0.24𝑥2 + 3.1𝑥−2.2 A1 1.1 2 terms correct A1 1.1 All correct in the required form in terms of x, with y = seen cao [4] 1.1 3 (a) 43.2 2.145+2.175 or 43.2 2.135+2.165 M1 1.1a For either expression correct, may be implied by 10 or 10.047 10 < P ≤ 10.047 A1 1.1 10.047 to at least 5sf (10.0465116279=432 43 ) Allow < or ≤ for either, or interval notation / words. ISW [2] 3 (b) 43.2 2.175−2.135 or 43.2 2.165−2.145 M1 1.1a For either expression correct, may be implied by 1080 or 2160 1080 < Q < 2160 A1 1.1 Allow < or ≤ for either, or interval notation / words. ISW [2] 3 (c) There is such a big difference because the denominator of Q involves the subtraction of nearly equal numbers B1 2.4 allow eg division by subtraction of nearly equal numbers [1] Y414/01 Mark Scheme June 2023 12 Question Answer Mark AO Guidance 4 (a) 0.002 × 0.576 M1 1.1 Use of g(1.802) ≈ g(1.8) + 0.002× g′(1.8), may see 0.001152 error ≈ 0.00115 A1 2.2b Mark the final answer, must be rounded correctly to 3sf SCB1 for 0.00115 without working Alternative Method: g(1.8) ‒ g(1.802) calculated (±) 0.00115 M1 A1 May be implied by sight of ‒0.00115088631 (from using given expression for g(x) from later part of question) Mark the final answer, must be rounded correctly to 3sf [2] 4 (b) 𝑥0 =1.8 𝑥1 =1.80082266922 𝑥2 =1.80129614714 M1 1.1 𝑥1, 𝑥2 seen to 5 or more dp (ignore labelling) 𝑥3 =1.8015686022 𝑥4 =1.80172536548 𝑥5 =1.80181555739 𝑥6 =1.80186744644 𝑥7 =1.80189729856 𝑥8 =1.80191447248 𝑥9 =1.80192435258 𝑥10 =1.80193003654 𝑥11 =1.80193330648 𝑥12 =1.80193518765 𝑥13 =1.80193626988 (𝑥14 =1.80193689247) M1 1.1 𝑥3, 𝑥4 or any 2 further correct iterations seen to 6 or more dp (ignore labelling) 1.80194 cao A1 2.2a Correct answer to 5dp, supported by at least 4 correct iterations. Must be seen separately from their list of iterations as a clear final answer. [3] 4 (c) Because the condition ‒1 < g′(0.445) < 1 is not satisfied B1 2.4 Allow eg because g′(0.445) > 1, must reference the point (x ≈)0.445 or 𝛽 (not just ‘because the gradient is more than 1’) [1] Y414/01 Mark Scheme June 2023 13 Question Answer Mark AO Guidance 4 (d) 𝑥𝑛+1 = (1 −(−0.258))𝑥𝑛+ (−0.258)g(𝑥𝑛) B1 1.1 Sight of the correct relaxed iteration formula with given values. May see 𝑔(𝑥𝑛) = √𝑥𝑛2 + 2𝑥𝑛−1 3 , may be seen in working. 𝑥0 =0.445 𝑥1 =0.44504176118 𝑥2 =0.445041867581 𝑥3 =0.445041867912 𝑥4 =0.445041867913 𝑥5 =0.445041867913 M1 1.1 𝑥2, 𝑥3 seen to 6 or more dp (ignore labelling) 0.44504187 cao A1 2.2a Correct answer to 8dp, supported by at least 3 correct iterations. Must be seen separately from their list of iterations as a clear final answer. [3] 5 (a) 1.30258554+1.28983372 2 M1 1.1 1.29620963 A1 1.1 Must be correctly rounded to 8dp. SCB1 for correct answer without working. [2] 5 (b) 2×1.30258554+1.28983372 3 or 2×1.29948881+their 1.29620963 3 M1 3.1a Or equivalent correct working [S2 =] 1.29833493… A1 1.1 Accept rounded correctly to 6dp or more SCB1 for correct answer without working. [S4 =] 1.29839575 A1 1.1 Accept rounded correctly to 6dp or more SCB1 for correct answer without working. [3] 5 (c) 1.298 (is secure) because S4 and S2 agree to 3dp B1 2.2b Allow ‘by comparison of S4 with S2’ or ‘1.2984 (is probable) because S4 is (more accurate / better) than S2’ but not referencing T/M values. [1] 5 (d) 16×1.29839575−1.29833493 15 M1 3.1a May be implied by 1.29839980467 Allow partial extrapolation for this mark only 1.298399 to 1.2984 A1 1.1 1.29840 by comparison with S4 A1 2.2b www, allow 1.298400 or 1.2984000 Accept ‘because extrapolation increases accuracy’ Y414/01 Mark Scheme June 2023 14 Question Answer Mark AO Guidance [3] 6 (a) t W ΔW ΔW² ΔW³ 0 35.90 ‒1.99 1 33.91 0.46 ‒1.53 ‒0.06 2 32.38 0.40 ‒1.13 ‒0.06 3 31.25 0.34 ‒0.79 4 30.46 M1 A1 1.1 1.1 Differences found, with at least three correct in first column All correct [2] 6 (b) The third differences are equal B1 2.4 Allow third differences are constant or fourth differences are zero [1] 6 (c) 35.9 + 𝑡× −1.99 + 𝑡(𝑡−1) 2! × 0.46 + 𝑡(𝑡−1)(𝑡−2) 3! × −0.06 M1 A1 3.3 1.1 Allow sign errors in substitution or one numerical error At least 2 terms correctly substituted 𝑊= −0.01𝑡3 + 0.26𝑡2 −2.24𝑡+ 35.9 A1 1.1 3 terms correct A1 3.3 All correct; A0 if different variable used or “W =” omitted [4] 6 (d) 𝑡= 6, 𝑊= 29.66 (which is ≈29.8) so model is a good fit B1ft 3.4 Ft 𝑊|𝑡=6 in their equation from (c) [1] 6 (e) In the long run model predicts weight decreases forever oe B1 3.5b Allow ‘modelled weight will eventually be negative’ [1] 7 (a) 16.17𝑥2 −4.34𝑥−1.11 seen M1* 3.1a Allow one slip in differentiation 𝑥− f(𝑥) their f′(𝑥) used M1 dep* 1.1 Need to see at least 3 iterates, including their 𝑥1 and 𝑥2 May be implied by correct iterates. Y414/01 Mark Scheme June 2023 15 Question Answer Mark AO Guidance (𝑥0 =‒0.5) 𝑥1 = ‒0.458598726 𝑥2 = ‒0.454582156 𝑥3 = ‒0.454545458 𝑥4 = ‒0.454545455 (𝑥5 = ‒0.454545455) M1 1.1 𝑥3, 𝑥4 seen to 6 or more dp (ignore labelling) ‒0.4545455 A1 3.2a Correct answer to 7dp, supported by at least 4 correct iterations. Must be seen separately from their list of iterations as a clear final answer. [4] 7 (b) (i) = B10 ‒ B9 B1 2.2a [1] 7 (b) (ii) = C12/C11 B1 2.2a [1] 7 (c) Ratio of differences is converging to 0.5 B1 2.2b Accept ‘approaching 0.5’ but not ‘is constant’ or ‘=0.5’ So convergence appears to be 1st order B1 2.2b Unusual because Newton-Raphson method usually has 2nd order convergence B1 2.3 [3] 8 (a) 1.4439304−1.3258177 0.8 M1 1.1 0.147640875 A1 1.1 Accept rounded correctly to 6dp or more. SCB1 for correct answer without working. [2] 8 (b) 1.4439304−0.9638087 1.6 M1 1.1 0.3000760625 (at least 6dp) A1 1.1 Accept rounded correctly to 6dp or more. SCB1 for correct answer without working. [2] Y414/01 Mark Scheme June 2023 16 Question Answer Mark AO Guidance 8 (c) Answer to part (b) likely to be closer to true value since central difference method is (generally) a 2nd order method, whereas forward difference method is (generally) 1st order method B1 2.4 Allow answer to part (b) probably closer to true value since central difference approximation takes values either side of 2 but forward difference (uses a step in the positive x-direction only) just takes a value to the right oe Explanation must mention both methods [1] 8 (d) 0.2355276 + (‒ 0.0007017) × 0.25 0.75 M1 A1 3.1a 2.1 allow slip in substitution 0.25 or any rounded version of 0.2484204 0.2352937 to 0.2352962 A1 1.1 allow 0.235, 0.2353, 0.23529 or 0.235294 because extrapolation improves accuracy significantly oe A1 3.2a www Allow 0.24 is certain by comparison of extrapolated value with 0.2355276 if M0 allow SC2 for 0.2355276 + (‒ 0.0007017) × 0.2484204 ≈ 0.235 as final answer; allow rounded versions of difference and ratio [4] Need to get in touch? 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